Delay Differential Equations: With Applications in Population Dynamics (Mathematics in Science and Engineering)

DELAY DIFFERENTIAL EQUATIONS WITH

APPLICATIONS IN

POPU LATION D Y N A M I cs

This is volume 191 in MATHEMATICS IN SCIENCE AND ENGINEERING Edited by William F. Ames, Georgia Institute of Tec-hnology A list of recent titles in this series appears at the end of this volume.

DELAY DIFFERENTIAL EQUATIONS WITHAPPLICATIONS IN POPULATION DYNAMICS YANG KUANG DEPARTMENT OF MATHEMATICS ARIZONA STATE UNIVERSITY TEMPE. ARIZONA

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

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Contents Preface

........................... . . . . . . .

1

. . . . . . . . . . . . . . . . . .

3

Part One: DELAY DIFFERENTIAL EQUATIONS

.

Chapter 1 Introduction

ix

1.1. Delay Differential Equations from Applications 1.2. Small Delays Can Have Large Effects . . . . 1.3. Concluding Remarks . . . . . . . . . . .

.

. . . . . . . . 3 . . . . . . . . 11 . . . . . . . . 12

. . 15 . . . . . . . . . . . 15

Chapter 2 Basic Theory of Delay Differential Equations

2.1. Preliminaries-Definitions and Notations 2.2. Existence, Uniqueness. Continuous Dependence. and Smoothing Property . . . . . . . . . . . . . . . . . . . . 2.3. Dynamical Systems and Invariance . . . . . . . . . . . 2.4. Local Stability Theory . . . . . . . . . . . . . . . . 2.5. The Method of Liapunov Functionals . . . . . . . . . . 2.6. Razumikhin-Type Theorems . . . . . . . . . . . . . . 2.7. Infinite Delay and Fading Memory Space . . . . . . . . 2.8. General Linear Systems . . . . . . . . . . . . . . . . 2.9. Hopf Bifurcation and a Periodicity Theorem . . . . . . .

.

. . . 3.1. Discrete Dela.ys-Preliminaries . . . . . . 3.2. Discrete Delays-First Order equations . . 3.3. Discrete Delays-Second Order Equations . 3.4. Discrete Delays-General Theory . . . . . Chapter 3 Characteristic Equations

V

. . . . .

. . . . .

. . . . .

18

. . . .

. 21 . 24 . 26 . 38

. . 46 . . 51 . . 58

. . . . . . 63 . . . . . . 63 . . . . . . 67 . . . . . . 74 . . . . . . 82

vi

3.5. 3.6. 3.7. 3.8. 3.9.

Contents

Distributed Delays-Special Cases . . . . . . . . . . . Reducible Systems-The Linear Chain Trick . . . . . . . Distributed Delays-First Order Equations . . . . . . . Distributed Delays-Higher Order Equations and Systems . Remarks and Open Problems . . . . . . . . . . . . . .

Part Two: APPLICATIONS IN POPULATION DYNAMICS

.

Chapter 4 Global Stability for Single Species Models

. . 90 . . 96 . . 98 . 106 . 113

. .

117

. .

119

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2. Wright’s Global Stability Result for Y(t)= -ry(t - 1)[1+ y(t)] . . . . . . . . . . . . . . . 4.3. Global Stability for General Delayed Nonautonomous Logistic Equations . . . . . . . . . . . . . . . . . . . 4.4. Asymptotic Theory for Nonautonomous Delay Equations with Negative Feedbacks . . . . . . . . . . . . . . . . 4.5. 3/2 Stability Results . . . . . . . . . . . . . . . . . . 4.6. A Model Exhibiting the Allee Effect . . . . . . . . . . . 4.7. Equations of Type i ( t )= f(zl) - g(z(t))-Preliminaries . . 4.8. Equations of Type i ( t )= f ( z t ) - g(z(t))-When f(x) Is Decreasing . . . . . . . . . . . . . . . . . . . . . 4.9. Equations of Type i ( t ) = f(zt) - g(z(t))-When f(z) Is Increasing or Has a Hump . . . . . . . . . . . . . . 4.10. Remarks and Open Problems . . . . . . . . . . . . . .

.

Chapter 5 P e r i o d i c Solutions. Chaos. S t r u c t u r e d Single Species M o d e l s . . . . . . . .

. . . . .

5.1. Global Existence of Periodic Solutions in i ( t ) = f ( z ( t - 1)) -g(z(t)) ....................... 5.2. Periodic Solutions in Delayed Periodic Lotka-Volterra-Type Equations . . . . . . . . . . . . . . . . . . . . . . . 5.3. A Model of Single Species Growth with Stage Structure . . 5.4. Reduction of Structured Population Models to Threshold Delay Equations and FDEs . . . . . . . . . . . . . . . 5.5. Chaos ........................ 5.6. Remarks .......................

119 119 126 133 140 143 146 150 158 170

173 173

. .

181 187 191 195 203

Contents

vii

Chapter 6 . Global Stability for Multi-Species Models 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7.

. .

Introduction . . . . . . . . . . . . . . . . . . . . . . Stability via Liapunov Functionals, I . . . . . . . . . . Stability via Liapunov Functionals. I1 . . . . . . . . . . Stability via Razumikhin-Type Theorems-Theory . . . . Stability via Razumikhin-Type Theorems-Applications . . When Nondelayed Diagonal Terms Do Not Exist . . . . . Remarks and Open Problems . . . . . . . . . . . . .

Chapter 7 . Periodic Solutions in Multi-Species Models 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . 7.2. Periodic Solutions in Delayed Gause-Type Predator-Prey Systems . . . . . . . . . . . . . . . . . . . . . . . 7.3. Periodic Solutions in Periodic Systems . . . . . . . 7.4. Remarks and Open Problems . . . . . . . . . . . .

205

. . . . . .

205 206 211 215 226 234 245

. .

247

.

247

. . . . . . .

247 263 269

Chapter 8 Permanence

. . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . Persistence in Infinite Dimensional Systems . . . . . . . . Permanence in Autonomous Lotka-Volterra-Type Systems . .

273

8.1. 8.2. 8.3. 8.4. Permanence in Nonautonomous Systems . . . . . . . . . . 8.5. Permanence in Nonautonomous Lotka-Volterra-Type Competition Systems . . . . . . . . . . . . . . . . . . 8.6. Permanence in Nonautonomous Lot ka-Vol terra-Type Predator-Prey Systems . . . . . . . . . . . . . . . . . 8.7. Remarks and Open Problems . . . . . . . . . . . . . .

273 274 280 285

.

.

Chapter 9 Neutral Delay Models 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7. 9.8.

. . . . . . . . . . . .

Models and Preliminaries . . . . . . . . Boundedness of x ( t ) in the System (1.6) . Boundedness Results for the System (1.6) . Convergence in Single Population Models, I Convergence in the System (1.6) . . . . Convergence in Lotka-Volterra Systems . Convergence in Single Population Models, I1 Boundedness in a Nonautonomous Neutral Logistic Equation . . . . . . . . . . .

297 302 309 311

. . . . . . . .

. . . . . .

311 315 317 321 327 333 335

. . . . . . . .

338

. . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

...

Contents

Vlll

9.9. A Periodic Neutral Logistic Equation 9.10. Remarks and Open Problems . . .

346 350

........................

353

. . . . . . . . . . . . . . . . . . . . . . . . .

375

...........................

395

References Appendix Index

. . . . . . . . . . . . . . . . . . . . .

Preface Ordinary and partial differential equations have long played important roles in the history of theoretical population dynamics, and they will no doubt continue to serve as indispensable tools in future investigations. However, they are generally the first approximations of the considered real systems. More realistic models should include some of the past states of these systems; that is, ideally, a real system should be modeled by differential equations with time delays. Indeed, the use of delay differential equations (DDEs) in the modeling of population dynamics is currently very active, largely due to the recent rapid progress achieved in the understanding of the dynamics of several important classes of delay differential equations and systems. Although the literature of DDEs in population dynamics is now very extensive, it is scattered in various journals encompassing different subject areas. Existing reference books are generally written for biologists and thus focus on local stability analysis of autonomous DDEs in population models. However, current research work tends to center around the more relevant global dynamics of the considered systems, such as the boundedness of solutions, persistence, global stability of positive steady states, and global existence of periodic solutions. To the best of our knowledge, there is no book that systematically covers these global aspects of the qualitative behavior of systems, many of which are nonlinear and nonautonomous. Indeed, many of these global qualitative results have been obtained quite recently, some of which are yet to be published in the literature. Consequently, time and effort are frequently wasted in rediscovering basic qualitative results of some well studied equations. There is thus an urgent need of a book that bridges the gap between local and global analysis and bring; readers to the forefront of current research in this prosperous field. The present book is intended to serve this role.

ix

X

Preface

As always, it is impossible to cover all of the relevant results in one book. Naturally, the selection of the material is largely influenced by my personal interests. In fact, the content of this book is predominantly of my own and my collaborator’s recent work, excluding those well-known basic results on DDEs. The basic mathematical prerequisite for most part of this book is a thorough understanding of Hirsch and Smale’s book (1974) Differential Equations, Dynamical Systems, and Linear Algebra, published by Academic Press, or equivalent texts (such as Waltman’s book (1986) A Second Course i n Elementary Differential Equations, also published by Academic Press). The book of Freedman (1980), Deterministic Mathematical Models in Population Ecology, published by Marcel Dekker, may serve as an excellent mathematical ecology background reference. Other than these, this book is largely self-contained. It thus should be useful for those who are interested in learning or applying the theory of delay differential equations in their studies regardless of their specific subject areas. In addition, this book can be used as text for graduate courses and seminars in applied mat hematics programs. In this book, the global analysis of the full nonlinear equations or systems are emphasized. Both autonomous and nonautonomous systems with various delays are treated. Specifically, I am interested in the possible influence of delays on the dynamics of the system, including such topics as stability switching for increasing delays, the long time coexistence of populations, and oscillatory aspects (such as the existence of periodic solutions and chaotic behavior) of the dynamics. It is hoped and expected that the analyses presented in this monograph will also be useful in the study of other types of DDEs in applications outside of the population dynamics area. A brief description of the organization of the book is as follows. The book is divided into two main parts encompassing nine chapters. The first three chapters constitute Part I, Delay Differential Equations. In the first chapter, various examples of delay differential equations arising in real life problems are presented. Naturally, the emphasis here is the delay models that have previously appeared in the population dynamics literature. The possible effects of time delays on some of these systems are briefly discussed. In particular, I would like to point out here that, contrary to intuition, small delays can have large influences. Chapter 2 contains the basic theory of DDEs, which is a small subset of J. Hale’s book (1977) Theory of Functional Differential Equations. The local stability of a steady state in a DDE is determined by the locations of the roots of the corresponding characteristic equation. Such a characteristic equation

Pmface

xi

is generally transcendental. In Chapter 3, I present several methods that are commonly used by researchers in their studies of various characteristic equations. These methods cover both discrete and distributed delays. Part I1 consists of the remaining six chapters. It deals with the analyses of various delayed population models. Chapter 4 focuses on the global stability analyses of positive steady states. Chapter 5 mainly concerns the oscillatory aspects of the dynamics. Other kinds of single species models are also discussed. Chapter 6 contains the global stability analyses of some multi-species models. Chapter 7 covers the oscillatory aspects of the dynamics in these models. Global existence of periodic solutions in both autonomous and periodic systems are established. Chapter 8 presents some recent results on permanence theory of delayed systems. Chapter 9 documents some initial attempts on the study of several neutral delay models. The book ends with an extensive bibliography and an effective algorithm written by Mr. Lo and Professor Jackiewicz at ASU for performing simulations of any delayed models. This book uses double enumeration for theorems, expressions, and so on, within the same chapter and triple enumeration when they are referred to in other chapters. It is impossible to thank everyone who has helped me over the years to understand differential equations and mathematical biology. However, two people must be singled out. I am forever grateful to my thesis advisor Herbert I. Freedman for his constant encouragement and invaluable suggestions, to my colleague and close collaborator Hal L. Smith for numerous discussions that produced many results, some of which are included here. Moreover, the presentation of this book has been improved considerably due to their many detailed and constructive comments. I would also like to thank my students and collaborators Baorong Tang and Tao Zhao for many fruitful interactions inside and outside classrooms that yielded some results presented in this book. I am deeply indebted to Edisanter Lo for performing simulations for me on several occasions and the writing of the appendix. My deepest thanks are due to Linda Arneson for her masterful job of typing most of my research papers and this manuscript. Special thanks are also due to Bruce Long for his work on all the drawings and to the Department of Mathematics at ASU and the National Science Foundation for their support of my research. I should also mention here that it was a great pleasure to work with the professional staff of Academic Press, especially Vice President and Publisher David F. Pallai, Executive Editor in Mathematics Charles B. Glaser, and Production Editor Nancy Priest.

xii

Preface

Last, but not least, I would like to thank my wife, Aijun, for her long lasting love. Yang Kuang Tempe, Arizona September, 1992

PART ONE DELAY DIFFERENTIAL EQUATIONS

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1 Introduction 1.l. Delay Differential Equations from Applications As has been asked by many students in many classrooms, “Why study this subject?’’ Why study differential equations with time delays when so much is known about equations without delays, and they are so much easier? The answer is because so many of the processes, both natural and manmade, in biology, medicine, chemistry, physics, engineering, economics, etc., involve time delays. Like it or not, time delays occur so often, in almost every situation, that to ignore them is to ignore reality. A simple example in nature is reforestation. A cut forest, after replanting, will take at least 20 years before reaching any kind of maturity. For certain species of trees (redwoods, for example) it would be much longer. Hence, any mathematical model of forest harvesting and regeneration clearly must have time delays built into it. Another example occurs due to the fact that animals must take time to digest their food before further activities and responses take place. Hence, any model of species dynamics without delays is an approximation a t best. So, while in many applications it is assumed that the system under consideration is governed by a principle of causality; that is, the future state of the system is independent of the past and is determined solely by the present: One should keep in mind that this is only a first approximation to the true situation. A more realistic model must include some of the past history of the system. In this chapter, further examples will be given where it is meaningless not to have dependence on the past. When a model does not incorporate a dependence on its past history, it generally consists of so-called ordinary differential equations ( O D E S )or partial differential equations (PDEs). Models incorporating past history generally include delay differential equations (DDEs) or functional differential equations (FDEs). Some other commonly used abbreviations include retarded functional differential equations (RFDEs) and neutral functional differential equations (NFDEs). 1.1.1. Delayed Logistic Equation with a Discrete Delay The simplest type of past dependence in a differential equation is that

4

Delay Differential Equations

in which the past dependence is through the state variable and not the derivative of the state variable, the so-called retarded functional differential equation ( R F D E ) or retarded differential difference equation ( R D D E ) .For example,

i ( t )= F ( t , x ( t ) , x ( t- r ) ) ,

. dx "=dt*

(1.1)

The well-known Hutchinson's (1948) equation is a special case of (1.1):

&(t)= yx(t)[l - x ( t - T ) / I t ' ] .

(1.2) Equation (1.2) is also frequently referred to as Wright's equation, or the deIayed logistic equation with a discrete delay. Lord Cherwell had encountered Eq. (1.2) in his application of probability methods to the distribution of prime numbers (Wright, 1955). It is also utilized as a single species growth model with time delay. One can show (Wright, 1955) that if 37 T T < -, x ( 0 ) > 0, 24 then x ( t ) + K as t + +oo; and if r r > ./2, then (1.2) has a nonconstant periodic solution oscillating with respect to It'.

1.1.2. Delayed Logistic E q u a t i o n w i t h a D i s t r i b u t e d Delay For a parasite that completes its life cycle within the same host and does not kill the host, immunological resistance by the host depends on exposure to the parasite population. We quote here from Michel (1969): "Characteristically the increase is exponential during early stages of infection when the host offers an ideal environment. Subsequently, when the host becomes resistant and represents a less suitable environment, the rate of increase declines to zero and the population then rapidly decreases." The following integro-differential equation seems to be an appropriate model for the parasite population growth (MacDonald, 1978): dN N ( s ) G ( t - S ) dS . dt Here the instantaneous self-crowding term is accompanied by a pollution term. It should be noted that in this situation it is more appropriate to take the integral from t = 0. This initial time represents the start of the experiment, or the time at which the naive host ingests parasites. It is therefore possible to adopt the simplest memory function, G ( t ) = constant. To illustrate the nature of the results, we ignore in the following the instantaneous self-crowding term and consider dN - = N [r N ( s )ds]. dt

1

cl

5

1 . Iniruduciion

Denoting

/ N(s) t

M ( t )=

ds,

0

(1.4) can be rewritten as

which implies that

dM C dt 2 where NO = N ( 0 ) . The solution of (1.5) is

- = rM - - M ~+ N O ,

and hence

Here a (> 0) and /3 (< 0) are the roots of

2r

2

C

C

M2 - - M 2 - -No = 0, and S = c(a - p ) / 2 > 0. Clearly, the population N ( t ) rises to a maximum and then exhibits a decay or order as e-6t. 1.1.3. Delayed Lotka-Volterra Predator-Prey Systems In his study of predator-prey models, Volterra (1928) investigated the equations

where x and y are the density of prey and predators, respectively, and all constants and functions are nonnegative. For similar interactions, Wangersky and Cunningham (1957) have also used the equations

i ( t ) = crx(t)

[

,(t’]

- bx(t)y(t),

6

Delay Differeniial Equations

for predator-prey models. More general delayed predator-prey models take the form .(t) = zc(t)F(t,X t , Yt), (1.9)

Y(t) = d W t , I t , Yt), where z t ( 0 ) = ~ ( t0), yt(0) = y ( t 0) for 0 5 0, and F and G satisfy appropriate conditions: e.g., dF/dxt 5 0, dF/dyt < 0; dG/dzt > 0, aG/ayt I0.

+

+

1.1.4. A N e u t r a l Delay Logistic E q u a t i o n It is easy to see that (Pielou, 1977) a modification of the well-known logistic single species population equation

i ( t ) = rz(t)[l - x(t)/K] leads to

(1.10)

+

.(t) = rx(t)[l - ( z ( t ) p i ( t ) ) / l ' ] , or, equivalently,

Such a modification is supported by F. E. Smith's (1963) investigation on laboratory populations of Daphnia magna. He argued that the per capita growth rate in (1.10) should be replaced by r[1 - ( ~ ( t )p i ( t ) ) / l ' ] . We may think of x as a species grazing upon vegetation, which takes time T to recover. In this case, it will be even more realistic to incorporate a single discrete delay 7 in the per capita growth rate, which results in the following neutral delay logistic equation:

+

i ( t )= r z ( t ) [ l - (x(t - T ) + p i ( t - T ) ) / K ] .

(1.11)

This equation was first introduced by Gopalsamy and Zhang (1988). Subsequently, it was studied by Freedman and Kuang (1991), Kuang and Feldstein (1991). Based on Eq. ( l . l l ) , Kuang (1991a,b) also introduced and studied the so-called neutral predator-prey and neutral competition models. An FDE is called neutral if the delayed argument occurs in the highest derivative of the state variable. 1.1.5. A Delayed E p i d e m i c M o d e l Mathematical biologist A. J. Lotka investigated, in a series of papers from 1912 on, a differential equation model of malarial epidemics due to Ross (1911). In particular (see Sharpe and Lotka (1923)), he examined the

7

1. Introduction

effect of incubation delays. Let us first look at the model without any delay. The equations are, as given by L o t h (1923), for the human population, rl(t) = b g m ( t ) ( p - h ( t ) ) / p - ~ h ( t-)r h ( t ) ;

and for the mosquito population, h(t)= b f h ( t ) ( q - m ( t ) ) / p - N m ( t ) - srn(t).

Here, p and q are the total human and mosquito populations, treated as constant quantities, which is a standard practice in simple epidemiological models. The functions h ( t ) and m(t) stand for human and mosquito populations carrying the malaria organism (the infected or diseased populations), respectively. The healthy populations are p - h ( t ) and q - m(t). A fixed proportion of each of these populations is assumed to be infective, with the infective populations being f h and gm,respectively. The quantities M and N are death rates, while r and s are recovery rates. It is assumed that each mosquito bites b people in unit time, and that each person receives a bites in unit time. Thus, bq = up, and as a result, only the ratio b / p appears in Eqs. (1.10) and (1.11). For our present purposes what is of most interest is the modification to include incubation delays, quoted from Ross (1911) to be u = 0.5 month in human and v = 0.6 month in mosquito. We thus have

+ +

i ( t ) = bgm(t - .)[p - h(t - u ) ] / p- ( M r ) h ( t ) , 7 q t ) = b f h ( t - v)[q- m(t - ? J ) ] / p- ( N S)m(t). The delay is from the time of a bite to the time at which the human or mosquito is infective.

1.1.6. Delay Models in Physiology: Dynamic Diseases There are many acute physiological diseases where the initial symptoms are manifested by an alteration or irregularity in a control system that is normally periodic, or by the onset of an oscillation in a nonoscillatory process. Such physiological diseases have been termed as dynamical diseases by Glass and Mackey (1979), who have made a systematic study of several important and interesting physiological models with time delays. The following are three examples of these models: aV,x(t)x~(t - T ) (1.12) i ( t )= x ' en

+q

t - ).

(1.13) (1.14)

Delay Differential Equaiions

8

Here, A, a,V,, n,7,6, Po, and y are positive constants. Equation (1.12) is used to study a “dynamic disease” involving respiratory disorders, where z ( t ) denotes the arterial C 0 2 concentration of a mammal, A is the C 0 2 production rate, V, denotes the maximum “ventilation” rate of C 0 2 , and T is the time between oxygenation of blood in the lungs and stimulation of chemoreceptors in the brainstem. Equations (1.13) and (1.14) are proposed as models of hematopoiesis (blood cell production). In these two equations, p ( t ) denotes the density of mature cells in blood circulation, and T is the time delay between the production of immature cells in the bone marrow and their maturation for release in the circulating bloodstream. Details for the derivation of these equations can be found in Mackey and Glass (1977). 1.1.7. System with Lossless Transmission Lines Neutral FDEs are frequently used for the study of distributed networks containing lossless transmission lines. We will consider an example of this type here. For more details, see Brayton (1966) and Kolmanovskii and Nosov (1986). Let the system consist of a long electrical cable of length 1, one end of which is connected to a power source E with resistance R, while the other end is connected to an oscillating circuit formed by a condenser C1 and a nonlinear element, the volt-ampere characteristic of which is i = g(v) (Fig. 1.1). Let L and C denote the linear inductance and capacitance of the cable, respectively, and assume it is lossless. The processes in such a system are described by the following hyperbolic PDEs:

Lit(z,t) = -vz(z,t), O
C vt(z, t )= -iz(z,t), t>0,

(1.15)

with boundary conditions

E - v(0,t) - Ri(0,t)= 0, C l ~ ( 1 , t=) i(l,t) - g(v(1,t ) ) . (1.1.6) / ~ z = ( J ~ / C ) ’It/ ~is .known that the general solution Let s = ( J ~ C ) - ’and of (1.15) can be represented as

v(z, t ) = $(z - S t ) - 4(z

where f$ and

+ st), +

i ( z ,t ) = z-’[f$(z - S t ) + G(z s t ) ] , are arbitrary smooth functions. From the expressions, it

9

1. Introduction

FGC

1

L, c

0T E

i

Figure 1.1. System with lossless transmission line. Using these expressions in the general solution and using the first boundary condition at t - l / s , one obtains

where Ii‘ = ( z - R ) / ( z + R ) ,a = 2E/(z+R). Inserting the second boundary condition and letting u ( t ) = v(1, t ) , we obtain

h ( t )- I(h

(t -);

= f ( u ( t ) , u( t -

t)) ,

(1.17)

where C l f ( U ( t ) ,U(t

1

I<

Z

Z

+

- r ) )= a - - u ( t ) - -u(t - r ) - g (u (t )) Kg(u(t - T ) ) ,

and all constants are positive and depend on the parameters in the original equations. Also, if R > 0, then It‘ < 1. Clearly, (1.17) is a nonlinear NFDE. Thus, the investigation of linear partial differential systems (1.15) with nonlinear boundary conditions (1.16) is reduced to the study of a nonlinear NFDE. This reduction may also be used in other problems with distributed lines. 1.1.8. An Electrodynamics Problem Imagine two charged particles, each moving under the sole influence of the retarded fields of the other. This problem can be modeled in terms of direct interparticle “action at a distance” with delays representing the times required for electromagnetic effects to travel from one particle to the other. (The model considered here does not include the so-called radiation

L

Delay Differential Equations

10

x.(t J

Figure 1.2. A two body problem. reaction force of a particle on itself.) For more details about this and related problems, see Driver (1984b). Let two particles, with charges el and eg and rest masses ml and m2, be located with respect to some inertial reference frame at positions z l ( t ) and z z ( t ) in R3 a t instant t (see Fig. 1.2). Then, the field acting on particle i at the instant t must have been produced by particle j a t an earlier time t - r;(t). If c is the speed of light, the delay r;(t)must satisfy the equation cr;(t)= Iz;(t) - " j ( t - T i ( t ) ) I ,

i # j,

(1.18)

where I I is the Euclidean norm in R3. The equations of motion for particle i under the influence of the retarded effects from particle j are expressed in the following. The notation used includes normalized velocities v;,where

x., - cv;

for i = 1,2,

(1.19)

and a unit vector and a scalar quantity: u; =

XI

-"j(t mi

- Ti)

and

-

where indicates the dot or scalar product in R3. From the Lorentz force law, we have

where vp stands for and Ej is the retarded (vector valued) electric field arriving at z; at the instant t from particle j . This field is found from

1. Introduction

11

the Lidnard-Weichert potentials to be

where k > 0 is a constant depending on the units used and x indicates the vector cross product in R3. (The Lorentz force law and the Lidnard-Wiechert potentials are given in standard texts on electrodynamics.) 1.2. Small Delays Can Have Large Effects As we know from the delayed logistic equation with a discrete delay in Section 1.1.1, a large delay will destabilize its positive steady state, while a small one will not influence the qualitative behavior of solutions. Indeed, researchers tend to ignore delays in their models when they think delays are small. Such a common practice generally works. However, it is not true without qualification. Consider first the following simple example that shows small delay can have large effects: We know that the trivial solution of i(t)

+24t) = -x(t)

(2.1)

is asymptotically stable. However, the trivial solution of

i ( t )+ 2 i ( t - 7)= - z ( t )

(2.2)

is unstable for any positive delay 7. (The proof of this can be found in Hale (1977), p. 28.) While the above example is somewhat artificial, the following one is more realistic. (For more details, see Kolmanovskii and Nosov (1986).) Figure 1.3 shows an ideal predictor described by the equations v(t) G K,[u(t) ~ ( t ) ~] ,( t=) K2v(t) v(t - 1). These imply, for Klli'2 = 1, that u(t 1) = - v ( t ) . Thus, the output at moment t depends on the input a t a future moment (t 1). This is somewhat absurd and contradicts the principle of causality. Such a paradox can be explained by the effects of small delays. For example, if one allows a little delay 7 in unit I(z, then the predictor is 1 described by the equations

+

+

+

+

+

v(t) = I G [ u ( t ) 4 ) ] , E ( t ) = K2v(t - h ) v(t - 1).

+

(2.3)

This implies that V(

t ) = Ii'1 [u(t ) + ~t ( h)+u( t-2h)] +Kf [u(t - 1 ) + 2 ~t(- 1- h ) ]+** *

.

(2.4)

Delay Differential Equations

12

Figure 1.3. “Ideal” predictor. Therefore, the output is determined by values of the input at preceding moments. Another example of this type is absolutely invariant systems studied intensively by Schipanov (1939). An absolutely invariant system loses its remarkable properties if one takes small delays into account. Finally, as we can see from the delayed logistic equation

i ( t )= TZ(t)[l - Z ( t the critical value z ( t ) Ii’ is

=

TO

-

4/14,

(2.5)

that destabilizes the local stability of steady state 7r

To

= -, 27-

(2.6)

which depends on the value of r . For large r (this depends on the measure of time t ) , TO can be viewed as small. Therefore, even though frequently small delays can be neglected, we need to know how small is really “small” in each individual equation. This important problem will be studied in subsequent chapters. 1.3. Concluding Remarks As mentioned earlier, a time delay occurs naturally in just about every interaction of the real world. The original motivations for studying DDEs mainly come from their applications in feedback control theory. In his study of ship stabilization and automatic steering, Minorsky (1942)

1. Introduction

13

pointed out very clearly the importance of the consideration of the delay in the feedback mechanism. The great interest in such kinds of problems in these and later years has certainly contributed significantly to the rapid development of the theory of differential equations with dependence on the past state. For more specific examples of engineering applications of DDEs, the reader is referred to the book of Kolmanovskii and Nosov (1986), which also contains various applications in other disciplines. Although the application of DDEs in population dynamics dates back to the 1920s, when Volterra (1927) investigated the predator-prey model (1.3), the momentum did not pick up until the last three decades. The now classical book of Bellman and Cooke (1963) is certainly to be credited for the current interest in the study of DDEs in the context of population dynamics and other areas of mathematical biology, and the authoritative book of Hale (1977) pushed the study of these models to the present level of depth. To be sure, the excellent books by Pielou (1977), May (1974), and J. M. Smith (1974) provided much of the biologica1)notivations necessary for the modeling and theoretical analysis of these population dynamics problems in the setting of DDEs. More concrete examples of DDEs in mathematical biology can be found in books by Cushing (1977), MacDonald (1989), and Gopalsamy (1992).

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2

Basic Theory of Delay Differential Equations 2.1. Preliminaries-Definitions and Notations As we can see from the examples included in Chapter 1, initial data for delay differential equations are generally continuous functions on a finite interval. This is distinct from the initial value problem for ordinary differential equations (ODEs) where initial data are points of Euclidean space. For ODEs, we can view solutions of initial value problems as maps in the Euclidean space. In order to establish a similar view for solutions of delay differential equations, we need some definitions. We adopt the following notations: R" is an n-dimensional real Euclidean space with norm I I, and, when n = 1, we simply denote it as R; for b > a , we denote C([a,b],R")the Banach space of continuous functions mapping the interval [a,b] into R" with the topology of uniform convergence; i.e., for 4 E C([a,b],R"),the norm of 4 is defined as 11411 = SUPa<&b ld(O)l, where 1.1 is a norm in R". Here, we use sup instead of max to allow the possibility of a = -m. When [a,b] = [ - T , 01, where T is a positive constant, we generally denote C = C([-T,O],Rn).For (T E R, A 2 0, z E C([o- T , ( T A],R"),and t E [ u , ~A], we define zt E C as zt(O) = z ( t O ) , O E [ - T , 01. Assume R is a subset of R x C , f : R + R" is a given function, and " * " represents the right-hand derivative; then, we call

+

+

+

k ( t ) = f(t, I t )

(1.1)

a retarded delay d i f e r e n t i a l equation (RDDE(f)) or a retarded functional d i f e r e n t i a l equation (RFDE(f)) on R. A function z is called a solution of Eq. (1.1) on [u- T , ( T A) if z E C([o- r , ( ~ A),R"), ( t , z t ) E R and zt satisfies Eq. (1.1) for t E [(T,(T A). For given (T E R, 4 E C, we say z(o,4) is a solution of (1.1) with initial value 4 at (T, or simply a solution through (0, #), if there is an A > 0 such that z(u, 4) is a solution of (1.1) on [(T- T , (T A) and z,,((~, 4) = 4. If the solution is unique, then, for each t 2 0, we may define

+

+

+

+

zt(u, 41,

T ( t ): 4 15

(1.2)

Delay Differential Equations

16

and clearly T ( t )maps C into C. We call T ( t )the solution map of

tL

q t ) = f(t,.t), 26

6,

= 4.

Also, it is easy to see that (1.1) includes ODES, differential difference equations

q t ) = f(t, +), 4 t - T l ( t ) ) , - ,4 t - . p ( W with 0 5 T j ( t ) 5 r , j = 1 , 2 , ... , p , as well as the integro-differential *

*

euuation

as special cases. We say Eq. (1.1) is linear if f ( t , x i ) = L ( t , x i ) h ( t ) ,where L ( t , q ) is linear in xi; linear homogeneous if h ( t ) 0 and linear nonhomogeneous if h ( t ) $ 0. We say (1.1) is autonomous if f ( t , q ) = g ( q ) where g does not depend on t ; otherwise, we say (1.1) is nonautonomous. In order to define a general class of neutral delay differential equations (NDDEs) (or neutral functional differential equations (NFDEs)), we need the definition of atomic. For Banach spaces X and Y , we denote by L ( X ,Y ) the Banach space of bounded linear mappings from X to Y with the operator topology. Let L(X) E L(C,R"), X E A; then, the Riesz representation theorem implies that there is an n x n matrix function q on [-r,O] of bounded variation such that

+

where do indicates that the integration variable is 8. For such an q , we always regard it as extended to R so that q(X,O) = v(X, - r ) for 8 5 - r , V(X, e) = V(X, 0) for e 2 0. Definition 1.1. If p E R and the matrix

A(X, P, L ) = V(X, P+) - V(X, P-) is nonsingular a t X = XO, we say L(X)is atomic at ,B a t XO. If A ( X , P , L )is nonsingular on a set Ii' A, we say L(X) is atomic a t ,8 on Ii'. The preceding definition applies only to linear mappings. The following one applies t o nonlinear mappings. Definition 1.2. Suppose R C_ R x C is open with elements ( t , $ ) . A function D : R + R" is said to be atomic at o n R if D is continuous

a

2. Basic Theory of Delay Differential Equations

17

together with its first and second FrCchet derivatives with respect to $: and Db, the derivative with respect to 4, is atomic at /3 on R. If D ( t , 4) is linear in $ and continuous in ( t ,4) E R x C,

then A(t,$, P ) = A(t,P) is independent of

4 and 4 4 P ) = dt,P+) - 77(t,p-1. Thus, D ( t , 4) is atomic at P on R x C if det A ( t , P ) # 0 for all t E R. For example, if P E [-.,( DI ( t) , 4) , = $ ( O ) + B(t)$(P),then A ( t , P ) = B ( t ) , and D ( t , 4) is atomic at P on R x C if det B ( t ) # 0 for all t E R; also, A(t,0) = I , and D ( t , $) is atomic at zero for all t E R. Now we are ready to define a large class of neutral delay differential equations. Definition 1.3. Suppose R R x C is open, f : R -t R", D : R -+ R" are given continuous functions with D atomic at zero. The equation

is called the neutral delay diferential equation N D D E ( D , f). The function D is called the diference operator for N D D E ( D , f ) . For a given N D D E ( D , f ) , a function x is said to be a solution of the N D D E ( D , f ) if there are a E R, A > 0 , such that

+

+

x E C ( [ a- T , 0 A ) ,R"), t E [a,0 A ) , ( t ,21) E R, D ( t , z t ) is continuously differentiable and satisfies Eq. (1.4) on [u,a A ) . For a given a E R, q5 E C, and (a,$)E R, we say z(u,$)is a solution of Eq. (1.4) with initial value $ at a, or simply a solution through (a,d), if there is an A > 0 such that ~ ( a4),is a solution of (1.4) on [a - T , u A ) and z6(a,4) = $. IfD(t,$) = D o ( t , d ) - g ( t ) , f(t,$) = L ( t , $ ) + h ( t ) ,where DO@,$)and L ( t ,4) are linear in $, then NDDE(D, f ) is called linear. It is called linear homogeneous if g 0 , and linear nonhomogeneous if otherwise. 0, h If both D ( t , $) and f ( t , 4) do not depend upon t , we call NDDE(D,f ) autonomous; otherwise, we call it nonautonomous. The following are some examples of NDDEs. Example 1.1. If T > 0, B is an n x n constant matrix, D ( 4 ) = +(O) - B$(-T), and f : R -+ R" is continuous, then the pair ( D , f ) defines an NDDE, d - [ X ( t ) - BX(t - T ) ] = f ( t , Xi).

+

+

dt

18

Delay Differential Equations

Example 1.2. If r > 0, z is a scalar, D($)= $(O) - sin$(-r), f : R + R is continuous, then the pair (D,f) defines an NDDE, d dt

-[z(t)

- sin z ( t - r ) ] = f ( t , z t ) .

and

(1.6)

Remark 1.1. Note that when z is continuously differentiable, (1.6) is equivalent to i ( t )- (cosz(t - T))i(t - T ) = f(t,zt). (1.7) This shows that our definition of NDDE requires that the derivative of x enters the equation in a linear fashion. In fact, the terms involving z that multiply 5 must occur with the same delay. Example 1.3. Consider Eq. (1.11) in Chapter 1:

i ( t )= r z ( t ) [ l - (x(t - T ) + p i ( t - . ) ) / I { ] . It is equivalent to

If we let

u ( t ) = lnz(t), then (1.8) reduces to

which is an NDDE according to Definition 1.3.

2.2. Existence, Uniqueness, Continuous Dependence, and Smoothing Property We consider first the basic theory for Eq. (1.1). Throughout this section, we assume that f is continuous. The ideas used in establishing the basic theory for Eq. (1.1) are simple. a E R, $ E C is Observe that finding a solution of (1.1) through (a,$), equivalent to solving the integral equation z ( t ) = $(O) 2,

+J

t

f(s,zs) ds,

tL

0,

(2.1)

= $.

To prove the existence of the solution through a point (a,$) E R x C, we consider an a > 0 and all functions z on [a- T , a a] that are continuous and coincide with $ on [a- T , a].The values of these functions on [a,u + a ] are required to satisfy Iz(t) - q5(O)l 5 p. The usual mapping T obtained

+

2. Basic Theory of Delay Dafferenlial Equations

19

from the corresponding integral equation is defined, and it is then shown that (Y and p can be so chosen that T maps this class into itself and is completely continuous (that is, the mapping T is continuous and takes closed bounded sets into compact sets). Thus, Schauder's fixed point theorem implies existence. Results in this section are taken from Hale (1977), where detailed proofs are given. Theorem 2.1 (Existence). In (l.l), suppose R is a n open subset in R x C and f i s continuow on R. If (a,$)E R, then there i s a solution of (1.1) passing through (a,$). We say f(t, 4) is Lipschitz in $ in a compact set Ii' of R x C if there is a constant k > 0 such that, for any ( t ,4;) E It', i = 1,2,

Theorem 2.2 (Uniqueness). Suppose R is an open set in R x C , f : R + R" is continuous, and f ( t , $ ) is Lipschitz in $ in each compact set in R. If ( a ,4) E 0, then there is a unique solution of Eq. (1.1) through (U,

4).

R x C is open, Theorem 2.3 (Continuous dependence). Suppose R (u,$)E R, f E C(R,R"), and x is a solution of Eq. (1.1) through (a,$), which exists and is unique on [o- r,b], b > a - r . Let W 5 R be the compact set defined b y

w = { ( t , x t ) : t E [.,bI), and let V be a neighborhood of W on which f i s bounded. If (ak,4',f k ) , k = 1,2,..., satisfies ok + u, dk + 4, and I f k - flv + 0 as k + 00, then there is a It' such that, for k 2 Ii', each solution x k = x ' ( o ~ , $ ~f k, ) through ( u k $, k ) of

4 t ) = f k ( 45 1 ) exists o n [ok- r, b] and z k + x uniformly on [o- r , b]. (Since some x k m a y not be defined o n [a - r,b], b y x k + x uniformly on [o- r,b], we mean that for any E > 0, there is a k I ( c ) such that x k ( t ) , Ic >_ I c ~ ( c ) ,is defined on [a - r E , b] and x k t x uniformly on [a - r E , b].) Let z be a solution of Eq. (1.1) on an interval [o,a), a > u. We say 2 is a continuation of x if there is a b > a such that 2 is defined on [a- r, b), coincides with x on [a-r, a ) , and x satisfies Eq. (1.1) on [a,b). A solution x is noncontinuable if no such continuation exists; that is, the interval [a, a) is the maximal interval of existence of the solution x . The existence of a noncontinuable solution is a consequence of Zorn's lemma.

+

+

20

Delay Differential Equations

Theorem 2.4. Suppose R i s a n open set in R x C , f : R --t R” is completely continuous, and x i s a noncontinuable solution of Eq. (1.1) o n [a- r, b). T h e n f o r a n y closed bounded set U in R x C , U c R, there is a t U s u c h that ( t , x t ) 4 U f o r tu 5 t < b. In other words, the preceding theorem says that solution of Eq. (1.1) either exists for all t 2 a or becomes unbounded (with respect t o R) at some finite time. Next, we consider the questions of existence, uniqueness, and continuous dependence of solutions of neutral delay differential equations. Theorem 2.5 (Existence). If R is a n open s e t in R x C and (a,d)E 0, t h e n there exists a solution of the NDDE(D, f ) through (a,$). Theorem 2.0 (Uniqueness). If R R x C is open and f : R + R” as Lipschitz in 6 o n compact sets of R, then, f o r a n y (u,6)E R, there exists a unique solution of the NDDE(D, f ) through (a,$). Theorem 2.7 (Continuous dependence). Suppose R R x C i s open, A i s a subset of a B a n a c h space, and D : R x A + R”, f : R x A + R” satisfy the following conditions: (i) D(t,$,X) is atomic at zero f o r each ( t , 6 ) E R u n i f o r m l y with respect t o A. (ii) D(t,$,X) and f ( t , $ , X ) are continuous in ( t , 6 ) E R f o r each X E A and continuous at ( t ,6,XO) f o r ( t ,6)E R. (iii) T h e NDDE(D(., Xo), f (., Xo)) has a unique solution through ( a ,$) E R that esists o n a n interval [a - r, b]. T h e n , f o r a n y (o’,$’,X’) E N(u,q5,X0), there i s a neighborhood N g f~ A 0 1 of [ D f Xo) such that all solutions x ( d , $’, A’) of the N D’E b(-, I ) , f 1’)’through (a’,6’)esist o n [ d - r , b], and xt(a’, A’) i s continuous at (t,a‘,d’, Xo) for t E [a,b],(a’, 6’,A’) E N(a,+,Xo). A continuation result similar to Theorem 2.4 also exists for neutral delay differential equations; see Theorem 12.2.5 in Hale (1977) for details. For the ordinary differential equation

+’,

a,

if f is of class Ck,k 2 0, then solution ~ ( tof) (2.3) is also of class C kon its maximum interval of existence. For Eq. (l.l),this is true only for a restricted value o f t . Theorem 2.8 (Smoothing property). Let ~ ( tbe) the solution of

;.(t) = f ( t , z t ) , 26 = 6, 6 E c, (2.4) where f is of class Ck,k 2 1, and I = u,tz the m a x i m u m interval of esistence f o r x ( t ) . T h e n ~ ( tis) of class C o n [a+lr,t,) for 1 = 0 , 1 , . . .,k.

I ’

2. Basic The0y of Delay Differential Equations

21

In other words, z ( t ) is getting smoother and smoother as t increases. In particular, if f is in C', @ c C is closed and bounded, and T ( t ) @ UdEaXt(a,4) is bounded for t 2 a+r, then T ( t ) @is compact for t 2 a+r. Note that the smoothing property is not valid for neutral delay differential equations. For example, the solution zt of $ [ z ( t ) - z ( t - r ) ] = 0 satisfies ,z, = $ c, where zo = $ and c is a constant function. Thus, zt is always as smooth as $.

+

2.3. Dynamical Systems and Invariance For autonomous ordinary differential equations, bounded solutions have R-limit sets that are nonempty, compact, connected, and invariant. Similar statements are also true for delayed differential equations. In order to define dynamical systems in the context of general delay differential equations, we introduce first the concept of process. Definition 3.1. Suppose X is a Banach space, R+ = [O,oo), u : R x X x R+ --+ X is a given mapping; and define U ( a , t ) : X --+ X for a E R, t E R+ by U(a,t)a:= u ( a , z , t ) . A process on X is a mapping u : R x X x R+ --+ X satisfying the following properties: (i) u is continuous; (ii) U(a,O) = I , the identity; (iii) U ( o s, t)U(a,s) = U ( a ,s t ) . A process u is said to be a p-periodic process, p > 0, if U ( u + p , t ) = U ( a ,t ) for u E R, t E R+. Suppose f : R x C --+ R" is completely continuous, and let x ( u , $ ) denote the solution of the RDDE(f),

+

+

qt)= f ( t , . t ) ,

= 4.

(3.1) We assume that z is uniquely defined for t 2 a. Theorem 2.3 implies that z ( a ,$ ) ( t )is continuous in u,4, t for a E R, 4 E C , and t 2 u. Define u(u, 47.) = %+T(O,

41,

2,

(a,4, ). E R x

c x R+;

then, u is a process on C. Indeed, let T ( t , a )be the solution operator for (3.1) defined as T ( t ,a)$ = a:t(u,4); (3.2) then, U(a,.r) = T ( u T , u ) , where U ( a , t ) 4 = u(a,+,t).We refer to u ( a , + , ~as ) the process generated by the RDDE(f). If f(o p , $ ) = f(a,$), p > 0, for all (u,$)E R x C , then the process generated by the RDDE(f) is a p-periodic process. For neutral equations NDDE(D, f ) ,one can obtain a process similarly. In essence, u(u,a:,t) is the state of a system at time a t if initially the state at time u was z.

+

+

+

Delay Differential Equafions

22

Definition 3.2. A process is said to be a ( c o n t i n u o u s ) dynamical s y s t e m (or an a u t o n o m o u s process) if U ( a , t ) is independent of a;that is, if

t 2 0, T ( t )= U(O,t), then T ( t ) zis continuous for ( t , e )E R+ x X, T(t + T) = T ( ~ ) T ( T ) ,

T(0)= I ,

t,T

E R+.

We also call T ( t ) ,t 2 0, a (continuous) dynamical s y s t e m . If S : X -+ X is a continuous map, the family {Sk, k 2 0) of iterates of S is called a discrete dynamical s y s t e m . If u is a pperiodic process and S = U ( O , p ) , then Sk = U ( 0 ,k p ) . We refer to this discrete dynamical system as the dynamical s y s t e m generated by t h e period m a p of the p periodic process. Definition 3.3. Suppose u is a process on X . The trajectory ~ + ( a , z ) through (a,z) E R x X is ~ + ( a , z ) ={ ( u + t , U ( a , t ) z ) : t ~ R c+R }

xX.

The orbit 7+(a,z) through (a,z) is r + ( o , x )= { U ( a , t ) x: t E R+} C X.

If H is a subset of X, then T+(.,H)

=

u

T+(O,X),

7 + ( 0 , H )=

u 7+(u,z).

xEH

ZEH

A point C E X is called an equilibrium (or steady state, critical point, etc.) of a process u if U(a,t)C = C for t E R+. If there is a a E R, p > 0 , z E X such that U ( a , t p ) z = U ( a , t ) z for all t E R+, then ~ + ( a , z )is said to be p-periodic. Definition 3.4. Suppose u is a process on X . y E X is said to be in the w - l i m i t s e t w ( u , z ) of an orbit 7+(a,x) if there is a sequence t , + 00 as n + 00 such that U(a,t,)z -+ y as n -+ 00. y E X is said to be in the a - l i m i t s e t a ( a , z ) of an orbit 7-(a,z) = UtCoU ( a , t ) x if U ( a , t ) z is uniquely defined for t 5 0 and there is a sequence t , ---t -00 as n + 00 such that U ( a , t , ) z -+ y as n -+ 00. Equivalently, we have

+

w(u,z) =

n c1( u w , +), n c1( rstu U ( ~ A ~ ) ,

t>O

a ( a , z )=

t
rzt

(3.3) (3.4)

23

2. Basic Theory of Delay Differential Equations

where C1 ( A ) means the closure of set A. For any subset H C X , we define the w- and a-limit sets of a set H , w ( u , H ) , a(a,H ) , by replacing z by H in (3.3), (3.4), respectively. If {T', k 2 0) is a discrete dynamical system of X , and H C X ,then the w-limit set of H is defined as w ( ~=)

nci( u T ~ H ) , j20

nlj

and the a-limit set of H is defined as

nci( uT ~ H ) ,

a ( ~= )

jlo

nlj

provided that TnH is well (uniquely) defined for all n For autonomous RDDE(f),

& ( t )= j ( Z t ) ,

2,

5 0.

= 4,

(3.5)

we have that w ( u , x ) = ~ ( z is) independent of a. y E w ( z ) ( a ( z ) )if and only if there is a sequence tn + oa (-00) as n + 00 such that n-ca lim

xtn(u,4) = y .

(3.6)

Definition 3.5. If u is a dynamical system on X , then Q C X is said to be a positively (negatively) invariant set of u if for every point x E Q, r+(O,x) c Q (if U ( u , t ) is well defined for t 5 0 and r-(O,z) = { V ( u , t ) z: t 5 0) c Q); and an invariant set of u if T(t)Q U I ET~( t ) z= Q, for all t 2 0, where T ( t )is defined above. Note that the definition of invariance does not require negative invariance. Indeed, it does not even require the backward trajectory being well defined. However, if a set Q is both positively and negatively invariant, then it is easy to show that Q is invariant. Theorem 3.1. If a n a u t o n o m o u s RDDE(f) generates a dynamical s y s t e m and r+($) is a bounded orbit, t h e n w ( 4 ) is n o n e m p t y , compact, connected and invariant. If H C C is connected and r + ( H ) is bounded, t h e s a m e conclusion is true f o r w ( H ) . In order t o present a similar result for NDDE(D,f), we need the concept of a stable D operator. Definition 3.6. Suppose D : C + R" is linear, continuous, and atomic a t zero; and let CD = {+ E C : D 4 = 0). The operator D is said to be stable if the zero solution of the homogeneous "difference" equation Dyt = 0,

t 2 0,

Yo = 4 E

CD,

Delay Differential Equations

24

is uniformly asymptotically stable. Theorem 3.2. Suppose that R C C i s open, D i s stable, and f : R t R" i s completely continuous. If r+(+)i s bounded, t h e n w(+) is n o n e m p t y , compact, connected and invariant. If H C C is connected and r + ( H ) is bounded, t h e s a m e conclusion i s true f o r w ( H ) . Clearly, Theorem 3.1 is a special case of Theorem 3.2. 2.4. Local Stability Theory We consider first a linear autonomous NFDE(D, L ) ,

where D, L : C --t R" are continuous linear functionals on C. For example,

where p, a, b are constants and k ( 0 ) is continuous. We assume that D is atomic at zero (see Definition 1.2). The characteristic equation of (4.1) is det A(X) = 0,

A(X) = XD(e"I)

- L(ex'I).

(4.3)

For example, the characteristic equation of (4.2) is

Note that when Dxt = x ( t ) ,(4.1) becomes an RDDE, and (4.3) reduces to det (XI - L ( e x ' I ) )= 0.

(4.5)

We call the roots of (4.3) the eigenwalues (or characteristic roots) of (4.1). In order to state local stability results, we need the following definition on stabilities. We state the definition in terms NDDE(D, f ) , since it includes RDDE(f) as a special case. Suppose the NDDE(D, f ) is as defined in Definition 1.3:

d -D(t,.t) dt

= f(t,xt).

In addition, we assume that f is completely continuous (a condition almost always satisfied in population dynamics models or other real systems), f : (a, oa)x 52 --t R", where R is an open set in C. However, for convenience,

2. Basic Theory of Delay Differential Equations

25

we simply assume in the following that the domain of definition of f is taken to be R x C. We say 4 E B(0,6) if 4 E C and 11411 5 6, where

11411 = -r<@ 0, there is a 6 = 6 ( € , 0 ) such that 4 E B(0,6) implies q ( u ,4) E B(O,E)for t 2 a. Otherwise, we say x = 0 is unstable. (ii) The solution x = 0 of Eq. (4.6) is said to be asymptotically stable if it is stable and there is a bo = b(a) > 0 such that q5 E B(0,bo) implies z ( u , + ) ( t )-+ 0 as t -+ 00. (iii) The solution x = 0 is said to be uniformly stable if the number 6 in the definition of stable is independent of u. (iv) The solution x = 0 is said to be uniformly asymptotically stable if it is uniformly stable and there is a bo > 0 such that, for every q > 0, there is a t o ( q ) such that 4 E B(0,bo) implies q ( u ,4) E B(0,q)for t 2 o+to(q), for every u E R. If ?(t) is a solution of Eq. (4.6), then % ( t )is said to be stable if the solution y ( t ) = 0 of the equation

is stable. The other stabilities are defined similarly. Theorem 4.1. In Eq. (4.1), let D 4 = +(O) - J!r $ ( O ) d p ( B ) , lim,,o ldP(e>l = 0. If sup{ Re X : det A(X) = 0) < 0,

Jf8x

where A(X) is defined by (4.3), t h e n the zero solution of Eq. (4.1) is u n i f o r m l y asymptotically stable. If ReX > 0 for s o m e X satisfying detA(X) = 0, t h e n Eq. (4.1) is unstable. Moreover, if detA(A) = 0 has a nonsimple pure root, t h e n Eq. (4.1) is also unstable. Consider now the perturbed equation

where D , L are as in Eq. (4.1) and F,G : C -+ Rn are continuously differentiable, F ( 0 ) = G(0) = 0 , F4(0) = G4(O) = 0. Definition 4.2. Let G : C + Rn be continuous. We say G ( 4 ) is independent of +(O) if there is an E E [-r,O) such that G(4) depends only on the values of +if?) for 0 E [ - r , ~ ] .

26

Delay Differential Equations

We present in the following the local stability theorem with respect to the first approximation. Note that when G ( q ) = 0, Dxt = x ( t ) , (4.7) reduces to the retarded delay differential equation. The first part of the following theorem is proved in Theorem 12.9.1 in Hale (1977), and the proof of the second part of it can be obtained by adjusting the proof of Theorem 12.11.1 in Hale (1977). Theorem 4.2. Suppose D, L : C t R" are linear and continuow and the zero solution of NFDE(D, L ) is uniformly asymptotically stable. If G(4) is independent of 4(0),then the zero solution of Eq. (4.7) is also uniformly asymptotically stable. If Re/\ > 0 f o r some X satisfying detA(X) = 0 , where A(X) is defined by (4.3), then the zero solution of Eq. (4.7) is unstable. In other words, local uniform asymptotical stability of a steady state solution of a nonlinear autonomous equation follows from that of its variation equation (or linearized equation) a t that steady state. In order to determine the stability of an autonomous linear equation, we need to analyze its characteristic equation, which is the subject of Chapter 3. Theorems 4.1 and 4.2 are also true when r = 00. 2.5. The Method of Liapunov Functionals As can be seen from the next chapter, the analysis of characteristic equations of linear autonomous delay differential equations is often a formidable task even for equations with just two discrete delays or systems with just one discrete delay! This is in contrast to ordinary differential equations, where one has the powerful and efficient Routh-Hurwitz criterion. However, one can often bypass this difficulty by using the method of Liapunov functionals to obtain sufficient conditions for stability and instability of steady states of delay differential equations in a way similar to the second method of Liapunov for ordinary differential equations. The stability results obtained in this way are often global, in contrast to the local stability results obtained from the analysis of characteristic equations. In the following, we present the method of Liapunov functionals in the context of a general RDDE(f), q t ) = f(4I t ) , (5.1) where f : R x C --t Rn is completely continuous and f ( t , O ) = 0. Let V : R x C t R be continuous and x(u,4) be the solution of (5.1) through (u,4). We denote

27

2. Basic Theory of Delay Diflerentiol Equations

Sometimes, V ( t ,4) is replaced by the notation q 5 . 1 i ( t ,4) to emphasize the dependence on the equation. The next theorem contains general stability results of the method of Liapunov functionals. Since this theorem is very important, we present a detailed proof. Theorem 5.1. Let u(s),v(s),w(s) : R+ -+ R+ be continuous and nondecreasing, u ( s ) > 0, v(s) > O f o r s > 0, and u(0) = v(0) = w(0) = 0. The following statements are true: (i) If there is a V : R x C -+ R such that

414(0)l) I V ( 44) I ~(llq5ll)? V ( t ,4) I

-4?wl),

then x = 0 is uniformly stable. (ii) If, in addition to (i), lim,,+, u ( s ) = +m, then solutions of(5.1) are uniformly bounded (that is, ~ O Tany a > 0, there as a p = p(a) > 0 such that, f o r all u E R, q5 E C, 11q511 5 a, we have 1x(u, $)(t)l5 p f o r all

t 2 u.) (iii) If, in addition to (i), w(s) > 0 for s > 0, then x = 0 is unzformly asymptotically stable. Proof. (i) (Uniform stability). For any E > 0, there is a 6 = 6 ( ~ ) , 0 < 6 < c, such that v(6) < u ( E ) .If q5 E B(O,6),~7E R, then v ( t , q ( u ,4)) I o

for all t 2 u.

Hence, V(t,xt(o,q5))I V(o,$)5 v(6) < u(t), which leads to

4lz(a, W)l)I V ( t ,4

0 ,

$1) < u(t),

or, equivalently,

for all t 2 u. Ix(o,$)(t)l < E (ii) (Uniform boundedness). Since u ( s ) -+ 00 as s + 00, we have for any a > 0 that there is a /3 = P(a) such that

.(PI

= .(a).

If 11$11 5 a,then the proof of part (i) implies that u ( l x ( u , $ ) ( t ) l ) 5 u(P) for all t 2 u,and hence 1z(u,q5)(t)l 5 /3 for all t 2 u. (iii) (Uniform asymptotical stability). Let c = 1, 60 = 6(1), where 6(.) as defined in (i). For any 0 < E < 1, we wish to show that there is a t o = to(So,E) > 0 such that, for 11q511 < 60, ) ) x t ( u , < E for t 2 u to. Let 6 = S ( E ) ; then, for 11$11 5 6, ~ ~ x ~ ( uI , E$for ) ~t ~2 u,u E R. Suppose

+

28

Delay Differential Equations

+

that a solution x = x(a,q5), 11q511 < 60, satisfies 11xtll 2 6 for t E [a,a TI, T > 2r. Since each interval of length r contains an s such that Ix(s)I 2 6, there is a sequence { t k } such that I Z ( t k ) l 2 6, where

a

+ (2k - l ) r It k Ia + 2kr,

k

T

5 -.

2r

Since f is completely continuous, there is a positive constant L such that Ik(t)l < L for t E [a,a TI. Therefore, we have

+

and hence

v ( t , x t ) 5 -w(%),

t E Ik.

Note that t k + l - tk 2 r ; therefore, we may assume that L ensures that I k does not overlap. Hence,

> 6 / r . This

Let Ii‘ = Ii‘(60, L ) be an integer satisfying

Then, if k

> 1 +I<, we have

(-)

6 6 Lv(60) V ( t k ,x t , ) < v(60) - W 2 L 6w(6/2) - O7

+

which is impossible. This shows that if t o = 2 r ( K l ) , then, for 11q511 < So, 11xt(a,q5)II < 6 for t 2 a to, proving the uniform asymptotic stability. 0

+

We say V : R x C + R is a Liapunow functional for Eq. (5.1) if it satisfies Theorem 5.1, statement (i). The following theorem gives sufficient conditions for the instability of the solution x = 0 of Eq. (5.1). Theorem 5.2. Suppose V(q5)is a completely continuous scalar functional o n C and there exists a y > 0 and a n open set U in C such that (i) V(q5) > 0 o n U , V(q5) = 0 o n the boundary of U ; 0 E c1(U n B(O,7 ) ) ; (4 V ( 4 >I .(lq5(0)l) o n n B(O,y);

u

2. Basic Theory of Delay Differential Equations

29

and where u ( s ) , w(s) are continuous, positive, and increasing for s > 0, u ( 0 ) = w(0) = 0. Then z = 0 i s unstable. Proof. Let 4 E U f l B(0,y), 0 E R. Then V ( 4 ) > 0 (since U is open). By (ii), I4(0)\ 2 u - ' ( V ( + ) ) ,and (ii) together with (iii) imply that zt = zt(0, 4) satisfies

Thus,

Hence,

+

V ( x t ) 2 V ( 4 ) ~ (-ta) + +m,

as t

-+

+a.

Therefore, for large t , zt must reach the boundary of U n B(0,y). In fact, V ( q ) > 0 for t 2 a implies that zt cannot reach the boundary of U ; instead it must reach the boundary of B(O,r)in a finite time. 0

For autonomous ordinary differential equations, the Liapunov-LaSalle theorem is a very effective tool in establishing sufficient conditions for the (global) stability of steady states or general attractors. Such a theorem also exists in the context of delay differential equations. Consider now

where f : C -+ Rn is completely continuous and solutions of (5.2) are unique and continuously dependent on the initial data. We denote by z(4) the solution of (5.2) through (O,$). For a continuous functional V : C + R, we define

the derivative of V along a solution of (5.2). In order to state the Liapunov-LaSalle type theorem for RDDE(f) (Eq. (5.2)), we need the following definition.

30

Delay Diflerential Equations

Definition 5.1. We say V : C + R is a Liapunov functional on a set G in C for Eq. (5.2) if it is continuous on C1G (the closure of G ) and V 2 0 on G. We also define

E = {4 E C1G : $5.21(4) = O}, M = the largest set in E which is invariant with respect to Eq. (5.2). The following result is the Liapunov-LaSalle type theorem for Eq. (5.2). Theorem 5.3. If V i s a Liapunov functional o n G and xt($) i s a bounded solution of (5.2) that stays in G, then w ( 4 ) c M ; that is, zt(4) + M as t + +oo. Proof. Assume that xt(q5) E G, t 2 0, and bounded. Then, { x t ( d ) : t 2 0) is a subset of a compact set in C (since f is completely continuous implies . that i ( t ) is bounded) and therefore has a nonempty w-limit set ~ ( 4 ) Since V is a Liapunov functional, V ( x t ( 4 ) )is nonincreasing and bounded from below, and thus it must approach a limit V, as t + +oo. Since V is continuous on ClG, we must have V ( $ )= V, for every $ E ~ ( 4 ) Hence, . V ( $ ) = 0 and w ( 4 ) c E . Since w ( 4 ) is also invariant, we must have 0 w ( 4 ) c M . This completes the proof. Cor ol l a r y 5.1. If V is a Liapunov functional o n Ul = {4 E C : V ( $ )< I } for Eq. (5.2) and there is a constant Ii'= Ii'(I) such that 4 in Ui implies that I+(O)l < Ii', then, f o r 4 E 171, w(4) C M . Proof. For 4 E Ul, we have V 5 0; hence, x i ( $ ) E U1 for all t 2 0. Therefore, Ix(4)(t)l 2 1' for t 2 0, which is equivalent to saying that zt(4) 0 is bounded. The corollary now follows Theorem 5.3. Corollary 5.2. Assume that a ( . ) and b(.) are nonnegative continuous, a ( 0 ) = b ( 0 ) = 0, lim,,+,a(s) = +oo, and that V : C + R is continuous and satisfies

T h e n the solution x = 0 of Eq. (5.2) is uniformly stable, and every solution is bounded. If in addition, b(s) > 0 f o r s > 0, then x = 0 is globally asymptotically stable; that is, every solution of Eq. (5.2) approaches x = 0 as t + +oo. Proof. The stability follows Theorem 5.3 since (5.2) is autonomous. That the solutions are bounded follows from

2. Basic Theory of Delay Differential Equations

31

and the fact that lim,,+, u ( s ) = +00. If, in addition, b ( s ) > 0 for s > 0, then the conditions of Corollary 5.1 are satisfied for all 1 > 0. Clearly, E = {4 : 4(0) = 0). If $ E M , then q($)E M c El which implies that 0 = q ( $ ) ( O ) = x ( $ ) ( t ) , t E R; in particular, 0 = x ( t ) = $ ( t ) for t E [-r,O]. Hence, M = {0}, proving the corollary. 0 For NDDE(D, f ) satisfying f ( t , 0) = 0 , ( d / d t ) D ( t ,0 ) = 0 for all t E R, Theorem 5.1 takes the following form: Theorem 5.4. Suppose D i s stable, u(s),v(s),w(s): R+ 4 R+ are continuous and nondecreasing, u(s),v(s) > 0 f o r s > 0, u(0) = v(0) = w(0) = 0. T h e following statements are true: (i) If there is a V : R x C + R s u c h that

t h e n x = 0 of NDDE(D, f ) is uniformly stable. (ii) If, in addition t o (i), lim,,+, u ( s ) = $00, t h e n solutions of the NDDE(D,f) are uniformly bounded. (iii) If, in addition t o (i), w(s) > 0 f o r s > 0, t h e n t h e solution x = 0 of the NDDE(D, f ) is uniformly asymptotically stable. We omit the proof here, since it is similar t.0 that of Theorem 5.1. Note that V ( t ,4) 5 -w(lD+1) can be replaced by V ( t ,4) I -w(Iq5(0)[). Definition 5.1 applies verbatim to NDDE(D,f). Theorem 5.3 and Corollary 5.1 can be extended as follows: Theorem 5.5. If D is stable, V is a Liapunov functional o n G , G C C, f o r the a u t o n o m o u s NDDE(D, f), and xi($), t 2 0 , stays in G , and either M as t -+ +00. If xi($) or D x t ( 4 ) i s bounded f o r t 2 0, t h e n xi($)

and there i s a constant I- = K(1) s u c h that 4 in Ui implies either I$(O)l < I - or ID+[< I-, then, f o r 4 E Ui, q ( 4 ) + M as t + +00. We again omit its proof since it is very similar to that of Theorem 5.3 and Corollary 5.1. In the following, we illustrate the preceding results by several examples. The following simple observation is frequently used in the construction of Liapunov functionals. Lemma 5.1. Let u ( t ) , b ( t ) be diflerentiable, and let p ( 0 ) be a f u n c t i o n

32

Delay Differential Equations

with bounded total variation such that p ( 0 ) - p(-r) = 1. Then

ddt

lo + e) z(t

-r

dB = z ( t )- z ( t - r ) ,

(5.3)

Proof. We know that, for a constant a,

and

whence the first part of (5.3). Note that

whence the second part of (5.3). Clearly,

0

= z ( t )-

J-, z ( t + e) d p ( e ) . 0

Example 5.1. Consider the scalar equation

i ( t )= -a(t)z(t) - b(t)z(t- ?-(t)),

(5.5)

where a ( t ) , b ( t ) , and r ( t ) are bounded continuous functions, a ( t ) > 0, r ( t ) > 0, +(t)< 1. If b(t) = 0, then (5.5) becomes an ordinary differential equation; a trivial Liapunov function is V l ( z ( t ) )= z 2 ( t ) / 2 ,or equivalently, if viewed

2. Basic Theory of Delay Differential Equations

33

as a functional, then V l ( + )= b 2 ( 0 ) / 2 .Its derivative along the solution of

(5.5) is vl(.(t)) = z(t)[-a(t)z(t)- b ( t ) z ( t - r(t))l = -u(t)z2(t) - b(t)z(t)z(t- r ( t ) ) . We cannot determine the sign of V l , since we do not know the ratio of z ( t ) / z ( t - r ( t ) ) . In order to find a Liapunov functional V , we want to generate a term like -z2(t - r ( t ) )in the V(5.5).For this purpose, we make use of Lemma 5.1. We try

or, equivalently,

v(zt)= ~

1

( t ,=~- z 2~( t )> + a J 2

0

z2(t

-r(t)

+ e) de.

We have V ( z t ) = -(a

- a ) z 2 ( t) b(t)z(t)z(t- r ( t ) )

- a(l - +(t))zZ(t - r(t)),

since

Jzr(t)z2(t+ 0 ) dB =

J:-r(t)

z2(6)dB and (5.3). Clearly, if

b y t ) < 4 ( a ( t ) - a)(1 - +@))a,

(5.6)

then V ( z t ) < 0. Let r ( t ) 5 T , where r is a positive constant; u ( s ) = s 2 / 2 ; v ( s ) = ((1/2) a r ) s 2 . Then,

+

.(l+(O)l)

L V ( t > +L) 4"4l)*

If a > 0 satisfies (5.6), then there may be a positive constant that V ( q )5 - € 2 2 ( t ) .

6

> 0 such

Thus, we may take w(s) = a2. By Theorem 5.1, we know z = 0 is uniformly asymptotically stable. Indeed, since (5.5) is linear, we see that all solutions of (5.5)tend to z = 0 if (5.6)is true for some positive constant a. When a , b and r are constants, (5.6)reduces to

b2 < 4(u - a)a 5 a 2 , which implies that if Ib[ < a, then z = 0 is globally asymptotically stable; i.e., limt,+,zt($) = 0 for q5 E C. Note that the length of delay r is not restricted.

Delay Differential Equations

34

Example 5.2. As an example for the instability result Theorem 5.2, we consider i ( t )= a(t)2!3(t) 6 ( t ) z 3 ( t - r ) , (5.7)

+

where a ( t ) and b ( t ) are continuous and bounded with a ( t ) 2 S > 0, Ib(t)l < q6, 0 < q < 1, and r, 6,q are constants. In order to find a functional V satisfying Theorem 5.2, we try first &(z) = z4/4; we have

Pl(z(t))= a ( t ) z 6 ( t )+ b ( t ) z 3 ( t ) z 3 ( t - r).

(5.8)

We select &(z) = x4/4, because z3(t- r ) appears in (5.7) and we intend to make V a quadratic function of z 3 ( t )and z3(t- r ) . Clearly, we need a term like z6(t - r ) to complete the square of the quadratic function V l . This suggests that we try (see Lemma 5.1)

V(4) = W4(0))+ or, equivalently, V(z1)= z4(t)/4

Q(4) = V(5.7)(4) = b(t)

-r

46(qd8

(5-9)

+ a J f rz6(t + 8) do. We have

+ (.146(o) + b(t)+3(o)43(-.)

- a&-.).

(5.10)

Equation (5.10) is a positive definite quadratic form of 43(0), and 43(-r) if we choose a = -6/2. With a = -S/2, we have V ( 4 )5 $4(0)/4, which implies that it is completely continuous. Denote

then U is an open subset of C satisfying (i), (ii), and (iii); therefore, z = 0 is unstable by Theorem 5.2. If a ( t ) 5 -S < 0, Ib(t)l < q6, we can choose

to show that z = 0 is (globally) uniformly asymptotically stable. Example 5.3. We consider the general scalar Lotka-Volterra equation with distributed delays

where p,(O), i = 1,2 are nondecreasing and 0

J-,

d P i ( e ) = pi(o+) -

= 1,

i = 1,2;

(5.12)

2. Basic Theory of Delay Diflerenlial Equaiions

35

y, a , b, c, and r are nonnegative constants. We have the following:

Theorem 5.6. Equation (5.1 1) is globally asymptotically stable for all p i ( e ) , i = 1,2, and for all initial conditions satisfying +(d) > 0, q5 E C, if b - c < U. b + c 5 a,

+

That is, lim++m z ( + ) ( t = ) y/(u + c - b ) . Remark 5.1. Equation (5.11) has the following discrete delay equation as a special case:

(5.13) which was studied by Lenhart and Travis (1986). Their main theorem states that (5.13) is globally asymptotically stable for initial conditions qi satisfying +(O) 2 0 and for all rj 2 0 if and only if

c n

c n

lbil

5 --a

and

i=l

bi < --a.

i=l

However, the proof of “only if” part is not correct. They made use of the false statement that “global stability of the nonlinear equation (5.13) implies the asymptotic stability of its linearized equation”; since their result in Lenhart and Travis (1985) applies only to asymptotic stability. Therefore, Theorem 5.6 generalizes the main result of Lenhart and Travis (1986). Proof of Theorem 5.6. Since b - c < a , Eq. (5.11) has a unique positive steady state ~t = I*,where I*

= y/(a

+ c - b)

refers to the constant as well as the constant function in C. Since the equation is of Lotka-Volterra type, we can show that z(+)(t)> 0 for t 2 0, as long as qi(0) > 0, E C. We define

+ 2 1; (1‘ ( x ( s )- x*)2ds) dp2(8). t+e

(5.14)

36

Delay Differential Equations

We have 0

qs.ll)(~t - z*)= -a(x(t> - x*12 + b ( x ( t )- x*) J ( z ( t + e) dpl(e) b - c ( x ( t )- x*)J ( x ( t + e) - z*) dp2(e) + 5 ( x ( t )2*)

-r

0

x*)2

-r

- iJo (x:(t+ e) - . * ) 2 d p l ( e ) + ;(.(t) - ,*)2 2

-C 2

-r 0

J

-r

(x(t

+ e) - x * ) 2 d p 2 ( e )

5 -(a

- b - c)(z(t)- x * ) ~ b 0 - -2[ ( x ( t ) - x*) - - T (+

2

+ e) - x*)d p l ( e ) ] + J ( x ( t + e) - d p 2 ( e ) ] , J

2

0

C

- - [ ( x ( t )- I*)

2*)

2 -r since, from the well-known Schwarz inequality (Hardy e t al., 1952, p. 132), we have

[ J-,(x(t + 0) 0

2

- x*) d p i ( e ) ]

0

I J-r

+

( z ( t 0) - z * ) 2 d p i ( d ) J-ro d p i ( 6 )

0

= J-,(x(t

+ e) - z*12d p i ( 0 ) .

Hence, V is a Liapunov functional on C. Let

G = {q5 : q5 E C, +(O) > 0). If a - b - c > 0, then l&l)(x(t) - x*) 0 if and only if x ( t ) G x*; that is, M = {x*} (recall that q5 E M if and only if V(q(q5))= 0, t E R, q5 E CIG). Assume now that a = b c, but a > b - c. Then, i / ( s . l ) ( x t - x*) 0 if

+

0

z ( t )- 2* = / J x ( t

+ e) - x*)clpl(e)

(5.15)

and

x ( t )- z* = -

f p ( t + o) - x*) dp2(e).

However, (5.15) and (5.16) and a = b

+ c imply that

q t ) = x ( t ) { b [ J o ( z ( t+ e) - x*) dpl(e) -T

- ( x ( t )-

(5.16)

.*)I

2. Basic Theory of Delay Differential Equations

37

Hence, z ( t ) must be a constant 5. However, (5.16) implies that

F - 2* = -(% - z*), and hence z = z*. Therefore, we again have M = {z*}. The conclusion of the theorem now follows from Theorem 5.3. 0

Example 5.4. For a simple example for Theorem 5.4, we consider the neutral scalar autonomous linear equation k ( t ) - c3(t - r ) = - a z ( t ) ,

(5.17)

where (cI < 1 and a > 0. For this equation, we have Di$ = d(0) - ci$(-r), which is stable since 1. < 1. Since 3(t)now depends on i ( t - r ) , a function of the form V(z(t)) will have its derivative q 5 . 1 7 ) involve both z ( t ) and k ( t - r ) , which complicates the task of determining the sign of q 5 . 1 7 ) . To get rid of 3(t - r ) , we try V l ( D z t ) = ( 0 ~ 1 =) (~z ( t )- c z ( t - T ) ) ~ We . have v1(5.17)

= 2 ( z ( t ) - cz(t - r ) ) ( - - a z ( t ) ) = -2u22(t)

+ 2aco(t)e(t- r ) .

Again, using Lemma 5.1, we arrive a t the following candidate: (5.18) We have V(zt)=

-2a22(t)

+ 2 a c z ( t ) z ( t - r ) + a c 2 2 ( t ) - ac222(t - r )

= - a ( z ( t ) - c z ( t - r ) ) 2 - a(1 - C 2 ) 2 2 ( t ) .

It is easy to see that conditions of Theorem 5.4, part (iii) are satisfied by V ( z t ) , and we conclude that the trivial solution is (globally) uniformly asymptotically stable. From the preceding examples, we see that finding a proper Liapunov functional for a given equation (or system) involves some guessing, as we have experienced in the ordinary differential case. The following are some effective rules to follow. (A) In the case of RDDE(f): (i) View first the equation as an ODE; that is, set the delays to be zero, or replace z(t 0) by z ( t ) ; then, find a Liapunov function for the resulting ODE. (ii) Next, view the Liapunov function found in step (i) as a Liapunov functional, and find its derivative along the solutions of the original equation RDDE(f); use Lemma 5.1 to

+

38

Delay Differential Equations

find proper additional terms, usually in the form of a single or double integral of z2(t 0) over the interval [-r, 01. (B) In the case of NDDE(D,f): (i) Almost the same as (i) in (A) except that think of Dst as ~ ( t )(ii) . Replace z ( t ) by Dzt in the Liapunov function found in (i), and then follow (ii) of (A). For more examples on the use of Liapunov functionals, one can find them in Gopalsamy (1989), Burton (1985), and Hale (1977). In Part 11, we will use this technique to derive sufficient conditions for global asymptotical stability in population dynamical systems.

+

2.6. Razumikhin-Type Theorems Another effective method of analyzing the stabilities of delay differential equations involves the application of Razumikhin-type theorems. This technique makes use of functions rather than functionals. When using functionals, the theorems stated in the previous section generally require that their derivatives along solutions of the considered equations decrease monotonically. Such a requirement makes finding a Liapunov functional a rather difficult task, since the space C is much more complicated than R" and one has no control of the relationship of Iz(t)l and Iz(t+0)1,8 E [ - r , 01. For these reasons, it is interesting to try functions and study their rate of change along solutions. Let V : R" + R be a positive definite continuously differentiable function. Its derivative along a solution of RDDE(f) takes the form

V(z(t)= ) av(x) . f ( X t ) . dX

To have V ( z ( t ) )be nonpositive often requires that ~ ( tsomehow ) dominates ~ (+t0). From the definition of uniform stability, we know that if zt is initially in a ball B = B(0,S) in C, then, for it to escape B , it has t o reach the boundary of B a t some time t*. At time t*, we have Iz(t*)l = 6, and I x ( ~ * + 0)l < S for 0 E [-r,O), and we must have d / d t l z ( t * ) l 2 0. Therefore, if we can show this is impossible, then we arrive at the stability conclusion. This intuitive observation leads to a series of stability results, called Razumikhin-type theorems. , If V : R x R" + R is a continuous function, then V ( t , ~ ( t ) )the derivative of V along the solutions of an RDDE(f), is defined as -1

V ( t , ~ ( t )=) lim -[V(t t-O+

h

+ h , z ( t + h ) )- V ( t , z ( t ) ) ] ,

where ~ ( t=) zt(o,$), t 2 o,is the solution of RDDE(f) through (o,$). We always assume that f ( t , 0) = 0.

2. Basic Theory of Delay Differential Equations

39

Theorem 6.1. Suppose f : R x C + Rn maps R x (bounded sets of C ) into bounded sets of R", u , v , w : R+ t R+ are continuous, nondecreasing functions satisfying u ( 0 ) = v(0) = w ( 0 ) = 0, and u ( s ) , v ( s ) are positive for s > 0. Assume that there i s a continuous function V : R x R" --$ R such that ~ ( 1 x 1I ) V ( t , x ) I v(Ixl),

t E R,

z E Rn.

(6.2)

The following statements are true: (i) The solution x = 0 of RDDE(f ) i s uniformly stable if

(ii) The solution x = 0 of RDDE(f) is uniformly asymptotically stable if W ( S ) > 0 for s > 0 and there is a contznuous nondecreasing function p ( s ) > s f o r s > 0 such that i . ( t , 4 t ) ) 5 -w(lx(t)l)

fOT

v(t+ 8, x ( t + 0)) < p ( v ( t ,x ( t ) ) ) ,

e E [-r, 01.

(6.4)

If u ( s ) + 00 as

s ---f 00, then x = 0 is globally asymptotically stable. Proof. (i) For any given E > 0, we can find a 6 > 0 such that v(6) < u ( E ) . Let $ E B(0,S) c C and q ( u ,$) be the solution of the RDDE(f ) through (a,$).If, for some t* > u, Ix(t*)l > E , then

V ( t * , x ( t * ) )2 U(I.(t*)l) L 4

6 )

> 46) 2 V(.,$).

Hence, there must be a E (a,t*]such that

V(t,x(t))> o

+

while v ( t , x ( t ) ) 2 ~ ( te,x(t

+ e)),

e E [-T,o].

This contradicts (6.3), and we must have I z ( t ) ( 6 6 for t 2 u. (ii) Let 6 > 0, p > 0 satisfy v(6) = u(p). By (i), we have 11$11 5 6, which implies that 11xt(u, $)[I I p for t 2 u and V ( t ,x(u,$ ) ( t ) )5 v(6) for t2u-r. Let 0 I 7 5 p. We need to show that there is a Z = 2(7,6) such that, for u E R, ll$ll 5 6 implies that x(u,$ ) ( t )5 7 for t 2 u 1. This is true if we can show that

+

V ( t ,x(u,4 ) ( t ) )5 u ( 7 )

for t 2 u

+ t.

There is u > 0 such that p ( s ) - s > a for u(r])I s 5 v(6). Let N be the first nonnegative integer such that u ( q ) N u .(a), and denote 7 = inf,ss
+

40

Delay Differential Equations

We show first that

+

for t

V ( t , z ( t )I ) ~ ( 7 ) ( N - 1).

u

+

If V ( t ,z ( t ) ) > u(7) ( N - 1)" for all t E [u - r, u

+ v(6)/7.

+ v(6)/7), then

p(V(t,z(t)> ) V ( t , z ( t ) +) a = u(v)+ N u 2 v(6)

v(t+ e,z(t + e)), e E [-r,o]. t E [u - r,u + v(q/7], L

Thus,

v I -+(t)l)

L -7,

and, consequently,

a contradiction to the fact that u ( s ) t* E [u,u v(6)/7] such that

+

>0

V ( t * z, ( t * ) )= u ( 7 )

for s

> 0. Thus, there is a

+ ( N - 1)a.

However, this implies that

V ( t , z ( t )5 ) u(7)+ ( N - 1). by (6.4). Denote t i = u preceding, we can show

+ i v ( 6 ) / y ; then,

for t 2 t*, by similar arguments to the

which by (6.4) implies that

V ( t , z ( t ) )5

U(7)

for t 2 t N = 0

+ "U(6)/7.

The theorem is proved by letting 1 = N v ( G ) / y .

0

In order to determine the local stability of a scalar autonomous ordinary differential equation (at z = 0)

i ( t >= f(z(t>),

f(t,O) = 0,

(6.5)

2. Basic Theory of Delay Differential Equations

41

one can almost always choose V(z) = z2. Since if (6.5) is locally asymptotically stable, then we must have f(z(t)) 5 0 for x > 0 and f(z(t)) 2 0 for z < 0. For a scalar autonomous RDDE(f), this is also almost always true. The reasons are the following: If x = 0 is asymptotically stable and V(z) is a Liapunov function for RDDE(f), then it must also be a Liapunov function for the ODE

4 4 = f(z(t>),

(6.6)

where j ( z ( t ) ) = f(&), itt(0) = z ( t ) , 0 E [-r,O]. This is implied by (6.3); therefore, V'(z)j(z) < 0 if z # 0. However, since z = 0 is also stable for the ODE (6.6), we have zf(z) < 0 for z # 0, as mentioned previously. Hence, zV'(z) > 0 for z # 0 and, thus, V(z) is strictly monotone on (-a,O] and [O,p) for some a > 0, p > 0. Denote Vcl and V . ' as the inverses of V on (-a,01 and [0, p), respectively. Then the condition V(z(t

+ 0)) I V(z(t)),

E

[-.,OI,

is equivalent to 4t

+ 0) E [Vl-'(V(4t))),

VT1(V(4t)))l,

8 E [-r, 01.

(6.7)

C learly, max{lVl-'(V(4t)))l, V.1(V(4t>))) However, if we let V ( z )= z2, then

2 l4t)l.

(6.8)

+ e)) 5 V ( z ( t ) )is equivalent to Iz(t + e)l 5 1z(t)l, 8 E [-r, 01, which generally restricts more on Iz(t + 0)l than (6.7) does. Observe that V(z(t

V ( z ( t ) )5 -w(z(t)) is equivalent to V ( z ) k ( t )5 -w(z(t)), which implies

that

22

V ( z ( t ) )= 2 4 t ) I --w(z). V(X) This roughly explains that, for a scalar autonomous RDDE(f), V(z) = x2 is usually a good candidate as a Liapunov function. Indeed, it is generally also a good choice for a nonautonomous RDDE(f), as illustrated by the following example. Example 6.1. Consider again Eq. (5.5): k ( t ) = -a(t)z(t) - b(t)z(t- r ( t ) ) ,

(6.9)

where a ( t ) ,b ( t ) , and r ( t ) are bounded continuous, a ( t ) > 0, r ( t ) > 0. Note that we now do not assume that +(t)< 1. Assume that r ( t ) 5 T i r is a

Delay Differential Equations

42

positive constant. Let V ( z )= x 2 . If v ( x ( t then Iz(t +@)I5 z ( t ) , and we have

+ 6 ) ) 5 V ( z ( t ) )6, E [-.,el,

V ( z ( t ) )= - a ( t ) z 2 ( t ) - b(t)z(t)z(t- T ( t ) )

+

I -a(t)z2(t) lb(t)lx2(t) = - ( a @ ) - lb(t)l)z2(t). Hence, if a ( t ) 2 Ib(t)l, then V ( z )is a Liapunov function for (6.9), and x = 0 is uniformly stable. If a ( t ) 2 6 > 0 and there is a constant Ic E ( 0 , l ) such that Ib(t)l 5 k6, then Theorem 6.1, part (ii) implies that z = 0 is uniformly asymptotically stable. Indeed, we can choose p ( s ) = q 2 s , where q > 1 satisfying qk < 1; then, V ( z ( t ) )5 -(1 - q k ) 6 2 ( t ) . It is interesting to compare the preceding result with that found in Example 5.1 using Liapunov functionals. The result here is independent of delay ~ ( t while ) , in Example 5.1 the stability condition is

P ( t ) < 4 ( a ( t ) - a)(l - ;.(t))a

(6.10)

for some constant a > 0. Since a ( t ) and b ( t ) are constants, and f ( t ) > 0, then (6.10) implies that b ( t ) < a ( t ) ; while if + ( t )< 0, then b ( t ) < a ( t ) implies (6.10). So these two results overlap, but somehow they are complementary. Let V = V(z) be a Liapunov function for an autonomous RDDE(f),

q t ) = f(.t).

(6.11)

For a given closed set G, we denote

E = {+ E G : max V ( z t ( + ) ( s )=) max V ( + ( s ) for ) all t 2 0}, -r<s
-r<s
M = the largest set in E that is invariant with respect to Eq. (6.11).

+

It is easy to see that M is the set of functions E G for every element of which there exists a solution q(+)through such that

+

for all t E R. Note that if E M and max-,<,
2. Basic Theory of Delay Diffemniial Eguaiions

43

The following Liapunov-LaSalle-type result for Liapunov functions for Eq. (6.11) is due to Haddock and TerjCki (1983). Theorem 6.2. Suppose there is a Liapunov function V = V ( x ) and a closed set G c C that is positively invariant with respect to (6.11) such that V(q5)5 0

for q5 E G such that V(+(O))= max V ( + ( s ) ) . (6.12) -r<s
Then, f o r any q5 E G such that xt(q5) is bounded for t

q ( 4 )+ M

2 0, we

must have

as t + 00.

c c

Equivalently, w ( 4 ) M E G. Proof. Let 4 E G be such that x(4)(.) is bounded on [-r,00). Then, x t ( 4 ) E G for all t 2 0, and w ( $ ) is nonempty. Since (6.12) holds, we see that the function max-r<s
lim { -r<s
t+w

for some constant c. Let II,E w ( $ ) ; then, we have max-r.+.
) c = max V ( + ( s ) ) max V ( z t ( $ ) ( s )=

-r<s
-r<s
for t 2 0.

This implies that II,E E , and hence w ( 4 ) E. Since w(q5) is invariant and M is the largest invariant set in E , we have w ( 4 ) M . This proves the 0 theorem. The following are two examples illustrating the application of the preceding theorem. Example 6.2. The following equation is studied in detail in Cooke (1977) as an infectious disease model:

c

i ( t ) = bx(t - r)[l - x ( t ) ]- c z ( t ) , where b,c and r are positive constants. convenient to write (6.13) in the form

(6.13)

For our purpose, it is more

i ( t )= c z ( t ) + bz(t - r ) - bx(t)x(t- r ) .

44

Delay Differential Equations

Assume first that b = c. Denote G = {q5 E C : +(s) 2 0, -r 5 s 5 0). It is easy to show that G is positively invariant. Let V(z) = s2/2; then,

V(4) = -c42(o)

+ b(b(0)qq-r) - b42(o)$(-r).

(6.14)

If 4 E G, 11q511 = I+(O)l (= 4(0), since $(s) 2 0 in this problem). Then, V(4) 5 -b$2(0)q5(-r) 5 0. Thus, for any 4 E G, .(+)(.) is bounded on [-r, m). We wish to find E and M . First of all, we see that 0 E M . 4 E E if and only if 4 E G and Ilxt(4)ll = Ilq5ll1 for all t 2 0. Let ~ ( t ) x ( + ) ( t ) , q5 E E. If Ix(2)l = 11qll,t > r , then (6.14) implies that x2(t)x(r- r ) = 0. Hence, x(t) = 0 or x ( 1 - r ) = 0. ~ ( 2 = ) 0 leads to lldll = 0 and ~ ( t ) 0 for all t 2 0. z(T - r ) = 0; then, V ( z ( t ) )= - b x 2 ( t ) = 0 again leads to x ( t ) = 0 for all t 2 0. Hence, E = M = {0}, and x(q5)(t) + 0 as t + 00 for any 4 E G. If c > b, then the preceding argument can be applied directly to arrive at the same conclusion. In fact, the sets {q5 : 0 5 4 5 l}, {c$: 0 < $(s) 5 l}, and {4 : 0 5 $(s) < 1) are all positively invariant. Assume now that b > c. Let Go = {$ : $(s) > 0, -r 5 s 5 0). Then, Go is positively invariant. It is easy to see that x ( t ) 1 - c/b is a positive steady state. Let x ( t ) = (1 - c/b)(l y(t)); then, (6.15) y(t) = -by(t) cy(t - r ) - ( b - c)y(t)y(t - r ) .

=

=

+

+

Let p = c/b E ( 0 , l ) and V(y) = y2/2. Then,

V(4) = b[-42(o)

+ p4(0)4J(-r) - (1 - P)42(0)4(-7-)1.

Let Gy = {$ : +(s) > -1,-r 5 s 5 0). Then, V ( 4 ) 5 0 for 4 E Gy such that 11$11 = I+(O)l. This shows that Gy is positively invariant (and so is {$ : -1 < 4(s) < p / ( l - p ) } ) . Again, we can argue that yt(4) + 0 in C as t + m for any 4 E Gy. Thus, for any 4 E Go, z(q5)(t)-+ 1 - c/b as

t

+ 00.

Example 6.3. Consider again Eq. (5.11):

where p , ( 6 ) , i = 1,2, are nondecreasing and pi(O+) - p i ( - r ) = 1, i = 1,2; y , a , b,c, and r are nonnegative constants. As in Section 5, we denote z* = y / ( u c - b). Let

+

1 V(z) = -(. 2

- x*)2

2. Basic Theory of Delay Differential Equations

45

and z ( t ) = z(q5)(t). We have

- C x ( t ) ( z ( t )- z*) If 1z(t) - z*1 2 Iz(t

J'-r ( z ( t + e) - z*) dp2(e).

+ 0 ) - z*1, 8 E [-r,O], then

V ( z ( t ) )5 - ( a

- b - c ) ( x ( t )- z * ) ~ .

+

Hence, if a > b c, then z ( t ) G x* is globally asymptotically stable for all p ; ( e ) , i = 1,2, and for all initial conditions q5 satisfying $(O) > 0, q5 E C. However, when a = b+c, and c # 0, the preceding analysis yields only uniform stability. It is very difficult, if possible, to obtain a global asymptotical stability result as we succeeded in doing in Theorem 5.6. For this example, we see that the Razumikhin-type theorem is easy to apply, while the Liapunov functional approach yields a sharper result. Applications of Liapunov functions to systems of delay differential equations will be given in subsequent chapters. The argument made in the preceding example can be extended to general delay single species growth equation

k ( t ) = x ( t ) g ( x ( t ) ,Z t ) , where g is continuously differentiable with respect to z ( t ) and that there exists a positive constant z* such that

(6.17) zt.

Assume

g(x*,x*) = 0.

Let G

c C be a positively invariant set of (6.17) such that q5

$(e) 2 0, $(O) > 0, 8 E [-r,O].

E

G if

Denote

Theorem 6.3. For Eq. (6.17), if a + b < 0 , then x* is globally asymptotically stable with respect to G . We say that solutions of an RDDE(f) are uniformly ultimately bounded if there is a /3 > 0 such that, for any (Y > 0, there is a constant t o ( c r ) > 0 such that II((T, $)(t)l /3 for t (T to(a),for all (T E R,

<

> +

46

Delay Differential Equations

q5 E C, and 11411 5 a. The proof of the following result on uniform ultimate boundedness is essentially the same as that of Theorem 6.1, so we omit it. Theorem 6.4. Suppose f : R x C + Rn takes R x (bounded set of C ) into bounded sets of R", and consider the RDDE(f). Suppose u , v , w : R+ + R+ are continuous nondecreasing functions, u ( s ) + 00 a3 s -+ 00. If there is a continuous function V : R x R" + R; a continuous nondecreasing function p : R+ + R+, p ( s ) > s f o r s > 0 ; and a constant H 2 0 such that

and

if Iq5(0)l

L H,

V ( t + 8,4(8)) < P(V(t,4(0)),

8E

[-T,

01;

then the solutions of RDDE(f ) are uniformly ultimately bounded.

2.7. Infinite Delay and Fading Memory Space Frequently, we see models of real systems involving infinite delays in the literature. For example, the well-known Lotka-Volterra models (Volterra, 1931) were first introduced with infinite delays. Stability results in the previous sections are established for equations with finite delays, and they are not obviously true in general for infinite delays from the proofs given for the finite delay situation. The common and main difficulty is that the interval (-00, A ) is not compact, and the images of a solution map of closed and bounded sets in C([-oo,O],R") with uniform norm may not be compact in the same space. Indeed, the Banach space BC of bounded and continuous functions from (-00,0] into R" with uniform norm may cause problems for even the usual well-posedness questions related to delay differential equations of unbounded delay (see Sawano (1982)). To overcome these difficulties, we choose the following more friendly space, often referred to as fading memory space:

$ ( s ) / g ( s ) is uniformly continuous on

(-00,

O]},

where g : (-m,O] + [1,00) satisfies (gl) g : (-00,0] + [1,00) is a continuous nondecreasing function on (-00,0] such that g ( 0 ) = 1;

2. Basic Theory of Delay Diflerenfial Equations

47

+

($2) g ( s u ) / g ( s ) 4 1 uniformly on (-00,0] as u -,0-; (g3) g ( s ) + 00 as s -+ -00. For example, g ( s ) = e--as, Q > 0, satisfies (g1)-(g3). It is easy to see that UC, is a Banach space with norm

Also, UC, is a strong fading memory space in the sense of Hale and Kato (1978) and Haddock (1985). This implies that bounded solutions of an autonomous system of RDDEs corresponding to initial data 4 E BC have precompact orbits in UC,. Thus, positive limit sets are nonempty and have their usual properties (nonempty, connected, compact, and invariant). Another good reason for choosing UC, over BC is that BC can be continuously imbedded in UC, for any g satisfying (g1)-(g3). Results in this section are adopted from Haddock and TerjCki (1990). We present them without giving detailed proofs. We say {q : t 2 0) is precompact in a space if C1 (Ut>o - 21) is compact in the same space. Lemma 7.1. L e t g be a f u n c t i o n satisfying (g1)-(g3), a n d suppose z : (-0O,0O) -+ ~n is s a c h that 2o E uc,. rf

x is bounded a n d u n i f o r m l y continuous o n [0,00),

(74

a n d I z o ( s ) I / g ( s ) -+ 0 as s -+ -00, t h e n {xt : t 2 0) i s precompact in UC,. If xi is a solution of a n a u t o n o m o u s RDDE(f) o n (-00, 00) s u c h t h a t {xt : t 2 0) is precompact in UC,, t h e n t h e O m e g a l i m i t s e t W ( Q ) is n o n e m p t y , compact, connected, and invariant. We consider the autonomous RDDE(f),

4 E uc,, (7.3) for some g satisfying (g1)-(g3). Let D C UC,, W : D ---t R be a continuous j.(t) = f ( 4 ,

20

= 4,

functional and G a closed positive invariant subset of D. Assume that

if z is a solution of (7.3) such that z ( t ) is bounded on [0, w), then lim W ( q )exists.

(7.4)

t+m

For such a functional W and a constant c in R, define

E(W,C)= {4 E G : there exists a solution rt = q ( 4 ) such that W ( q ) c for t 2 0}, and let M ( W , c ) denote the largest invariant set (with respect to (7.3)) in EtW, 4.

48

Delay Differential Equations

Theorem 7.1 (An invariance principle). Let W : D + R be a continuous functional, and let G be a closed positive invariant subset of D UC, such that (7.4) holds. Then, for any solution zt of (7.3) such that zo E G and the positive orbit {zt : t 1 0 ) is precompact in UC,, w ( q ) C M ( W , c ) for some c in R. Thus, ~t + M ( W , c ) as t + +m. Proof. Define c = lim W ( z t ) . t+oo

Since W is continuous, we must have W ( $ ) = c for all $ E w(z0). The theorem now follows from the invariance of ~ ( $ 0 )and the definitions of M ( W , c ) and E(W,c). 0 Remark 7.1. For general nonautonomous RDDE(f) or NDDE(D, f) with infinite delay, all the results stated in the previous two sections remain valid provided (i) the space C is replaced by BC,, that is, the BC space with norm ( ( . 1, for some g satisfying (g1)-(g3); (ii) one modifies Definition 4.1 according to (i), namely, B(0,c) is the €-ball in BC,. The proofs of these generalized results are largely the same as those provided for finite delay cases. In places where changes are required, one can make use of the observation that if 11zt11, E [O,E], for t 2 0, 20 E BC. Then, for any 6 E (O,c], there is an rg > 0, such that llztll, > 6 for t 2 0 leads to i i ~ t i i 5g max{iiL(t e)ii : e E [-r6,01). (7.5) This observation comes from the fact that g(s) + 00 as s + -m. We thus omit the statements and proofs of the mentioned results for equations with infinite delays in order t o avoid repetition. Definition 7.1. A pair (V,W ) is said to be a Liapunov-Razumikhin pair for (7.3) if V : R" + [O,m) and W : D + [O,m) are such that V has continuous first partial derivatives, W is continuous, and (LR1) for every p > 0 and 4 in D such that qLP E D and q5 is continuous on [-p,O], we have

+

V(d40)) 5 W(4) 5 (LR2)

max V ( W ) : W(4 - p ) > ;

--p<s
4 E D and 0 < V(+(O))= W ( 4 )imply ?7.3)(4)

50.

Of importance in applications is the pair

V ( X )=

I4

and

W(4) = Il4llg.

2. Basic Theory of Delay Differential Equations

49

It is easy to show that this pair satisfies (LR1) and (LR2). Lemma 7.2. Let z be a solution of (7.3). Then (LR1) and (LR2) imply that W ( z t )is a nonincreasing function o f t . The next result follows immediately from Theorem 7.1 and Lemma 7.2. Thus, it can be regarded as an updated invariance principle. Theorem 7.2. Suppose (v, is a Liapunov-Razumikhin pair f o r (7.3). Further, upp pose G is a closed, positive invariant subset of D UC,, and z is a solution of (7.3) o n [0,00) such that zo E UC, and { z t : t 2 0 ) is precompact in uc,. Then w ( z 0 ) E(W,c) f o r some constant c, and zt + M ( W , c ) as t t 00. Theorem 7.3. Suppose in (7.3), 0 E D BC,, f(0) = 0, and (V,W) is a Liapunov-Razumilchin pair such that (i) V ( 0 ) = 0 and V ( z )> 0 f o r 1x1 # 0; (ii) if q5 E D and V(q5(0))= W(q5) > 0 , then

w)

(iii) q5 E D implies

(iv) there is a neighborhood N of 0 in BC, such that q5 in N implies 2 0) is precompact in UC,. Then the zero solution of (7.3) is asymptotically stable. Proof. Lemma 7.2 indicates that W ( z t )is nonincreasing; hence V ( z ( t ) )5 W ( q )5 W(z0). Since V ( 0 ) = 0 and V ( z ) > 0 for 111 # 0, we see that z ( t ) 0 is stable. Therefore, for all E > 0 there exists 6 > 0 such that sups 0 such that M ( W , c ) is not empty. Let 1c, E M ( W , c ) ; then, W ( z t ( 4 ) )= c. (LR1) implies V ( z t ( 4 ) )5 c, t E R. (iii) implies that sups
+

Delay Differential Equations

50

As a simple example of illustrating the application of UC, or BC, space, we consider the infinite delay version of Eq. (5.11):

where pi(O), i = 1,2, are nondecreasing and pi(O+)-p;(-00) = 1, i = 1,2; 7, a , b, c are nonnegative constants. Again, assume that a c - b > 0, and denote t* = 7 / ( a c - b). The following theorem is essentially proved in

+

+

the proof of Theorem 5.6 by using proper Liapunov functionals. Here, we present a proof using UC, space and Liapunov functions. Theorem 7.4. Assume that, in Eq. (7.6), u

+c - b > 0,

b+ c < U .

Then z ( + ) ( t )--t z* as t --t 00 for 4 E BC((-m,O],R+). Proof. Let V(z) = ( z - ~ * ) ~ / Observe 2. that there exists ag(s), satisfying (g1)-(g3), such that

We thus have

+

V ( z ( t ) )= - a z ( t ) ( z ( t ) - x * ) ~ b z ( t ) ( z ( t )- z*)

So, we denote W ( z t )= l l zt l l ~/ 2.Then (V,W) is a Liapunov-Razumikhin pair for (7.6). It is easy to see that conditions of Theorem 7.3 are satisfied. Then N in (iv) can be taken as BC((-m, O],R+). Theorem 7.4 now follows 0 from Theorem 7.3. More application examples of UC, space will be provided in subsequent chapters.

2. Basic The0y of Delay Differential Equations

51

2.8. General Linear Systems The common type of linear systems with finite delay known to be useful in the applications take the form

where 0 I rk I r , IAk(t)l 5 A,, A ( t , e ) is integrable in 8 for each t , and there is a function a ( t ) E L ~ ( ( - c o , c Q ) )(that is, for any a < b, J: a ( t ) dt < +CO) such that

1 /",

A ( t , e ) N e )del I a(WI,

9 E c.

t E R,

(8.2)

A more general linear system takes the form

+

k(t) = L(t,zt) h(t), where L ( t , 4) : R x C --+ Rn is linear in

(8.3)

9. In the following, we assume

where q(t,O) is an n x n matrix function measurable in ( t , e ) E R x R, normalized so that

q ( t , e ) = 0 for 8 2 0,

? ( t , e ) = q ( t ,- T )

for

e 5 -r,

roc

q t , 8 is continuous from the left in 0 on (-r, 0), and there is an m(t) E L , ( ()- C O , C O ) ) such that

IL(t,9)I I m(t)l9l,

t

E

R,

$E

c.

(8.5)

Theorem 8.1 (Global existence). Suppose the preceding hypotheses o n L are satisfied. For a n y given a E R, 9 E C , and h ( t ) E L p ( [ a , c o ) ) , there is a unique f u n c t i o n x(a,$)defined and continuous o n [a - r, CO) that satisfies the s y s t e m (8.3) o n [a,00). Proof. Local existence and uniqueness are known from Section 2. Clearly,

for all t for which ~ ( texists. ) Thus,

52

Delay Differential Equations

Hence, by Gronwall’s inequality (see Lemma 8.1 following), we have

0 ) for all t 2 0 . Theorem 2.4 implies we must have ~ ( texist Inequality (8.10) below is often referred to as Gronwall’s inequality, Bellman’s inequality, or the Bellman-Gronwall inequality. L e m m a 8.1. If u and (Y are real-valued continuous functions o n [u,b], and p 2 0 is integrable o n [ ~ , bwith ] (8.8) then

(8.9) If, in addition, a is nondecreasing, then

(8.10)

Proof. Let R(t) = J: P ( s ) u ( s ) ds. Then R ( t ) = Pu

5 Pa + PR.

Multiplying both sides by exp(- J: P ( s ) d s ) yields (8.11) Integrating (8.11) from a to t , one obtains

which, together with (8.8), implies (8.9). If a(t) is nondecreasing, then u ( t ) I a(t)[l

+

J t P ( s ) (exp

ItP(.)

An integration by parts yields (8.10).

d r ) ds] 9

t

E [ a ,bl.

0

53

2. Basic Theory of Delay Differential Equations

In the following, we consider the linear autonomous equation

Proofs of results to be presented in this section can be found in Hale (1977). For 4 E C , z($) denotes the solution of (8.12) with initial function 4 a t zero. Define T ( t ): C + C as

4 4 )= T(t)4. It can be shown that T ( t )satisfies (i) T ( t T ) = T ( ~ ) T ( Tfor) ,all t 2 0, T 2 0; (ii) T ( t ) is bounded for each t 2 0, "(0) = I , and T ( t ) is strongly continuous on [O,oo); that is,

+

(iii) T ( t )is completely continuous (compact) for t 2 r; that is, T ( t ) , t 2 r , maps bounded sets into precompact sets. Therefore, for t 2 0, T ( t ) is a strongly continuow semigroup of linear operators on C on [O,m); and the infinitesimal generator A of T ( t ) is defined as

1

A$ = t-O+ lim -[T(t)4- 41, t

in norm of C,

(8.13)

whenever this limit exists. The domain D ( A ) of A is dense in C , and the range R(A) is in C. If q5 E D ( A ) , then

It can be shown that

For a linear operator A : B + B , B a Banach space, we define the resolvent set p ( A ) of A as the set of values X in the complex plane for which X I - A , I the identity operator, has a bounded inverse with domain dense in B. The complement of p ( A ) is called the spectrum of A and is denoted by o ( A ) . In what follows, we consider A as defined in (8.13). The constant X E p ( A ) if and only if ( A - X I ) $ = 1c,

(8.15)

54

Delay Differential Equaiions

has a solution 4 in D ( A ) for every $ in a dense set in C and the solution depends continuously upon 4. Hence,

&(el - x4(e) = +(e),

-r 5

e 5 0;

that is,

+

+(O) = eXeb

1

e

d<,

5 8 5 0,

6 = q5(0).

(8.16)

+ J”-r le e ’ ( e - E ) 4 ( < ) d< dq(e) = -(a, G),

(8.17)

eX(e-t)$(()

q5 will be in D ( A ) if and only if

-r

& E C and

Hence,

Thus, A(X)b = -4(O) where A(X) = XI -

J

0

e’ d l l ( e ) ,

-r

( a ,4 ) = (.(0)4(0)-

/” J e -Y

(8.18)

o 5 s 5 r,

a ( s )= e - ” ~ , 0

- WG)4 dll(8).

(8.19)

Equation (8.19) is called the bilinear form of Eq. (8.12). Equation (8.15) has a solution for every 1c, in C only if detA(X) # 0, since the mapping (a,.) covers Rn.Also, detA(X) # 0 implies a solution of (8.15) exists for every 1c, in C and it depends continuously upon 4. Therefore, p ( A ) = { A : det A(X) # 0). If det A(X) = 0, then (8.16) and (8.17) imply there exists a nonzero solution of (8.15) for 4 = 0. For this reason, we call X satisfying det A(X) = 0 an eigenvalue of A , and the nonzero solution of (8.15) for this X an eigenfunction (or eigenvector) of X of A. The null space N ( A - XI) of A - X I is the set of all 4 in C for which ( A - XI)4 = 0. For X E a ( A ) , the generalized eigenspace of X will be denoted by Mx(A), and it is defined as the smallest subspace in C containing all 4 E C such that ( X I - A ) k $ = 0 for some k = 1,2,. . . . The function det A(X) is an entire function and hence has zeros of finite order. Therefore, the resolvent operator ( A - X I ) - ’ has a pole of

2. Basic Theory of Delay Differential Equations

55

order k at Xo if A0 is a zero of det A(X) of order k . Since A is a closed operator, it follows from Hille and Phillips (1957), p. 320, that Mxo(A), the generalized eigenspace of Xo, is finite dimensional and there is a k such that

Mxo(A)= N ( A - Xol)' and

C = N ( A- Xol)'

@J R ( A - X o l ) ' ,

where @ means the direct sum. It can be shown that the following decomposition theorem holds for Eq. (8.12). Theorem 8.2. Suppose A = { A l , . . . ,X p } is a set of eigenvalues of (8.12), and @A = (Oxl,. . . ,Ox,), BA = diag (Bxl,. . . ,Bx,), where axj as a basis f o r the generalized eigenspace of X j and Bxj is the m a t r i x that satisfies AOxj = Oxj Bx,, j = 1,2,. . . , p . T h e n the only eigenvalue of Bxj is Xj, and for a n y vector u of the s a m e dimension as @ A , the solution T ( t ) @ A u satisfies T ( t ) a A a = OAeBAta, t 2 0, ~ p h ( ~=) OA(0)eBAe, 5 e 5 0. Furthermore, there exists a subspace QA of C s u c h that T ( t ) Q A G QAfor all t 2 0 and C = PA@J QA, where PA = {q5 E C : q5 = @ h a f o r s o m e vector a } . To obtain more details of such decomposition, we should introduce the so-called formal adjoint equation. Formulas (8.16), (8.17), and (8.19) show that ( A - X I ) $ = 1c, has a solution if and only if for every nonzero row vector u for which uA(X) = 0, it follows that ( u e b A ' l ,1c,) = 0. This reminds us of the Fredholm alternative theorem in the setting of ODES. We therefore need the notion of adjoint operator A* of A. We would like to have

where D ( A * ) C* = C([O,r],Rn*), and Rn*is the n-dimensional vector space of row vectors.

56

Delay Differential Equations

If a has a continuous first derivative and

+ is in D(A), then

provided -da( s ) / d s ,

O<s
Obviously, D(A*) is dense in C*. We call this A* the f o r m a l adjoint of A relative t o the bilinear f o r m (8.19), and we call Y(T) = -

/

0

-r

Y(‘ - 0)

(8.21)

the f o r m a l adjoint equation of (8.12). Notice that (8.21) must be solved backward and existence and uniqueness of solutions holds (letting t = -T in (8.21)). For Eq. (8.1), its formal adjoint equation takes the form

It is not surprising that A* has similar properties to those of operator A. Theorem 8.3. X is in a(A) if and only if X E a(A*), and f o r a n y X E o(A*), its generalized eigenspace i s finite dimensional; in f a c t , dimMx(A) = dimMx(A*) = multiplicity o f X as a root of detA(X). A necessary and s u f i c i e n t condition f o r (A - X I ) k + = $,

$ E C,

57

2. Basic Theory of Delay Differential Equations

to have a solution $ in C OT, equivalently, $ E R(A - XI)' is that (a,$)= 0 for all a in N(A* - XI)'. More precisely, N ( A - XI)' consist3 of

Also, N(A* - XI)' consists of functions $ of the form

where p = TOW (Pi,. . . ,pk) satisfies PA' = 0. For X E .(A), let Qx = col(41,. . . ,?,hp) and @ A = ( $ 1 , . . . ,&,) be bases for Mx(A*) and Mx(A), respectively, and denote (Qx,@x) = (($i, $j)), i , j = 1,2,. . . , p . Then, (Qx, @A) is nonsingular and thus may be taken as I by properly selecting $; or d;, i = 1,. . . , p . Theorem 8.4. FOT X E .(A), let I = (Qx,@x). Then, f o r any $ E C,

6 = dPx + $ Q x , $ P x

E

PA,

$Qx E Qx,

(8.23)

This detailed decomposition result can easily be extended to cover E .(A), and let PA be multi-eigenvalue cases. Let A = {XI,. . . ,A,} the linear extension of the Mxj(A), Xj E A; we refer to this set as the generalized eigenspace of Eq. (8.12) associated with A. Similarly, we

58

Delay Differential Equations

define P i as the generalized eigenspace of the adjoint equation (8.21) associated with A. If @ and Q are bases of PA and P i , respectively, such that ( Q , @ )= I , then

C = PA@ QA, PA = {+ E C : 4 = @ b for some vector b } , Qx = {4 E C : (Q,4) = 01, and, for any

4 E C, (8.26)

Such a decomposition is needed in using certain fixed point theorems to establish the global existence of periodic solutions in delay differential models. Examples of this kind will be given in later chapters. The following theorem provides a useful estimate of solutions on the complementary subspace QA. Theorem 8.5. Let A = A(p) = { A E o ( A ) : ReX 2 P } and C = PA@&A. T h e n there exist positive constants (I and y such that

In particular, we have the following local stability result: Corollary 8.1. If a = max{ReX : X E o ( A ) } < 0, then, f o r any 6 there i s a constant I( = K ( 6 ) such that

IIT(t)4ll 5 I(e(a+6)tI1411, t 2 0,

> 0,

4 E c.

2.9. Hopf Bifurcation a n d a Periodicity Theorem

In this section, we present two well-known methods of establishing the existence of periodic solutions in autonomous delay differential equations. As we know in ODES, one of the simplest ways in which a nonconstant periodic solution can arise is through Hopf bifurcation. This occurs when, as a real parameter a in the equation passes through a critical value cro, two eigenvalues cross the imaginary axis from left to right. Generally speaking, Hopf bifurcation theorems only assure the local existence of

t. Basic Theory

of Delay Differential Equations

59

periodic solutions when they arise. To establish global existence, one often resorts to some kind of fixed point theorems in a cone shaped subset of space C. We present below a general Hopf bifurcation of delay differential equations due to De Oliveira (1980). It generalizes the corresponding result for RDDE(f) in Hale (1977) to NDDE(D, f). We consider a one parameter family of NDDEs: d -p(", - g ( a ,4 .t)

1 = L(a,21) + f(0, It),

a E R,

(9.1)

where D , L , f, and g are continuously differentiable in a and zt, f(a,0) = g(a,O)= 0, af(a,O)/azt = ag(a,O)/drt= 0, D ( a , z t ) and L ( a , z t ) are linear in z t , and

for zt E C,a E R, where ro(a) = 0, r k ( a ) E (0,1], and A k ( a ) , Bk(a), A ( a , 0 ) , and B ( a ,0 ) satisfy

It is easy to see that the characteristic matrix

is continuously differentiable in a E R and A ( a ,A) is an entire function of A. We will assume the following in Eq. (9.1): (Hl) There exist constants a > 0, b > 0 such that, for all complex values X such that I R e X l < a and all a E R, the following inequalities hold:

60

Delay Differential Equations

(H2) The characteristic equation detA(a,X) = 0 has, for (Y = (YO, a purely imaginary simple root Xo = iwo, vo > 0, and no root of A((Yo,A) = 0, other than fiwo, is an integral multiple of Xo. (H3) R e v # 0. Note that (Hl) is easily satisfied by Eq. (9.1) if IA(O)( dominates in D ( ( Yq), , that is, if

Often, if (9.6) fails, then the steady state xt = 0 is unstable. (H2) and (H3) are standard assumptions for Hopf bifurcation theorems. Now we are ready to state the generalized Hopf bifurcation theorem for (9.1). The detailed proof for an even more general result can be found in De Oliveira (1980). Note that the delay itself can be chosen as the bifurcation parameter. Theorem 9.1 (Hopf bifurcation theorem). In Eq. (9.1), assume that (Hl)-(H3) hold. T h e n there is a n E > 0 such that, for a E R, la1 5 E , there are functions a(.) E R, w ( a ) E R, a(0) = cro, w ( 0 ) = 2n/vo, such that Eq. (9.1) has a n w(cr)-periodic solution x * ( a ) ( t ) ,that is continuously diferentiable in t , a with x*(O) = 0. Furthermore, for I(Y- ( Y O [ < E , Iw - (2n/vo)I < E , every w-periodic solution x ( t ) of (9.1) with Ix(t)l < E must be of this type, except f o r a translation in phase; that is, there exists a E ( - E , E ) and b E R such that x ( t ) = x * ( a ) ( t b) f o r all t E' R. In Chapter 9, complete local stability analyses of some neutral delay population models are carried out, and conditions for the stability switching are given. These conditions imply (Hl)-(H3), and thus establish the occurrence of Hopf bifurcation when the considered steady state becomes unstable. In the following, we present a fixed point theorem that is very useful in obtaining global existence of periodic solutions for autonomous retarded delay differential equations. Again, we do not include proofs of the results. They can be found in Hale (1977) and the references cited therein. We need the following well-known definition due to Browder (1965). Definition 9.1. Suppose X is a Banach space, U c X , and x E U . The point x is called an ejective point of a map A : U\{x} + X if there is an open neighborhood G c X of x such that if y E G n U , y # x, then there is an integer rn = rn(y) > 0 such that Amy E G n U . For any M > 0, we let SM = {x E X : 1x1 = M } and BM = {x E X : 1x1 < M } , where 1 . I is the norm of X . Clearly, Shi = 8Bw.

+

2. Basic Theory of Delay Differential Equations

61

Theorem 9.2. If Ii' as a closed, bounded, convex, and infinite dimensional set in X , A : K\{so} --+ 'I is completely continuous, and so E K i s ejective, then there is a fixed point of A i n K\{so}. If Ii' is not bounded, then the following theorem applies. Theorem 9.3. If Ii' as a closed, convex, and infinite dimensional set in X , A : I'\{O} --+ 'I i s completely continuous, 0 E Ii' is ejective, and there is an M > 0 such that A s = A s , s E 'I n S M , implies A < 1, then A has a fixed point in 'I n BM\{O}. In the application of Theorems 9.2 and 9.3 to autonomous retarded delay differential equations, the mapping A is usually similar to the Poincard map in ODES. More precisely, one obtains a set Ii' C C such that every solution z($), q5 E I<, of the RDDE(f) returns to K in some time T($) > 0; that is, s r ( 4 ) ( $ ) E It' if $ E K . The mapping A : K + 'I is then defined by A$ = s r ( b ) ( $ ) . If Ii' is chosen as closed, convex, and bounded (usually infinite dimensional), and A is proved to be completely continuous, then there is a q5 such that A$ = 4, which implies that s r ( b ) ( $ ) = $, and s t ( $ ) is a periodic solution of RDDE(f) of period T. If, further, one can show that all constant solutions in Ii' are ejective, then that periodic solution is nonconstant. Therefore, it is important to have efficient methods to determine when a constant solution of an RDDE(f) is ejective in I<. There are two methods often used in the literature. One of them was first used by Nussbaum (1973), which directly proves the ejectivity according to the definition, using an idea due to Wright (1955). Examples of this method will be given in subsequent chapters. A somewhat more efficient method, due to Chow and Hale (1978), makes use of the powerful decomposition result for C. We describe it in what follows. Suppose we consider k ( t ) = Lst

+ f(.t),

(9.7)

where L : C + R" is linear and continuous, f : C R", f(0) = 0, f'(0) = 0. That is, Y(t) = LYt (9.8) is the variational equation of (9.7) at st = 0. For any eigenvalue X of (9.8), there is a decomposition of C as C = PA @ QA, where PA and QA are invariant under the solution operator Eq. (9.8). Let the projection operators defined by this decomposition of C be 7 r ~and I - T A ,with the range of XA equal to PA. Theorem 9.4. Suppose the following conditions are satisfied (i) There is a characteristic root X of Eq. (9.8) satisfying ReA > 0 .

62

Delay Differential Equations

(ii) There i s a closed convex set I( C C, 0 E I{, and 6 > 0 such that a = a(6) E inf(llaxq5ll : 4 E K,II4II = 6) > 0.

(iii) There is a completely continuous function a 2 0 , such that the m a p A defined by

T

: I{\{O}

+ [cr,m),

takes K\{O} into I{ and is completely continuous. T h e n 0 i s a n ejective point of A . In application, the complete continuity of T can be weakened to: T : U n I{\{O} + [ a , m ) ,a 2 0, is completely continuous, where U is a neighborhood of 0 in C. Combining Theorems 9.2, 9.3 and 9.4, we have the following: Theorem 9.5. Suppose that K is a closed convex set in C , 0 E I(, such that (i)-(iii) of Theorem 9.4 are satisfied. Suppose either of the following conditions holds: (1) 'I is bounded and infinite dimensional; (2) There i s a n M > 0 such that Az = Xz, x E Ii'n S M , implies x < 1. T h e n Eq. (9.7) has a nonzero periodic solution with initial value in K \ { O } .

3 Characteristic Equations 3.1. Discrete Delays-Preliminaries

Most studies on delay differential equations start from the local stability analysis of some special solutions. For this purpose, the standard approach is t o analyze the stability of the linearized equations about the special solution. If the delay differential equations are autonomous and the special solution is constant, then the linearized equations take the form of linear autonomous delay differential equations. The stability of the trivial solution (i.e., the zero solution) of the linearized equations depends on the locations of the roots of the associated characteristic equation. When delays are finite, the characteristic equations are functions of delays, and hence the roots of these characteristic equations are also functions of delays. As the lengths of delays change, the stability of the trivial solution may also change. Such phenomena are often referred to as stability switches. In the next three sections, we consider the question of stability switches in general linear neutral delay differential equations. The key technique utilized here was developed by Cooke and Grossman (1982) mainly for retarded equations with one discrete delay. Later on, Cooke and van den Driessche (1986) extended this technique to retarded equations with several discrete delays, and Freedman and Kuang (1991) extended it to neutral delay differential equations with one discrete delay. The material of this and the next two sections is adopted from Freedman and Kuang (1991). Throughout this chapter, the stability of a delay differential equation is referred to as the stability of its trivial solution. We consider the following general linear real scalar neutral differential difference equation with a single delay T (T > 0):

do A where ;iSaz(t) = ~ ( t ) . It is well known that if the characteristic equation associated with a linear neutral equation has roots only with negative real parts, and if all the

63

Delay Differential Equations

64

roots are uniformly bounded away from the imaginary axis, then the trivial solution of the linear neutral equation is uniformly asymptotically stable. Thus, the stability analysis of Eq. (1.1) is very much equivalent to the problem of determining conditions under which all roots of its characteristic equation n

bkXk

z a k X k +

k=O

e-Xr

(k10

=o

(1.2)

lie in the left half of the complex plane and are uniformly bounded away from the imaginary axis. In the remainder of this section, we denote

k=O

k=O

Without loss of generality, we assume an = 1. Theorem 1.1. If lbnl > 1, then, f o r all 7- > 0, there is a n infinite number of roots of P(X) Q(X)eWxr= 0 , (1.4) whose real parts are positive. Proof. Consider the equation

+

V(X,7-)= 1

- 0.

+ bne-Xr

(1.5)

The roots of (1.5) are given by Xk

1 = -(h(-bn)

+2 k 4 ,

7-

k = O , f l , f 2 , .. . .

Since lbnl > 1, Re(ln(-b,)) = In lbnl > 0. Equation (1.4) can be rewritten as

+ bnXne-Xr + R ( X ,

7-1

where R(X,T) =

n-1

c akXk +

= 0,

e-Xr. w e consider

k=O

It is clear that V(X,7-)is analytic when X lim

Ikl-00

# 0, and

v ( X k , 7 ) = 0.

(1.6)

3. Characteristic Equations

65

Let 0 < T < In lbnl /(2r), and let B ( X k , T ) be the circle centered at X t with radius r . We observe that

U ( X+~ r e i0 ,r ) = U ( X+~reie, T I , where U ( X , r ) = 1

+ bne-Xr.

(1.10)

Hence, we can denote

(1.11) Since R(A,r) contains only terms ajXj or bjXje-",

Therefore, there exists I( > 0 such that, when llcl

which implies that, when llcl

where j

< n , we have

> I(,

> It', (1.13)

By RouchC's theorem, we conclude that, when llcl > It', there is a X k such that Xr; lies inside B(Xk,r), and V ( X k , r ) = 0. Since 0 < r < In l b n 1 / ( 2 ~ ) , 0 we see that ReXk > 0. This completes the proof. An immediate consequence of this theorem is the following: Theorem 1.2. If lbnl > 1, then the trivial solution of Eq. (1.1) is unstable f o r all r > 0. By virtue of the proof of Theorem 1.1, we have proved the following more general result: Theorem 1.3. Let f(X, r ) = A" ctXne-Xr g(X, r ) , where g ( X , r ) is a n analytic function. Assume IcrI > 1, and

+

1 lim -g(X,T) R e b o An

+

=0

l+m

Then, for all r > 0, there zs a n infinate number of roots o f f ( X , r ) = 0 whose real parts are positive. In fact, there is a sequence { X i } of the roots of f(X,r)= 0 Such that l X j l + 00, and limj-.w ReXj = $ In IayI > 0, when r > 0.

66

Delay Differential Equations

The next theorem is of fundamental importance for our stability switching analysis of the neutral delay equation (1.1). g(X,T), where g ( X , T ) 2s a n analytic Theorem 1.4. Let ~ ( X , T ) = A" function. Assume

+

a = lim sup lA-ng(A,T)l Re ~ > o

< 1.

(1.14)

I+ca

Then, as T varies, the s u m of the multiplicities of roots of ~ ( X , T ) = 0 in the open right half-plane can change only if a root appears o n or crosses the imaginary asis. Proof. We note first that, since f(X, T ) is an analytic function, it can have only a finite number of zeros in any compact set of the complex plane. If f ( A , T ) = 0 has infinite number of roots in the open right half-plane, then there is a sequence { X j } such that f ( X j , ~ ) = 0, ( X j l + 00 as j -+ 00, which in turn implies (1.15) Therefore,

which is a contradiction. Hence, we conclude that the total multiplicity M ( T ) of roots of f(X, T ) = 0 in the open right half-plane is finite. Let X = X ( T ) be any root of f ( X , T ) = 0. If we place a small disk around X ( T ) , then for T' sufficiently close to T , the total multiplicity of roots in the disk equals the multiplicity of X(z). This, again, follows from Rouchd's theorem, which implies that a root X ( T ) cannot suddenly disappear or appear, or change its multiplicity at a finite point in the complex plane. Suppose that M ( T ) changes, but no roots appear on or cross the imaginary axis. This can only occur due to the appearance of a root a t infinity. That is, there would exist T * and a root X ( T ) such that I X ( T ) ~-+ 00 as T -+ T* 0 (or T + T* - 0), with Re X ( T ) 2 0. However, since le-rA(r)l 5 1, when ReX(7) 2 0 we have

+

X-"f(X,

T) =

1

+ X--"g(X, T),

and l X - " f ( X , ~ ) l 2 i ( 1 - a) > 0, when T is close enough to contradicts the fact that ~ ( X ( T ) , T ) = 0, completing the proof.

T*.

This

0

9. Charnclerislic Equations

67

3.2. Discrete Delays-First Order Equations In this section, we consider the following first order real scalar linear neutral delay equation:

d z ( t ) a dx(t - T ) dt dt +

+ P x ( t ) + yz(t -

where r , a, P, 7 are real constants. equation is

T)

= 0,

Its corresponding characteristic

x + aXe-Ar + P + ye-Ar

= 0.

(2.2)

By Theorem 1.2 we see that, when la1 > 1, the trivial solution x ( t ) = 0 of (2.1) is always unstable for all T > 0. Therefore, we assume la1 c 1 in the following discussion. The case of la1 = 1 will be treated as a critical case. In case la1 < 1, by Theorem 1.4, we know if the stability of the trivial solution z ( t ) E 0 of (2.1) switches at r = 7 , then (2.2) must have a pair of pure conjugate imaginary roots when r = 7 . In fact, because of Theorem 1.4, we can think of the roots of (2.2) as continuous functions in terms of the delay r , i.e.,

+

+ +

~ ( r )a ~ ( r ) e - ’ ( ~ P ) ~ ye-A(r)r = 0.

Therefore, in order to understand the stability switches of (2.1) in detail, it is crucial to determine the value of 7 at which (2.2) may have a pair of conjugate pure imaginary roots. We assume X = iw, w > 0 is a root of (2.2) for r = 7 , 7 2 0, which implies the following assumption: (HI) P + r # O . We will treat P y = 0 as a critical case. From (2.2), we have

+

+ +

awsinwr p ycoswr = 0, w awcoswr - ysinwr = 0.

+

(2.3)

By moving P and w to the right hand side of Eqs. (2.3), squaring them, and adding them together, we obtain

+

a2w2 y2 = w2 Hence.

+ p2.

Delay Differential Equations

68

From (2.2), we have (1

+ [a-

dA + y ) ~ e - ' ~ )d r = X(OX + ~

T ( ~ x

) e - ~ ~ .

(2.6)

We assume

(H2) y 2 > P2. Under this assumption, we see that purely imaginary roots of (2.2) exist and are simple. By (2.6), we have

Thus, s i g n { v }

=sign{Re($)-'} k i w

k i w

= sign

{ -}

1- "2

w2

+

p2

{

= sign 1 - a 2 }= 1.

Equation (2.4) was used in the second to last step. The last step is valid since w # 0. Therefore, if la1 < 1, then

This implies that all the roots that cross the imaginary axis a t iw cross from left to right as T increases. We consider the following two cases.

69

3. Characteristic Equations

Case 1. P + y < 0. Let 7 = 0 in (2.2). Then, we have that X(0)(1 + a)= -(P and hence X(0) = -(P

+ Y),

+ y ) / ( l + a) > 0;

i.e., the trivial solution of the neutral delay equation (2.1) is unstable when there is no delay, and by (2.8) it will remain unstable for all r > 0. Case 2. P + y > 0. In this case, X(0) = < 0; i.e., Eq. (2.1) is asymptotically stable when there is no delay. From (2.3), we have

-%

-(ad2

cos w r =

y2

+ PY)

+ a2w2

and (2.10) Hence, there is a unique 8, 0 < 8 5 2 ~such , that w r = 8 makes both (2.9) and (2.10) hold. Note y 2 > p2, p y > 0; hence, y - /3 > 0, and therefore y > [PI 2 0 + sinwr > 0. Hence,

+

8 = arccot and 0

();apP:$ -

< 8 < T . We denote 70

=

0

-.

(2.11)

(2.12)

W

Then the preceding arguments together with the proof of Theorem 1.4 show that when 0 < r < 70, the trivial solution of Eq. (2.1) is uniformly asymptotically stable; and when T > T O , it is unstable. Remarks 2.1. It is easy to see that if y2 < P2, then there are no pure imaginary roots for Eq. (2.2). In other words, there are no roots of (2.2) crossing the imaginary axis when r increases. Therefore, there are no stability switches, no matter how the discrete delay r is chosen. In the case that y = /3 # 0, we see that w = 0 is the only solution of (2.5). However, X = 0 is not the root of (2.2) by assumption ( H l ) ; hence, there is no stability switch as well.

70

Delay Differential Equations

With the exception of some critical cases, we have obtained the following complete stability switching analysis for the neutral delay equation (2.1).

Theorem 2.1. In (2.1), a s s u m e la1 # 1; t h e n the following s t a t e m e n t s are true. (1) If la1 > 1, t h e n (2.1) is unstable f o r all positive delay r . (2) 'I la1 < 1, y2 < p2, o r y = p # 0, t h e n increasing r does n o t change the stability of (2.1). (3) ~f IQI < 1, y2 > p2, and (i) /3 y < 0 , t h e n (2.1) is unstable f o r all positive delay r ; (ii) p y > 0 , t h e n (2.1) i s uniformly asymptotically stable w h e n r < 70 and unstable w h e n T > 70, where 70 = O/w, and

+ +

The two main critical cases that remain unsettled are the following: Critical Case 1. I-QI = 1. In this situation, Theorem 1.4 is no longer valid. Hence, the previous analysis no longer works. To analyze the stability of (2.1), new techniques need to be developed. Critical Case 2. IcxI < 1, P + y = 0. In this case, X(r) = 0 is always a root of (2.2) for all r 2 0. Therefore, (2.1) can never be asymptotically stable. The questions of stability and instability of (2.1) are, however, still of interest. A partial analysis of these critical cases follows. Assume X = u iu is a root of (2.2); then, (2.2) implies

+

u

+ -Que-rucos r u + -Que-'u sin rv + P + ye-'' u - aue-r" sin r u + sue-'" cos r u - ye-"'

cos r u = 0, sin rv = 0.

(2.13)

Analysis of Critical Case 1. I - Q I = 1. We assume first (i) cx = -1. In this case, we have three subcases to be discussed. (a) A s s u m e y P = 0. Then, (2.2) is equivalent to

+

(2.14)

9. Characteristic Equations

71

If /3 < 0, then (2.1) is always unstable for T 2 0; whereas if /3 2 0, then (2.1) is stable but not asymptotically stable. (b) Assume /3 > 171. In this case, (2.2) does not make sense when T = 0. Assume (2.2) has a root X = u iv, where u 2 0, for some T > 0. From (2.13), we have

+

(u

+ /312 + v2 = e-2ru ((au+ y)2 + a2v2).

(2.15)

+ /3)2 + v2 I(au+ $2 + a2v2.

(2.16)

Hence,

(u

Since a = -1, we have (2.17) + 7 ) Ir2- P2. In the case /3 > 171, we have /?+ y > 0, y2 - P2 < 0. Hence, (2.17) implies 2u(P

+

2u(P y) < 0, which contradicts the assumption that u 2 0. Hence, all the roots of (2.2) must have negative real parts for T > 0, which implies in this case (2.1) is asymptotically stable for all positive delays. (c) Assume 7 > 1/31. Suppose in this case (2.2) has a root X = u iv, where u 5 0, for some T > 0. Then, by (2.15), we see that

+

(u

+ P)2 + v2 2 (au+ r)2+

Hence, 2u(P

+

+ 7 ) 2 y2 - P2.

2 2

.

(2.18) (2.19)

+

Since y > IPI, we have /3 y > 0, y2 - p2 > 0, which implies 2u(P y) 2 y2 - P2 > 0. This contradicts our assumption that u I 0. Therefore, in this case all roots of (2.2) have positive real parts when T > 0, which implies that (2.1) is unstable for all positive delays. Now, we assume (ii) a = +l. In this case, we also have three subcases to be discussed. (a) Assume /3 = y. Then, (2.2) is equivalent to (2.20)

Hence, if /3 2 0, then (2.1) is stable for all T 2 0 (but not asymptotically stable); while if /3 < 0, then (2.1) is always unstable. (b) Assume P > 171. Suppose in this case (2.2) has a root X = u iv, where u 2 0, for some T 2 0. Then, by (2.15), we have

+

2

0 - 7 ) I Y2 - P2.

(2.21)

Delay Differential Equations

72

Since ,B > 171, we see y2 - ,B2 < 0, ,5 - 7 > 0; therefore, (2.21) contradicts the assumption that u 2 0. This implies that all the roots of (2.2) have negative real parts for all T 2 0. Hence, (2.1) is asymptotically stable in this situation. (c) Assume y < -[PI. Suppose in this case (2.2) has a root X = u + i v , with real part u 5 0 for some T 2 0; then, by (2.15), we have (2.22) But y2 - P2 > 0, ,8 - y > 0, u 5 0 makes (2.22) impossible. This contradiction implies that in this case, for all T 2 0, all the roots of (2.2) have positive real parts; therefore, (2.1) is always unstable. Analysis of Critical Case 2. (a1< 1, P + y = 0. In this situation, we have two subcases to be discussed. (i) Assume P 2 0. Suppose (2.2) has a root X = u iv, where u > 0, for some T 2 0. Then, (2.16) applies to our case, from which we obtain

+

+

(1 - a2)u2 sup( 1

+ a)+ v2(1 - a 2 )< 0.

(2.23)

+

However, this contradicts the fact that (1 - a2)u2> 0, 2up(l a) 2 0, v2(1 - a 2 ) 2 0. Hence, in this case, (2.1) is always stable (but not asymptotically stable). (ii) Assume P < 0. Since /3 y = 0, we see that y = -P > 0. Equation (2.2) is equivalent to

+

(2.24) + a~ + y = 0. Denote ~ ( X , T ) = ( A - y)eXr + aX + y. Here, ~ ( X , T ) is considered as a

(A

- y)exr

function of real A. We have

h ( 0 , T ) = 0, and

dh -(A, dX

T)

h ( Y , T ) = y(1 + a ) > 0.

= a + eXr + (A - y)~e".

(2.25) (2.26)

(2.27)

+

Hence, if T > ~ - ' ( l a),then ~ ( O , T )< 0, which implies that there is a S > 0 such that, when 0 c X 5 S, h(X,7) < 0. Since h ( y , ~ )> 0, we see there is at least one 6 5 < y, such that h ( x , ~ =) 0; i.e., in case T

> y-'(l

x,

x

+ a),(2.2) always has a positive real root, which implies that

(2.1) is unstable.

9. Characteristic Equations

73

Example 2.1. Consider the following neutral delay logistic equation:

q t ) = T l ( t ) [ l- 3(t - T ) + p 3 ( t - T ) ) / K ] ,

(2.28)

which was introduced in Section 1.1.4. Let u ( t ) = 1 - z(t)/Ii';then, we have

q t ) = r ( u ( t )- l ) [ u ( t- T ) + p q t - T ) ] .

(2.29)

We are interested in the stability of the zero solution u ( t ) 0 of Eq. (2.29), which is equivalent to the stability of z ( t ) E K in Eq. (2.28). Linearizing (2.29) at u ( t ) = 0, we obtain

Y ( t ) = -ry(t

- T ) - rpY(t

- T).

(2.30)

Its corresponding characteristic equation is

x + ~ r p e - ~ +' re-"

= 0.

(2.31)

Comparing (2.31) with (2.2), we see, in this case, cr = rp,

p=

0,

(2.32)

y = r.

From (1) of Theorem 2.1, we conclude that if rc > 1, then the zero solution of (2.30) is unstable for all T 2 0. If r p < 1, then, since y2 > p2 and p y = r > 0, part (ii) of (3) in Theorem 2.1 applies, from which we obtain

+

w = (r2/(1 - a2))1'2 = r ( l - r 2 p 2 ) -112

8 = arccot(-pw),

,

(2.33) (2.34)

and

Since the trivial solution y ( t ) = 0 of (2.30) has the same stability as x ( t ) = Ii' in (2.28), we have proved the following theorem. Theorem 2.2. In Eq. (2.28), if r p > 1, then the steady state solution x ( t ) = I( is unstable f o r all T 2 0. If rp < 1 and T < TO, where TO is defined as (2.35), then x ( t ) = Ii' is uniformly asymptotically stable; while if T > 70, it is unstable.

74

Delay Differential Equations

This result improves Theorem 2.1 in Gopalsamy and Zhang (1988), which is obtained via an algebraic approach. 3.3. Discrete Delays-Second

Order Equations

In this section, we consider the following second order real scalar linear neutral delay equation: d2z(t)

dt2

+a

dz(t) d2z(t- r ) + a -+b dt2 dt

dz(t

-T)

dt

+cz(t)+dl(t-r)

= 0 , (3.1)

where 7,a, a, b, c, d are real constants. Its corresponding characteristic equation is X2

+ aX2e-Ar + aX + bXe-Ar + c + de-"

= 0.

(3.2)

As in the first order case, we see that, when la1 > 1, the trivial solution, i.e., the zero solution z ( t ) = 0 of (3.1), is always unstable for positive delay T . Again, we assume l a1 < 1 in the following discussion, and the case of la1 = 1 will be regarded as a critical case. Suppose X = iw, w > 0, is a root of (3.2)for some r . Assume (Hl) c + d # 0. This assumption implies that w # 0. We have c

- w2 + bu sin w r

+

(d - a w 2 )cos w r = 0, a w + b u c o s w r - ( d - a w 2 )sinwr = O .

Thus,

+

(w2 - c ) ~ a2w2 = b2w2

Hence, (1 - a2)w4

+ (d -

0

~

~

(3.3)

)

+ (a2 - b2 + 2da - 2c)w2 + c2 - d2 = 0.

~

.

(3.4)

(3.5)

Its roots are w 2f = -(1 1 - a')-'{ 2

(b2

+ 2c - a2 - 2 d a ) 112

f [ ( b 2 + 2c - a2 - 2da)2 - 4(1 - a 2 ) ( c 2- d2)]

}.

(3.6)

If c2 5 d2, then there is only one imaginary solution, X = iw+, w+ > 0. If > d2, there are two imaginary solutions, Ah = iwh, with w+ > w- > 0, provided that the following are true: (a) b2 2c - a2 - 2da > 0 ; and (b) (b2 2c - a2 - 2 d ~ >) 4(1 ~ - a 2 ) ( c 2 - d2); and no such solutions otherwise. c2

+ +

3. Characteristic Equations

75

Again, we need t o determine the sign of the derivative of ReX(r) a t the points where X(r) is purely imaginary. From (3.2), we have {2X + a

+ [b - 7(bX + d ) + aA(2 - X T ) ] ~ - ~ ‘ }

+ +d )e-?

d X ( r ) = A (d2 bX dr

(3.7) If A(?) = iw is not simple, then dX(T)/dr = 0 a t T = 7. Since w # 0, e-lwr # 0. Hence, cr(iw)’ + biw d = 0 , which implies b = 0, d = aw’. From (3.4), we see that a = 0, w’ = c. Therefore, a = b = 0 , d = a c in this case, and (3.2) is equivalent to

+

(A’

1 = o’

+ c ) (1 +

(3.8)

+ +

which is easy t o analyze. Hence, we assume u’ b’ (d - ac)’ # 0, which guarantees the simplicity of X = iw. Again, for convenience, we study ( d X / d r ) - l instead of d X / d r . We have

(2)

+

+ +

r (2X a ) e X r b 2aX - = X(aX2+bX+d) A’

(3.9)

and (3.10)

Therefore,

{

(g)} -1

sign T} d(ReX) = sign {Re X=iw

X=iW

+ aX + c ) ] +Re [ X(aX2+ bX + d ) w’) - aw’) - 6’ = sign + (2a(d { ( ca’-- 2(c +- a2w2 d-~ w ’ + ) b2w2 ~

=sign{Re[

-(2X+u)

X(X2

A=lw

w2)2

= sign {a’

+ 2ad - 2c - 6’ + 2w2(1 - a’)}.

(3.11)

Equation (3.4) was used in the last step. By inserting the expression for w:, it is seen that the sign is positive for w i and negative for w:. In the case of c’ < d2, only one imaginary root exists, X = iw+; therefore, the

76

Delay Differential Equations

only crossing of the imaginary axis is from left to right as T increases, and the stability of the trivial solution can only be lost and not regained. In the case of c2 > d2, crossing from left to right with increasing r occurs whenever T assumes a value corresponding to w+, and crossing from right to left occurs for values of T corresponding to w-. From Eq. (3.3), we obtain the following two sets of values of T for which there are imaginary roots: 81 2n?r (3.12) Tn,1 = - -, w+

where 0

+ w+

5 81 < 27r, and ah:

+ ( c - w:)(d

- aw:)

cose1=

-

sin81 =

(d - aw:)aw+ - h + ( c - w:) b2w: ( d - CYW:)~

and

+ (d - aw:)2

b2w:

+

62

Tn,2 = -

w-

+ -,2w-n

’

(3.13) 9

~

(3.14)

(3.15)

where 0 5 82 < 2a, and

+

a h ? ( c - w2_)(d- awZ) b2w: ( d - ~ w L ) 7~ ( d - CIWZ)UW- h - ( c -wZ) sin82 = , b2wz ( d - C X W ? ) ~

cos82 = -

+

(3.16)

+

(3.17)

w h e r e n = 0 , 1 , 2 ... . In the case that c2 < d2, only TO,J need be considered, since if (3.1) is asymptotically stable for T = 0, then it remains asymptotically stable until TOJ,and it is unstable thereafter. At the value of T = TOJ, (3.2) has pure imaginary roots, f i w + . In the case that c2 > d 2 , if (3.1) is stable for T = 0, then it must follow that TO,J < 7 0 , 2 , since the multiplicity of roots with positive real parts cannot become negative. We observe that (3.18) Therefore, there can be only a finite number of switches between stability and instability. Moreover, it is easy to see that there exist values of the parameters that realize any number of such stability switches. However,

77

9. Characteristic Equations

there exists a value of r , r = i,such that at T = .ia stability switch occurs from stable t o unstable, and for r > .i the solution remains unstable. If (3.1) is unstable for r = 0, then a similar argument as before can be made. Equation (3.1) can either be unstable for r > 0, or exhibit any number of stability switches as in the preceding case. As r is increased, the multiplicity of roots for which R e x > 0 is increased by two whenever T passes through a value of rn,l, and it is decreased by two whenever r passes through a value of rn,2. When the zero solution is stable for r = 0, k switches from stability to instability to stability occur when the parameters are such that

or k switches from instability to stability to instability m a y occur when 70,2

< r0,l < 71,2 <

’* *

< 7k-1,2 < Tk-l,l < Tk,1 < 7k,2

* * * 7

when the zero solution is unstable for r = 0. The conditions on the parameters in order that the preceding orderings be valid can be formulated directly from (3.12)-( 3.17). We can summarize the preceding results as the following theorem. Theorem 3.1. In (3.1), assume IayI < 1, c+d # 0, and a2+b2+(d-ac)2 # 0. T h e n u m b e r of different imaginary roots with positive (negative) i m a g i n a r y parts of (3.2) can be zero, one, o r t w o only. (1) If there are n o s u c h roots, t h e n the stability of the zero solution does n o t change f o r a n y r 2 0. (2) If there i s one imaginary root with positive imaginary part, a n unstable zero solution never becomes stable for a n y r >_ 0. If the zero solution is asymptotically stable f o r T = 0, t h e n it is uniformly asymptotically stable f o r r < TO,J,and it becomes unstable f o r r > 7 0 , ~ . (3) If there are t w o imaginary roots with positive imaginary part, iw+ and iw-, s u c h that w+ > w- > 0 , t h e n the stability of the zero solution can change a finite n u m b e r of t i m e s at m o s t as r is increased, and eventually it becomes unstable. As in first order case, we have the following two critical cases remaining. Critical Case 1. la1 = 1. In this case, Theorem 1.4 is no longer valid. Critical Case 2. la1 < 1, c d = 0. In this case, X(r) = 0 is always a root of (3.2) for all r 2 0. As before we are able to obtain partial results in these critical cases.

+

Delay Diffemniial Equations

78

+ iv is a root of (3.2); then, we have + 2iuv + a(u2 - v2 + 2 i u ~ ) e(cos - ~ v~r - i sin v r )

Assume X = u u2-v2

+ QU + iav + b(u + iv)e-ur(cos v 7 - i sin vr) + c + de-ur(cosvr - i sin vr) = 0.

(3.19)

Therefore,

+ au + c + ((r(u2- v2) + b~ + d)e-ur cos + ( 2 a u v + bv)e-ur sin v r = 0, 2uv + av + (2~x212) + bv)e-ur cos vr - (a(u2- v2) + b ~ l + d)e-ursin 07 = 0.

u2 - v2

VT

(3.20) (3.21)

From (3.20) and (3.21), we obtain

( 2- v2 + au + c)2 + (2uv + av)2 = e-2ur ( ( a ( u 2- v2)

+ bu + d ) 2 + (2cruv + bv)2)

(3.22)

Analysis of Critical Case 1. IQI = 1. (i) Stability result. Assume c > Id1 and a 2 Suppose (3.2) has a root X = u iv, where u 2 0, for some T 2 0. By (3.22), we see, for Q = f l , the following is the consequence of the assumption of u 2 0;

.-/,

+

2au3

+ 2auv2 + ( a 2 + 2c)u2 + ( a 2 - 2c)v2 + 2acu + c2 < 2cubu3 + 2bauv2 + (b2 + 2da)u2 + (b2 - 2da)v2 + 2bdu + d 2 .

(3.23)

However, it is easy to see that this contradicts the fact that c > [dl and a 2 Therefore, we have shown that if la1 = 1, c > Id[,

.-/,

and a 2 T > 0.

d

m

, then (3.1) is always asymptotically stable for all

(ii) Instability results. (a) Assume Q = -1, d 5 - 1 ~ 1 , and b 2 Suppose (3.2) has a root X = u iv, where u 5 0, for some 7 2 0. From (3.22), we

+

.-/,

have

2au3

+ 2auv2 + (.2 + 2+2 + (.2 - q v 2 + 2acu + c2 2 -2bu3 - 2buv2 + (b2 - 2d)u2 + (b2 + 2d)v2 + 2bdu + d2.

(3.24)

9. Characteristic Equations

79

However, this cannot be true with the assumption that d 5 -1cI and b2 Therefore, (3.1) is unstable in this case for all T 2 0. (b) Assume a = 1, d > IcI, and b 5 - d 2 d - 2c u2. Suppose (3.2) has a root X = u iv, where u 5 0 for some T 2 0. Then, from (3.22), we have

d

m

.

+

+

2au3

+ 2auv2 + (.2 + z C l u 2 + (.2 - 2c)v2 + 2acu + c2 2 2bu3 + 2buv2 + (b2 + 2d)u2 + (b2 - 2d)v2 + 2bdu + d2.

(3.25)

Again, it is easy to see (3.25) contradicts the assumption that d > IcI and b 5 -2d - 2c a2. Hence, all roots of (3.2) must have positive real parts, which implies that (3.1) is unstable for all T 2 0. Analysis of Critical Case 2. la1 < 1, c d = 0. (i) Stability result. Assume c 2 0, a 2 d w .Suppose (3.2) has a root X = u iv, where u > 0, for some T 2 0. Then, from (3.22), we have

+

+

+

+

+ +

(1 - a2)(u2- v2)2 (1 - a)4u2v2 2(" - ba)(u3 ( a 2 2c - b2 - 2da)u2 ( a 2 - 2c - b2 2(ac - bd)u c2 - d2 < 0.

+ +

+

+

+ u.2) + 2da)v2 (3.26)

+

This is impossible, since we are assuming 101 < 1, c 2 0 , c d = 0, a 2 d w ,and u > 0. Hence, in this case, all roots of (3.2) have nonpositive real parts; this implies that (3.1) is stable (but not asymptotically stable, since X ( T ) = 0 is always a root of (3.2)). (ii) Instability result. Assume c < 0, and T > d-'(a b). Consider the following real function:

+

[(A,

T) =

X2

+ aX2e-Xr + aX + bXe-" + c + de-".

(3.27)

P(0,T) = 0

(3.28)

We observe

and lim ~ ( X , T ) = 00. x++m By the assumption that la1 < 1 , we see there exists a Ad X 2 M , !(X,T) 2 0. We also have

-d C ( X ' T ) - 2X dX

(3.29)

> 0 such that, if

+ a + (2aX - aX27+ b - ~ A -T dT)e-Ar.

(3.30)

80

Delay Differeniial Equaiions

+

+

Therefore, al(O,~)/aX= a b - dT < 0, since T > &'(a b). This implies that, when T > d-'(a b), there exists a 6 ( ~ )> 0 such that when 0 < X 6 b ( ~ ) ~, ( X , T )< 0. Therefore, there must exist at least a A, 6 ( ~< ) 5 M , such that C ( ~ , T )= 0; i.e., (3.2) has at least a positive root. Hence, (3.1) is unstable. Example 3.1. Freedman and Rao (1983) studied the following model of predator-prey interactions with mutual interference and time lag in gestation:

+

d

-dt4 t ) = + ( t ) g ( x ( t )) Y(t)"P(X(t)),

where 3, T , and m are positive constants, 0 5 m 5 1. z ( t ) represents prey population at time t , and y ( t ) represents predator population at time t. The following properties are assumed: (i) g ( 0 ) > 0, dg(x)/dx 5 0. There is a I(, I< > 0 (which is referred to as the carrying capacity of the environment), such that g ( x ) ( z- I<) < 0 when x # I<. (ii) ~ ( 0 = ) 0, d p ( x ) / d x > 0, q(0) = 0, d q ( y ) / d y 2 0. Assume the system (3.31) has a positive equilibrium E(x*,y*). The variational system associated with (3.31) about E is

d -u(t) = H u ( t ) N v ( t ) , dt d -v(t) = Qu(t - T ) R v ( t ) , dt

+

+

(3.32)

where

M Y * ) < 0. R = ( m - l)cy*"-'p(x*) - y* dy System (3.32) is equivalent to the equation

d2 -u(t) dt2

d dt

- ( H + R ) -u(t)

+ HRu(t)- N Q u ( t -

T)

= 0,

(3.33)

9. Chamcierisiic Equations

81

whose characteristic equation is X2 - ( H R)X HR - NQe-" = 0. (3.34) Comparing (3.34) with (3.2), we see in this case that a = 0, a = ( H R ) , b = 0, c = HR, d = - N Q . From (3.6), we have 1 (3.35) w: = - (-(IT2 R2)f d ( H 2 - R2)2 4N2Q2) . 2 If lHRl > -NQ, then the right hand side of (3.35) will be always negative. By (1) of Theorem 3.1, we see there are no stability switches. In the case H R < N Q < 0, we see E(z*,y*)is unstable when T = 0; therefore, it remains unstable for all T > 0. In the case H R > - N Q > 0, since R 5 0, we have H < 0; therefore, E(z*,y*)is asymptotically stable when T = 0; hence, it remains asymptotically stable for all T > 0. In the case lHRl < - N Q , we get one imaginary root with positive imaginary part from (3.35). It is

+

+

+

+

w : = f ( -(H2

+

+ R2)+ d ( H 2 - R 2 ) 2- 4N2Q2

Since we can never have two imaginary roots with positive imaginary part (w- is always negative), we see that the stability of E ( z * ,y*) can only be lost and will not be regained. In the case H + R > 0 , E ( z * ,y*) is unstable when T = 0; therefore, it will remain unstable for all T > 0. If H R = 0 and H R - N Q < 0, we see it is unstable when T = 0; therefore, it remains unstable for all T > 0. If H R - N Q > 0, then w: = H R - NQ, and dReX = sign { ( H R)2- 2NQ) = +l.

+

+

kiw+

+

Therefore, E ( z * , ~ *will ) be unstable for all T > 0. If H R < 0 and H R - N Q < 0, then E(z*,y*)is unstable when T = 0 as well. In the case H + R < 0, and H R = - N Q , E(z*,y*) is stable when T = 0, and w: = 0. However, X = 0 is not a root of (3.34). Therefore, E(z*,y*) will not change its stability. This result is not included in Freedman and Rao (1983). In the case H R < 0 , and lHRl < -NQ, part (2) of Theorem 3.1 applies. It is clear that E(z*,y*)is asymptotically stable when T = 0. From (3.13) and (3.14), we obtain 1 (3.36) cos81 = - ( H R - w 2 ) , + NQ 1 (3.37) sin81 = -((H R)w+), NQ

+

+

(3.38)

Delay DijJereniial Equations

82

Therefore, we have shown the following: R < 0 , lHRl < - N Q . Then Theorem 3.2. In (3.31), assume H E ( z * , y * )is uniformly asymptotically stable when r < T O J , where r0,1 is computed b y formula (3.38). E(z*,y*)is unstable when 7 > TOJ: This improves Theorem 4.1 in Freedman and Rao (1983), where, under the same assumptions, they conclude only that E(z*,y * ) is asymptotically stable when 0 5 r < ( H + R ) / N Q . Obviously, ro,~= 81/w+ > sin81/w+ = (H +R)/NQ. In the case H + R 5 0 , and HR - N Q = 0 , our instability result for Critical Case 2 applies; by which we conclude that if r > (H R ) / N Q , then E ( z * , y * )is unstable. This result is also not included in Freedman and Rao (1983). We also observe that the G. J. Butler's Lemma stated at the end of Freedman and Rao (1983) is a simple corollary of our Theorem 1.4.

+

+

3.4. Discrete Delays-General

Theory From previous sections, we know that, for a scalar delay differential equation

the characteristic equation takes the form

+

P(A) Q(A)e-Xr = 0 ,

(4.2)

where k=O

k=O

For equations with several delays and systems with one or more delays, their characteristic equations typically take the form

c Pj(A)e-Xrj + Q(A)e-" I

= 0,

(4.4)

j=O

where 0 = 70 < r1 < < TI and r 2 0. The term containing e-Xr has been separated for special consideration. If we let I

P ( A )=

C Pj(A)e-Arj,

j=O

(4-5)

83

9. Charuclerislic Equalions

then (4.4) reduces to the form of (4.2). In this case, P(A) is an analytic function and may not be a polynomial. The following general result is a corrected version of the main theorem of Cooke and van den Driessche (1986). The correction was proposed by denotes complex conjugate. Boese (1992). In the following, U-n Theorem 4.1. Consider Eq. (4.2), where P(A) and Q(A) are analytic functions in ReA > 0 and satisfy the following conditions (i) P(X) and &(A) have n o c o m m o n imaginary root; (ii) P ( - i y ) = P ( i y ) , Q(-iy) = Q(iy) f o r real y; (iii) P ( 0 ) Q(0) # 0; (iv) limsup{lQ(A)/P(A)I : 1x1 + 00, ReA 2 0) < 1; (v) F ( y ) IP(iy)12- IQ(iy)I2 f o r real y has at m o s t a finite number of real zeros. T h e n the following statements are true: (a) If F ( y ) = 0 has n o positive roots, t h e n n o stability switch m a y occur. (b) If F ( y ) = 0 has at least one positive root and each of t h e m is simple, t h e n a3 r increases, a finite number of stability switches may occur, and eventually the considered equation becomes unstable. Remark 4.1. (1) Note that hypotheses (i)-(iii) are not strong restrictions. For hypothesis (i), if A = i y is the only common imaginary root of P and Q , then P ( A ) Q(A)e-Xr = ( A - i y ) k ( P 1 ( A ) Q1(A)e-Ar), where k is an integer and PI and Q1 have no common imaginary root, and the preceding theorem then can be applied to Pl(A) Ql(A)e-Ar = 0. If P and Q are functions of real coefficients, then (ii) is always true. (iii) simply states that A = 0 is not a root; otherwise, the considered equation is not uniformly asymptotically stable. (2) In Cooke and van den Driessche (1986), the condition (iv) in Theorem 4.1 was replaced by the following: (iv)' There are at most a finite number of roots of (4.2) in the right half-plane when T = 0. The following example of Boese (1992) indicates that if (iv) is replaced by (iv)', then Theorem 4.1 is false. Consider

+

+

+

+

+ (2 + A),-" = 0. (4.6) In (4.6), P(X) = 1, & ( A ) = 2 + A, and F ( y ) = 1 - (4 + Y ) ~ Equation . (4.6) 1

is stable when r = 0. Clearly, F ( y ) has no zero. If Theorem 4.1 remains true after replacing (iv) by (iv)', then (4.6) should be stable for all r > 0. However, this is manifestly false. Let A = x + i y be a root of (4.6). Then, 1

+ (2 + x + iy)e-"(cos

YT

- i sin y r ) = 0,

(4-7)

Delay Differential Equations

84

which is equivalent to

+ +

ezr = -(2 z)cos(yr) - y sin(yr), 0 = (2 z)sin(yr) - y cos(y7).

(4.8) (4.9)

Hence,

+ z)2. (4.10) Clearly, if z 2 0 is large enough, e2zr > (2 + z ) ~hence, ; y is well defined in Y - e 2zr - (2

(4.10). This implies that the right half-plane ReX > 0 contains infinitely many roots of (4.6). Proof of Theorem 4.1. Note first that (iv) implies (iv)' and that no roots of (4.6) can bifurcate from infinity at r = 0 (see the proof of Theorem 1.4). Assume that X = iy # 0 is a root of (4.2). Because of (ii), we may choose y > 0. Equation (4.2) implies that IP(iy)I = IQ(iy)l*

(4.11)

This clearly implies (a). We set

P(~Y = )&(Y)

+ ;Pr(y),

Q ( ~ Y )= QR(Y)

+ ~QI(Y),

(4.12)

where pR,4, QR and QI are real-valued functions. Then (4.2) implies QR cos r y

+ QI sin r y = -&,

QI cos r y - QRsin r y = - f i .

(4.13)

Hence,

+ Q; + Q?

sinry = -PRQI Q R , ~

cosry =

-&QR

Q;

- PIQI.

+ QT

(4.14)

+

Clearly, Q; Q? # 0; otherwise, P(iy) = Q(iy) = 0. For each root y of (4.11), it may be possible to determine values of T that satisfy (4.14). Now assume that we have found values of iy, r that satisfy (4.2), (4.11), (4.13), and (4.14). We regard the root X(r) = z(r) iy(r) of (4.2) as a function of 7, and we try to determine the direction of motion of Z ( T ) as r is varied. That is, we determine

+

Since the left side of (4.2) is an analytic function of X and r , a root X(r) will be a differentiable function of r , except at points where the root is multiple. At a multiple root, we have

P'(X) + [&'(A) - T Q ( X ) ] ~ - ~ ' = 0,

(4.16)

85

9. Characterisiic Equations

or, since e-Ar = -P(A)/Q(A),

+

P'(A)Q(A)- P(A)&'(A) .P(A)Q(A) = 0.

(4.17)

Assuming that (4.16) is not true, we may consider A = A(T) to be a differentiable function, and then differentiating (4.2) with respect to s yields dA _ .\&(A) ds P'(A)eAr+&'(A) - .&(A)' Using (4.2), we obtain

(2)-'=-m

P'(A) +--&'(A) AQ(A)

s

A'

(4.18)

Note that (4.18) holds at any simple root i y of (4.2). Hence,

Since IP(iy)I = IQ(iy)I, we thus have, for y > 0,

S = -sign Im[P'(iy)P(iy) - &'(iY)&(iy)].

(4.20)

Clearly, if S # 0, we can determine the direction in which A(s) crosses the imaginary axis at iy. Note that S is independent of s . Observe that

+

- Im [P'(iy)P(;y) Q'(iy)Q(iy)] = P R P ~PiPf - Q R Q ~- &I&;,

and

F'(Y)= ~ ( P R+PPipi ~ - Q R Q ~- &I&!).

Hence,

S = sign F'(y).

-

Assume -y1 > y2 > > y, > 0 are constants such that iyk, k = 1 , . .. , p , are simple roots of F ( y ) = 0 that cross the imaginary axis at iyk at values 'rk,,, (rn = 1 , 2 , . . . ) determined by (4.14). It is easy to see that crossing at two adjacent simple roots yj and yj+l must be in opposite directions, since F'(yj)and F'(yj+l) must have -

a

86

Delag Dzffemntial Equations

opposite signs. Furthermore, the difference between the crossings at a given point iyk is

T

values for

Tk(rn+l) - Tkm = 2r/Yk*

Therefore, crossings occur most frequently at iyl , next most frequently at iy2,. . . , and least often at iy, as T + 00. Suppose that F(y) = 0 has at least one positive root, and that each positive root is simple. As T + 00, more roots cross at iyl than at iy2, and more at iy2 than at iy3, and so on. We claim that the crossings at iyl must be to the right. For if they were to the left, then for sufficiently large T there would be a negative number of roots in the right half-plane, which is impossible. We thus conclude that, if y1 > y2 > y,, > 0, and iyk are simple, then crossings at iY2k+l must be to the right, and crossings at iy2k must be to the left. At each crossing, the number of roots in the right half-plane changes by two, since roots occur in conjugate pairs by 0 hypothesis (ii). This clearly implies (b) of our theorem. Remark 4.2. It should be pointed out that, in case (b), it is possible that increasing delay T may stabilize an unstable equation. For example, assume that F(y) = 0 has two positive roots y1 and y2 with y1 > y2. If, when T = 0, (4.2) has one or two roots with positive real parts and 721 < 711, then, for T E (721,711), (4.2) has no roots with positive real parts. The following examples illustrate the applications of Theorem 4.1. We consider the first order equation with two delays

---

x + ae-'lA where 7 1 2 0, 72 2 0, and

a

+ be-QA = 0,

(4.21)

and 6 are positive constants. Denote

P(X) = X

+ ae-r'A ,

Q(X) = b .

(4.22)

We have

F ( y ) = liy + ae-i'1Y12 - b2 = a2 - b2 - 2ay sin q y + y2, 1 - ~ ' ( y )= y - a sin q y - a q y cos T1y. 2 Observe that F ( y ) = 0 is equivalent to 2asinqy = y

+ ( a2 - b2 )/y

g(y).

(4.23)

(4.24)

3. Chamcierisiic Equaiions

87

Consider first the case a = 6. In this case, y = 0 is a root of F(y) = 0. However, X = 0 is not a root of (4.21). Therefore, we only need to consider other zeros of F(y) that satisfy y = 2asin71y. If 2a71 5 1, there are no positive roots, and therefore, by (a) of Theorem 4.1, no stability switches occur as 72 increases. In this case, the stability is that of the equation with 72 = 0: stable for 71 E [0,1/2a]. On the other hand, if 2a71 > 1, then there are an odd number of positive roots of F ( y ) = 0, and stability switching may occur as 72 increases. Next, consider the case 0 < a < 6, so g(y) is monotone increasing and concave down for y > 0, is zero at y = (b2 - a2)lI2, and y - g(y) tends to 0 as y -+ 00. It is easy to see that the graphs of g(y) and rasinsly intersect at an odd number of positive values of y. For 72 = 0, (4.21) reduces to X 6 a e - r l A = 0, and this equation is stable for 0 5 a < 6 and all 71 2 0. Thus, for fixed 7 , (4.21) is stable at 72 = 0. If there is only one positive root of F ( y ) , then all root crossings of the imaginary axis are destabilizing, and so there exists a 72'(depending on 71) such that (4.21) is stable for 0 5 72 5 72' and unstable for 72 > 7;. Further analysis becomes complicated if F ( y ) has more than one positive root. In fact, we have the following sharp result when a = 6 > 0, 71 72 > 0 in (4.21). It is stated and proved in Stkpin (1989), pp. 65-66 by a very different method (the proof there seems to be incomplete!). Theorem 4.2. The trivial solution of

+ +

+

.(t)

+ a z ( t - 71) + a z ( t - 72) = 0

(4.25)

is uniformly asymptotically stable if and only if (4.26)

Proof. The characteristic equation of (4.25) is D(X)

x + ae-rlA + ae-"'

= 0.

(4.27)

Clearly, when 71 = 72 = 0, (4.25) is uniformly asymptotically stable. If the increasing of 71 + 72 leads to instability of (4.25), then there is an w > 0 such that D(iw) = 0 for some 71 and 72. Note that

+

R(w) = ReD(iw) = a[cos(qw) COS(T~TZW)] = 2a cos( F w ) cos( 2 71 - 7 2 w ,

)

I ( w ) = Im D(iw) = w - a sin(qw) - a sin(r2w) = w - 2a sin( v w ) cos( 7 71 - 7 2 w .

)

88

Delay Differential Equations

Observe that R ( w ) = 0 if and only if

If the latter of (4.28) is true, then Z(w) = 0 would imply w = 0, which contradicts our assumption. Hence, we must have

+1 71 + 7 2 2n

w=-

n = 0 , 1 , ....

?r,

(4.29)

Also, R ( w ) = 0 and I ( w ) = 0 together imply

(

w = 2a lcos y

w

)

1,

which implies that

2n 71

+1

+ 72

?r

I

= 2 a cos

--- + 5

(rl

(4.30)

2

or, equivalent 1y,

Hence, if, for n = 0 , 1 , 2 , . . . ,

I

2u(71 + 7 2 ) cos

-7*

+

1-72 72

(rl

2 n + 1 x ) l <(2n+1)7F,

(4.32)

2

then D(X) has no pure imaginary roots. However, one can show that (4.32) holds if and only if (4.26) holds. This is because for 8 E [0, $1, n = 0 , 1 , 2 , . . . , we always have that n sin 8 2 I sin n8l. If we denote

(4.33) then 8 E [ O , ; ] , and = I sin(2n = (2n

+ l)dl 5 (2n + 1 ) sin 8

+ l)cos(

71

- 7 2 r)

2(71

+72)

.

This shows that if (4.26) is true, then (4.25) is uniformly asymptotically stable.

9. Characteristic Equations

89

Finally, we note that when (4.26) fails, there exists ak, 0 < ak 5 a , k = 0 , 1 , . . . ,I( (It' is a finite number dependent on a ) , such that = 0, where

(2k

wk =

71

Also note that

ax

I

(4.34)

=b k ,

(2) I -1

=

=signRe i

~

= sign

+

+ T2

Ma

Hence, when a = ak and sign Re da ~

+ 1).

{

~

-(TI

A=iwc

sin T1Wk

k:

+ ~2 sin

T2Wk)

+

since T1Wk T2Wk = (2k 1)" implies that s i n q w k = sinr2wk, and I ( w k ) = O implies that sin r1wk = Wk/2a > 0. This implies all crossings of characteristic roots of (4.27) when a = ak are from left to right. This shows that if (4.26)fails, then (4.25)is not uniformly asymptotically stable. 0 Remark 4.3. Note that in (4.34) the partial derivative is taken with respect to a , not with respect to delays as discussed in the preceding theory. We have chosen the delay r as our parameter in the presentation of the general theory only for the sake of convenience. The preceding approach can also be applied to equations with several delays. For example, simple sufficient stability conditions can be obtained for n

Z ( t ) + a l i ( t ) + a o ~ ( t= ) C bjZ(t - T j ) ,

(4.35)

j=l

where ao, a l , and rj are nonnegative constants. The following statements are proved in Stdpin (1986) pp. 78-80, by a different method. Theorem 4.3. The following statements are true. (i) The trivial solution of (4.35)as uniformly asymptotically stable for all values of rj if

90

Delay Differential Equations

(ii) The trivial solution of (4.35) is uniformly asymptotically stable

if a0 >

5

and

lbjl

>

a1

j=1

5

1bjl.j.

(4.37)

j=l

Proof. (i) It is easy to see that if rj = 0, j = 1 , . .. ,n, then the trivial solution of (4.35)is uniformly asymptotically stable. If as rj, j = 1,. .. ,n, increase, the stability changes, then there exists a set of rj such that D ( i w ) = 0 for some w > 0, where

(4.38) D ( i w ) = 0 implies n

w2 = a0 -

C bj cos

W T ~ ,

(4.39)

j=1

n

alw

+ C bj sin wrj = 0;

(4.40)

j=l

(4.39)implies

and hence

a contradiction to (4.40). (ii) We need only to observe that, for w a1w

> 0,

n

n

j=l

j=1

+ C bj sin wrj 2 alw - C IbjIwrj > 0,

since 8 2 sin8 for 8 2 0.

0

3.5. Distributed Delays-Special Cases In this section, we present three examples that will illustrate that the

method described in the previous section can be applied to some equations with distributed delay to obtain meaningful stability results.

3. Characteristic Equations

91

Example 5.1.

The following linear equation with both discrete and distributed delays is obtained by linearizing a structured population model in Metz and Diekmann (1986) p. 237:

+ As(t - 1) - q

i = -As(t)

0

l z ( t - 0) d6,

(5.1)

-1

where A > 0 and q > 0 are constants. This equation was thoroughly studied in Smith (1992d), and results to be presented here are stated and proved there via a somewhat different approach. The characteristic equation of (5.1) is 0

A

+ A - Ae-', + q / -1 eAed6 = 0.

The following theorem is a part of the main result in H. L. Smith (1992b). Theorem 5.1 (H. L. Smith, 199213). There ezists a curve I' = {(A,q) : q = ij(A), A > 0}, where i j : (0,O) --+ (0, m) satisfies dij/dA > 0, ij(m) = 2x2. (5.3) If (A,q) belongs to I', then (5.2) has ezactly one pair of purely imaginary roots Ah = * i f i and all other roots X of (5.2) satisfy ReA < 0. The root A+ is simple and satisfies i j ( O + ) = n2/2,

dReX+/aq

> 0,

aReX+/aA

< 0,

f o r (A,q) E

I',

(5.4)

where we view A+ = A+(A,q) as an analytic function of q and A. FOT (A,q) such that q < ij(A), all roob of (5.2) satisfy ReA < 0, and f o r those such that q > ij(A), there esist roots of (5.2) with positive real part. Proof. Note that when q = 0, (5.2) reduces to

A

+ A - Ae-',

= 0.

(5.5)

It is easy to see that except X = 0, all other roots of (5.5) have negative real parts. Also, note that roots of (5.5) are nonzero roots of

D ( A )= x2 + AA + q - (AA+ q)e-', = 0.

(54 Clearly, roots of (5.2) with positive real part (if any) are bounded by 2A q . So if (5.1) changes stability as q increases from zero, then there is a ij(A) > 0 such that, when 7) = ij(A), (5.2) has a pure root iw, w > 0. Since iw is also a root of (5.6), we have

+

w 2 - q = -qcosw - Awsinw, Aw = -qsinw -k Awcosw,

(5.7) (5.8)

Delay Differeniial Equaiions

92

Unstable Region pure imaginary root f@i

/

I

3zJ4

A

Figure 3.1. Stability diagram for (5.1)

which implies that w = q # k2n2/2, k = 0,1,. . ,

.

A.

Also observe that f ( z ) = z(1 z E (kn,(k l ) n ) , since

+

Equation (5.7) thus implies that, for

+ cosz)/sinz

f’(z) = -(sin z - z)( 1

is strictly increasing for

+ cos z)/ sin2 z.

I

(5.10)

It is straightforward to find that lim A(q) = 0

q-r+

and

lim A(q) = +oo.

q-2a-

Hence, we may define the inverse function of A(q) as q = V(A) for A which is also strictly increasing. Clearly, q(A) satisfies (5.3). Observe that if D(X) = D’(X) = 0, then

AX3 + (q + A2

+ A)A2 + 2q(1+ A)X + q2 = 0.

> 0,

(5.11)

93

3. Charncteristic Equations

Applying the well-known Routh-Hurwitz test to (5.11) shows that Re X < 0. Hence, roots of (5.6) with R e X 2 0 are simple. Also, if we view A+ = X+(A,q) as the function of A and q , such that i f i = X+(A,V(A)), then straightforward differentiation yields

aReX+ aA

=X+-I aReX+ all

ltl=ij(~)

X+(e-X+ - 1)

-

By using e-'+

= (A$

v=ii(A)

+ A - Ae-x+ + (AX+ + r))e-X+ A+=,-' + AX+ + q)/(AX+ + q), we obtain 2X+

Equation (5.12) indicates that as 7 passes the value ij(A) from left to right, Re A+ changes from negative to zero to positive. Since w = fiis the only positive value such that D(iw) = 0, we conclude that, for q E [O,V(A)), roots of (5.2) satisfy ReX < 0, and, for 7 > ij(A), (5.2) has roots with 0 positive real part. This proves the theorem. Figure 3.1 should be helpful in understanding Theorem 5.1. Remark 5.1. As we can see from the preceding example, when delay effect is evenly distributed along a finite interval, the resulting characteristic equation of the linearized distributed delay equation is in fact the same as that of a discrete delay equation. Our next example also illustrates this special phenomenon. Example 5.2. Consider the following second order delay equation with evenly distributed delay over a finite interval:

where € , a , and b are nonnegative constants. When 6 = 0, the following theorem is reduced to Theorem 3.24 in StCpin (1989), pp. 81-82. We present in the following a different proof and will describe StCpSn's proof in Section 3.9 to illustrate the theory adapted in his book. Theorem 5.2. W h e n 6 = 0 , the trivial solution of (5.14) is uniformly asymptotically stable i f and only i f a > b > 0 and a # 4n2r2, n = 1 , 2 , . . . . W h e n 6 > 0, the trivial solution of (5.14) i s always uniformly asymptotically stable.

Delay Differeniial Equations

94

Proof. Assume first that c > 0. The characteristic equation of (5.14) is D(X) = X2

+ €A + a - b P ( l - e-A)

= 0.

(5.15)

When b = 0, the real parts of the roots of (5.15) are negative, which implies that x = 0 is uniformly asymptotically stable. If, as b increases, the trivial solution becomes unstable, then there is a b* = b*(a,c) > 0 such that, when b = b*, D(iw) = 0 for some w > 0. This implies that -w3

+ au = b* sin w, + b' = b*

cu2

Thus, (w2 - a)2

COSW.

+ c2w2 + 2€b*= 0,

(5.16) (5.17) (5.18)

which is impossible. Therefore, for c > 0, b > 0, and a > 0, I = 0 is uniformly asymptotically stable. If c = 0 and (5.15) has roots with nonnegative real parts, let A = a + i w be such a root, a 2 0, w 2 0. It is easy to argue that a = 0, since otherwise, for small 6 , (5.15) will continue to have roots with positive real parts, a contradiction to our earlier findings. If X = iw, then (5.17), (5.18) imply that cosw = 1, w2 = a . Hence, a = 4n27r2for some n = 1,2,. . . , since a > b > 0 implies that X = 0 is not a root of (5.15) when c = 0. However, this is impossible as we have assumed that a # 4n27r2,n = 1,2,. . . . Also, the form of the function D(X) indicates that (5.15) cannot have sequence of roots X j = aj iwj, j = 1,2,. . . , such that aj + 0 as j + +OO. This proves the theorem. 0 Our final example in this section is an equation with an infinite distributed delay. Example 5.3. Consider

+

qt)

+ asp) = b j

0

I(t -W

+ e) dv(e),

(5.19)

where a 2 0, b > 0; v ( 8 ) is a nonconstant and nonincreasing function. When a = 0, b = 1, it reduces to Eq. (3.36) in StCpdn (1989), pp. 8788. The following theorem generalizes Theorem 3.28 in StCpin (1989) and Theorem 1 in Stech (1978). Theorem 5.3. The triviaI solution of (5.19) i s uniformly asymptoticaIly stable af (5.20)

3. Characteristic Equations

95

and there e z i ~ t sa constant v

> 0 ~ u c hthat

Proof. Assume first that a > 0. Clearly, when b = 0, (5.19) is uniformly asymptotically stable. The characteristic equation of (5.19) is D(X) = X

+a -b

(5.21)

D(iw) = 0 implies that 0

a - b l , cos(w8) d p ( 8 ) = 0,

(5.22)

0

w - b l , sin(w8) d p ( 8 ) = 0.

However, for w

(5.23)

> 0, we have, from (5.20), sin(w8)dv(6) > w ( l - b /

0 -m

8 d p ( f l ) ) > 0.

Hence, for a > 0, (5.20) is uniformly asymptotically stable. By continuity, when a = 0, (5.21) has no roots with positive real parts. The same argument as before also shows that it cannot have pure imaginary roots. If X j = aj + iwj, j = 1,2,. . . , are roots of (5.21) such that aj 4 0 as j + 00. Then, 0

aj

+a - bJ_,

eaje cos(wj8) d p ( e ) = 0,

(5.24)

w j - b L , eajesin(wj8) d p ( 8 ) = 0.

(5.25)

0

Idp(8)l < +m, there are T Since, !$ (8e-ve/21< e-"' for 8 < -T, and

Hence, for

aj

> -6,

> 0,

v/2

>

6

> 0 such that

(5.25) fails to hold. This proves the theorem.

Delay Differeniial Equations

96

3.6. Reducible Systems-The

Linear Chain Trick

Fargue (1973) observed that the integro-differential equation

i ( t )= H ( z , t ) +

1, t

k(t

- B)G(z(B))dd

(6.1)

with initial condition z ( t ) = $(1), t E (-oo,O], is equivalent to an ordinary differential system with initial condition if and only if k is a linear combination of functions eat, teat,

... ,tmeat,

(6.2)

where a is a real or complex constant. In the following, we denote e -a@ ( m - I)! ’

amem-l ka,m(e)

=

(6.3)

and assume that k(0) is a linear combination of ka,m(8). For example, we consider the following integro-differential equation:

A straightforward computation shows that the characteristic equation of (6.4) . , is x b ca*/(a x)m = 0,

+ +

+

which is equivalent to

(A

+ b)(a +

x)m

+ cam = 0,

+

a polynomial of order (rn l)! Alternatively, we can introduce some new variables z,(t) as zo(t) = z ( t ) , t

z l ( t )=

J-ca ka,l(t - e)z(e)de,

I = 1 , 2 , . .. ,rn.

Differentiating under the integral sign and noting that ka,/(O) = 0, 1 > 1;

k a , ~ ( O )= a;

k a , l ( o o ) = 0;

it follows that these new variables satisfy

i o ( t ) = -bzo(t) - cz,(t), i l ( t ) = a[zl-l(t) - z l ( t ) ] ,

1 = 1 , . .. , m .

97

9. Characteristic Equations

Equation (6.10) replaces the original integro-differential equation (6.4) by an ordinary differential equation linking zo(t) and zm(t), and (6.11) constitutes the so-called linear chain; they correspond to a number of variables, each of which follows, or is driven by, the previous one. Thus, some change in zo(t) is propagated down the chain until it reaches zm(t), at which stage it feeds back to the derivative of zo(t) as indicated by (6.10). So instead of the effect of the history of z ( t ) acting on its derivative, as indicated by (6.4),we have a delay caused by intermediate processes! The characteristic equation for system (6.10)-(6.11) is X+b u 0 det

0

u+X

0

0 0

a

a+X

0

0

0 ..* 0 0 0

C

0

=o,

0

(6.12)

a+X

which reduces to (6.6). Note also, that if the initial condition for (6.4)is z ( t ) = 4 ( t ) , t E (-m,O], then the initial condition for (6.10)-(6.11) is

In the following, we consider a population of p interacting species. We assume that the intraspecific growth rate as well as the interaction terms may depend on the population densities in the past. Let z i ( t ) denote the population density of the ith species at time t . The following system of integro-differential equations may model such a population growth process:

i = 1,... , p , (6.13) where Pi, a i j , rij, i , j = 1,. . . , p , are real constants and F , j : [0,m) + R are normalized convex combinations of ka,m(0), m = 1,2,.. . . We define and introduce further distinct functions yp+l,. .. ,yn of the form

where ka,p runs through all density functions (6.3) appearing in Fij, i = 1,. . . , p , for j = 1 ,... , p . The new functions yp+l ,... ,yn satisfy

98

Delay Differeniial Equalions

a system of linear ordinary differential equations in y l , . . . ,yn. Together ,yp we get a system of ordinary with the differential equations for y l ,

. ..

differential equations of the form i i = yi(pi

+2

i = 1 , . -. , p ,

Gijyj

(6.14)

j=l

1=p+l,

... ,n.

(6.15)

j=l

The parameters pi, a;jli , j = 1 , . . . , p , are those of system (6.13) and a i j , i = 1 , . .. , p , j = 1,. . . ,n, are elements of the set { r i j : i , j = 1,. . . , p } . The a l j , 1 = p 1,. . . , n , j = 1,. . . , n , are related to the parameter a appearing in Ej,i , j = 1, ... , n . There have been numerous applications of very short chains, say p = 1 or 2, particularly to population models, such as (6.13). Perhaps the earliest clear statement of one rationale for the linear chain was given by Smith and Mead (1974) in a discussion of some prey-predator models. They point out that for passage of organisms through a life stage S (which could, for example, be the stage vulnerable to predation) neither the “pipeline” assumption of a fixed passage time T nor the “random passage” assumption that the time spent in S is exponentially distributed have much claim to realism. They suggest that the stage S be considered as made up of p stages S, with random passage through each. The overall distribution of the total passage time is then a gamma distribution (see Cox (1962)). From this point of view, we may have to look for an overall picture of the dynamics for general p rather than committing ourselves to a specific p.

+

3.7. Distributed Delays-First Order Equations In this section, we consider general retarded linear first order autonomous differential delay equations with distributed delay:

where T > 0 or T = 00, and f r Idq(0)l < 00. It is easy to see from Theorem 2.1 that

k ( t ) = -CrZ’(t - T ) , is uniformly asymptotically stable if stability of

Cr QT

> 0, T > 0, (7.2) < ~ / 2 . We consider first the

3. Chamcierisiic Equaiions

99

where r j : [ - ~ , 0 ] --f R is increasing and has total variation V(rj) not exceeding unity. Obviously, (7.2) is a special case of (7.3). The following theorem is due to H.-0. Walther (1975). Theorem 7.1. Let a > 0 and 7 > 0 . Assume that Y,I : [ - ~ , 0 ] 4 R is a nonconstant increasing function such that rj(0) - ~ ( - 7 )5 1 and ar < 7rJ2. Then Eq. (7.3) is uniformly asymptotically stable. Proof. The characteristic equation of (7.3) takes the form of

Let X = u + iv be a root of (7.4), where u and v are real. If u 2 0, then Eq. (7.4) implies that

Also, since u = -a J f r euecos vO dq(O), we have 0 2 (-a)-lU =

0

J_, eu8cosvOdg(8) = V(q)eU8*cosvO*,

where O* E [-.,O], from the mean value theorem. impossible, since from (7.5) we have

(7.6)

However, this is

ve* 5 vr < T / 2 , 0 and this clearly contradicts (7.6). In a subsequent paper, H.-0. Walther (1976) gave other conditions on a,7,and q ( O ) that ensure that all roots of (7.4) have negative real parts, or that there are roots with positive real parts. Most conditions were placed on V ( O ) . Without loss of generality, we assume for the next theorem that 7

= 1,

In other words, we consider

q(-1) = 0,

q ( 0 ) = 1.

(7.7)

100

Delay Differential Equations

Our next theorem, due to K. P. Hadeler (1976), provides conditions for (7.8) to be unstable. Theorem 7.2. Let 1/2 < c 5 1 and suppose q(8) = 1 f o r 8 > --E.If

> a/%, (7.9) with 0 < Re X < a, n/2 < Im X < a/€.

asin(a/26)

then Eq. (7.8) has a root X If we denote T = l/c, & = a€,we then have the following equivalent formulation: Theorem 7.2'. Let 1 5 T < 2 and & s i n ( ~ a / 2 )> a/2. Then, for arbitrary n o n d e c r e a h g left continuous ij with ij(8) = 0 f o r 8 < -T and ij(8) = 1 f o r 8 > -1, the equation

(7.10) has a solution j\ with

0 < R e x <&,

a/27

< ImX < a.

Our next theorem, also due to K. P. Hadeler (1976), gives a bound for the root with positive real part of

X + tie-'

(7.11)

= 0.

Theorem 7.3. For ii > n/2, Eq. (7.11) has ezactly one root ReX 2 0, 0 5 ImX 5 a. This root satisfies & - a/2 ln& 5 ReX 4ii a

+

5 &,

-

Jr

e-60 cos(8y) dfip),

with

(7.12)

Proof of Theorems 7.2' and 7.3. We denote y are real. Then, (7.10) is equivalent to

x = -a

X

X = x + i y , where x and

y = & J r e-0' sin(8y) d i p ) .

+

Suppose &-, = xo iyo is a solution of (7.10) with zo 2 0, 0 5 yo 5 a. From the mean value theorem, we see that there exist s,t E [l,T ] such that (xo,y o ) is a solution of

x = -tie-'=

cos s y ,

y = tie-tZ sin t y ,

(7.13)

9. Charncierisfic Equations

101

or, equivalently,

y = s-l arccos(-xeSz/&) = f(x), x = t-lln((& sin ty)/y) = h(y).

(7.14)

Let x, be the solution of xea2 = ir; then, x, < 13. Note that when x increases from 0 to x,, f(z) increases from n/2s to n / s . Thus, the function f : [0, x,] + [ n / ( 2 s )n, / s ] is well defined. Since i r s i n ( ~ n / 2 )> n/2 implies ir > 7r/2, this, together with (7.10), implies that yo # 0. By (7.13), we see that yo # ?r. Let 7t = ln(&t/t); when y runs from 0 t o n / t , h(y) decreases from Tt to -m. Hence, h : [O,n/t)+ (-oo,"y] is well defined, as well as is the inverse function g : (-00,7t]+ [O,n/t). A straightforward computation shows that f and -g are increasing and convex. The function k : [O,?t] + R, rt = min(yt,z,), defined by k(z) = f(z) - g(x) is increasing and convex. If = z iy is a solution of (7.10) in 0 5 y I n, then k(x) = 0. Since k is strictly increasing, there is at most one zero. Since &J = xo iyo is a solution, zo is the only zero of k. Let yt = g(0); then, yt is the unique solution of

x

+

+

tisinty = y,

0

< y 5 n.

Clearly, yt 2 yr, and, by assumption, y,. > n/2. Suppose xo = 0. Then, n/(2s) = yt 2 yr > n/2 is a contradiction. Note that xo 5 x, < & and n/(2r) 5 n/(2s) 5 yo < n. Hence, we have the following: The boundary of

R = { z + i y : 0 5 x 5 &,

0 5 y 5 n}

does n o t contain a n y solutions of (7.10). The next two paragraphs constitute the proof of Theorem 7.3. We replace f and -g by their linear majorants

such that the zero of the linear majorant is a lower bound for the real part xo: (7.15)

102

Delay Differeniial Equations

Consider now the case .r = 1. Then, s = t = reduced t o (7.11), which is

z = -tie-'

cosy,

T

= 1, and (7.10) is

y = iie-' siny.

+

For ii 2 e, we have 71 = 21 lnzl L 21, since q e z 1 = ti,and z1 2 1 for ii 2 e. Thus, for ii 2 e, k(T1) = f(z1) - g(z1) = T - g(z1) > 0. Similarly, z1 > 71 for ti c e, and hence k(T1) = f(71) - g ( n ) = f ( n ) > 0. On the other hand, k(0) = f(0) - g(0) = 7r/2 - y1 c 0. It follows that k has a unique zero in (0,Tl). For y E (n/2, A), we have sin y 1 2(7r - y ) / q hence, y1 2 .r&r/(.rti + T ) . Substituting this into (7.15), we obtain (7.12). T h e solution is unique because k has a t most one zero. This proves Theorem 7.3. To finish the proof of Theorem 7.2', we define F : R + R2 and Fo : R + R2 by z + ii J; e-" cos(y8) d q e ) y - zu J; e-" sin(y8) dij(e)

x -+ ZUe-' cos y

(7.16) (7.17)

and Fp : R + R2 by

The previous argument shows that Fo has a unique zero in 0. Since the Jacobian of FO is (1 -

cosy)2

+

sin y12

>o

at this zero, we see that the index of the zero is +l. Similarly, we can show that Fp has no zero on dR for p E [0,1], since Fp has the representation (7.16) and we can replace Q ( 8 ) by Qp(8) in our previous arguments,

where

0 for 8 5 1, 1 for 8 > 1. By the invariance of the topological degree d(R, F0,O) = d(R, Fl, 0) = 1, we see that F = F1 has at least one zero in R (see Cronin (1964)). 0

9. Characteristic Equations

103

In the rest of this section, we consider equations with infinite distribution delays. More specifically, our results are motivated by the single species logistic equation with infinite delay,

where 9 is bounded, monotone increasing, and normalized so that ~ ( 0 = ) 0, J!,dq(B) = 1. (Y and NO are positive constants. Letting z ( t ) = N ( t ) / N o - 1, (7.18) reduces to

+ 111

0

q t ) = -+(t)

z(t

-00

+ e) drl(e).

(7.19)

We consider here how the stability of the zero solution depends on when the self-regulating mechanism acts in (7.19), rather than on the particular function (v(.)) defining that mechanism. We therefore introduce in (7.19) the parameter It' 2 0 to obtain

+ 111

-h'

q t ) = -+(t)

-W

x(t

+ e) d v ( e + 10.

(7.20)

For It' > 0, the function 7 in (7.19) is seen to be shifted It' units to the left. Thus, the self-regulating mechanism starts after a time lag of K units. The discussion of May (1974) indicates that as the self-regulating mechanism acts after a longer time lag (i.e., It' increases), the zero solution of (7.20) should exhibit decreasing stability. The following three theorems, all due to Stech (1978), lend support to that conjecture. Linearizing (7.20) about x = 0, we obtain r-

h'

Its characteristic equation is (7.22) Our goal is to show that (7.22) has roots with positive real parts if K is large enough. An immediate consequence of Theorem 5.3 is the following: Theorem 7.4. Equation (7.22) ha3 no root with p o ~ i t i v ereal part if (7.23)

Delay Differential Equations

104

Proof. Note that (7.24)

Denote (7.25)

Then (7.22) is equivalent to

X

+ aJO

2 0

dfi(8) = 0.

(7.26)

--M

Condition (5.20) of Theorem 5.3 implies that (7.22) has no root with positive real part if - C Y J !8~d i j ( e ) < 1, which is equivalent to (7.23).

0 We now consider to what degree an increase in K can lead to the existence of roots with positive real parts. In the special case when T > 0, g ( 8 ) = g( -T) for 8 5 -T, (7.22) reduces to (7.27)

The following result follows from Theorem 7.2’. Theorem 7.5. Let cr > 0, T > 0, g ( 8 ) = g(-7’) I( > T such that

8 5 -T. FOT all

~ O T

(7.28) (7.27) has a root with positive real part.

Proof. Let & = (YKand

T

=1

+ T / K . Then (7.27)

reduces t o

x

if one defines = KX and t = 8 / K . By Theorem 7.2’ we conclude that (7.27) has a root with positive real part if tisin(m/2) which is equivalent to (7.28).

> x/2, 0

9. Charucteristic Equations

105

Finally, we consider (7.22) without assuming the delay is bounded. Motivated by the proof of Theorem 7 . 5 , we make a similar change of variables tu = a K , =AK, Kt =0 K;

x

+

then, (7.22) becomes

J_",el' dq(Kt) = 0.

j, + tueJ

(7.30)

Clearly, (7.30) has roots with positive real part if and only if (7.22) does. Theorem 7.6. For tu > lr/2 and all sufieiently large K, (7.30) has rookr with positive real parts. Proof. For Rex 2 0, K > 0, we define

We claim that as K + 00, H ( x , I<) --t 1 uniformly on compact subsets of R e x 2 0. Let E > 0. We may choose T > 0 such that J I ,dq(6) < E . We have

since R e x 2 0. Hence,

I H ( ~ , K-) 11 5 26

+ max{le" - 11

T < t 5 01. K -

: --

Clearly, max{leX' - 11 : T / K 5 t 5 O} tends to zero uniformly on compact subsets of Rex 2 0 as K -+ +00. This proves the claim. The preceding claim implies that the analytic function x+tue-'H(x, K ) approaches tue-A uniformly on compact subsets of Rex > 0. From Theorem 7.3, we know that has roots with positive real part

+

x+

Delay Differential Equations

106

when & > ~ / 2 .By RouchC's theorem, we conclude that, for & > a/2 and 0 sufficiently large K , (7.30) has roots with positive real part. The preceding theorem indicates that in some sense that (7.21) indeed exhibits decreasing stability as I< + +00. 3.8. Distributed Delays-Higher Order Equations a n d Systems The general form of the characteristic equation of systems of delay differential equations (which include higher order equations as special cases) is

(

D(X)= det X I -

J_", ex'

dV(0)) = 0,

where V ( 6 ) is an n x n matrix whose entries are functions of bounded total variations. Equation (8.1) may be expanded as D(X) = A"

+ fl(X)X"-' + + * * .

fn(X),

(8.2)

where the fj(X) are bounded, and D(X) is analytical in ReX >_ -7 for some 7 > 0. Letting X = iw in (8.2), we have

D(iw) = .(W)

+

i.(W),

(8.3)

where U ( W ) and V ( W ) are real. We assume in the following that (8.1) has no imaginary roots. A direct application of the argument principle may result in the following well-known Michailov criterion. For more details, see Kolmanovskii and Nosov (1986). Theorem 8.1. Equation (8.1) i s uniformly asymptotically stable if and only if the variation of arg D ( i w ) i s n ~ / when 2 w varies f r o m 0 to 00; z.e., 00

arg ~ ( i w ) = l ~nn/2.

(8-4)

Recall that we say a characteristic equation is uniformly asymptotically stable if all real parts of its roots are negative. The condition (8.4) is difficult to verify in practice. We define

d R ( w ) = -[argD(iw)]

du

-

.(W).'(W)

.2(w)

- .(W).'(W)

+

.2(w)

9. Chamcierisiic Equations

107

It is easy to see that Theorem 4.1 is equivalent to the following so-called integral criterion of stability (see Kolmanovskii and Nosov (1986)). Theorem 8.2. Equation (8.1) i s uniformly asymptotically table if and only if

where R(w) is defined as in (8.5). The integral stability criterion is more convenient for computer calculations. Choose a number s such that

lW< R(w) &

1.

Then, it is sufficient to verify the inequality

1’

~ ( wd~ )

> (n - 1)n/2

in order to establish (8.6). Consider, for example, the system with characteristic function D(X)= 0.1X2

We have

+ 0.3X + 0.5 + (0.1X + 0.2)e-’11 + (0.2X + 0.3)e-’PL. (8.7)

+ + 0 . 1 sin ~ q w + 0 . 2 ~ 0 r1w s + 0 . 2 sin ~ 72w + 0.3 cos 72w, v =0 . 3+ ~ 0.lw cos r1w - 0.2 sin r1w + 0 . 2 cos ~ 72w - 0.3 sin 72w.

u = - 0 . 1 ~ ~0.5

Computing the integrals, we obtain

L L

20

20

R(w) & = 3.0455,

for

71

= 3,72 = 1.5,

R(w)& = -1.0834 < ?r

for

71

= 2 . 5 , ~= 1.7.

Hence, (8.7) is stable when 71 = 3, 7 2 = 1.5 and is unstable when 71 = 2.5, = 1.7. For more details on this example, see Kolmanovskii and Nosov (1986). Consider the following Volterra-type predator-prey model with delays:

72

&l(t)/Nl(t) = bl - a12

J

0

-W

N2(t

+ fl)k2(6)do, (8.8)

108

Delay Differential Equations

where ,!J Ik,(O)l dB < +m, i = 1,2; all parameters are positive. Initial functions are assumed to be nonnegative. Observe that (8.8) has a positive steady state N* = ( N : , N z ) , N,* = b/a21, N,* = bl/a12. Linearizing (8.8) at N * results in

Its characteristic equation is D(X) = X2

+ b1b2/ 0

e A e k l ( 8 ) d B / o eXek2(e)d6= 0.

-W

(8.10)

-W

In the following, we assume that

Then, an application of Theorem 8.1 yields Theorem 8.3 and Corollary 8.1 following, both of which can be found in Cushing (1977) Section 4.1. Theorem 8.3. Assume that D(X) of (8.10) has no imaginary roots. Then (8.9) is uniformly asymptotically stable if and only if

Corollary 8.1. Suppose, i n addition to (8.11), kj(f3) E C2((-co,0],R+), such that kY(0) 2 0 , k;(e) 2 0 f o r 8 5 0, and Icj(-oo) = k;(-m) = 0 . Then N * i n (8.8) i s unstable. Proof. Denote

Then, for w > 0,

s.

- -w-l

I -

0

I-,

k;(e)(i - coswe)de < 0.

Since ImD(iw) = b l l ~ ( c 1 s 2+ czsl), it follows that ImD(iw) < 0 for > 0. Note that D ( 0 ) = blb2 > 0, and ReD(+im) = -m. Therefore,

w

3. Charucterisiic Equaiions

109

argD(iw)[r = - A # A . By Theorem 8.3, we conclude that N* in (8.8) is unstable. 0 This approach has been extended in Cushing (1977), Section 4.9, to the discussions of (local) stability of n species population models with general distributed delays. We would like to point out here that many stability results obtained previously and in the literature for discrete delay differential equations can be extended to distributed delay differential equations. The following are two such examples. Consider first the following second order equation with distributed delay: qt)

+ a l i ( t ) + a o z ( t )= J

0

-r

z(t

+ e) d v ( e ) ,

(8.12)

where J!r Idq(8)l = q < +oo, and there is a v > 0 such that f r e-ueldq(8)1 < +oo. We denote

The following result generalizes Theorem 4.3. Theorem 8.4. The following statements are true. (i) The trivial solution of (8.12) is uniformly asymptotically stable (8.13)

(ii) The trivial solution of (8.12) is uniformly asymptotically stable if a0

>q

and

a1

> u.

Proof. We may consider

q t ) + a l i ( t ) + ao+)

=

(8.14)

where a E [0,1] is a constant. When cr = 0, the trivial solution of (8.14) is uniformly asymptotically stable. If its stability changes as a increases, then, for some 6 E (0,1], we have D(iw) = 0 for some w > 0, where (8.15)

Delay Differential Equations

110

D(iw) = 0 implies that (8.16) (8.17) Equation (8.16) implies

w2 > a0

- "'I 2 a0 - 'I.

This together with the second inequality of (8.13) yield 0

a1w

+ cy J-r sin

ciq(e) 2 a1 (a0 - q ) 1 / 2 - crq

> 0,

a contradiction to (8.17). This proves (i). Observe that, for w > 0,

= w(a1 - 00)> 0,

since I sin 81 5 0. This proves (ii). 0 Clearly, (8.12) reduces to (4.35) when q ( d ) = C;=, qj(O), r = max{Tj : j = 1,. . . , n } , where

In Freedman and Gopalsamy (1988), sufficient conditions of nonoccurrence of stability switching in the following system with discrete delays are established:

pp-

+

- a21u(t) a22v(t)

m

m

+ jC c l j u ( t - t l j ) + C ~ 2 j v (t t2j)r =l j=l

(8.18) where the coefficients are real constants, and the q j , r 2 j , ( l j , & j , j = 1,. . . ,m, are nonnegative constants. Clearly, (8.18) is a special case of

(8.19)

3. Characteristic Equations

where there is a w > 0 such that J!,e-"'1dq,(O)l convenience, we denote

111

< +oo, i = 1,2,3,4.

For

Our objective here is to establish sufficient conditions for the nonoccurrence of stability switching in (8.19). We adopt a similar approach to that of Freedman and Gopalsamy (1988), which exploits the fact that stability switching is possible only when the corresponding characteristic equation has pure imaginary roots. The characteristic equation associated with (8.19) is

We let A = iw in (8.21) and separate the real and imaginary parts of (8.21):

(8.22)

112

Delay Differenfial Equations

For convenience, we denote

+ a227 P = 171 + 6 = 171174 + 17273.

Ql = all

174,

Q2

= a11a22 - a12a21,

7 = la111174 4-la221171 4- la121173

+ b211172,

If we denote by f ( w ) the sum of the squares of the right side of (8.22) and (8.23),then we have, after some algebraic simplifications,

+

f(w)I P2w2 4P(7

+ S)w + 476 + r2+ b2,

(8.24)

where we make use of the observation (from the Schwarz inequality)

2ab 5 u2

+ b2.

Hence, we have w4

+ (Q:

- 2a2)w2

+ a; 5 P2w2 + 2 P ( y + S)w + (7 + S)2.

(8.25)

A sufficient condition for there to be no stability switches is that (8.25) not be satisfied for any real w. This is equivalent to g ( w ) = w4

+ (a:- 2~12- /3 2 )w 2 - 2 P ( y + 6)w - (7 + S)2 +

Q:

> 0, (8.26)

9. Chamcierisiic Equaiions

113

for all real w. Note that (8.26) can be written as

> 0, (8.27) for all real w. Therefore, we have proved the following theorem. Theorem 8 . 5 . The trivial solution of (8.19) has the same stability for all V ; ( O ) such that

1, 0

qi =

I+i(d)l,

if

( i ) a; - 2a2 - p2 > 0, (ii) (a!- (7 ~ ) ~ ) ( a2 a;2 - S2)> p2(r 6)2. The preceding result generalizes the main theorem of Freedman and Gopalsamy (1988). Observe that the absolute value of the right hand side of (8.23) is less than or equal to

+

+

I w I ( 7 7 1 + 7 7 4 + l a l l 1 ~ 4 + + ~ 2 2 l+~7l 7 1 ~ 4 + 1 / 4 ~ 1 + ~ 2 ~ 3 + 7 7 3 ~ 2 + l a l 2 l 0 3 + l a 2 l l 0 2 ) .

(8.28) We thus have the following: Theorem 8.6. The trivial solution of (8.19) has the same stability for all v i ( O ) satisfying (8.20) if la11

+ a221 > 771 + 774 + [all104 + la22101 + 77104 + 77401 + V2O3 + 77302 + la12103 + la21102.

(8.29)

Note that 0, 5 777,. We have the following corollary. Corollary 8.2. The trivial solution of (8.19) has the same stability for all v i ( O ) such that 77; = f T ldqi(e)1, if la11

o < T <

2771774

+2772773

+ a221 - 771 - 774

+ b i i l 7 7 4 + la221771

la12173

+ b211772'

3.9. Remarks a n d O p e n Problems This chapter described in detail the method developed by Cooke and Grossman (1982) and other authors for the study of characteristic equations of delay differential equations. This method exploits the fact that for

Delay Differeniial Equations

114

retarded delay differential equations, stability switching becomes possible if the characteristic equation has pure imaginary roots. Under reasonable conditions, the same statement is true for neutral delay differential equations. Other well-known methods include the Pontryagin criterion for the investigation of characteristic equations of discrete delays, the Laplace transformation, and the Nyquist criterion. All of these are well documented in the classical text book by Bellman and Cooke (1963). In StCpin (1989), a less well-known but equally general and effective approach is presented in great detail. In essence, the whole book by StCpln documents the proof and various applications of the following theorem: Theorem 9.1. Consider the n dimensional linear autonomous retarded delay differential equation ~ ( t =)

J

0

-m

z(t

+ e) w e ) ,

and suppose that there exists a scalar v

(9.1)

> 0 such that

0

1,

j , k = 1,.. . ,n.

e-")dgjk(8)1 < t o o ,

The characteristic equation of (9.1) is

D(X) = det(XZ -

-

lo

-m ex'

d g ( 0 ) ) = 0.

(9.2)

-

2 pr 2 0 and u1 2 - .2 us = 0 denote the nonnegative real Let p1 2 zeros of R(w) and s ( w ) , respectively, where R ( w ) = Re D(iw),

S ( w ) = Im D(iw).

(9.3)

The trivial solution of (9.1) is uniformly asymptotically stable if and only if n = 2m,

S ( p k ) # 0,

k = 1,... , r ,

OT

n = 2m

R ( o ~#) 0,

+ 1,

k = 1,. . . ,S - 1,

R(0) > 0,

(9.4)

9. Charncterislic Equalions

115

where m i s a nonnegative integer.

For example, we consider 0

q t ) + a z ( t ) = b J-1

z(t

+ e) a,

(9.8)

which is a special case of (5.14) (c = 0). We assume a > b # 4n27r2, n = 1,2,. .. . The characteristic equation of (9.8) is

> 0, and

Q

D(A) = A2 + a - b(l

- ~ z - ~ ) / A= 0,

A

# 0.

(9.9)

Thus, R(w) and S(w) take the form

+

R(w) = -w 2 a - bsinw/w, S(w) = b( 1 - cos w)/w,

w w

# 0, # 0.

We have here n = 2, m = 1. a > b implies R(0) > 0. This implies that R(w) has positive roots of odd number. S(w) is positive in ( 0 , ~ if) b > 0 and w # 2n7r for all n = 1,2,. .. . Thus, r

C(-1lksignS(pk) = -1, k=l

and (9.5) is satisfied. The condition a # 4n27r2 is equivalent to (9.4). If R(0) < 0 or b < 0, then it is easy to see that (9.5) cannot be satisfied. Therefore, we have the conclusion of Theorem 5.2. Open problems and research questions are numerous on the subject of characteristic equations of delay differential equations. Here we list four classes of such problems. O p e n P r o b l e m 3.1. Systematically study stability switching of equations with three discrete delays. Such a study for equations with two discrete delays was carried out successfully in Hale and Huang (1991). Open P r o b l e m 3.2. Systematically study stability switching of (9.10) Open P r o b l e m 3.3. Obtain sufficient conditions for ~ ( e that ) make Eq. (9.1) stable or unstable. An example of such an effort is Corollary 8.1. Also, see Walther (1976).

116

Delay Differential Equations

Open Problem 3.4. Analyze some critical cases of characteristic equations of the form (9.11) k=O

k=O

Examples of such analyses are documented in Sections 3.2, 3.3 and Theorem 5.1.

PART TWO APPLICATIONS IN POPULATION DYNAMICS

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4

Global Stability for Single Species Models 4.1. Introduction

In Chapter 2 we have considered the stability problem of scalar Lotka-Volterra equations (2.5.11) by methods of Liapunov functionals and Razumikhin-type theorems. Examples of local stability analysis of scalar equations are given in Chapter 3. Population models of scalar delay differential equations are numerous, and they are often nonlinear and quite different from the Lotka-Volterra equation (2.5.11). The detailed analysis of these models generally demands more than just the theories presented in the previous chapters. Different nonlinearities often require different treatments. A large portion of the existing models assume the so-called negative feedback effect and often appear as nondelayed forces, such as the term --ax(t) in Eq. (2.5.11). When such a term dominates other terms (delayed and nondelayed), the model tends to have a globally asymptotically stable steady state with respect to some set of suitable initial functions. This is exactly the case for Eq. (2.5.11). Our focus in this chapter is global stability analysis of steady states in various scalar delay differential population models. Along with this main objective, we also study the boundedness of solutions and perform local stability analyses when needed. The techniques used in this chapter should be applicable to delay models in areas other than population dynamics. 4.2. Wright’s Global Stability Result for y(t) = -ry(t - 1)[1

+ y(t)]

The delayed logistic equation with one discrete delay (l.l.l),which is also called Hutchinson’s equation,

can be rewritten as

119

120

Delay Differential Equations

by assuming y ( t ) = -1

+z(t)/K.

Letting t = rt, y(t) = y ( t ) , we then have

Denote (Y = rr. We may thus consider (dropping the bars from 2, y) i ( t ) = -(Yy(t

- 1)[1 + y ( t ) ] .

(2.4)

A direct application of Theorem 3.2.1 yields that the trivial solution of (2.4) is asymptotically stable for (Y < 7r/2 and unstable for cy > r / 2 . The derivation of (2.1) in the context of population dynamics is as follows. Without considering time delay, the per capita growth rate is often assumed to be

i ( t ) / z ( t )= 1.p - z ( t ) / K ] . If we think the gestation period is r , then the per capita growth rate function should carry a time delay r , which results in (2.1). In analyzing a nonlinear delay differential population model, it is often desirable to obtain sufficient conditions for the global asymptotical stability (with respect to suitable initial functions) of its positive steady states (often unique). In case of Eq. (2.1), the only positive steady state is z ( t ) = K , which is reduced to the trivial solution of Eq. (2.4). The following global stability result is due to Wright (1955). Theorem 2.1. Let a 5 3/2, E C([-l,O],R), +(6) 2 -1, 4(0) > -1, and y(t) = y(+)(t) be the solution of (2.4) with initial function 4. Then limt-.+oo y ( t ) = 0. We say a function y(t), t 2 a for some a E R, is oscillatory with respect to y* if there is a sequence tn 2 a, t , + 00 as n -+ 00, y ( t n ) = y * . Otherwise, we say it is nonoscillatory. If y* = 0 , we simply call it oscillatory or nonoscillatory, respectively. To prove Theorem 2.1, we need the following lemma. Lemma 2.1. Under the conditions of Theorem 2.1, y(t) > -1 and is bounded for t 2 0. Proof. Indeed, we have

+

1- 1

1

+ Y(t) = (1 + Y(to))exPl-a~o-l Y ( t ) 4

which implies that 1

5 7

+ y ( t ) > 0 as long as y ( ( ) exists on [-1, t - 11.

(2.5)

4 . Global Stability for Single Species Models

121

If y(t) is nonoscillatory, then y(t) > 0 or y ( t ) < 0 for some t 2 t o 2 0. Assume first that y(t) > 0 for t 2 t o ; then, y(t) < 0 for t 2 to 1 from (2.4) (since 1 y(t) > 0). Hence, y(t) is strictly decreasing for = c. And we t 2 t o 1. There is a c 2 0 such that limt,+,y(t) must have lirnt++, y(t) = 0 = -ac(l c ) . Therefore, c = 0. The same conclusion holds for y(t) < 0 for t 2 lo. Assume now that y(t) is oscillatory. Let t2 > tl > 0 be arbitrarily two consecutive zeros of y(t) such that y(t) 2 0 for t E [ t l , t 2 ] ,and assume that y(t) attains its maximum at t * . Then, i ( t * ) = 0, which implies that y(t* - 1) = 0. Letting t o = t* in (2.5), we have

+

since y(t)

+

+

+

> -1. Hence, y(t*) < em - 1. This proves that -

lim y(t) 5 ea - 1,

t+cn

and the lemma follows. We now prove Theorem 2.1. Proof of Theorem 2.1. Since y(t) is bounded, we may define 21

= l!-lim y(t), 00

2,

0

= - !ig y(t). t-oo

We assume that y(t) is oscillatory; otherwise, we have u = v = 0 from the proof of Lemma 2.1. Let 6 be a positive constant such that, for t 2 t l = t l ( c ) > 0,

-v - E < y(t) < 21 t €.

(2.7)

If y(T) is a (local) maximum or minimum with T > t1+2, then y(T-1) = 0 and

T-1

+ €1 < I n ( l + y ( ~ ) =) - a / T-2

-a(u

y(t) dt

< a ( v t c),

which leads to

-1

+ exp{-a(u

t E ) } < y(T) < -1

+ exp{a(v +

E)}.

(2.8)

From the definition of u,v, we see that there is a T > 0 such that y(T) is a local maximum and y(T) > u - 6, and a T' > 0 such that y(T') is a local minimum and y(T') < -v E . Hence,

+

u

- 6 < exp{a(v

+

E)}

- 1,

v -E

< 1 - exp{-a(u

+ c)}.

(2.9)

Delay Diflerential Equations

122

> 0, this leads t o v 5 1 - e--au. u 5 eaW - 1,

Since (2.9) is true for all e

(2.10)

It follows that v < 1 and that, if one of u and v is zero, then so is the other, and the theorem is proved. Therefore, we assume in the following that u>o, O
5 1, then from (2.10) we have 1 + u 5 ew 5 exp( 1 - e-"). However, since u > 0, we have If a

1

+ u - exp( 1 - e-")

=

1' A''(

1 - e-uz) exp( 1 - e-uz - u2) du2 dul

> 0,

a contradiction. This proves the theorem when a 5 1. Now we assume that a > 1. Let T be a maximum or minimum point such that T 2 t l c. Hence, y ( T - 1) = 0 and, for t > 0,

+

For

tl

+ 1 5 t 5 T - 1, we thus have -a(v + E)(T- t - 1 ) < ln{l + y ( t ) ) < a ( u + c)(T - t - l ) ,

because of (2.7). Hence,

-1

+ exp{-a(v + C )(T- t - 1)) 5 y(t)

I -1

+ exp{a(u + €)(T- t - 1)).

(2.12) Let T E [0,1] be an arbitrary constant. The inequalities (2.7) and (2.12) yield T-1

h { l + y(T)} = - a /

T-2

T-1-r

y(t)dt = -a/

c a(l - 7)(v + €)

5 a(1 - T ) ( V

T-2 T-1

+ a JT-1-r

y(t)dt-

~(t)dt

{ 1 - e-a(v+c)(T-t-l) 1dt

+ + a7 - [I - exp{-a.r(v + c)}]/(v + c), t)

(2.13)

and

x (exp{a(u

T-1-r

+ €)(T- t - 1)) - 1) dt

4 . Global Siabiliiy for Single Species Modeb

123

Again, we can find T such that y ( T ) > u - c. Using it in (2.13) and letting c + 0, we obtain In(l+ u ) 5 a(1Letting

T

T)V

+ QT - v-l(l-

e--ar").

(2.15)

= 1, (2.15) reduces to

- e--Q"). (2.16) 2 - ln(1 - v), we may let T = - ln(1 - v ) / ( ~ v )in (2.15) and obtain I n ( l + u ) 5 a v - v-l(l- v ) l n ( l - u ) - 1. (2.17) I n ( l + u ) 5 Q - v-'(1

If av

Next, we choose a minimum point T for which y ( T ) < -v in (2.14), letting c + 0, and setting T

= (au)-' In(1

+u) 5 1

we have

- ln(1 - v) 5 cru - u-l(1 We make the following claim: Claim. If 1 < a 5 3/2, u > 0 and v ln(l+u)
+ c.

Using this

(since a > I),

+ u ) In(1 + u ) + 1.

(2.18)

> 0 as before, then -ln(l-v)
This claim follows at once from (2.16) and (2.18) if we show that av - 1

+ e-aw 5 v2 - v3/6

(2.19)

whenever a

5 3/2,

av

< - In( 1 - v),

(2.20)

and that

32 v + ( ly- v) l n

3, 2 - (--)

+

1 1 2 (G) - 1 < v - g" ln(l+ u)

7

1 + 1 < u + -u2. 6

Since u > 0, we have (l+u)ln(l+u)-u=

LU(AU12) > JOY 1 "dul

(1

u2) du2 dui

(2.21) (2.22)

Delay Diflerential Equations

124

< v < 1, we have

which is (2.22). Since 0

1' (1''A) > iv J," +

v + (1 - v)ln(I - v) =

0

dvi

1-v2

0

(1

"2)

dv2 dvi

1 + -v3,

1 = -v2 2

6

which gives (2.21). It remains to prove that (2.19) is true whenever (2.20) is true. Denote W = 1 - e-w. We have from (2.20) that QV

av - 1

+ e-av = J,

(1 - e-w)dw

dW <

<

(1 - e-'") dw

-1 1

v

1-v

0

WdW=

02

2(1 - v)'

~

If 0 < v 5 0.45, we have V2

2( 1 - v)

< 0 . 9 2 5 ~5~v2(1 -

and so we have (2.19) for such a v. Since a QV

av - I

+ e-Qv = J,

(1 - e-'")dw 5

5 L3'l2( w - y 1

1

5 3/2, we have

3v/2

0

i),

(1 - e-")dw

+ -w3 dw = -v2 9 - -v3 9 2 61 ) 8 16

+ -v27 128

4

'

and the last expression is less than v2 - v3/6

provided that 81v2 - 152v

+ 48 < 0,

which is equivalent to

1888

43.45

(.-;)2<s=(F)' and which is true when

1 2 v > (76 - 43.45)/81 = 0.402, and so certainly for v

> 0.45. This proves the claim.

4 . Global Stability for Single Species Models

125

We now suppose that the conditions in the claim are satisfied, and define 03 = v3(u) as the smaller root of ln(1 Clearly,

03

+ u) = v3 - v32/6.

(2.23)

> 0 (since u > 0). We also define v4 by 1 6

- In(1 - v1) = u + -u2. Equation (2.24) yields by the claim, we have

v4 =

0

1 - exp{-u - (l/6)u2}

(2.24)

< u, if u > 0. Hence,

< 213 c v < v4 < u.

(2.25)

Thus, 1

1

du

+ u d ~ -31 - 3 ,

v3

and so

5 = (1 + u ) ( l + dv3

1-d ~ 4U -1+s 1 - 214 du

i)(l

-v4)

Hence, by (2.23) and (2.24), we have =ln(l+u)+ln

+ln(l-v4)+ln

21; u u2 v3 2 1 , 2 1 < - v3 - - + - - 21 - - - - = -v3 - -v3 - -u -

6 2 5 -{ln(l 3

3

6

3

3

6

3

6

+ u) - u } < 0.

Therefore, d(v4 - v ~ ) / d u < 0 and so, since v4 + 0 and 03 t 0 as u t 0, we have v4 < v3 for u > 0. But this contradicts (2.25) and so, for 1 < a 5 3/2, we must have u = v = 0. The proof is thus complete. 0 As pointed out by Wright (1955), the preceding method, a t the cost of considerable elaboration, can be used to extend the result to (Y 5 37/24 (= 1.5416) and, probably to a < 1.567 (compare with n/2 = 1.571.. . ). Gopalsamy (1986) claimed that Wright’s conjecture-Theorem 2.1 is true for a < 7r/2-can be proved by applying Tychonoff’s fixed point theorem. However, his proof is not correct, since the set he constructed is A = {u E B : limt,, u(t) = 0; Iu(t)l 5 ae@},where a and p are constant, B is the real linear space of all bounded continuous real valued functions Iu(t)l, which is obviously not closed, on [O,m) with norm llull = since [0, m) is not compact! Thedoore, Wright’s conjecture remains open.

126

Delay Differenlial Equalions

4.3. Global Stability for General Delayed Nonautonomous

Logistic Equations When Eq. (2.1) is employed as a model for single species growth in the real world, we often expect that r and K , and even T are time dependent, since the environment is constantly changing. It is thus necessary and interesting to study the following general delayed nonlinear nonautonomous logistic equation:

where f ; ( t , O ) = 0, r ( t ) > 0 , p , ( t , ~ ) i, = 1 , . . . ,n, are nondecreasing. Clearly, (3.1) has (2.4) as a special case. In this section, we modify the method used in the previous one to establish conditions under which the trivial solution of (3.1) is globally asymptotically stable with respect to appropriate initial functions. More specifically, we assume in what follows that f ; ( t , z ) and r ( t ) are continuously differentiable with respect to their arguments; p ; ( t , 3) are continuous with respect to t , nondecreasing with respect to s, and are defined for all ( t ,3) E R2. In addition, we always assume that the following hold: ( H l ) For 2 > -1, z f i ( t , x ) 2 0, and for all t 2 0, C&1f i ( t , z ) = 0 if and only if z = 0; (H2) r ( t ) > 0, t - r ( t ) is nondecreasing, and limt++m(t-r(t)) = +oo; (H3) p i ( t , t ) > p i ( t , t - r ( t ) ) ; (H4) For any c # 0, there exist continuous functions a ; ( t ) 2 0 , b;()1. 2 0, b ; ( l ~ I= ) 0 if and only if c = 0, such that Ifi(t, c)] 2 a i ( t ) b i ()1. and t++w lim

J'0 ( 5l:r(r) a i ( T ) dpi(T,

3)) dT

=

Let r = r(0);then, the initial value problem for (3.1) takes the form = $(%

fl E [-.,Oh

(3.2)

where $(a) E C. Motivated by its applications in mathematical ecology, we assume that $(O) 2 -1, +(O) > -1. Clearly, (3.1) and (3.2) have a unique local solution. We denote it as z(+)(t), or simply z ( t ) . Notice that the delay length r ( t ) is not necessarily finite. More specific examples of Eq. (3.1) that appear in the literature will be considered at the end of this section. We treat them as application examples of the main result, Theorem 3.2, to be presented later. The results included in this section are taken from Kuang (1991~).

4 . Global Stability for Single Species Models

127

From ( H l ) and (H3), we see that (3.1) has exactly two steady states, = -1 and z ( t ) E 0. The following lemma is very useful. Lemma 3.1. A.wume there i s an M > 0 such that, for z > -1, f i ( t , z ) > -M, i = 1, ... ,n. Then z ( + ) ( t )ezi& for all t 2 0. Moreover, z ( + ) ( t ) > -1 for t 2 0, and afz(+)(t)is nono~ciZZatory,then limt--r+ooz(+)(t) = 0. Proof. We note (3.1) can be rewritten as z(t)

from which one can see that z ( t ) > -1 for t 2 0. Thus, - f i ( t , z ) < M for t 2 0, i = 1,. . . ,n, which yields n

k ( t ) L ~ ( 1 z+( t ) )Cbi(t,t ) - pi(t, t - ~ ( t ) ) ] .

(3.4)

i=l

Hence, 1

+ z ( t ) L (1 + +(o))exp{ M J

t "

cbi(T,T ) - pi(T,T - r(T>)]d T } .

(3.5)

0 i=l

This proves that z ( t ) exists for all t 2 0. If, for some 1 > 0, z(+)(t) = -1, and z ( + ) ( t ) > -1 for t < 2, then there is an &I > 0 such that Iz(+)(t)l 5 &I for t 5 t. Thus, there is an N > 0 such that

for t 5 t. Hence,

k ( t ) 2 -(1 which leads to

+ z(t))N

1 + z(t) 2 (1

for t 5 t,

+ $(o))e-Nt

> 0,

a contradiction to our assumption. This proves that z ( + ) ( t )

> -1 for

t 2 0. Suppose z ( t ) is nonoscillatory; then, there is a T > 0 such that, for t > T, z ( t ) does not change sign. Without loss of generality, we may assume that z ( t ) > 0 for t > T. Then Eq. (3.1), together with assumptions

128

Delay Differential Equations

(Hl)-(H3), implies that i ( t ) < 0 for t > 2'1, where Ti - r ( t 1 ) = T . Thus, there is a c 2 0 such that

lim z ( t ) = c. t++m If c > 0, then (3.3) together with assumption (H4) yields lim ln(1 t-+m

+ z ( t ) )=

-00,

which leads to limt++m z ( t ) = -1, a desired contradiction. Hence, c = 0. The case of z ( t ) < 0 for t > T can be dealt with similarly. This proves the lemma. 0 The first theorem gives conditions under which solutions of Eq. (3.1) with initial functions satisfying q5(0), 2 -1, +(O) > -1, possess a uniform supremum. Theorem 3.1. In (3.1), assume that there ezzst M > 0, N > 0 such that, for x > -1, f , ( t , x ) > - M , i = 1,. . . , n , and, for large t ,

+

Then, for large t , x(q5)(t) 5 f?fN - 1. i s an M > M such that, for -1 for all t 2 0, i = 1 , . . . , n , then, for large t ,

If, in addition, there Ifi(t,x)l

cM

x(4)(t) 2 exp(-MN) - 1.

<

x

<

(3.7) eMN, (3.8)

Proof. By Lemma 3.1, x ( + ) ( t )exists for all t 2 0, and if limt,+, x(q5)(t)# 0, then x ( + ) ( t )must be oscillatory. We assume x ( t ) = z(q5)(t)is oscillatory, and let t l = r ( t 1 ) (by (H2), this t l exists), t* > t l be such that x ( + ) ( t ) assumes a local maximum at t*. Then, i ( t * ) = 0. Since t* - r(t*) 2 t l - r ( t 1 ) = 0 , by Lemma3.1, we see that x ( $ ) ( t ) > -1 fort E [t*-r(t*),t*]. Hence, from (3.1), we obtain J:.*+'*) f i ( t * , x ( S ) ) d P i ( t * , S )= 0 . By ( H l ) and (H3), we see this indicates that there is a t o E [t*- r(t*),t*]such that x ( t 0 ) = 0. Thus,

4 . Global Stability for Single Species Modekr

129

Now, by (3.6), it is easy to see that, for large t , ln(1

+ z ( t ) )I M N ,

which yields

z(t)IP

N

- 1.

Thus, there is a t 2 > t i such that, for t 2 t 2 - r ( t 2 ) , z ( t ) 5 e M N . Let f > t 2 be a local minimum of z ( t ) ; then, i ( f ) = 0. Again from (3.1), we see that there is a t 3 E [z - r(f), $1 such that z ( t 3 ) = 0. Hence, by a similar argument t o that preceding, we have I n ( l + z(f)) 2 - m N , which leads to

z ( i ) 2 exp(-mN) - 1. This proves our theorem. 0 The following theorem is the main result on global stability of solutions in Eq. (3.1). Theorem 3.2. W e consider Eq. (3.1). In addition to the assumptions made i n Theorem 3.1, assume further that there are continuous functions ai(t) such that If,(t, c)I 5 cr;(t)lcJ, i = 1,2,. . . ,n, and, for large t ,

Then

lim z ( 4 ) ( t )= 0.

t-+a,

Proof. We assume z ( t ) = z ( 4 ) ( t ) is oscillatory. Otherwise, by Lemma 3.1, limt,+, z ( t ) = 0. Denote u = lim sup z ( t ) ,

v=

t++w

- lim inf z ( t ) .

(3.10)

t-bO

By Theorem 3.1, we have

o Iu <_ e M N - 1,

o 5 5 1 - exp(-MN).

Let e be any positive number, and

t4

(3.11)

be so big that, for t 2 t 4 - r(t4),

130

Delay Diffemniial Equations

and

-v Thus,

+

-ai(T)(u

E)

- E < z ( t ) < 21 + E. 5 -fi(T,z(s))

(3.12)

Iai(T)(v

+

6).

Assume z ( t * ) is a maximum or a minimum such that t* - r ( t * ) 2 there is a t 5 E [t* - r ( t * ) , t * ]such that z(t5) = 0. Thus,

t4.

(3.13) Then,

Clearly, (3.12) implies


Hence, z ( t * ) < eu+'

Similarly, we have z ( t * ) > -1

- 1.

(3.15)

+ e--(u+f).

(3.16) t6

> t 5 , t 7 > t 5 such

- E < 1 - e-(U+c).

(3.17)

By the definition of u and v , we see there always exist E. Therefore, that Z ( t 6 ) > u - 6 and z(t7) < -v

+

u -6

< eu+' - 1,

By letting 6 + 0, we obtain u<eu-l,

v
(3.18)

From (3.18), we see that u = 0 if and only if v = 0. Thus, we may suppose that u>o, O
+ u 5 eu 5 exp(1 - e+').

(3.20)

> 0, we have

+ u - exp(1-

e-u) =

J0u J0u1 (1 - e-uZ)exp(l-

e-uz

- u2) du2 d u l > 0,

4. Global Slabiliiy for Single Species Models

131

0 a contradiction to (3.20). This proves the theorem. We now apply Theorem 3.2 to several frequently used delay equations modeling single species growth. We consider first the well-known nonautonomous logistic equation with several variable delays:

c n

3j(t) = ~ ( t ) y ( [I t )-

ai(t)y(t

- ri(t))]

(3.21)

i=l

where R(t),ai(t),r i ( t ) are nonnegative continuous functions. We assume that there is a positive constant M such that 0 < R(t) < M , C $ l ai(t) = 1, and r ( t ) = max{r;(t) : i = 1 ,... ,n } > 0. Letting z ( t ) = y ( t ) - 1, a i ( t ) = R(t)ai(t),we see (3.21) can be reduced to n

i(t) = -

C a i ( t ) ( z ( t ) + l)z(t - r i ( t ) ) .

(3.22)

i=l

We assume the same initial function for (3.22) as

.(el

= 4(e)

2 -1,

e E [ - r ( 0 ) , O ~ , 4 E c,

4(0) > -1. (3.23)

Notice that in Eq. (3.21) we did not assume the existence of a nondelayed term a ( t ) y ( t ) as in Eq. (2.5.11). This makes it very difficult to apply the method of Liapunov functionals or Razumikhin-type theorems. By applying Theorem 3.2 to Eq. (3.22), we obtain the following: Theorem 3.3. I n (3.22), assume limi,+, C?=l a i ( 7 ) d~ = +00, and assume, for large t ,

Ji

n

-4

(3.24) where r ( t ) satisfies (H2). Then the solution z ( + ) ( t ) of (3.22) and (3.23) tends to zero as t $00. Proof. Comparing (3.22) with (3.1), we have the following correspondences: (i) f i ( t , ~ ( 3 ) )= ai(t).(s); (ii) ~ & c ) ~ ( 3d)p i ( t , 3) = z ( t - r i ( t ) ) * Thus, (3.9) reduces to (3.24). Clearly, (Hl)-(H4) are satisfied. Since R ( t ) is bounded and Cr=lq ( t ) = 1, we see a ; ( t )is bounded, which implies that all conditions of Theorem 3.1 are satisfied, too. Therefore, by applying 0 Theorem 3.2, we conclude that lirn~++, z ( 4 ) ( t ) = 0. Assume in (3.22) that a i ( t ) = ai > 0 , r i(t ) = ri 2 0, r ( t ) = r > 0. Then, (3.22) reduces to the autonomous logistic equation with several

Delay Digemdial Equaiiona

132

constant discrete delays:

(3.25) In this case, Theorem 3.3 reduces to the following: Corollary 3.1. Consider (3.25) with initial condition (3.23). rCb1 ai 5 1, then lim z ( + ) ( t ) = 0.

If

t++w

R e m a r k 3.1. By constructing a nice but complicated Liapunov functional for (3.25) in the case n = 2, Gopalsamy (1992), Theorem 1.4.3, has shown that, for positive al,a2,7-1, and r2, if (a1

+

+

a 2 ) ( ~ 1 r2) exp[(ai

+

+

a 2 ) ( ~ 1 TO)]

< 1,

(3.26)

then lim~_.+wz ( + ) ( t ) = 0. Clearly, the conclusion of our Corollary 3.1 is much more general. Next, we consider the so-called Michaels's-Menton single species growth equation with several discrete delays. For simplicity, we focus only on the autonomous equation

(3.27) where 7 > 0, a; > 0, cj > 0, r; 2 0, T > 0, and Letting z ( t ) = y(t) - 1, we see (3.27) reduces to

E b l ai(1

+

Cj)-'

= 1.

By applying Theorem 3.2 to (3.28), we obtain the following: z ( $ ) ( t ) = 0. Corollary 3.2. In (3.28), if.7 5 1, then lirnt,+, Proof. We note that (3.9) reduces to

i=l

0 which is equivalent to ry 2 1. Finally, we consider an autonomous single species growth equation with distributed delays:

4 . Global Stability for Single Species Models

133

0 where J ! , d p l ( s ) = J-,.dpz(s) = 1, and p2 are nondecreasing, and r, a1, a2, and c are positive constants, a1 = 1. Again, by the change of variable x ( t ) = y(t) - 1, (3.29) can be reduced to

+2

Again, an application of Theorem 3.2 leads to a corollary: Corollary 3.3. In (3.30), if r y 5 1, then limt+m z ( d ) ( t ) = 0. 4.4. Asymptotic Theory for Nonautonomous Delay Equations

with Negative Feedbacks The results of this section are adapted from that of Haddock and Kuang (1992). We consider the following general first order real scalar delay differential equation

where f , ( t , x) and r ( t ) are continuous with respect to their arguments, and p i ( t , s ) is continuous with respect to t , nondecreasing with respect to s, and is defined for all ( t ,s) E R2. In addition, we always assume the following: f;(t, x) = 0 if and only if x = 0; (Hl) x f i ( t , x) 2 0 , and (H2) r ( t ) > 0, t - r ( t ) is nondecreasing, and limt++,(t-r(t)) = +oo; (H3) p : ( t , t ) > p d t , t - r ( t ) ) . Let r = r ( 0 ) ;then the initial value problem for (4.1) takes the form

cbl

40) = +(%

+(4E c.

6 E [-.,OI,

(4.2)

As in the previous section, r ( t ) may be unbounded as t + +m. Observe that Eq. (4.1) is more general than Eq. (3.1). This can be seen by a transformation of the form 4 t ) = -1

+ exp(y(t)),

which reduces (3.1) to

where

j;(t,y(s)) = f;(t, ey(3)

- 1).

Delay Differrntial Equations

134

A solution z ( t ) of (4.1) is called global if it is defined for all t 2 0. In the following, we restrict our attention to global solutions. Lemma 4.1. Assume z ( t ) is a global solution for (4.1)-(4.2). Then either x ( t ) is bounded or it is oscillatory. Proof. Suppose x ( t ) is nonoscillatory; then, there is a T > 0 such that, for t > T , z ( t ) does not change sign. Without loss of generality, we may assume that z ( t ) > 0 fort > T . Then, Eq. (4.1), together with assumptions (Hl)-(H3), implies that z ( t ) is strictly decreasing. Thus, x ( t ) must be bounded. 0 We assume first that the right hand side of (4.1) is linear with respect to x ( s ) . Without loss of generality, we may assume fi(t,x(s)) = x ( s ) . Thus, Eq. (4.1) reduces t o

p i ( t , s ) . We have the following boundedness and where p ( t , s ) = C7='=1 global stability result. Theorem 4.1. In (4.3), assume that t

limsup t - + ~

Jt-r(t)

[p(T,T ) - p ( T , 7 - r('r))ldT = p.

(4.4)

(i) I f p < 1, then all solutions of(4.3) are bounded. (ii) If, in addition to (i), J 0 " [ p ( ~ , 7 )- ~ ( T , T-.(.)I d~ = +m, then lirnt-+" s ( t ) = 0. Proof. We note first that solutions of (4.3) are defined and continuous for all t 2 0. This can be proved by a similar argument as the proof of Theorem 6.1.1 in Hale (1977), where global existence and exponential estimates are established for general linear systems with bounded delays. Proof of Boundedness. Suppose x ( t ) is a solution of (4.3) that is not bounded. By Lemma 4.1, we know x ( t ) must be oscillatory about zero. Let E > 0, p E < 1, and T > 0 be such that, for t 1 T ,

+

Since z ( t ) is oscillatory and unbounded, there exists a t* > T such that Iz(t)l Iz(t*)l for t < t*. Without loss of generality, we may assume z ( t * ) > 0. This implies that i ( t * )10, which is equivalent t o

<

Jt* t*-r(t*)

4.)d p ( t * , s ) < 0.

4 . Global Stability for Single Species Models

135

By our assumptions on r ( t ) and p ( t , s ) , we see that there exists t o ,

E [t* - r ( t * ) t, * ] , such that z(to) = 0. Integrating (4.3) from to to t* yields z(t*)= -

It*(Jr

r-r(r)

to

z(s) d p ( 7 , s ) ) d7.

(4.7)

Hence,

t*

* [ P b ,T ) -A . 7 I Iz(t*)l J. t -r(t ) I ( P + +(t*)l;

7 - 7-(41dT

i.e., Iz(t*)l I ( P + 4lz(t*)l, (4.8) which contradicts our assumption on z ( t * ) . This proves the boundedness

of x ( t ) . Proof of Global Stability. Now, assume p < 1. Thus, solutions of (4.3) are bounded. If the conclusion of (ii) is not true, then there is a solution z ( t )

of (4.3) such that limt++oo z ( t ) # 0. Then, either (A) z ( t ) is oscillatory, or (B) z ( t ) is nonoscillatory. Assume first that z ( t ) is nonoscillatory; then, there is a 21' > 0 such that z ( t ) z ( T ~>) 0 for t 1 7'1. Without loss of generality, we may assume z ( t ) > 0 for t 2 7'1. Then, S ( t ) I 0 for such t . Hence, there is an z* > 0 such that t++m lim z ( t ) = z*. (4.9) Thus, there exists a T* such that, for t 2 T * , z ( t ) 2 z*/2. Integrating (4.3) from t o to t yields

which is equivalent to (4.11)

136

Delay Differential Equations

For t 2 t o , and t o - .(to) L T*, we have z(t0) - z ( t )

5 1*/2.

However,

which leads t o

(4.12) This clearly contradicts (4.11). Therefore, we must have limt,+, z ( t ) = 0. Now, suppose z ( t ) is oscillatory. Choose T2 > 0, c > 0 such that p c < 1 and, for t 2 T2,

+

(4.13) Denote z = limsupt,+, Iz(t)l. We claim that z = 0. Clearly, there = +00, such is a sequence ti > 0, i = 1,2,..., ti+l > ti, limi++,ti that ? ( t i ) = 0, 1z(t;)l + ii as i + +oo. Without loss of generality, we may assume that this sequence can be chosen so that z(ti) > 0. Let 6 = iii(l-p-c), and let T3 > T2 be so large that, for t 2 T3, Ix(t)l < z+6. Clearly, we can choose ti such that ti - .(ti) - .(ti - .(ti)) > T3 and z(ti) > F - 6. Since ?(ti) = 0, from Eq. (4.3), we see there exist t*, t* E [ti - T ( t i ) , t i ] and z ( t * ) = 0. Hence, we have

(4.14) This leads to

with the last inequality yielding

z - 6 < ( p + €)(Z+ 6).

(4.15)

4 . Global Stability for Single Species Models

137

Thus, (4.16)

which contradicts our selection of 6. This proves our claim, and hence our theorem. 0 For a proper choice of p ( t , s ) , (4.3) can be reduced to n

i ( t )= - C Ui(t)z(t - T i ( t ) ) ,

(4.17)

i=O

where a ; ( t ) 2 0, ri(t) are continuous, 0 < r,(t) < T i + l ( t ) 5 T ( t ) , i T O ( t ) = 0. The following corollary is an immediate result of Theorem 4.1. Corollary 4.1. I n (4.17), assume

# 0,

(4.18)

If p < 1, then all solutions of (4.17) are bounded. If, in addition,

then, for every solution z(t) of (4.17), limt+ooz ( t ) = 0. We now return to Eq. (4.1). The following result generalizes Theorem 4.1.

Theorem 4.2. I n (4.1), for c # 0, assume l f i ( ~ , c ) iIs nondecreasing with respect to c, and

T h e n all global solutions of (4.1) are bounded. If, in addition,

fOT

c # 0,

then all global solutions of (4.1) tend to zero as t + +oo. Outline of the proof. By Lemma 4.1, if a global solution z ( t ) of (4.1) is not bounded, then it must be oscillatory. By a similar argument as in the proof of Theorem 4.1, one can show z ( t ) must be bounded. The

138

Delay Differeniial Equaiions

condition (4.20) together with the monotonicity of l f i ( T , c)I (with respect to c ) ensures that z ( t ) cannot tend t o a nonzero constant. Thus, z ( t ) either tends t o zero or is oscillatory. In the latter case, the assumption (4.19) leads to limt,+, z ( t ) = 0. 0 Denote t o = min{t : t = r ( t ) } .We have the following local result. Theorem 4.3. Assume, for (4.1), l f , ( ~ , c ) is l nondecreasing with respect to c, and there is an M > 0 such that, for IcI 5 M , we have

(i)

t 2 to;

/1 2 / c t-r(t) i=l

c - l f i ( T , c ) b i ( T , 7) - p i ( T , T

- .(.))I

dT

Ip

c 1,

for

(4.21)

bon

(ii)

l f i ( 7 , c ) I [ p i ( T ,T )

0

- pi(T,T

i=l

- .(.>)I dT =

c

+

0;

(4.22)

(iii) I f i ( ~ , c ) II a i ( T ) l c l , where u ; ( T ) is continuous and nonnegative for T 2 0. (4.23) Then, for any 0 < c 5 M , there is a 6 ( c ) > 0 such that, for q5 E C, 11q511 < a(€), Iz(t)l 5 c, and limt++,r(t) = 0, where z ( t ) is the solution of (4.1) with initial function q5. Proof. First, we claim that, for small enough 6 > 0, if 11q511 c 6, then z ( t ) , the solution of (4.1) with initial function q5, satisfies Iz(t)l < M for t E [O,tol. Integrating (4.1) from 0 to t leads to

], Assume Iz(s)I 5 M for s E [ - - f , t ~ where we have

T-

= r(0). By assumption (iii),

Let E ( t ) = max{lz(s)l : s E [-r,t]}; then, (4.25) leads to

Thus, by Gronwall's inequality, we have

E ( t ) IE(O)exP{

1

t " ~ai(T)[pi(T,T)-pi(T,T-r(~))]dT}.

(4.26)

4 . Global Siobiliiy for Single Species

Models

139

Hence, if we choose 2(0) < 6, where

{ J,

"

to

6 = ~ e x p-

Cai(T)[pi(T, i=l

- pi(T,T -

.(.)>I dT},

(4.27)

then Iz(t)l I Z ( t ) < M for t E [ O , t o ] , proving the claim. In the following, we prove that, for any 0 < 6 5 M , we can choose

-J,

6(c) = € ~ X P {

to

"

C a i ( ~ ) [ ~ i ( ~ , ~ ) - ~ i ( ~ , ~ (4.28) - r ( ~ ) ) ] ~ ~ i=l

such that, for 11~~511< 6(e), Iz(t)l < c and limt++mz(t) = 0. Clearly, by the proof of the preceding claim, we have Iz(t)l I c for t E (O,to]. Assume there is a t* 2 t o such that, for s 5 t*, Iz(s)l I c, z ( t * ) = c, and k ( t * ) 2 0 (the case of z ( t * ) = - 6 , k ( t * ) I 0 can be dealt with similarly). Then, Eq. (4.1) indicates there is a t' E [t* - r ( t * ) t, * ] such that z(t') = 0. Clearly, t* - r ( t * )2 t o - r ( t 0 ) = 0. We have

l , have By the monotonicity of l f , ( ~ , z ) we

which leads to

a contradiction to assumption (i).

The rest of the proof is similar to the proof of Theorem 4.2; we omit 0 the details here. It should be pointed out here that when Theorem 4.3 is applied to Eq. (3.21), the result obtained is not as sharp as Theorem 3.3. Nevertheless, Theorem 4.3 applies to general equations of the form (4.1). We also have the following boundedness result. Theorem 4.4. Assume, for (4.1), l f , ( ~ , c ) I is nondecreasing with reJpect to c, and there is an M > 0 such that, for IcI 2 M , we have

(i)

t 2 to;

Jt f--r(t)

c - 1 5 fi(T,c)[pi(T,T) - pi(T,T i=l

- .(.>)I d7 I p < 1, for

Delay Differrnlial Equations

140

(ii) l f i ( T , C ) l 5 U i ( T ) l C l , U i ( T ) i s COntinUOu3. Then, the solution z ( t ) of (4.1) i s bounded, and limsupt,+, Iz(t)l 5 M . Outline of the proof. Since (ii), we see every solution of (4.1) must be global. The condition (i) implies the boundedness and limsupt-++oo1z(t)l< M , the proofs of which are similar t o proofs of previous theorems. 0 4.5. 3/2 Stability Results

In the previous two sections, we allowed the delay length r ( t ) to be unbounded and imposed a few restrictions on the nonlinearity of fi(t,z). Results presented in those two sections appear to be not so sharp compared with the results of Wright in Section 2. However, as can be seen from the examples t o be given here, there are significant differences between Eq. (2.4) and Eq. (4.1). Indeed, we will see that for any a > 3/2 we can find a ( t ) such that a ( t ) 5 a for all t 2 0, and the trivial solution of

i ( t ) = - a ( t ) z ( t - 1)[1 + z ( t ) ]

(5.1)

is not stable! This is in contrast to the fact that if a < 7r/2, then the trivial solution of the autonomous equation

i ( t ) = -az(t - 1)[1

+ z(t)]

(5.2)

is uniformly asymptotically stable. In all the equations t o be considered in this section, we assume that the delay lengths are bounded, and this restriction is essential. We will also impose rather restrictive conditions on the nonlinearities of the functions in those equations. In Yorke (1970), the equation

i ( t ) = f(t, is studied, where f : [0, m) x C,(P)

--t

21)

(5.3)

R is continuous, and

C,(P) = {+ E C([-q,OI,R) : 11411 < PI. The following result was proved in Yorke (1970). Theorem 5.1 (Yorke). Suppose that there is an a 2 0 such that (A) 3/2 and (B) - a M ( + ) 5 f ( t , + ) 5 aM(-+),for all 4 E Cq(P), where M ( + ) = max{O,SUP,E[-Q,O] 4(4L Then (i) the zero solution of (5.3) is uniformly stable;

4 . Global Slabiliiy for Single Species Models

141

(ii) if, in addition, 0 < aq < 3/2, then limt++m z ( t , to, Q) ezists for any t o 2 0 and Q E Cq(2p/5). Note that (i) refers to uniform stability, not asymptotic stability. Condition (B) is often referred to as the Yorke condition. Consider now the nonautonomous delayed logistic equation

i ( t )= - a ( t ) z ( t

- 1)[1 + z ( t ) ] ,

(5.4)

where a : [0,00) + R is a nonnegative (piecewise) continuous function. We have the following. Corollary 5.1. Suppose that there ezists a constant 7 > 0 such that a(t)I 7 < 3/2

for all t 2 0.

(5.5)

Then (i) the trivial solution of (5.4) i s uniformly stable; (ii) i f , an addition there is a positive constant c such that a(t) 2 c f o r all t 2 0, then the trivial solution is a b o asymptotically stable. Proof. Let 0 < p < 1/2 be a constant such that 70

+ P ) < 3/2.

+

Denote a = ~ ( lp), and f ( t , zt) = - a ( t ) z ( t - 1)[1

-aM(4) I f(t,$1

+ z ( t ) ] . Then

IaM(-Q)

for Q E Cl(p). The uniform stability thus follows from Theorem 5.1. If also a ( t ) 2 c for t 2 0, then (ii) of Theorem 5.1 applies, and limt++m z(t,to,4) exists for 4 E Cl(2p/5). Assume the limit is z*. If z* > 0, then, for large t , z ( t - 1)[1+ z(t)]2 z*/2, and (5.4) implies that k ( t ) 5 -z*c/2, and hence, z ( t ) + -00, a contradiction. Similarly, we can prove that z* cannot be negative. Hence, z* = 0, proving the corollary. 0 In Sugie (1992), an example is given to show that if 7 > 3/2 in (5.5), then the zero solution is not stable. This indicates that, for nonautonomous equations, the preceding results are sharp. The following result of Yoneyama (1987) partially generalizes Theorem 5.1. Theorem 5.2. Suppose that there is a continuous function a : [0,00) + [0,w) such that

142

Delay Diffemntial Equations

and, f o r all t 2 0, 4 E C,(p), -a(W(4)

If(t,$1 Ia(t)W-4).

Then (i) the zero solution of (5.3) is uniformly stable; (ii) zf X = SUP^>^ - J;+'u(s) ds < 3/2 and p = infrlo $+,u ( s ) ds > 0 , then limt,, z ( t ,t o , 4) ezists for all t o 2 0 and 4 E C , ( e - 2 A p ) . Note that when q = 1, K a ( s ) = Q < 3/2, we have A > 1 and, thus, ,-2x < e-2 < 2/5. Therefore, (ii) of Theorem 5.2 does not include that of Theorem 5.1. The following simple example is given by Yoneyama (1987) to show that 3/2 is the best possible value for Theorems 5.1 and 5.2. However, the equation is of Caratheodory type (for the definition, see Hale (1977), pp. 55-56). Example 5.1. Let a : [0, co) + [0, co)be a piecewise continuous function defined by Q for t E [3k, 3k 1 a(t)= 0 fortE[3k+1+cr-1,3k+3), where Q > 0 and k = 0 , 1 , 2 , . . . . Let 4(t) 6 for t E [-1,0] and x ( t ) = x ( t ,0,4) be the solution of

+ +

{

i ( t )= -a(t)x(t - 1). Then, for k = 0 , 1 , 2 , . . . , '

x(t) =

(-l)k€(Q

- 1/2)k{l - Q ( t - 3k)},

+

for t E [3k,3k 1) ( - l ) k c ( ~- 1/2)k{l - ~ (- t3k) a2(t- 3k - 1)2/2}, for t E [3k 1,3k 1 a - l ) (-1)k+lc(a - 1/2)k+' for t E [3k 1 a-l, 3k 3).

+

+

+ +

+ +

+

Thus, for Q = 3/2, x ( t ) is periodic with period 6 ,and if Q > 3/2, then z ( t ) is unbounded! Yoneyama and Sugie (1988) considered the following perturbed form of equation (5.3): i ( t ) = f(t,.t) 9 ( t ,z ( t ) ) , (5.6) where g : [0, co) x S ( p ) t R is continuous and S(p) = {x E R : 1x1 < p}. A typical example of (5.6) is

+

i ( t ) = - a ( t ) z ( t - q ) + b(t)x(t).

(5.7)

4 . Global Siabilily for Single Species Models

143

The following result is proved in Yoneyama and Sugie (1988). Theorem 5.3. In Eq. (5.6), assume that ( H I ) - a ( t ) M ( + ) If ( t ,4) I a ( t ) M ( - 4 ) f o r t 1 0 and

d E C9(P);

where a(t),b(t): [O,oo) + [O,oo) are continuow; and (H2) limsupt,, $+9 a ( s ) ds < 3/2, liminfl,, J:+9 a ( s ) ds > 0, and Jo" b(s) ds < 00. Then ( i ) the trivial solution of (5.6) i s uniformly stable; (ii) i i in addition, (H3) for all sequences t , + oo and 4, E C9(p) converging to g ( t n , & ( O ) ) does not a nonzero constant function in C,(p), f ( t n , 4,) converge to zero; then the trivial solution of (5.6) is uniformly asymptotically stable. Thus, the asymptotic stability conclusion (ii) of Corollary 5.1 can be strengthened to uniform asymptotic stability.

+

4.6. A Model Exhibiting the Allee Effect The so-called Allee effect refers to a population that has a maximal per capita growth rate at intermediate density. This occurs when the per capita growth rate increases as density increases and decreases after the density passes a certain critical value. This is certainly not the case in the delayed logistic equation (2.1), where per capita growth rate is a decreasing function of the density. As is known, in nature some species often cooperate among themselves in their search for food and/or to escape from predators. For instance, some predators form hunting groups to enable them to capture larger prey; fish and birds often form schools and flocks as a defense against predators; some parasitic insects aggregate so that they can overcome the defense mechanism of a host. A number of social species, such as ants, termites, bees, humans, etc., have developed complex cooperative behavior involving division of labor, altruism, etc. Processes such as these provide individuals with a greater chance to survive and reproduce as density increases. In sexual populations, cooperation among individuals is necessary for mating, nest building, rearing the young, etc. Aggregation and associated cooperative and social characteristics among members of a species had been extensively studied in animal populations by Allee (1927, 1931).

144

Delay Differential Equations

When the density of a population becomes too large, the positive feedback effect of aggregation and cooperation may then be dominated by density dependent stabilizing negative feedback effect due to intraspecific competition due to excessive crowding and the ensuing shortage of resources. To study these processes, Gopalsamy and Ladas (1990) introduced the following delayed Lotka-Volterra type single species population growth model: i ( t )= z ( t ) [ a 6z(t - T ) - cz2(t- T ) ] , (6.1)

+

where a, 6, c are real constants, and a > 0, c > 0. When T = 0, the per capita growth rate is g(z) = a bz - cx2. If 6 > 0, then g’(0) = 6 > 0, and g(z) achieves its maximum value at z = 6/2c, thus exhibiting the Allee effect. When 6 < 0, then g(z) is a decreasing function of z for z 2 0, and thus there is no Allee effect. One can interpret (6.1)as a single species model with a quadratic per capita growth rate, which represents a nonlinear approximation of more general types of nonlinear growth rates with single humps. As usual, we consider only solutions of (6.1)that correspond to initial functions of the form

+

~ ( t=)# ( t ) 2 0,

-T

5 t 5 0,

#(O)

4 E C ( [ - T 01, , R).

> 0,

(6.2) Clearly, Eq. (6.1)has a unique positive equilibrium

The transformation

+

x ( t ) = x*[1 y ( t ) ] reduces Eq. (6.1)to

$(t)= -A(t)y(t - T),

t 2 0,

(6.5)

where A (depends on y) is given by

+ y(t)].

+

A ( t ) = [(2ct*- 6 ) ~ * ~ ( ~ * ) ~-y.)][(I t

+

(6.6)

Since y ( t ) > -1, we see that A ( t ) 2 (cz*- b)x*[l y ( t ) ] > 0 if cz* > 6. Clearly, limt,, z ( t ) = I* if and only if limt,, y ( t ) = 0. Lemma 6.1. Assume that cx* - 6 > 0. Then every nonoJcillatory solution of Eq. (6.5)tends toward zero as 2 --$ 00.

4 . Global Siabiliiy for Single Species Models

145

Proof. We assume in the following that y(t) is eventually positive; the negative case is similar. We have $(t) < 0, and so L = 1imidw y ( t ) 2 0. If L > 0, then

lim $ ( t ) = - L [ ( ~ U *- b)x*

i+w

+ c ( x * ) ~ L<] 0,

0 which implies that lim:-,w y(t) = -00, a contradiction. Let L = 2cx* - b, M = [Lx* c ( z * ) 2 ( e L z * r- 1)](eLZ*'- 1 ) . The following lemma gives global lower and upper bounds for oscillatory solutions of Eq. (6.5). Lemma 6.2. Assume that cx* - b > 0 and y ( t ) is an oscillatory solution of Eq. (6.5). Then there is a T = T ( y ) > 0 such that

+

edMr 5 1

+y(t) 5

Lz* r

for t 2 T .

(6.7)

Proof. Let t 2 > tl > 27. be any t w o zeros of y ( t ) ;that is, y(t1) = y ( t 2 ) = 0. Let t E (tl,t2) be a point where y(t) achieves its maximum or minimum value in (tl,t2). Since y(t)= 0, it follows from (6.5) that y ( t - r ) = 0. We have ln[l

+ y(()] = - /c-rc [Lx* + 5-

i:r

C(X')~Y(S

L x * y ( s - r )ds

+

- r)]y(s - r )ds

5

i:r

Hence, 1 y ( t ) 5 e L z * r , which implies that 1 Similarly, we have

- ln[l + y(()] =

/

5[

€

c-r

[Lx*

Lx* ds = L x * r .

+ y(t) 5 eL2*' for t 2 ti.

+ ~ ( x * ) ~ -y (r)]y(s s - r )ds

+

~ x * c ( x * ) 2 ( e L z *r ~ ) l ( e -~ 1). ~ *=~~

r

,

+

which implies that 1 y(t) 2 e - M r for t 2 t i . This proves the lemma. 0 An immediate consequence of this lemma is that, for t 1 t l , (cx* - b)x*eVMr 5 ~ ( t5) h,

where

M = [ ~ x +* c(x*)2(eLz*r- l ) l e L Z * r .

(6.8)

(6.9) The following result, due to Gopalsamy and Ladas (1990), provides sufficient conditions for the global asymptotical stability of the steady state

146

Delay Differeniial Equaiions

z ( t ) = I* of Eq. (6.1). The proof provided below is very different from and much simpler than that of Gopalsamy and Ladas (1990), where a nice Liapunov functional was constructed for the proof. Theorem 6.1. In Eq. (6.1), assume that a*> b and f i <~1. Then all positive solutions of Eq. (6.1) satisfy

lim z ( t ) = z*.

i+m

Proof. We will show that oscillatory solutions of Eq. (6.5) satisfying 1 + y ( t ) > 0 tend to zero. We define u = limsup I y ( t ) l .

(6.10)

t-m

Clearly u < +oo by Lemma 6.2. If u fiT(21

+

> 0, then there is an > 0 such that

€)

< u - €.

(6.11)

The definition of u implies that there is a 1 > t i such that Iy(t)l

Let

<

Iy(<)l

< u + c,

for t 2 1.

(6.12)

> 1 be a local maximum or minimum point of y(t) such that > u - 6. Then y(<) = 0, and hence y(< - T ) = 0. This leads

to u

- 6 < Iy(<)l =

1/

€

€-r

+

A(s)y(s - T ) dsl Ifi(u f ) ~ ,

a contradiction to (6.11). Therefore, u = 0, and the theorem is proved.

0

Note that in order to have cz* > b and fir c 1, we must have z* > bc-' and T < k-'. This amounts to saying that the steady state must be reasonably large and the delay small. 4.7. Equations of Type i ( t ) = f(zt) - g(z(t))-Preliminaries In order to describe some physiological control systems, Mackey and Glass (1977) proposed the following three first order nonlinear differential delay equations as their appropriate models:

4 . Global Siabiliiy for Single Species Models

147

and

Here, X , a , l / , , n , ~ , O , & and 7 are positive constants. Equation (7.1) is used for studying a “dynamic disease” involving respiratory disorders, where z ( t ) denotes the arterial C02 concentration of a mammal. In (7.1), X is the C02 production rate, Vm denotes the maximum “ventilation” rate of COO,and T is the time between oxygenation of blood in the lungs and stimulation of chemoreceptors in the brainstem. Equations (7.2) and (7.3) are proposed as models of hematopoiesis (blood cell production). In these two equations, P ( t ) denotes the density of mature cells in blood circulation, and T is the time delay between the production of immature cells in the bone marrow and their maturation for release in the circulating bloodstream. Details for the derivation of these equations can be found in Mackey and Glass (1977). Subsequently, these equations are studied by many authors from various angles. Local asymptotic stability analysis, existence of periodic solutions through Hopf bifurcation, and numerical simulation are presented in Glass and Mackey (1979), and Mackey (1979, 1981). Mackey and Heiden (1984) proposed and studied several related models consisting of differential delay equations; the numerical simulations presented there indicate the existence of apparently aperiodic (“chaotic”)solutions (see also the references cited therein). More theoretical explanations of this “chaotic” behavior for Eq. (7.3) can be found in Hale and Sternberg (1988), Heiden and Walther (1983), and Walther (1981). Global existence of periodic solutions in (7.2) and (7.3) was first established by Hadeler and Tomiuk (1977). Mallet-Paret and Nussbaum (1986) studied the singularly perturbed differential delay equation & ( t ) = --5(t) f ( z ( t - l ) ) ,

+

where existence of periodic solutions was shown using a global continuation technique based on degree theory. By letting ~ ( t =) y-l(t), we see that Eq. (7.1) reduces to

We notice that Eq. (7.4) resembles the well-known logistic equation Y(t) = Y(t”

- XY(t)ll

(7.5)

where in (7.4) y is replaced by a delayed term. Motivated by the mathe matical forms and the physiological or biological backgrounds of the previously mentioned equations, we thus propose to study the following general

Delay Differential Equations

148

nonlinear delay equations:

and

4 t )=

[f(J--r

z(t

+ 4 w )- 4 5 ( t ) ) ] .

(7.7)

Here, 0 < 0 < T are constants, f(z) and g(z) are continuously differentiable, p ( s ) is nondecreasing and normalized t o satisfy p ( - u ) - p ( - ~ ) = 1. We always assume that (7.6) and (7.7) have a unique positive steady state. Clearly, Eq. (7.6) has Eqs. (7.2) and (7.3) as special cases, while Eq. (7.7) is more general than Eq. (7.4). When f (s-;” z(t s) d p ( s ) ) = p z ( t - T ) e - a z ( l - r ) , g(z) = 6z, where p , a , and 6 are positive constants, (7.6) reduces to

+

i ( t ) = p s ( t - r ) e - a z ( t - r )- 6s(t).

(7.8)

This is the model proposed by Gurney et al. (1980) in order to describe the population dynamics of Nicholson’s blowflies. For various special forms of Eq. (7.7) that have appeared in the literature as mathematical population models, the readers are referred to the recent monograph of Gopalsamy (1992). One apparent advantage of using equations of form (7.6) or (7.7) as models of real systems is that we can allow the delay effects to be distributed in a period of time, which is more realistic than assuming that the delay is of the discrete type. As pointed out earlier, Eqs. (7.6) and (7.7) are capable of producing “chaotic” solutions. Therefore, it is very important for us to know when these equations are less “chaotic” in the sense that no chaotic solutions may appear. An obvious approach for this problem is to derive sufficient conditions for its unique positive steady state to be globally asymptotically stable. These sufficient conditions are the main objectives of this paper. The most relevant works along this direction are documented in Gopalsamy et al. (1989, 1990b). In these works, Gopalsamy et al. obtained sufficient conditions for the positive steady states in (7.1)-(7.3) t o be global attractors. Their method is to compare these nonlinear differential delay equations to a nonautonomous linear one, for which the asymptotic stability condition for its trivial solution is available. Roughly speaking, their main finding is that, if r is small enough, then the positive steady states in (7.1)-(7.3) are global attractors. Sufficient as well as necessary and sufficient conditions for all positive solutions of (7.1)-(7.3) to oscillate about their positive steady states are also given in those works.

4 . Global Stability for Single Species Models

149

The methods to be used in the subsequent sections are entirely different from the one used by Gopalsamy e t al. (1989, 1990b). To some extent, our approaches are related to the method of Razumikhin-type function. We are able t o obtain sufficient conditions for the positive steady states in (7.6) and (7.7) to be global attractors regardless of the length of delay. We also establish a global existence result of periodic solutions in Eqs. (7.6) and (7.7) when they have only one discrete delay (which may be state dependent) in the next chapter. Throughout this and the next two sections, we always assume that f ( x ) and g(x) are positive functions that are continuously differentiable for x > 0; g ( 0 ) = 0 , g ( s ) is strictly increasing, limz++oog(z) = +m, and lim,,+, f(x)/g(x) = 0; f ( z ) is either monotone or has only one hump; 0 < u < T are condants, p ( s ) is nondecreasing, p ( - u ) - p ( - ~ ) = 1; there ezists a unique x* > 0 such that g ( x * ) = f(x*). Motivated by the physiological or biological backgrounds of these equations, we assume that the initial conditions for (7.6) and (7.7) are of the following type: z(s) = qqs)

> 0,

s

(7.9)

E [-501,

c.

E We say z ( t )iz I* is globally asymptotically stable (G.A.S.) in (7.6) (or (7.7)), if, for fixed T , all solutions of (7.6) and (7.9) (or (7.7) and (7.9)) tend to x* as t + +w. We say x ( t ) x* is absolutely globally asymptotically stable (A.G.A.S.) if it is globally asymptotically stable for all T 2 0. If a function f(x) is strictly monotone, we denote f-'(x) as the inverse function of f(z). The following facts are used implicitly in the subsequent sections. Proposition 7.1. The solution z ( t ) of (7.6) ( O T (7.7)) and (7.9) is positive and bounded for t 2 0. Proof. If z ( t ) is not positive for t 2 0, then there is a t > 0 such that z ( t ) > 0 for t E [O,t), z(t) = 0. Thus, we must have i ( f ) 5 0, which requires that 4(s)

Clearly, this is impossible if f ( x ) > 0 for z 2 0. Our assumption of f(x) implies that this can be true only when f(0) = 0, and if

1-z(t +

s) d p ( s ) = 0.

However, this is a contradiction to the definition of

1.

150

Delay Differential Equations

Assume first that f(z) is decreasing or has exactly one hump. Then, > 0 such that f(z) < for z 2 0. If z ( t ) is not bounded, there is an then there is a 1 > 0 such that 5(S) > 0, z(S) > g-'(x), and z ( t ) < z(S) for -T 5 t c 1. Clearly, k(3) > 0 implies that

f(LT4 2 +

s)

W))> g W ) ) .

However, since .(I) > g - ' ( x ) , we have g(z(t)) >

(7.10)

x;thus,

This is a contradiction. Assume now that f(z) is increasing. Since limz++oof(z)/g(z) = 0, we see that there is an N > 0 such that, for z 2 N , f(z) < g(z). If z ( t ) is not bounded, then there is a t > 0 such that i(t) > 0, z(t) > N , and z ( t ) < z(S) for -T 5 t < 1. Again, we have that (7.10) is valid. Since J--,P z(t + s) d p ( s ) < ~ ( t )this , leads to

f(.(t)) > s(.(t>), .(I) > N . This proves the proposition. 0 If g(z) 5 Mx for some constant M , 0 5 z 5 1, then one can show that the solution of (7.6) (or (7.7)) with the following more general initial function is positive for t 2 0: 4.)= 4(s) 2 0, E [-T,OI, 4(0) > 0, (7.11) 4(s) is piecewise continuous on [ - T , 01. This can be seen easily from the fact that, for z ( t ) 2 0, a contradiction to the fact that

? ( t ) 2 -g(z(t)) 2 - M z ( t ) , which implies that z ( t ) 2 x(0)e-J: M ~ =Jz(0)e-M'

> 0.

The material of this and the next two sections is adapted from Kuang (199213). 4.8.

Equations of Type k ( t ) = f ( z t ) - g(z(t))-When Decreasing We consider first the following autonomous equation:

f(z) Is

4 . Global Siabiliiy

151

for Single Species Modeb

where T > u > 0, p ( s ) is nondecreasing and JI-," d p ( s ) = 1. We assume throughout this section that (Hla) f(z) is strictly decreasing, f(0) > 0, lim,,+, f(z)= 0; (H2) g ( z ) is strictly increasing, g ( 0 ) = 0, limz-r+wg(z) = +00. Clearly, there exists a unique z* > 0 such that f(z*)= g ( z * ) . Proposition 8.1. Assume that z ( t ) is a solution of (8.1) and that it is not oscillatory about z*. Then limt++w z ( t ) = z*. Proof. Assume first that z ( t ) - z* is eventually positive; then, there is a T > 0 such that z ( t ) > z* for t 2 T - T , and hence, for t 2 T, i ( t ) = f(S--," z(t 3) 4 / 4 3 ) )- g ( z ( t ) ) If(z*)- g ( z ( t ) ) < 0 (since f(2) is strictly decreasing and g ( z ) is strictly increasing). Thus, there exists.an 5 2 z* such that lim z ( t ) = 3.

+

t++m

If 5 > z*, then we have lim i ( t ) = f(z)- g ( 5 ) < 0 .

t-+m

This implies that lim z ( t ) = -00,

t++m

a contradiction to (8.2).

The case that z ( t ) - z* is eventually negative can be dealt with similarly. 0 Our first result presents sufficient conditions for the unique positive steady state z ( t ) = z+ to be absolutely globally asymptotically stable. Theorem 8.1. In (8.1), assume that I g - ' ( f ( y ) ) - 2*1 < IY - z*I,

Y

> 0,

Y # z*.

(8.3)

Then the steady state x ( t ) = x* is absolutely globally asymptotically stable. Proof. Let z ( t ) be a solution of (8.1). Since Proposition 8.1 holds, we may assume that z ( t ) is oscillatory about I*. Denote

I.

u = lim sup ( z ( t )- z+ t-+w

Since z ( t ) is bounded, we see that u < +00. If u # 0, then, by the continuity of g-'(e) and f(.),we see that there exists 6 > 0 such that Ig-'(f(ztfu+6))-z*I

< u-6,

for 6 E [-€,el and z * f u + 6

> 0. (8.4)

Delay Differential Equations

152

Clearly, there exists T > 0 such that, for t 2 T , I+(t) - 2+1< u

Let Z > T

+

€.

(8.5)

+ 27 be such that

Assume first

~(> t )z*. Since k ( t ) = 0, we have

The relations (8.6),together with (8.7),yield u - €
-2+.

The inequality (8.8), together with (8.3), gives

This, together with (8.4), implies that

Otherwise, by (8.4), we have that

a contradiction to (8.8).

Since

1:

2(t + s) d p ( s ) - z*= /-l(x(t -r

we conclude from (8.10) that there is a 8 E

+ s) - 2+)d p ( s ) ,

[-7,-01

such that

I2(t + 8 ) - $*I > u + €. Clearly, this is a contradiction to (8.5). The case that .(I) < 2* can be dealt with similarly. This proves the theorem. 0

4. Global Stability for Single Species Model3

153

It is easy to see that condition (8.3) is equivalent to Ig-'(z)

- z*1< lf-'(z)

- z*1,

0
I f(O),

z

# g(z*).

(8.11)

Intuitively, this implies that if we denote A, B , and C as the intersection points of z = zo, 0 < zo If(O), with z = f(z), z = z*, and z = g(z) in the zz-plane, respectively, then the distance from A t o B is greater than the one from B to C. It is also easy to see intuitively that, if max{lf'(z)l : z E [0,2x*]} < min{g'(z) : z E [0,2x*]}, then (8.11) holds. Let f(s) = pen/(P+sn), J - - , " z ( t + s ) d p ( s ) = z(t--7),a n d g ( z ) = yz, where /3,8,n , and y are positive constants. Then Eq. (8.1) reduces to

(8.12) By applying Theorem 8.1 to (8.12), we obtain the following corollary. Let x ( t ) = z* be the unique positive steady state in Corollary 8.1. (8.12). Assume that one of the following two conditions holds; then, x ( t ) = x* i s absolutely globally asymptotically stable: (i) n I 1 and /3 < 2yz*; (ii) n > 1, +pe-ln-l(n ~ ) ~ + l /-~ ( n < 7. Proof. Clearly, Eq. (8.12) satisfies ( H l a ) and (H2). Therefore, Theorem 8.1 can be applied. We assume first that (i) holds. We have

+

+

f ' ( s ) = - n n p ~ s " - ~ ( ~ "sn)-2

< 0,

for s > 0,

and

Denote A = (O,p), B = ( x * , p ) ,C = (@y-',P). Since f"(s) > 0 for s > 0, it is easy to see that, if AB > BC, then condition (8.3) holds (see Fig. 4.1). This is equivalent to

x* > p7-1 - 2*, i.e.,

p < 27x*. Assume now that n

at

> 1. For s > 0, f ( s ) has a unique inflection point

Delay Differential Equations

154

Y

A

/

/

X

X*

Fig. 4.1. The graph for the proof of Corollary 8.1.

Denote

F = max{lf'(s)l : s 2 0). Then we see that F = lf'(s)/. A straightforward calculation shows that

F = -PO 1 4

-1

n -1 ( n

Now, it is easy to see that, if F proves the corollary.

+ l ) " + l / n ( n- 1)"-'jn. <

7, then condition (8.3) holds. This

Lemma 8.1. Let ~ ( tbe) a solution o f ( 8 . 1 ) . Then there is a T that, f o r t 2 T , ~ ( t <) g-'(f(O)). Proof. We note

s-'(f(o))> s - ' ( f k * ) = ) s-l(s(X*))= Z*.

0

> 0 such

4 . Global Stability for Single Species Models

155

If z ( t ) 2 z*for all large t , say t 2 T, then we see that S ( t ) 5 0 for t 2 T, and hence limi,+,z(t) = z*. Thus, we may assume z ( t ) is oscillatory about z*. Let 1 > 27 be such that z(t) > z* and i ( 1 ) = 0. Then, (8.1) implies that -7

Therefore,

This completes the proof. 0 In order to state our next result, we need the following definitions: 5 = g-'(f(o)),

1= max{-f'(z)

:3 E

3 = max{g'(z) : z E

[o,s]},

[0,2]}.

(8.13) (8.14) (8.15)

Theorem 8.2. In (8.1), if ( j + g ) < ~ 1, then z ( t ) = z* is globally asymptotically stable. Proof. Since Proposition 8.1 holds, we may assume z ( t ) is a solution of (8.1) that is oscillatory about z*.By Lemma 8.1, we see that there exists a T > 0 such that, for t 2 T, z ( t ) c z. Let 2 > T be such that z(1) > x* and z(t) is a local maximum. Then, &(I) = 0, which leads to

Since g ( z ( 1 ) ) > g ( z * ) = f(z*),and f ( - )is strictly decreasing, we have

[T

Z(t

+

3) d / A ( S )

< Z*.

This implies that there is a 6 E [-T,O] such that

z(t + 6 ) = z*. Similarly, we can show that, if ~ ( t<) z* and z(t) is a local minimum, then there is a 8 E [-T,O] such that z(t 6) = z*. We denote u = limsup I z (t) - z*1.

+

i++w

Delay Differential Equations

156

If

u

> 0, then there exists c > 0 such that u

Let To

-€ > (J

+ j ) 7 (u +

(8.16)

€).

> T be such that, for t 2 To - 27, IZ(t)

- Z*1< u + €.

We assume first that there is a 5 maximum such that Z(5)

> TO with ~ ( t>) Z* and ~ ( 5 a) local

- x* > u - €.

Let B E [--7,O] be such that z(5+ 0) = z*.By integrating (8.1) from 5 to t , we arrive at

I j e ( u + 6 ) + g q +~6 ) 5 (7 + g)+ That is, -€

+

+0

6).

< (f+ 9)T(u + f ) ,

a contradiction to (8.16). Hence, u = 0. Similarly, we can show that u = 0 if there is a 5 > TO such that ~ ( t<)z*,~ ( tis) a local minimum, and Z* - ~ ( t >) u - 6 . T h e proof is complete. 0 Corollary 8.2. In (8.12), assume n > 1 and [y

1 + -pB-'n-'(n + l)"+'/"(n - 1)"-'I"]7 4

< 1.

(8.18)

Then its unique positive steady state i s globally asymptotically stable. Proof. From the proof of Corollary 8.1, we see that

f 5 :pe 1

-1

-1

.(

+ iy+*/yn- i)n-l/n.

Clearly, 3 = y. Hence, (8.18) implies that follows from Theorem 8.2.

(f + 9)' < 1. The conclusion 0

4 . Global Stability for Single Species Models

157

In the rest of this section, we consider the following Lotka-Volterra type delay equation

, T are the same where f , g satisfy (Hla), (H2), respectively, and p , ~ and constants as those that appear in (8.1). By a similar argument as in the proof of Theorem 8.1, we obtain the following. Theorem 8.3. In (8.19), assume that the assumptions of Theorem 8.1 are satisfied. Then its unique positive steady state W absolutely globally asymptotically stable. A simple modification of the proof of Theorem 8.2 yields the following. Theorem 8.4. I n (8.19), i f i t ( f + g ) ~< 1, then its unique positive steady state i s globally asymptotically stable. Proof. We note that, in order to use the proof of Theorem 8.2, we need ~ it(? S)T and change (8.17) to only replace (f g ) by

+

z(j)

- z(j

+

+ 0) = j! i

t+e

z(t)

[f(lT + z(t

s) d p ( s ) )

- f(z*)]dt

Obviously, similar statements to Corollaries 8.1 and 8.2 can be estab-

(8.20) To conclude this section, we consider the delay differential equation k(t) = x -

av,z(t)zn(t - T ) 6"

+ Xn(t -

T )

'

(8.21)

where A, a, V,, o,and T are positive constants. This is the model proposed by Mackey and Glass (1977) for studying a "dynamic disease" involving respiratory disorders. A nice discussion on the oscillations and global attractivity of this equation is given in Gopalsamy et al. (1989). It is easy to see that the initial value problem of Eq. (8.21) has a unique solution. If z(0) > 0, then z ( t )> 0 for t > 0. In the following, we assume that z(0) > 0. Denote y ( t ) = l/x(t); then, we have

(8.22)

Delay Differential Equations

158

Let ~ ( t=) u y ( t ) ; then,

[

i ( t ) = Z ( t ) zn(t:v:)

+ 1 - Xu-'z(t)] .

(8.23)

Clearly, (8.23) is equivalent to equation (8.21). Therefore, Theorems 8.3 and 8.4 can be applied to obtain the following result. C o r o l l a r y 8.3. Let z ( t ) = z* be the unique positive steady state in (8.21). Assume that one of the following two conditions holds: (i) n 5 1 and aV, < 2z*Xu-'; (ii) n > 1, $av,n-'(n + ~ ) ~ + ' / ~ (I)"-'/" n c XK'. Then at is absolutely globally asymptotically stable. I f n > 1 and

then x ( t ) = x* is globally asymptotically stable. Proof. The first half of the corollary follows from Theorem 8.3 and Corollary 8.1, where one notes that 8 = 1, /3 = OVm, 7 = Xu-'. The second half follows from Theorem 8.4 and Corollary 8.2, where one notes that = uX-'aV,. 0 4.9. E q u a t i o n s of T y p e X ( t ) = f ( q )- g ( z ( t ) ) - W h e n f(z) Is I n c r e a s i n g or H a s a H u m p In this section, we assume first that the function f ( . ) in Eq. (8.1) satisfies the following condition. (Hlb) f(.) is strictZy increasing, f(0) = 0; there is a unique z* > 0 such that f(x) > g(z) f o r z E (O,z+) and f(z)< g(z) f o r z > I*. All other assumptions remain the same as in the previous section. It is easy to verify that the following equation meets all these requirements:

where /3,8,~,n and 7 are positive constants, 0 < n 5 1, and p > 7. Lemma 9.1. Let z ( t ) be a solution of (8.1) such that 0 < z(0) < x*. Then, for t 2 0, z ( t ) > min{z(t) : t E [0,7]}. Proof. Denote 2, = min{z(t) : t E [0,7]}. (94 By Proposition 7.1, we see that xm > 0. Assume that the conclusion of the lemma is false; then, there is a t 2 T , x ( 2 ) = xm, z ( t ) 2 xm for 0 5 t 5 1,

4 . Global Stability for Single Species Models

159

and i(1)5 0. This leads to

By (Hlb), this implies

a contradiction to the definition of zm. The proof is thus completed.

We are ready t o state our main result in this section. Theorem 9.1. Assume (Hlb) and (H2) in (9.1). Then z = z* is absolutely globally asymptotically stable. Proof. Denote u = limsup Iz(t)- z*1. (9.3) t-+m

Since z(t) is bounded, we see that u < t o o . If u > 0, then at least one of the following two statements is true: = +OO, and (i) There is a sequence {ti}, ti > ti-1, limi++,ti lim,++, z(ti) = X* 21; = +oo, and (ii) There is a sequence {ti}, ti > ti-1, lim;,+,ti lim,,+, z(t;) = z* - u , provided that u 5 z*. Assume first that (i) is true. Then there is an E > 0 such that

+

f(u

For this

E,

+ + €

2*)

< g(u - € + z*).

there exists a T = T ( E > ) T such that, for t 2 T - T, we have z(t) < u

+ + 2*. €

We have two subcases to consider: (ia) z(t) is not monotone; (ib) z(t) is monotone. Suppose first that z ( t ) is not monotone. Then there is a that i ( 1 ) = 0, s(1) - z* > u - 6. This implies that

Thus.

(9.4)

(9.5)

t > T such

Delay Differential Equations

160

By (9.4), we have

Hence, there is a 0 E

[-T,

-u] such that

z(t

+ 0) > u + + z*, €

a contradiction to (9.5). Suppose now that z ( t ) is monotone. Thus, we must have k ( t ) 5 0 for large t , and l-+m lim z ( t ) = z* u. (9.6)

+

However, this leads to

which implies that lim z ( t ) = -00,

i-+m

a contradiction to (9.6). Assume now that (ii) is true. Without loss of generality, we may assume that 0 < z(0) < x*. We can thus choose 0 < 6 < X m , where xm is defined as in (9.2), such that f(z* - u - €)

> g(z* - u + €).

The rest of the proof is similar to the one for case (i). We omit it here to avoid repetition. 0 An immediate consequence of the preceding theorem is the following. Corollary 9.1. In (9.1), i f 0 < n 5 1, a n d ,B > y, t h e n its u n i q u e positive steady state is absolutely asymptotically stable. It is easy to see that the same argument as in the proof of Theorem 9.1 can be applied to

where f , g satisfy (Hla) and (H2), respectively. We have the following: Theorem 9.2. A s s u m e (Hla) and (H2) hold in (9.7). T h e n its u n i q u e positive steady state x ( t ) = x* i s absolutely globally asymptotically stable.

4 . Global Stability for Single Species Models

161

In the rest of this section, we always assume that the function f(.) in Eq. (8.1) satisfies the following. (Hlc) f(0) = 0; there i3 an ZM > 0 , m c h that f(.) i s strictly incwa.9ing i n [0,x ~ and ] strictly decreusing i n [ Z M ,+oo); limz++oof(x) 2 0. There i s a unique x* > 0 8uch that f ( x ) > g(x) for x E (o,z*) and f ( x ) < g(x) for x > x*. We keep the rest of the assumptions made in the previous section. It is easy to verify that, if n > 1, then Eq. (9.1) meets all these requirements. Another interesting example is the following model, which has been used in describing the dynamics of Nicholson's blowflies in Gurney et al. (1980):

i ( t ) = p z ( t - T)e-+r)

- 6x(t),

(9.8)

where p , T , a and 6 are positive constants. In order to state our next theorem, we need the following notations: f l = z$f(z(t)),

f2

Fl(Y) = f-1(Y)1[0,2M]' F2(Y)

= f-l(Y)l[zM,oo),

G(Y) = g-'(Y),

= f(2M);

Y E [O,f21;

Y E (f19f21;

Y E [O,00).

(9.9) (9.10) (9.11) (9.12)

We also need the following lemma. Lemma 9.2. Assume (Hlc) and (H2) hold in (8.1), and 0 < x ( 0 ) < x*. Then there are 6 > 0 , T > 0 such that, for t 2 T , z ( t ) 2 6. Proof. Since z ( t ) is bounded, there is an it > x* such that x ( t ) < it for t 2 0. Let 0 < z < x* be so small that f(z) is strictly increasing on [O,d, and max{f(z) : r E [O,xJ} = f(x) - = min{f(x) : x E k , ~ ] } . If the conclusion of the lemma is false, then there is a t > T , x ( t ) = min{x(t) : t E [O,?]}, z(?) < g-'(f(g)), and i ( t )5 0. However, this leads to

By the definition of x and the choice o f t , we see that

1; By (Hlc), we have

x(t

+ s) d p ( s ) < g.

162

Delay Differeniial Equaiionr

which implies that there is a 8 E

[--7,

such that

-01

Z(t+ e) c .(I), a contradiction to the definition of ~ ( 1 ) This . proves the lemma. Theorem 9.3. Assume (Hlc) and (H2) hold in (8.1). If

M Y ) - z*I < IF2(Y) - 2*1

for Y E ( f l , f 2 1 , Y

# g(z*),

0 (9.13)

then x ( t ) = x* i s absolutely globally asymptotically stable. Proof. Denote u = limsup Iz(t)- z*I; i-+m

again, we have u < $00. In the following, we assume that u > 0. It is easy to see that (9.13) is equivalent to Ig-'(f(z))

< Iz - 2*1,

-I*[

z E [r~,oo), z

# z*.

(9.14)

If X M < z*, then we have I g - ' ( f ( z ) ) - 2*1 < Iz - t*l,

since f ( z ) > g ( i ) for

I E

0

z E (O,Fl(f(2*))],

[0, x*). Also, we have, for y E

(9.15)

[f(~*), ~(zM)], (9.16)

5 2* - F2(y) 5 5* - F1(y).

Thus, by (9.13), we have, for y E [ f ( x * ) , f ( z ~ ) ] , 0

5 G ( y ) - 2* < I* - F l ( y ) ,

(9.17)

which is equivalent to

0 5 g - l ( f ( 2 ) ) - 2* < 2* - 2,

2

E [F1(f(st)),4.

(9.18)

Combining (9.14), (9.15), and (9.18), we arrive at I g - ' ( f ( z ) ) - 2*1 <

12 - 5*1,

z

> 0,

z

# 2*.

(9.19)

If ZM 2 I*,we can also arrive at (9.19) by a similar argument. If z ( t ) is monotone, then by an argument similar to the one included in the proof of Theorem 9.1 (dealing with the monotone case), we conclude that u = 0. In the case that ~ ( tis) oscillatory about r*,the rest of the proof follows from Lemma 9.2 and the proof of Theorem 8.1. We omit the 0 details here to avoid repetition.

4 . Global Stability for Single Species ModeLs

163

Clearly, the preceding theorem is equally true for Eq. (9.7). Let z ( t ) = B y ( t ) ; then, (9.1) reduces to

(9.20)

p > 7, n > 1, and

Corollary 9.2. In (9.20), assume

1 4

- 1)2 < 7.

-p.-'(.

(9.21)

T h e n its unique positive steady state is absolutely globally asymptotically s table.

Proof. Denote f(Y) =PY/U

+Y"),

Y 2 0.

We have

(9.22) and

f " ( y ) = pnyn-1

( n - 1)y" - ( n ( 1 Y")3

+

+ 1)

(9.23)

It is easy to see that f ( y ) achieves its maximum value a t y~ = ( n - l)-"-l. Equation (9.23) indicates that the minimum of f ' ( y ) for y 2 y~ is achieved a t ym, where ( n - l ) y L = n 1 . A simple algebraic computation yields

+

If'(Ym)I

1 =Z

P -1~( n - 112.

Therefore, if (9.21) is satisfied, then (9.13) holds for Eq. (9.20). The 0 conclusion now follows from Theorem 9.3. We now consider Eq. (9.8). By letting y ( t ) = a z ( t ) , it reduces to

3j(t) = p y ( t - T)e-g(t--r) - 6 y ( t ) .

(9.24)

Corollary 9.3. In (9.24), assume pe-2 < 6 < p . T h e n its unique positive steady state i s absolutely globally asymptotically stable. Proof. Let f ( y ) = pye-9. We have f ' ( y ) = pe-g(l - y ) and f " ( y ) = pe-y(y - 2). Hence, its maximum is achieved at y = 1, and its minimum

Delay Diffeerential Equations

164

derivative for y 2 1 is achieved at y = 2; lf'(2)I = pe-2. Now the conclusion follows from Theorem 9.3. 0 The next, simple, theorem deals with the situation when (9.13) fails. Theorem 9.4. I n Eq. (8.1), assume that (Hlc) and (H2) hold and X M 2 x*. Then z ( t ) = x* is absolutely globally asymptotically stable. Proof. If x ( t ) is monotone, then by a similar argument as in the proof of Theorem 9.1, we can show that limt++mz(t) = x*. Thus, we assume in the following that z ( t ) is not monotone. Assume first that ZM = z*. One can show easily that, if, for some t o > 0, to) 5 z*, then z ( t ) 5 z* for t 2 t o . If z ( t ) > z* for t 2 0, then we see that k ( t ) < 0, which implies that z ( t ) is monotone and, therefore, limt++m z ( t ) = 2'. Hence, without loss of any generality, we may assume that z ( t ) 5 z*, t 2 0. Using the notations as defined in (9.10) and (9.12), we thus have

Now a similar argument as in the proof of Theorem 9.3 can be made to show that limt--.+m z ( t ) = z*. Suppose now that X M > z*. Denote

u = limsupz(t).

(9.26)

t++m

Assume first that u 2 X M . Since X M > z*, we see there is an that 9(" - c)

6

> 0 such (9.27)

>f(XM).

Clearly, by the definition of (9.26), we see that there is a 1 > T such that > u - c and ?(?) 2 0. This implies that

z(t)

a contradiction to (9.27). This proves that u < z ~ This . implies that there is a T > T such that, for t 2 T , z ( t ) < x ~ Again, . by using the notations as defined in (9.10) and (9.12), we have

M Y ) - z*I < IFl(Y)- z*L

Y E (O,f(zM)),

Y

# f(z*).

(9.28)

Thus, by a similar argument as in the proof of Theorem 9.3, we can show that limt++m z ( t ) = z*. This completes the proof. 0 Remark 9.1. For Eqs. (9.20) and (9.24), the results obtained from Theorem 9.4 are not as sharp as the ones derived from Theorem 9.3.

4. Global Stability for Single Species Models

165

However, Theorem 9.4 is very useful when f(z) decreases rapidly for Z

2 XM.

Our next result is similar to Theorem 8.2, which relates the delay length T to the global asymptotic stability of z ( t ) = z*. Because of Theorem 9.4, we assume in the following that XM < z*. We need the following definitions:

2 = Fl(g(z*));

5 =g-'(f(zM))i

7 = max{ If'(z)l : z E b,z]}; i j = max{g'(z) : 2 E [o,?]}.

(9.29) (9.30) (9.31)

We also need the following simple lemma: Lemma 9.3. I n (8.1), assume that (Hlc) and (H2) hold. Then limsupt-r+ooz ( t ) 5 5. Proof. Denote u = limsupz(t). t++w Assume u > 5; then, there is a 2 > T such that z(t) > it, $(I) 2 0. This implies that

a contradiction to (9.29). The proof is complete. Theorem 9.5. I n (8.1), assume that (Hlc) and (H2) hold, ZM < z*, and (f i j ) < ~ 1. Then z ( t ) = z* i s globally asymptotically stable. Proof. By the continuity of f'(-)and g'(-), we see there exists c > 0 such that (9.32) 9 € ) T < 1,

+

(5+

where

+ €11, +

= max{lf'(z)l : z E &,z gc = max{g'(z) : z E [O,Z c]}. fc

(9.33) (9.34)

By Lemma 9.3, we see that there is a T > T such that z(t) 5 5

+ c,

t 2 T.

(9.35)

u = limsup Iz(t) - z*I. t-+m

(9.36)

Denote

166

Delay Differeniial Equaiiona

We assume in the following that u > 0. We assume first that there is a sequence of ti, ti+l > ti > 7, limi+mti = +OO, &(ti) 5 0, and limi++oo z(ti) = z*- u. By Lemma 9.2, we must have u c z*in this case. Clearly, there exists 0 < 6 < c such that f,5

A =min{f(z* -u

+ 06) : 0 E [-l,l]} > g(z* - u + 6)

and u

Let TI > T

+ & ) 7 ( u + 6).

-6 > (j€

+ 27 be such that

z ( t ) > z* - u - 6,

t 2 TI

Let t = ti0 be such that t;o > TI, z(1) < z* - u

(9.37) (9.38)

- 27.

(9.39)

+ 6. Since i(1) 5 0,

S(J_Y 4 1 + .) 4 4 s ) ) I g(4)) < g(z* - u + 6).

(9.40)

Hence, we have, from (9.37), or, because o f the nonmonotonicity of

(ii) / - “ z ( t -7

f(.),

+ s) d p ( s ) > z*.

+ +

If (i) holds, then there is a 0 E [--T, -a] such that z(1 0) < z*- u - 6, a contradiction to (9.39). If (ii) holds, then we see that there is a 0 E [-T,O] such that z(1 0) = z*. By integrating (8.1) from 1 0 to 1, we obtain

+

I (7€+ sf)e(. That is, 21

+ 6). +

+

- 6 I (7f 9c)7(u 61,

a contradiction to (9.38). Assume now that there is a sequence of ti, t;+l > ti > 7,limi-.+oo ti = +OO, k(ti) 2 0, z(ti) > x’, and limi++ooz(ti) = z* u. In this case, we can show easily that there are 0, E [ - ~ , 0 ]such that z(ti 0,) = z*. By a similar argument as the one made in the previous case, we again arrive at a contradiction. Therefore, u must be zero, proving the theorem. 0 By the virtue of the proof of Theorem 9.5, we have the following. Theorem 9.6. I n (9.7), assume that (Hlc) and (H2) hold, z~ < z*, and %(f + 9 ) < ~ 1. Then z ( t ) = z* is globally asymptotically stable.

+

+

4 . Global Slabilily for Single

Species Models

167

X

Fig. 4.2. Here, h ( a ) 2 f ( z ) for 0 is A.G.A.S.

5 z5

I*,and

h ( z ) 5 f(z) for

I

2 z*;thus, z*

Clearly, both Theorems 9.3 and 9.4 are valid for Eq. (9.7) provided that (Hlc) and (H2) hold. Direct applications of Theorem 9.5 can easily lead to explicit conditions for the unique positive steady states to be globally asymptotically stable in both Eqs. (9.20) and (9.24). Our criteria for the absolute global asymptotic stability have a strong geometrical background, which makes them easy to use practically. Assume first that f ( x ) is strictly decreasing; then, the steps to apply the criterion are as follows: (i) Graph the functions f ( ~ and ) g(x); locate the positive intersection X*.

(ii) Graph the function h ( z ) = g(22* -x), which is symmetric to g(x) with respect to the line x = x*. (iii) Compare f ( x ) with h(x). If, for x E [O,z*), h ( z ) > f(x) and, for x > x*, f ( x ) > h ( ~ )then , X* is absolutely globally asymptotically stable (see Fig. 4.2). Otherwise, the criterion is inconclusive. If f(x) is strictly increasing and satisfies (Hlb), then, by Theorem 9.1, we know X* is absolutely globally asymptotically stable. If f(z) satisfies (Hlc) and its only hump lies at the right side of line x = z*,

168

Delay Diffemntial Equations

Fig. 4.3. h ( z ) 2 g(z) for 0 5 z 5

I’

implies that

E’

is A.G.A.S.

then the same conclusion as before is true. Otherwise, graph the function h ( z ) = f(2z* - z), the symmetric function of f(z) with respect to line z = z*. If we have h ( z ) > g(z) for z E (O,z*), then, again, z* is absolutely globally asymptotically stable (see Fig. 4.3). Otherwise, the criterion is inconclusive. It is easy to see from these criteria that, in order to have z* the global attractor, roughly speaking, it is sufficient that the slope of g(z) be steeper than that of f ( z ) for z > z*, i.e., Ig’(z)l > If’(z)l. Clearly, this requires that, after passing the steady state z*, the self-crowding effect becomes stronger than the growth mechanism in the described systems. We note that the absolute global asymptotic stability is independent of the delay length T and the distribution function p ( s ) . In these situations, the effects caused by the time delay can be ignored. We also note that whether or not f(z) has only one hump is not critical. What is really important here are the locations and shapes of these humps. When all the previously mentioned criteria fail, it generally implies that the delay effects play important roles in the dynamics of the considered systems. As the delay length T increases from zero, periodic and aperiodic (“Chaotic”) solutions may appear. In these cases, our criteria (including Theorems 8.2-8.4, Corollaries 8.2, 8.3; Theorems 9.5, 9.6) require that the

4, Global Stability for Single Species Models

169

delay length T be small in order to have z* continue to be the global attractor. These results are largely in agreement with the findings of Gopalsamy et al. (1989, 1990b). However, it may be difficult to judge which result is sharper mathematically, since the method used here is entirely different from the one used by Gopalsamy et al. Nevertheless, it is easy t o see that our results are relatively easy to use, since one needs only to estimate the values z,y, and g as defined in (8.13)-(8.15) or in (9.29)-(9.31). Corollaries 8.2 and 8.3 are just two simple examples of their applications. It should be pointed out here that our methods for the global attractivity can be applied to the so-called state dependent delay differential equations j.(t) = f ( 4 t - 4 . t ) ) )

- g(4t))

(9.41)

and d t ) = 4t)"t

- T(21))) - g("l9

(9.42)

) replaced by ~ ( t ) , where ~ ( qis )a continuous functional of 5 1 . If ~ ( 1 1 is a bounded nonnegative continuously differentiable function, our methods still apply. It should be mentioned here that our approaches fail to work for the following more general equation:

where 0 <

(T

< T < w are constants, p ~ ( s and )

p~g(s)are nondecreasing,

J--," d p l ( s ) = JI-," d p 2 ( s ) = 1, and h ( z ) has similar properties as that of f(z),which are stated in the previous sections. This equation is more general than a simple model proposed by Bilair and Mackey (1987) for the regulation of mammalian platelet production:

F(t)= - y ~ ( t + ) ,o(P(~- T)) - p ( ~ (-t T - o ) ) e - y a ,

+

(9.44)

where p ( P ) = p o P P / ( P P"), and p o , d , n , y , ~ ,and (T are positive constants. More realistically, one should probably replace g ( z ( t ) ) in (9.43) by g a!J( z(t s) d p 3 ( s ) ) , where 6 < (T,p 3 ( s ) is nondecreasing and p ( 0 ) - p(-6) = 1. One of the main difficulties involved in the analysis of (9.43) is that solutions of positive initial values may fail to be positive. However, in these cases, partial results for the convergence of positive solutions can be obtained by arguments similar to that of Kuang and Smith (1991c), which will be described in Section 6.6.

+

170

Delay Diflerenlial Equoiions

When the functions f ( z ) and g(z) in (7.6) and (7.7) are replaced by nonautonomous ones f(t,z) and g ( t , z ) , our methods may still apply, provided that we assume that there is a unique z* > 0 such that f ( t , z*) = g ( t , z * ) for all t 2 0. In this case, we will need an argument that, roughly speaking, combines the one presented in Kuang and Smith (1991b) and a perturbation argument presented in Kuang et al. (1991a) for systems of delay equations in population growth models, which will be discussed in Section 6.4. 4.10. Remarks and Open Problems In the first half of this chapter, we focused our attention on logistic type equations or equations with negative feedbacks. Other variations of logistic type equations considered in the literature include, for example,

(10.1) which is discussed by Gopalsamy et al. (1988, 1990a), and

i ( t ) = .(t)z(t)

1- z ( t - T ( t ) ) 1 - cz(t - T ( t ) ) ,

(10.2)

which is considered in Kuang e t al. (1991b). All the coefficients (functions) are positive, except c ( t ) or c may be zero. Equation (10.1) takes into account the fact that a growing population may consume more “food” than a saturated one, since a growing population needs food both for maintenance and growth. Equation (10.2) takes into account the phenomena observed in nature by Thompson (1952) that microorganisms, plants, and animals all show a maximum growth velocity when the population density is greater than half of the carrying capacity. Miller (1966) studied

i ( t ) = z ( t ) [a - bz(t)-

1

1

f ( t - s)z(s)

ds] ,

(10.3)

C

where c = 0 or -00. The generalization of his results will be given in Chapter 6. Gilpin and Ayala (1973) considered

i ( t ) = rz(2) [I -

,

(10.4)

where 0 # 1. Other general time delayed single species models are discussed in Freedman and Gopalsamy (1986). Delayed single species growth models with stage structure and/or state dependent delays will be studied in the next chapter.

4. Global Stability for Single Species Models

171

For some insect populations, delayed difference equations may be the more suitable models for their growths. Examples of such models are introduced and studied in Kocic and Ladas (1990, 1991), Kuruklis and Ladas (1992) and Rodrigues (1992). The following are four interesting open problems on the global asymptotical stability of delayed scalar equations. Open Problem 4.1. Is it true that, if a < 7r/2, then positive solutions of 5 ( t ) = az(t)[l - z ( t - l)] (10.5) tend t o the steady state z ( t ) = l ? Open Problem 4.2. Is it true that, if 0 < a(t) < 3/2, then positive solutions of i ( t ) = a(t)z(t)[l - z ( t - l)] (10.6)

l ? Here, a(2) is a positive continuous tend to the steady state z ( t ) function. Open Problem 4.3. Obtain sufficient conditions for the global asymptotic stability of the positive steady state of Eq. (9.43), which has Eq. (9.44) as a special case. Open Problem 4.4. Obtain sufficient conditions for the global asymptotic stability of the positive steady state of equations of the form

where f and g satisfy similar conditions as that for Eq. (8.1). Here, we assume that J,! z ( t s) d p z ( s ) # z ( t ) . Equation (10.7) allows delays to appear in both f(.) and g(.). This is important in applications since, in all real systems, changes take time, no matter how small. Convergence (but not global stability) results for Eqs. (9.43) and (10.7) will be given in Section 6.5 as special cases of similar results for systems of delay differential equations.

+

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5

Periodic Solutions, Chaos, Structured Single Species Models In this chapter, we consider global existence of periodic solutions in both autonomous and periodic equations. We also discuss chaotic behavior induced by time delays in some models. Moreover, we present a single species model with stage structure and describe an effective way of reducing a class of structured models to standard DDE models. 5.1.

Global Existence of Periodic Solutions in i ( t ) = f ( t ( t - 1)) S(X(t))

In this section, we establish the global existence of periodic solutions in

i ( t ) = f(4t - 7))- 9(4t)),

(1.1)

which is a special case of (4.7.6). Our approach can be easily modified to cover the equation

i ( t ) = z(t)[f(z(t- 7 ) ) - g ( W ) l .

(1.2)

Without loss of generality, we assume the unique positive steady state is 1, i.e., f(1)= d1). (1.3) As we have shown in the previous chapter, if f(z) is increasing or has a hump located at the right hand side o f t = 1, then ~ ( t3) 1 is A.G.A.S. Thus, in order t o have periodic solutions, it is necessary for us to assume that either j ( z ) is decreasing or has a hump to the left of I = 1. More precisely, we will assume, throughout the rest of this section, that ( A l ) f(x) satisfies (Hla) OT (Hlc) ( i n Sections 4.8 and4.9, respectively) W i t h X M < 2* = 1. For convenience, we denote X M = 0 when f satisfies (Hla). In the following. we also assume

173

Delay Differential Equations

174

Fig. 5.1. f(g-’(f(E))) implies that k , Z ] is invariant.

The accompanying Fig. 5.1 will be helpful in understanding the assumption (A2). The following simple lemma is necessary in our subsequent discussion. Lemma 1.1. Assume that f ( z ) in (1.1) satisfies ( A l ) and (A2), and z(s) E [.,it] for s E [ t o - 7,toI. Then z ( t ) E [ 3 ~ for]t 2 t o . Proof. Let 0 5 3 < z be such that, for z E [3,z], - f(g-’(f(z))) > g(z). If the lemma is false, then there are two cases to consider: (i) There is a > to, 3 < z(t) < 5, z(t) < z ( t ) 5 Z for to - T < t < t, and z’(t) < 0; (ii) There is a > t o , x ( t ) > F , g 5 z ( t ) < z(t) for t o - T < t < t , and ~ ‘ ( 1>) 0. We consider first case (i). Since .(I< )-z, z’(t) < 0, we have

fb(2 - TI) < 9 ( z ( t ) ) .

(1.4)

Since f(z) 2 g(z) for x E [0,1], and g(z) is strictly increasing, (1.4) thus

175

5. Periodic Solutions, Chaos, Structured Single Species Models

implies that z(1 - T ) > 1. If 1 < $(IT ) 5 5,then the monotonicity of f ( t ) for t 2 ZM together with (A2) imply

f ( 4 - T I ) 2 f(z) > g ( 9 . And the monotonicity of g ( t ) leads to

f ( 4 - 7))> d 4 > g M W , a contradiction to (1.4).

Assume now that (ii) holds. Clearly, i(’i) > 0 implies that

f ( 4 j- T I ) > 9 ( 4 W The monotonicity of g ( z ) thus implies that

f ( 4-

>4 5 )=g ( g W d ) )=

m.

Now the monotonicity of f ( z ) for t > I M clearly indicates that

t ( t - T ) < 2, which contradicts our assumption that 2 5 z ( t ) < .(I) for t o - T < t < 1. 0 This completes the proof of the lemma. In the rest of this section w e assume that the initial condition for (1.1) 3 atisfies t ( s ) = qqs), s E [-.,O], 5 5 4(s) I5, (1.9) and d(s) E C([-T,O],R). Lemma 1.1 thus implies that 2 5 z ( t ) 5 it for t 2 0. Clearly, in [ -z,5], both - f ( t ) and g ( z ) are strictly increasing. The readers thus, without loss of generality, may assume in the following that f ( x ) satisfies (Hla). Also, for convenience, we assume from now on that T = 1. Equation (1.1) thus reduces to

4 4 = f ( 4 t - 1)) - g

( N .

(1.10)

We denote a = -f’(l), p = g’(1). Then, the linearized equation of (1.10) a t t = 1 takes the form

i ( t ) = -at(t - 1 ) - Pz(t).

(1.11)

It has the characteristic equation

x +p +

= 0.

(1.12)

Delay Differential Equations

176

The following lemma can be found in Hadeler and Tomiuk (1977); it can also be obtained by applying Theorem 3.2.1. We note that cr > 0, p > 0. Lemma 1.2. Let a > crp, where crp is the smallest positive solution of p+acos\la2-p2 =o. (1.13) Then (1.12) has a solution X with Re X > 0, n/2 < Im X < n. In order to present our main results, we need the following notations: (1.14) v = max{g'(z) : z E k , ~ ] } ; K = { 4 ( s ) : 4 E C,4(-1) = 1 , 5~ 4(s) 5 Z,(~(S) - l)exp(vs) is a nondecreasing function of s E [-l,O]}; (1.15) zt'1 = K\{ l } , (1.16) where 1 is the function 4(s) e 1, s E [-1,0]. It is easy to see that K is a closed, bounded, and convex subset of the Banach space C([-l,O], R) with the standard supremum norm 11 .I[. This set is of crucial importance in the proof of the following theorem, which is the main result of this section: Theorem 1.1. Assume ( A l ) and (A2) hold, and cr > max{l,ap}. T h e n Eq. (1.10) has a nonconstant periodic solution z ( t ) with period greater than 2, and that satisfies z 5 z(t) 5 2. Roughly, our approach involves showing that initial conditions near 1 in It' are taken away from it, and those away from it tend in some sense to approach it (with the help of Lemma 1.1). This leads to the existence of nonconstant fixed points of an operator A of the form A 4 = zu(-,4), where o = o(4) is a nonnegative number (to be defined), 4 E Zt', and cY(s,4) = z(n s,41, s E [-LO]. The following lemma is needed to show that there is an operator A that maps It' into itself. Lemma 1.3. Assume that all conditions of Theorem 1.1 are satisfied and z ( t ) = z ( t , $ ) , 4 E Zt'1, is a solution of (1.10). Then the following hold: such that (1) There is a sequence { z i } z l , 0 < z1 < z2 < c ( z i ) = 1, zj+1 > zi, i = 1 , 2 , . . . (2) k ( ~ k - 1 )< 0, k(~2k)> 0, f o r k = 1 , 2 , . . (3) The function e U t ( z ( t ) - 1 )is nonincreasing o n each of the interval3 (z2k-1, z2~-1+ 1) and nondecreasing o n each of the intervals (22k, z2k l ) , k = 1 , 2 , 3,... . (4) .There is a constant q > 0 such that, for 4 E Zt', 2 2 5 q. Proof. Since 4 E K1, we must have 4(0) = z(0) > 1. We shall show that there is a finite time z1 > 0, such that z1 = inf{t : t 2 O , z ( t ) = 1). (1.17)

+

.

e . . ,

..

+

5. Periodic Solutions, Chaos, Siructured Single Species Models

177

Since -f‘(l) = a > 1, we can choose a constant 6 > 0 such that 6 < min{z - 1 , l - g}, and

If(.)

- f(1)l 1 12 - 11,

for Iz - 11 5 6.

(1.18)

Clearly, if z ( t ) 2 1 for t E [0, to], then k ( t ) I 0 for t E [0, t o ] . Let t l = inf{t : t

1 O,z(t) I 1 + 6).

(1.19)

If s(0) 5 1+6, then tl = 0. Now, assume that z(0) > 1+6. For 0 5 t 5 t l , we have (1.20) i ( t )= -[g(z(t)) - d1)1 [f(z(t - 1)) - f(1)l. Hence, (1.21) i ( t ) I - b W ) - g(1N I -b(l 6) - S(1)l. Thus, we must have

+

+

tl I (z - 1 - 6)/[g(l Suppose (‘1 > tl we have

+ 6) - g(1)I.

(1.22)

+ 1; then, 0 I z ( t ) - 1 5 6 if t E [tl,tl + 11. From (1.20),

k(t) I f ( z ( t- 1)) - f(1) 5 1 - ~ ( -t l), Since ~ (- t1) 2 1

x E [tl,tl

+ 6 for t E [tl,tl + 11, we have i ( t ) 5 -6 for t E [tl,tl + 11,

which leads to

+

+ 11.

(1.23)

(1.24)

(1.25) X(tl 1) I X(t1) - 6 = 0, a contradiction. Thus, we have shown that z1 5 t l 1. If zi 1 1, then i(z1) = f ( z ( t - 1)) - f(1) < 0. On the other hand, suppose that 21 < 1 and i(z1) = 0. Then, we must have +(t) 1 if -1 I tI -1 21 (since E Kl). For t E [O,zl],we have

+

+

+

k ( t >= -[g(s(t)) - s(t)l 2 - 4 4 t ) - 11,

(1.26)

which clearly implies that s(z1) - 1 2 ( ~ ( 0 ) l)e-” > 0,

(1.27)

= inf{t : t > zl,z(t) = I}, z* = min{zz,zl 1).

(1.28) (1.29)

a contradiction. We denote 22

+

Delay Differential Equations

178

For t E (zi,z*), we have z ( t ) c 1, and j.(t)

I --y(z(t) - 1) + [f(z(t - 1)) - f(1)l.

(1.30)

Hence,

d - l)eut]= e"'[f(z(t - 1)) - f ( l ) ] < 0, (1.31) dt which implies that ( z ( t )- l)eul is decreasing on ( z l , z * ) , and therefore 22 > z1 1 and z* = 21 1. Thus, we have proved that 5 5 z ( t ) < 1 for t E ( Z 1 , Z l 1). For t E (21 1 , 2 2 ) , we have i ( t ) 2 0 from (1.20). We shall show that 22 is finite. Let -[(z(t)

+

+

+

+

+ l , s ( t ) 2 -6+ 1). (1.32) If "(21 + 1) 2 1 - 6, then t2 = z1 + 1. On the other hand, suppose that s(z1 + 1) < 1 - 6; then, for z1 + 1 I t I t2, we have, from (1.20), t2 = inf{t : t 2

21

; ( t ) 2 -[g(l - 6) - g ( l ) ] = g(1) - g(1 - 6).

(1.33)

Hence, we must have

t2-21-1 Now, suppose that j.(t)

which leads to

22

5(1 -.-6)/[g(l)-g(l-6)].

2 t2 + 1. Then, for t

E [t2,t2

2 f ( s ( t- 1)) - f(1) L 1 - Z ( t

(1.34)

+ 11,

- 1) 2

6,

(1.35)

+ 1) L Z(t2) + 6 = 1, (1.36) Hence, we must have 5 12 + 1. By summarizing the s(t2

a contradiction. preceding argument, we have

22

2-1-6 l < z 2 5 3 + g(l + 6 ) - g(1)

+ g(1)1 -- 9g(-l6 - 6) = q.

(1.37)

+ 1).

(1.38)

Similarly, we can show that

d

- [ ( s ( t ) - l)e'']

dt

>o

for t E (z2,z2

+ +

Now, the function $1, where $l(s) = 4 2 2 s I), -1 5 s 5 0, is again in K1. Therefore, the argument can be repeated and the existence of zi7 i = 1,2,. . . , is thus established. This completes the proof. 0

5. Periodic Soluiions, Chaos, Siruciured Single Species Models

We are now ready to define the operator A : K

--+

179

K as

The following lemma is quite straightforward. Lemma 1.4. The mappings 4 + ~ ( 4 o)f K 1 into (1, +m) and A : K --+ (I are completely continuous. Proof. The continuous dependence on the initial data together with the fact that k ( z l ( + ) , $ ) < 0, k(z2(4),4) > 0 clearly indicates that if 119 - 411 is very small, then the function x ( t , 9 ) has two zeros 21,22 very close to z 1 ( 4 ) , z 2 ( 4 ) ,and k(21,cp) < 0, 4 2 2 , ~ > ) 0 , and cannot have any other zeros for t 5 22. The complete continuity follows from the fact that z2(+) 5 q for 6 E K1. The continuity of A follows from the continuity of 2 2 ( + ) and again the continuous dependence on the initial data. Since z2 : K1 + (1, +m) is completely continuous, we see that, for any bounded set B c K1,A ( B ) is bounded and equicontinuous (since z2 > 1) and, thus, compact. Therefore, A is completely continuous. 0

1 is an ejective fixed point of A. Finally, we shall show that $(s) Lemma 1.5. Let 6 = max(l,ap}, 6 = min(2 - 1,l - z}, and J be a compact set of ( 6 , ~ )Then . p = inf(r,q,,+

:

+ E Zt’, 114 - 111 = 6,a E J} > 0.

(1.40)

Proof. Let X = X(a) be the solution of (1.12) given by Lemma 1.2 (where p is fixed), + ( O ) = cAe/(l X p), O E [-1,0], $ ( s ) = e-”, s E [0,1], Cp = (4,$), Q = ($,$). The bilinear form of (1.11) is

+ +

(1.41) We thus have

since X

+ p = -ae-’.

Also,

Delay Differential Equations

180

and

1 p) = 0. A-A Hence, obviously, ($,$) = 1, (+,$) = 0. Therefore, (*,a) is the identity. Hence, for any E C,q q 5 = @(@,+). To show the conclusion of the lemma is true, it is thus sufficient t o show that

- -(-p

+

inf(l(rE,iP-l)1:~EI(,II~-111=6,aE J } > O .

Clearly, it is sufficient to look a t (+,+ - 1). If d(0) - 1 2 6e-' and ($34

- 1,a)

=

($34

4 E K, 114 - 111 = 6, then

- 1) = Re(($) + i Im(4), 0

Re(+) = d(0) - 1 - a / (q5(0) - l)e-Y(e+')cosc(f3 -1

+

+ 1)dB,

where A = 7 ia,7 > 0, c E ( A / ~ , A ) .Since J is compact, there is an 6 > 0 such that 6 < ~7= c(a)< A - 6 for a E J . Now, suppose that there exist sequences 4, E I(, & ( O ) , a n E J such that (+,& - l,a,) + 0 as n + $00. We may assume that 3 &, & ( O ) - 1 + 77 2 6e-l as n + +00, since J is compact and 6-'e 5 r$"(O) - 1 5 6. Since Im($,) + 0, we must have &(8) - 1 --t 0, -1 5 0 < 0. Thus, Re($,) + as n + +00. This is a contradiction to the fact that Re(&) + 0 as n + +m, and the lemma is proved. 0 Finally, we are ready to state the proof of Theorem 1.1. Proof of Theorem 1.1. From the definition of K , we see that it is a closed, bounded, and convex set of infinite dimension in the Banach space C. A as defined in (1.39) is completely continuous by Lemma 1.4 and 1 is an ejective fixed point of A by Theorem 2.9.4 and Lemma 1.5. Therefore, by Theorem 2.9.2, we conclude that A has a fixed point $ in K1, which clearly corresponds to a nonconstant periodic solution z ( t , 4) of period greater than 2. This completes the proof. 0 Remark 1.1. By making arguments similar to that of Smith and Kuang (1992), and modifying the previous discussion accordingly, one can

5. Periodic Soluiions, Chaos, Struciured Single Species Models

181

establish the global existence of periodic solutions in the following state dependent delay equation:

where T is Lipschitz on each bounded subset of C. The case of g(z) = vz is considered in Smith and Kuang (1992). The key observation that can be exploited is that if a solution z ( t ) of (1.42) satisfies ; ( t o ) = 0, then, for t > t o , the solution "forgets" its history prior to t o - .(.to) in the sense that t - ~ ( q>)t o -.(zt,) for t > to. Interesting examples of ~ ( q include ) the bell-shaped function T ( z t ) = ~ ( z = ) 1 - a aexp[-x2(t)], 0 5 a 5 1, and a special class of the generalized threshold delay T = ~ ( q defined ) implicitly by

+

(1.43) where k : R2 -+ (0,oo) is locally Lipschitz and rn > 0. The detail of such generalization is rather technical and tedious; we choose to omit it here. R e m a r k 1.2. By careful modification, one can also use the method adopted by Hadeler and Tomiuk (1977) to show that 1 is ejective. However, the procedure is slightly more involved and needs further adjustment if one would like to consider the more general state dependent delay equation (1.42). The key observation that should be used in such a modification is that, for small 6 > 0, we can choose small ~(6) > 0 such that if Iz(t)-ll < 6 for large t , then y ( t ) = e w z ( t ) is monotone on ( z ; , z; 1)) where i is large, w =p C, and z ( z ; ) = 1.

+

+

5.2. P e r i o d i c Solutions in Delayed Periodic Loth-Volterra-Type

Equations The material of this section is adapted from Tang and Kuang (1992a). Frequently, we observe that populations in the real world tend to fluctuate. There are three typical approaches for modeling such behavior: (i) introduce more species into the model, and consider the higher dimensional systems (like predator-prey interactions); (ii) assume that the per capita growth function is time dependent; (iii) take into account the time delay effect in the population dynamics. Generally speaking, approach (i) is rather artificial, while (ii) and (iii) emphasize only one aspect of reality. Naturally, more realistic models of single species growth should take into account both the changing environment and the effects of time delays. Therefore, it is important to study the following general nonlinear nonautonomous delayed Loth-Volterra type

Delay Differential Equations

182

equation:

When (2.1) has a positive steady state, it reduces to the Eq. (4.3.1). Indeed, the results to be presented in this section can be viewed as generalizations of those in Section 4.3. We show that, under reasonable conditions, Eq. (1.1) has only one asymptotic state, in the sense that if q ( t ) and ~ ( tare ) two solutions of (l.l),then we have lim ( z l ( t )- z 2 ( t ) )= 0.

t++m

In Eq. (2.1), we always assume that f , ( t , z )and r ( t ) are continuous with respect to their arguments, and p ; ( t , s )are continuous with respect t o t, nondecreasing with respect to s, and are defined for all ( t , s ) E R2.In addition, we also always assume the following: (Hl) For z > 0, z f , ( t , z )2 0, f , ( t , z )is nondecreasing with respect f i ( t , z ) is strictly increasing with respect to z; to z,and (H2) r ( t ) > 0, t - r ( t ) is nondecreasing, and limt,+,(t-r(t)) = +co; (H3) p i ( t ,t ) > p i @ , t - r ( t ) ) ; (H4) For any c # 0, there exist a , ( t ) 2 0, b,(lcl) 1 0, bi(lc1) = 0 if and only if c = 0 and b: 2 0 are such that

cR1

Ifi(t,cl)- f i ( t 7 ~ 2 ) IL ai(t)bi(lcl - ~ 2 1 ) and

(H5) There exist K1

> 0 and K2 > 0 with K1 IK2 such that

and

Let r = r(0);the initial value problem for (2.1) is assumed to take the form E .(e) = +(el 2 0, e E [-r,ol,

+

c.

183

5. Periodic Solutions, Chaos, Structured Single Species Models

Theorem 2.1. I n (2.1), assume that there eziskr M > 0 such that

/

t

t-r(t)

b(T)dT 5 M < +oo.

Then x ( + ) ( t ) eziskr for all t 2 0 . Also, for large t , z(+)(t) I 1(2eM.

(2.3)

If, in addition, there is an N > 0 and > 0 such that, for 0 KZeM, f i ( t , x ) < % f o r all t 2 0, i = 1 , . .. ,n, and

cxc

Proof. Let z ( + ) ( t ) be a solution of (2.1) with the maximal existence interval [0, Tmax). Then, we claim that .(+)(t) > 0 for t E [0, Tmax). If not, there is a to > 0 such that .(+)(to) = 0. Without loss of generality, taking to = min{t E [O,Tmax) : x ( + ) ( t ) = 0}, we have

Letting t --t to, we get the contradiction: The left side goes to -00, and the right side is bounded. It is easy to see that the existence of x ( + ) ( t ) for all t 2 0 follows from (2.3). Now we prove that (2.3) is true. If there is some to 2 0 such that, for all t 1 to, z(+)(t2 ) J(2,

then

? ( t ) = Z ( t > [b(t) -

Hence, limt,,

x ( t ) = c 2 I(2.

2 JI

i=l

t-r(t)

fi(t,

dPi(t,s)]

184

Suppose that we have

Delay Differential Equaiions

c

> K2; then, for all large t, z ( t ) > c. By (H5) and (H4),

By (H4), this inequality yields a contradiction. Thus, lim z ( t ) = K2.

t-+oo

Now, suppose that z ( t ) - I(z has infinitely many zeros {tn} with limn-ootn = +CO. If z ( t ) assumes a local maximum at t* with z ( t * )2 K 2 , then i ( t * )= 0, and ( H l ) implies that there is a t o E [t* - r ( t * ) , t * ]such that z(t0) = K 2 . Thus,

Hence, for large t , In z ( t ) - In K Z 5 M , which gives that z ( t ) Iz(2eM.

Equation (2.6) and the preceding inequality complete the proof for (2.3). 0 The proof of (2.5) is similar. In order to establish the global asymptotic behavior of the solutions of (2.1), we need the following lemma. Lemma 2.1. In (2.1), suppose that the assumptions made in Theorem 2.1 (conditions (2.2) and (2.4)) are true. Then, for any two solutions z ( t ) and y(t), zfz(t) > y(t) for all large t, then limt,+,(z(t) - y(t)) = 0. Proof. From (2.1), we have

185

5. Periodic Soluiions, Chaos, Structured Single Species Models

Hence,

x (t ) - c 2 1. lim -

t++m y(t)

Suppose c > 1; then, lim x : ( t )- Y ( t ) = c - 1 . y(t)

t++m

Theorem 2.1 implies that

x ( t ) - y(t)

>qK2eM

for all large t. Now, (H4) yields

Also, by (H4), we get a contradiction. Thus, lim - 1, y(t)

t++w

which, together with the fact that y(t) is bounded between two positive constants for large t , proves the lemma. We are now in a position to state and prove our main result. Theorem 2.2. In (2.1), in addition to the assumptions made in Theorem 2.1, assume further that there are continuous functions ai(t) such that Ifi(t,cl) - f i ( t 7 ~ 2 ) l

I ai(t)lcl - ~ 2 1 7

i = 172,.

. . ,n,

and for large t ,

Then, for any two solutions z ( t ) and y(t) t o (2.1), limt+m(z(t)-?j(t)) = 0. Proof. Let z ( t ) and y(t) be two solutions of (2.1). Lemma 2.1 implies t , = +OO; that z ( t ) - y(t) has infinitely many zeros { t n } with limn,+, otherwise, the proof is finished. Let ~ ( t=) x(t)/y(t); then, from (2.1), it follows that

186

Delay Differential Equations

In a similar argument to that in Theorem 2.1, we have

e-MN

I z ( t ) 5 emN

for all large t.

(2.9)

Let u = limt-r+oosup(z(t) - 1) and . -v = limsupt++m(l - ~ ( t ) )Then, 0 5 u 5 e M N - 1 and 0 5 v 5 1 - e - M N . Choose tl > 0 such that (2.7) and (2.9) are true for all t 2 tl - ~ ( t l ) . Also, for t 2 t l - r ( t l ) ,

-v i.e.,

-(v

- € < Z ( t ) - 1 < 21 + E ;

+4Y(t)

< 4 t ) - Y(t) < (21 + E)Y(t).

Hence, -(v

+ E ) a i ( t ) y ( s ) If i ( t , x ( s ) )- f i ( t , y ( s ) )I + E) a i ( t ) y ( s) (U

Assume z ( t * ) is a maximum or a minimum such that t* - ~ ( t *2) ti. Then, by (Hl) and (H3), there is a t 2 E [t* - r(t*),t*]such that z ( t 2 ) = 1. Thus,

I V + E ,

and so z ( t * )- 1

< eU+' - 1.

Similarly, we have z ( t * ) - 1 > e--(u+t) - 1. The definitions of u and v and the preceding inequalities lead to u - E < e"+' - 1, v - E < 1 - e-('+'). 0 The rest of the proof follows that of Theorem 4.3.2. In fact, our previous arguments have also proved the following result. Theorem 2.3. FOT (2.1), in addition to the assumptions (Hl)-(H4), assume further that there ezists a solutzon zo(t) to (2.1) with 0 < 12 5 x o ( t ) 5 11 for large t, where 11 and 12 are constants. Also, suppose that there are continuous functions a ; ( t )such that Ifi(t,cl) - fi(t,c2)l

I ai(t)lcl - ~ 2 1 ,

i = 1, - - - ,n,

187

5. Periodic Solutions, Chaos, Struciured Single Species Models

and for large t ,

Then, for any two 3olution3 z ( t ) and y ( t ) to (2.1), limt-.+w(z(t)-y(t)) = 0. Proof. By the arguments made in the proofs of Lemma 2.1 and Theorem 2.2, we have that, for any solution z ( t )of (2.1), limr-+oo(z(t)-zo(t)) = 0, 0 and so the conclusion follows. A simple application of Theorem 2.2 or 2.3 results in the following theorem which gives conditions for a periodic equation of form (2.1) to have a globally asymptotically stable periodic solution. Note that such a periodic solution is induced by the periodicity of the equation. Theorem 2.4. Suppose b(t T ) = b ( t ) , r ( t T ) = r ( t ) , f i ( t T , z ) = f i ( t , z), and P i ( t T , 8 ) = P i ( t , S ) for 30me T > 0, and the a ~ ~ u r n p t i o w made in Theorems 2.2 o r 2.3 are true. Then Eq. (2.1) ha3 a unique, globally asymptotically dable periodic solution. Proof. Let z ( t ) be an arbitrary positive solution of Eq. (2.1). Then znT E C , n = 1 , 2 , . . . . Since z ( t ) is bounded from above and the right hand side of (2.1) is completely continuous, the sequence {znT}~=O=l is precompact. Assume that this subsequence z n i converges ~ to z* E C, that is, lim z(n;T 0) = z*(8), 8 E [-r,O]. (2.11)

+

+

+

+

+

i-rw

Since Eq. (2.1) is periodic, y ( t ) = z(t limt,,(z(t) - y ( t ) ) = 0 implies that lim z(n,T

i-w

+ T) is also a solution of

+ T + 0) = z'(0).

(2.1). (2.12)

Let z * ( t ) be the solution of (2.1) with initial condition as z*(E C); then, (2.11) and (2.12) imply that z$ = z*,or equivalently,

z*(T + t ) = z * ( t ) ,

for all t E R.

The conclusion now follows from that of Theorem 2.2 or 2.3.

0

5.3. A Model of Single Species Growth with Stage Structure It is the purpose of this section to develop and analyze a model of single species growth where the individual members of the population have a life history that takes them through two stages, immature and mature.

188

Delay Differeniial Equations

There has been a fair amount of previous work on modeling single species population growth with various stages of life history using discrete models (e.g., Bulmer, 1977; Fisher and Goh, 1984), continuous models (Anderson, 1960; Barclay and van den Driessche, 1980; Landahl and Hanson, 1975), and even stochastic ones (Tognetti, 1975). Here, we develop a continuous model consisting of two equations with a discrete time delay T representing the time from birth to maturity. The material of this section is modified from Aiello and Freedman (1990). Let q ( t ) and zm(t) denote the density of immature and mature populations, respectively. We assume that these populations grow in a closed, homogeneous environment. The following system was introduced by Aiello and Freedman (1990) as a proper model for this:

where T , a,p, y are positive constants. This model is derived as follows. We assume that at any time t > 0 birth into the immature population is proportional to the existing mature population with proportionality constant a. We then assume that the death rate of the immature population is proportional to the existing immature population with proportionality constant 7. We assume for the mature population the death rate is of a logistic nature, that is, proportional to the square of the population with proportionality constant 0. Finally, we assume that those immatures born at time t - T that survive to time t exit from the immature population and enter the mature population. This would be computed as follows. If N ( t ) is a given population at time t , then the number that survive from t 1 to t i is In particular, we have N ( t ) = N ( t - T ) e - 7 ' . conditions, we require

For continuity of initial

the total surviving immature population from the observed births on -T 5 t 5 0. We assume that zm(e)is continuous and nonnegative on [ - T , O ] ; then, solutions of system (3.1) exist and are unique for all t 2 0. Theorem 3.1. Let zm(0) > 0 , zm(0) 2 0 o n -T 5 0 5 0. T h e n the solution of (3.1) with initial condition zm(0) and zi(0) given by (3.2) i s positive for all t 2 0.

5. Periodic Solutions, Chaos, Slruciured Single Species Models

189

Proof. Note that xm(t)is independent of z ; ( t ) and satisfies

i , ( t ) = ae-Y'z,(t

- T ) - pq,,(t). 2

Observe that the solution u ( t ) of

is u(t) =

pi

+

1

>0

fort

2 0.

x;l(O)

By comparison, we have x m ( t ) > 0 for t 2 0. Consider now the equation

We have xi(t) > u ( t ) on O

< t 5 T . Clearly,

Hence,

We thus have u ( 7 ) = 0 by (3.2), and therefore u ( t ) > 0 for t E [ O , T ) . By induction, we can show that q ( t ) > 0 for all t 2 0. There are two nonnegative steady states in system (3.1), namely, Eo(0,O) and E*(xr,I;), where

It is easy to show that Eo is a saddle point in the sense that it has eigenvalues with both positive and negative real parts. It is also easy to show that E* is locally asymptotically stable by analyzing its characteristic equation. However, we can in fact show that E* is globally asymptotically stable. Theorem 3.2. Let xm(8) 2 0 for 8 E [-T,O], and x m ( 0 ) > 0. Then lim [ z i ( t ) , x , ( t ) ] = ( x r , x k ) .

t-bo

190

Delay Diffemntial Equations

Proof. Observe that Eq. (3.3) is a special case of Eq. (4.8.1). By Theorem 4.9.1, we have lim z m ( t ) = .:x t+w

Observe next that, for any t q ( t ) = zi(T)e-y(t-T)

2 T > r , we have

+ e-yLJ,' eys[azm(s) - ae-'rzm(s

- T ) ] ds. (3.7)

Hence, Izi(t) - zr I 5 zi(T)e-Y(t-T)

t

+ 1e-7' J , aey5[ x ~ ( s-) e-yrzm(s-T ) ]ds - 3:

I.

Using the fact that if lim5-w f(s) = fo, then

we have

lim Ixi(t) -

a

5 -(zk - e-yr 2,)

- xrI = 0, I7 0 which proves that limt,, zi(t) = zr. Note that both x: and xk depend on the delay r. We may define the total carrying capacity K ( r ) as 1-02

K(T)

Then,

= x:(r)

a

+ zL(r).

+ y - ae-yr).

K ( r )= -e-Tr(a

Pr

K ( 0 ) = a/P and K(o0) = 0. We have d dr

-K(T)

= ap-'e-yr(2ae-yr

- a -7).

Clearly, if a 5 7 , then ( d / d r ) K ( r )< 0. If a > 7 , then K ( r ) is increasing on 0 5 r < r* and decreasing for r > r*, where

Also, we can obtain

+

K(.r*)= (a r)2/4p7.

5. Periodic Soluiions, Chaos, Siruciumd Single Species Models

191

Hence, we have

Thus, if the birth rate exceeds the death rate of immatures, we have an optimal time delay T * that maximizes the total population. This may be interpreted as an evolutionary strategy in changing the length of time to maturity so as to maximize the total stable steady state population values in a given environment. If the preceding interpretation is true, then it is more appropriate to assume the delay T is state dependent; that is, T is a function of ~ ( t=) zi(t) z m ( t ) . System (3.1) thus becomes

+

{

i;.;(t)= a z m ( t )- yzi(t) - a e - y r ( z ) z m ( t - T ( z ) > , im(t) = ae-yr(z)zm(t - ~ ( 2 )) ~&(t).

(3.9)

This model was studied in Aiello et al. (1992), Cao and Freedman (1992), and Cao et al. (1992a). 5.4. Reduction of Structured Population Models t o Threshold Delay Equations a n d F D E s In contrast to the two stage structure model studied in the previous section, most structured population models appearing in the literature take the form of hyperbolic partial differential equations (see the various examples documented in the important text by Metz and Diekmann (1986)). It is a well-known fact that many structured population models can be reduced to delay differential equations (e.g., Blythe et al. (1984), Nisbet and Gurney (1983), Sulsky e t al. (1989)). There are occasions, however, when this reduction does not yield a standard FDE (or DDE), but rather a delay differential equation of threshold type, which we refer to as a threshold delay equation (TDE). The following is an example of a TDE:

where w ( t ) is given on [O,TO],

TO

> 0, by initial data i i ) ( t ) satisfying

so that T ( Q ) = TO. Clearly, (4.1) is in fact a state dependent delay differential equation, since T depends on ID(-). Another good example of

192

Delay Differential Equations

this can be found in Nisbet and Gurney (1983). Other TDE models are derived and studied in a series of papers by Gatica and Waltman (1982, 1984, 1988) in the modeling of immune response. In these epidemiological models, a susceptible individual who is first exposed to the disease at time t - T will become infectious at time t provided that the individual accumulates a sufficient dosage of infection, usually modeled by some function of the infective population, during the period from t - T to t. The goal of this section is to show, by a simple example, that those structured population models that can be reduced to TDEs can be further reduced, in a biologically natural way, to standard FDEs. This may allow the existing theory of FDEs to be applied to determine the asymptotic behavior of the original system. Most previous attempts to deal with TDEs have proceeded by differentiating the threshold condition, which yields a larger system of variable delay-size equations. At best, these resulting equations can be numerically integrated. We will show that this approach is unnecessary and counterproductive. The material of this section is modified from H. L. Smith (1992c, d). Consider a single population consisting of two subpopulations, namely, mature and immature. We assume that the mature population consists of individuals that are identical insofar as mortality and fecundity rates and their interaction with the immature population. The immature individuals are assumed to differ in their “level of maturity,” x , which might represent age, size, or the accumulated amount of some chemical substance, for example. The maturity level varies between x = 0 (newborn) to x = m (mature). The principle assumption we make is that the rate, d x l d t , at which maturity is attained depends, in a possibly nonlinear way, on the current mature population size. The model takes the form

Here, z ( t ,x ) represents the density of the immature population with respect to the maturity level variable x . The variable w denotes the mature population size. The maturity level of an immature individual increases at the rate P ( w ( t ) ) . The per capita death rate of immature, p, and mature individuals, v , are constants, and 7 denotes the constant per capita reproductive rate of the mature population. The constant p may be 1 if it is assumed that the newly mature simply enter the mature population.

193

5. Periodic Solutions, Chaos, Structured Single Species Models

On the other hand, if the immature are single cells that divide on reaching maturity, then p = 2 (Metz and Diekmann, 1986, p. 237). If a larger mature population size is better able to facilitate maturation, then P increases with 20, and we have positive feedback. On the other hand, if the population consists of cells, such as stem cells living within a larger organism, then some control mechanism is required to maintain an appropriate number of these cells. In this case, negative feedback in the sense of P monotone decreasing would be appropriate. For t < TO, the only recruits into mature population come from those individuals of advanced maturity (z> 0), which were introduced at t = 0 as the initial density z(0, z). Integrating the first equation of (4.2) along its characteristic curve d z ( t ) / d t = P ( w ( t ) ) ,we obtain

which yields

+ /t o P ( w ( s ) ) d s5 m. t

z ( t , s ( t ) ) = z(to,s(to))e-P(t-'o),

Since ~ ( t=) s(0)

provided

s(t0)

+ G(t),we have s(0) = z ( t ) - G(t). Hence,

z ( t , r ( t ) )= z ( O , z ( t ) - G(t))e-pt,

t5

(4.4)

TO,

and, in particular,

z ( t , m )= z ( 0 , m - G(t))e-@,

tI

TO.

This implies that w(t) and G(t) must satisfy the following system of ODES:

{

; ( t ) = -vw(t)

+ pP(w(t))z(O,m- G(t))e-pt,

G t )= P(w(t)),

0 0

< t 5 TO, < t 5 TO,

(4.5)

~ ( 0= ) W O ,G(0) = 0.

We assume that z(0,z) is Lipschitz continuous, and hence (4.5) is well posed, barring blow-up, and determines a unique nonnegative function w(t) on [0,701. For t > T O ,the newly mature cohort at time t , z ( t , m),was the newborn immature cohort ( s = 0) at time t - T , where t

=

h,

P ( w ( s ) )ds = G ( t )- G(t - T ) .

194

Delay Differential Equations

Note also that, from the fourth equation of (4.2), we have Z(t,

0) = 7 w ( t ) / P ( w ( t ) ) ;

this together with (4.4) leads to

Therefore, w ( t ) must satisfy the TDE (4.1) given a t the beginning of this sect ion. Summarizing, the solution of the coupled system (4.2) requires the solution of the threshold delay equation (4.1) for t 2 TO, corresponding to the initial data w ( t ) , 0 5 t 5 T O , provided by the solution of (4.5). Following H. L. Smith (1992c), we make the change of variable

1 t

u(t)=

0

P ( w ( s ) )ds,

W ( u ( t ) )= w ( t ) .

(4.6)

Clearly,

du (> 0). dt Hence, if we view t ( u ) as the inverse function of u ( t ) , then - = P ( w ( t ) )= P ( W ( u ) )

du From

1'

= ( P ( W ( u ) ) ) - l> 0.

(4.7)

P ( w ( s ) )ds = m,

(4.8)

t-r(t)

we have ~ ( t - T = ) u(t)-r n, and hence, by (4.6), w ( t - T ) = W ( u ( t - T ) )= W ( u ( t )- m).Integrating (4.7), we obtain

If uo = u - m, then (4.8) implies that t(u0) = t ( u ) - ~ ( t )hence, ;

Denote W,, E C([-rn,O],R+) as usual, that is, Wu(8) = W ( u [-m,O]; then, (4.9) becomes

+ O),

8E

(4.10)

5. Periodic Soluiions, Chaos, Siruciured Single Species Models

The initial condition then changes from W ( 6 ) ,0 5 6

195

5 TO, to (4.11)

where i(u) is the inverse function of u ( t ) defined in (4.6). Therefore, Eq. (4.1) is equivalent to the following standard delay differential equation (or FDE):

where we define (following (4.10))

1

T*(wu) =

0

p(wU(e))-1 de.

-m

(4.13)

It is easy t o see that the preceding method of changing variables can be applied t o the more general threshold-type delay equation

- f ( z ( t ) , z(t - T ) , e - P r z ( t - T I ) , (4.14) J;-r k(z(~)) d~ = m, 2 0, m > 0, f and k are locally Lipschitz functions of their

{

j ( t )=

where arguments, and k is positive valued. The details of this process can be found in H. L. Smith (1992~). As we mentioned earlier, once we reduce the structured population model to a standard FDE such as (4.12), we can apply the extensive existing theory developed for FDEs to derive sharp asymptotic results, which can be converted back to the original structured model or the threshold-type delay equation. Such an effort was carried out in H. L. Smith (1992 a, b, c, d). 5.5. Chaos

Irregular behavior has been the subject of intensive studies by scientists in various areas since the mid-1970s. This interest was in part kindled by the work of Lorenz (1964), by the rediscovery of Li and Yorke (1975), and by the popularization of some of the analytic results of Sharkovski (1964) related to the regular and irregular behavior exhibited by the solutions of difference equation models of the form zn+l = G(zn),

TZ

= 1,2,... ,

(5.1)

for suitably defined nonlinear functions G. A simple example of such is the discrete logistic equation G(zn) = rzn(1-

zn),

(5.2)

196

Delay Differential Equaiiona

which exhibits chaotic behavior for large r (> 3.570) (see Marotto (1978)). In contrast to first order difference equations, scalar and two dimensional autonomous ordinary differential equations do not generate chaotic solutions. Nevertheless, three dimensional autonomous Lotka-Volterra systems of ODES are capable of producing strange attractors and thus displaying chaotic behavior (see Arneodo et al. (1980)). Surprisingly, the scalar delay differential equation of the form (4.8.1) has also recently been shown to be able to produce such kinds of chaotic solutions. Detailed analyses of such behavior are given in Walther (1981), Heiden and Walther (1983), Heiden and Mackey (1982). Despite the differences in the analyses and techniques, all approaches have in common the reduction of the problem to a discrete equation in order to apply known criteria, such as the period 3 condition of Li and Yorke (1975), and the snap-back repeller condition of Marotto (1978). For simplicity, in what follows we restrict our consideration to piecewise constant functions f in

k ( t ) = f ( ; t ( t - 1)) - cr.(t), (5.3) where a is a positive constant. We give conditions for (5.3) to exhibit chaotic behavior (to be defined in what follows). The restriction of f is not essential, since by techniques used in Walther (1981) and Heiden and Walther (1982), the results may be generalized to smooth nonlinearities. The material of this section is adopted from Heiden and Mackey (1982). Equation (5.3) is in some sense a continuous version of the difference equation (5.1). This may be seen when we rewrite (5.3) as

Eqt)

+ z ( t ) = g(z(t - l)),

where E = a-', g ( s ( t - 1)) = ~ f ( z (-t 1)). In Li and Yorke (1975), it is shown that (5.1) displays chaos if there is some r E R such that G3(r) 5 r < G(r) < G2(r),

(5.4) where G"(r) stands for the nth composition of G. Li and Yorke defined that chaos appears in (5.1) if it has infinitely many periodic solutions with different periods and if, in addition, there is an uncountable set S c R such that

limsuplG"(p) - G"(q)I > 0 n-w

if p E S and q is any periodic point of map G.

(LY)

5. Periodic Soluiions, Chaos, Struciured Single Species Models

197

It is precisely this type of chaos that we shall prove to exist for Eq. (5.3). For suitable f, there is a subset of initial conditions such that the corresponding solutions z ( t ) , t > 0 , oscillate around some level a. The values t , where z ( t ) = a , form a sequence (a,)?, 0 < a, < a,+1, such that (rn,)fD with m, = a2, - a2,-1 satisfies a difference equation exhibiting chaos in the sense of (LY). We choose f in (5.3) as

0 if<
d ifb<<, where the constants satisfy b > 1, d 5 c, and 7 = c / a > b. For technical reasons we allow d < 0. The transformation y = x - d / a leads to an equation with nonnegative nonlinearity. Theorem 5.1. Consider (5.3) with f given by (5.5). Let a and y = c / a satisfy z Y/(T < 1, (5.6) where z is the positive root of

+

z 2 - [2y - (y - l)e-Q - y 2 ] z- e-Qy(y - 1) = 0.

(5.7)

Then there are numbers p = p(cr,cy) > 0 and d* = d * ( a , y ,b) 5 1 with the following property: If (y - z)/(y - 1 ) < b < (y - z)/(y - 1)

+p

and

d

5 d*,

(5.8)

then (5.3) has infinitely many periodic solutions and uncountably many aperiodic solutions. More ezphcitly, we have the following: There is a sequence TI = { X k : k = 1,2,. . . } of periodic solutions and an uncountable set T2 of aperiodic solutions to (5.3) satisfying the following: (i) To each 5 E T1 U T2, there corresponds a sequence 0 < ti < t2 < ..*, lim,-,m ti = 00, such that x ( t , ) = 1, z ( t ) # 1 f o r all t # ti, i = 1,2, ...; (ii) There is a continuous map G : I t J of intervals I c J c [0,1] such that to every x E TI U T2 there corresponds a vz E I obeying

t2i - t2i-1 = 1 - G'(vZ),

i = 1,2,. . . ;

(iii) FOT x = X k E T I ,

Gk(vZ= ) v2,

G'(4

# vz,

l
198

and

Delay Differeniial Equaiions

- t l i s the (smallest) period of x t ; (iv) The sets S1 = {v, : x E T I } ,S2 = {v, : z E 2'2) satisfy - Gi(v')l > 0, IlimsuplGi(v) liminflGi(v) - Gi(v')l = 0, i+m

{

ifv,v' E S2, w

# d,

1-00

I

limsuplG'(v) - G'(v')l

>0

i+00

if v E S1,d E S2.

In order to prove the theorem, we need some lemmas. Define also

0 ifOI{Il, c if { > 1.

(5.9)

It is straightforward to show that the following statements are true. Lemma 5.1. Consider Eq. (5.3) with f given by (5.9). If 7 = c / a < 1, then x = 0 is globally asymptotically stable. If 7 > 1 , then both x = 0 and x = 7 are locally asymptotically stable. Lemma 5.2. Consider (5.3) with f given b y (5.9). If 7 > 1, then (5.3) has a n unstable periodic solution 2 with period between 1 and 2. There i s exactly one minimum within one cycle of 5. Proof. Define

D

=

{'p

: 'p(t)> 1 for

t

E [-1, -1

+ w),w = w(p) E [O,l],'p(t)< 1 for

t E (-1 +w,O),p(O) = 1 ) ; V : D + [0,1]; V ( 9 )= ~ ( 9 ) . It is easy to verify that a solution of (5.3) with

'p

E

D, V('p)= w ,satisfies

z ( t ) = 7 - (7 - l)e-a'f

for all t E [0, w],

x ( t ) = x(w)e-a("w)

for all t E [w,I ] .

(5.10) (5.11)

Observe that x depends only on w. We may thus denote it by x = The values

zw(w)= 7 - (7 - l)e-aw, z w ( l )= zw(w)e-a(l-w) - [7(eaw- 1 )

2,.

(5.12)

+ l]ewa

(5.13)

are increasing with respect to w. There is a unique w = wg E ( 0 , l ) satisfying x W 2 ( 1 )= 1. For all w 2 w2, we have x w ( t ) > 1 for all t E ( 0 , l ) . It is easy to see that, for t E [1,21, x ( t ) = 7 - (7 - x ( l ) ) e - a ( t - l ) ,

5. Periodic Soluiions, Chaos, Siruciured Single Species Models

199

since for w 3 w2, 1 5 z(1) < 7, and, by induction, we see that limt,,z,(t) = 7 for 20 2 w2. If w 5 w2, then there is a t l E [ w , l ] such that zw(tl) = 1. If w increases from 0 to w2, then ti = tl(w) increases from 0 t o 1. For w E [0, w2] and t E [ l , 1 t l ] , we have again

+

zw(t)

(5.14)

+ t l ) is an increasing function of w;

Note that z,(1 zo(1

= 7 - [7- Gu(1)1e - Q ( t - l )

+ t l ) = e-Q < 1,

zwz(l+ t l ) = 7 - (7 - l)e-Q

> 1.

+

There is a unique wl E (0,wi) such that zwl(l 1 1 ) = 1. For w E [O,wi], we have z ( t ) < 1 for all t E ( t 1 , t l 1). For these w, the corresponding solutions converge to 0 as t ---t 00. Assume w E [wl,wz]. Then there is a t 2 = t2(w) E [1,1+ t l ] such that zw(t2)= 1. Define another initial condition II, = $(w) by

+

&,(t) = zw(t2+ t ) Obviously, map

$w

F : [WI, wp]

for all t E [-1,OI.

E D and &,(-(t2(w) ---t

[0,1]

- tl(w)))

= 1. Define a continuous

by F ( w ) = V ( G w )= 1 - ( t 2 ( ~ )- t l ( ~ ) ) .(5.15)

It follows from (see (5.11)) W

and

( 1 ) = e-O('-'l)

that e-Q(wl)

1 = 7 - (7- z , ( 1))e-a(t2-1)

=zw(l)(7 -

- zw(1));

hence, eQF(4 =e zw(lN7 Q

- I)/(? - zw(l))l,

(5.16)

which implies that (using (5.13))

F(w)= w

+ [ln(y - 1 ) + In zw(w) - ln(7 - z,(

l))]/a.

(5.17)

Clearly, if F(w) 5 201, then z v ( t ) = s(O,cp,t) t 0 as t t 00, while if F ( w ) 2 w2, then z v ( t ) ---t 7 as t + 00. If F ( w ) E (w1,w2), then zp evolves for t > tg(w) just as z v ( t ) evolves for t > 0, where $ E D satisfies V ( $ )= F ( w ) . Therefore, the iterates of F determine the evolution of zv. The function F has the following properties:

F(w1) = 0, F(w2) = 1, F is differentiable, and dF/dw > 1 for all w E [w1,wz].

(5.18)

200

Delay Diffemntial Equations

It follows that F has a unique fixed point wp E (w1,wa). With respect t o the iterations F"(w), n = 1 , 2 , . .. , this fixed point is unstable. Thus, we have the following conclusion: If cp E D satisfies V(cp0)= wp then, for t > t2(wp), the solution zp is periodic with minimal period between 1 and 2. This periodic solution is unstable since, for cp E D, V(cp) < wp implies that z J t ) --t 0, and cp E D, V ( 9 )> wp implies that zp(t)+ 7 as t + 00. Equations (5.10) and (5.11) imply that the periodic 0 solution has a unique minimum within each cycle. Lemma 5.3. The minimum value of the periodic solution P i n Lemma 5.2 is the positive root of (5.7). Proof. According to the proof of Lemma 5.2, the minimal value of P is z = zwp(l). Equation (5.13) implies that z = ye--a('-wP)

that is, [z

- (7 - 1)e-",

+ (7 - ~ ) e - ~ ] /=y e-a(l-wp).

The fact that F(wp) = w p and (5.16) imply that [z

+ (y - ~ ) e - ~ l /=y

e--Q(l-wp)

= z(y - I)/(-, - z ) ,

implying (5.7). 0 It is easy to check that z = zwp(l)< 1 from (5.7). Proof of Theorem 5.1. Let i be the periodic solution of (5.3) with f given by (5.9). This solution also solves (5.3) in case f is given by (5.5) if b 2 i,,,,,,the maximum value of i. Let

E Db = (9E D : ~ ( t<) b,

t

E [-l,O]},

202

> 20

= V(p)

>~

p ,

and zw be the corresponding solution to (5.3) with f given by (5.5). Then, zw2(w2)>

zw(w)= 7 - (7 - l)e-aw > zwp(wp)= 3,,,.

Assume &max

< b < zwZ(w2).

Then, there is a unique W = W ( b ) E (wp,w2) satisfying zu(ii)) = b. The values c ( b ) and b are related by e-rrw

- (7 - b ) / ( 7 - 1).

Let w E [wp, W). Then, z w ( t ) < b for all t E [0,1], and (5.13) implies that z(, 1) is an increasing function of w; zc( 1) = be-"(y - l ) / ( y - b) < 1.

201

5. Periodic Soluiions, Chaos, Siruciured Single Species Models

+

For t E [ l , 1 + tl(w)], the solution zw increases, and zw(l tl(w)) is an increasing function of w ;z e ( l + t l ( 9 ) )= r2(6- 1)/(6(7 - 1)) e-Q > 6. For 6= 6 ( 6 ) = F-l(G), we have ze(l tl(;i)) = 6. Let 0 < tl(w) < t2(w) < . . denote the successive times such that zw(ti(w)) = 1 and 1 < q ( w ) < 4 w ) < the successive times with z,(~,(w)) = 6. For w E [wp,ii)], we have t z ( w ) - tl(w) = 1 - F(w). For w E [wp,;i), the inequality z w ( t ) < 6 holds for all t E (l,t4(w)), and t4(w) - t 3 ( ~ = ) 1 - F 2 ( w ) ;in particular, t4(6) - t 3 ( ; i ) = 1 - F2(;i) = 1 - F ( 9 ) . For w E [6,zO], it follows from (5.11) that

+

+

+

+

+

zw(t2(w) 1) = zW(tl(w) l)e-a(l-F(w)),

which is increasing with respect to w . For these w, we also observe that t 2 ( w) < T ~ ( w )< tl(w) 1. Hence, z w ( t ) increases as a function of t E (tz(w)+ 1 , q ( w ) l ) , and, since, e-a(rl(w)-tz(w))= (7 - 6)/(7- 1) by (5.10), we have

+

+

which increases as a function of w. Let 6 = 6 ( ~=) 5max 6 , 6 > 0. Then,

+

where z is the minimal value of i (see Lemma 5.3). To obtain an estimate of e-O'"P, note that wp as a fixed point of F satisfies ln(7- l)+lnzwp(wp)In(? - zwp(l))= 0 (from (5.17)); hence, 7 - (7 - l)e-aWP = zwp(wp)= (7 - zwp(l))/(y - 1) = (7 - z ) / ( 7 - 1). Therefore,

Condition (5.6) yields

Thus, there is a p = p ( a , r ) > 0 such that 6 E i m a , < 6 < xWz(w2)and

+

z w ( ~ l ( w ) 1)

<1

(imax,imax

+ p ) implies

for all w E [6,9].

202

Delay Differential Equations

With this p , let the first condition of (5.8) hold in the following. For w E (&,I%], the solution 2, satisfies

+

z w ( t )= d/a - ( d / a - z w ( q ( w ) l))e-a(t-n(w)-'),

+

(5.19) E (+) 1 , 7 2 ( 4 + 11, zw(72(w) + 1) = d / a - ( d / a - z w ( q ( w ) + l))e-a(n(w)-n(w))). 2

+ +

For these w ,there is a unique t3(w), 7 2 ( W ) < tg(w) < t2(w) 1, such that zw(t3(w))= 1. Hence, z, increases on (72(w) + 1, t3(w) 1) and, since e-a(ts(w)-Q(W)) = b-1, we have

+

zW(t3(w) 1) = y ( 1 - b-')

+ zw(72(w) + 1)b-l

for all w E (G,I%].

For each fixed w E (&,I%], it follows from the first part of ( 5 . 8 ) that zw(t3(w)+ 1 ) is an increasing function of d satisfying

if d = c, zw(t3(w)+ 1) > 1 lim z W ( t 3 ( w ) 1) = -m.

+

d+-m

There is exactly one d = d(w) such that zw(t3+l) = 1. For all d E [d(w),c], we have t3(w) < q ( w ) 1 < t4(w) 5 t3(w) 1 and t4(w) - t3(w) is a decreasing function of d;

+

t4(w) - t3(W)

+

=

1 - F 2 ( w ) if d = c, if d = d(w).

(5.20)

Define, for each d I c,

For w E [I%,F(I%)], define

Since the times t ; ( w ) ,q ( w ) depend continuously on w ,for fixed parameters a , b, c , d , the function Fd : [0,F ( w ) ]+ [0,1] is continuous.

+

Assume now that d / a 5 1. Then, zw(72(w) 1) < 1, and hence > 72(w) 1 for all w E (&,I%]. This implies that for these w the function t4(w)

+

5. Periodic Soluiions, Chaos, Siruciured Single Species Models

203

is an element of Db. Therefore, it follows from the construction of Fd that for every w E [wp, F ( w ) ]satisfying

F ~ ( wE)[wp,F(iir)],

k = 1,2,. .. ,

we have, by induction,

t2k - t2k-1 = 1 - Fdk (W). We can now apply the following theorem of Li and Yorke (1975): Let

I, J be two intervals, I c J. Let G : I + J be a continuous function. Assume there is an r E I such that G'(r) E I, i = 1,2,3, and (5.4) holds. Then there is a sequence S1 = { q , ~ 2 , ... } C I and an uncountable set S2 C I such that S1 and S2 are invariant with respect to G and

and (LY) holds. It should be pointed out here that Li and Yorke (1975) assumed I = J ; however, inspection of their proof shows that their result can be generalized as indicated. We denote

I = [O, F ( G ) ] , G ( w ) = F d ( W ) , r = 6 ( b ) = F;'(iir(b)). J = G(I), Condition (5.4) is true if G 3 ( r ) = Fd(iir) 6. It follows from the observation with (5.20) that there is exactly one dl = d l ( a , 7 , b ) such that F i (iir) = 6 and F d ( c ) < 6 for all d < d l . Therefore, the theorem is proved for all b and d satisfying (5.8) with p as given and d' = min(cr, dl

6

5.6. Remarks

In this chapter we presented some results on sufficient conditions for the global existence of periodic solutions in autonomous or periodic systems. Other related results include those of Kuang et al. (1991b) on the

global existence of a periodic solution for 1 - z(t - T ) 1 - cz(t - 7)'

5 ( t )= r s ( t )

where r , ~and , c are positive constants, a model used in forest management; the work of Walther (1975a) on the global existence of a periodic solution for k ( t ) = -cr(z(t) 1) J; x ( t - 3) dp(s); and the work of Zhang

+

204

Delay Diflerential Equations

and Gopalsamy (1990) on a periodic delay logistic equation with periodic solution induced by the periodic equation without delay. See also Busenberg and Cooke (1978), Cooke and Kaplan (1976), and Cushing (1977). Global existence of periodic solutions in state dependent delay equations are addressed in Kuang and Smith (1991e), Smith and Kuang (1992), and Mallet-Paret and Nussbaum (1991). Sharp results on period and shape of these periodic solutions are given in Mallet-Paret and Nussbaum (see also the references cited therein). For state-dependent delay models used in areas other than mathematical biology, see Bklair and Mackey (1989). The two stage structured model discussed in Section 5.3 is only an example showing how to set up and analyze models of single species population growth whose individuals exhibit several distinct stages in their life cycle. More and different examples of structured models can be found in Metz and Diekmann (1986). The reduction technique described in Section 5.4 presents an effective approach of analyzing such models. The chaos result presented in Section 5.5 is restrictive in the sense that important conditions are placed on both the equation and initial functions. Nevertheless, numerical works frequently show that chaotic solutions do appear even for equations such as (7.1)-(7.3) with appropriate choices of parameters; for details, see Heiden and Mackey (1982), and Mackey and Glass (1977). Such chaotic phenomena are essential for models of some human diseases, since they may be interpreted as the onset of the diseases being modeled. For general oscillation results on scalar delay differential equations, see the recent books of Ladde e t al. (1987), Gopalsamy (1992), and Gyori and Ladas (1991).

6

Global Stability for Multi-Species Models 6.1. Introduction In the real world, species interact and coexist in somewhat closed environments. It is thus more realistic to model population growths in system settings; that is, modeling the growths of interacting species by systems of equations. Suppose we are to model n interacting species in a closed environment, and the per capita growth rate for the ith species at , ( t )is the density of ith time t is F;(t,q),where z ( t ) = ( z i ( t ) , . ,z n ( t ) ) q species at time t ; then, we have a system of n delay differential equations

..

i = 1 , . . . ,n.

$,(t) = z ; ( t ) F , ( t , q ) ,

(1.1)

The type of interactions (competition, predator-prey, or cooperation) .). among species determines the sign of partial (FrCchet) derivatives of F;(t, For example, systems (1.1.7)-(1.1.9) are delayed predator-prey systems. As we can see from Chapter 4, the study of system (1.1) with the most general form of F;(.) can be very complicated and may not even be well defined. In applications, one often adopts the following general Lotka-Volterra type system as models of multi-species population growths in closed environments:

& j ( t )= bj(ui(t))Gi(t,~ t ( * ) ) , where

1

i = 1 , 2 , . .. , n ,

0

Gi(t,w(-)) r ; ( t )- a ; ( t )

-r

Ui(t

(1.2)

+ 0) 444 0)

+ 2 J_”,U j ( t + 0) + i j ( t , O ) ;

(1.3)

3=1

~ ( t =) ( q ( t ) ,. . . , ~ n ( t ) ) ,ri(t), u ; ( t ) are positive continuous functions; p , ( t , * ) ,p ; j ( t , . ) are of bounded variation; and r, are positive constants, . . 2 , j = 1 , . . . ,n. Note that (1.2) is a special case of (1.1). Our main concern

in this chapter is to establish sufficient conditions for the global asymptotic 205

206

Delay Differeniial Equaiions

stability of the positive steady state (assumed to be unique) in (1.2) with respect to its positive solutions. This chapter is organized as follows: In the next two sections, we describe two types of Lyapunov functionals that are quite effective in analyzing the global stability of system (1.2)-(1.3) when it is autonomous and has strong nondelayed negative feedbacks. In Section 4, we use Razumikhin-type theorems to study the global stability problem of (1.2)(1.3), and we apply the results obtained in that section to autonomous systems in Section 5 to obtain some sharp and easy to use theorems. In Section 6, we deal with the situations that occur when nondelayed diagonal terms do not exist, where both the Razumikhin-type theorem and Liapunov function methods are applied to establish regions of attraction of the positive steady states (assumed to be unique) in initial function spaces. We end this chapter with some remarks and open problems. 6.2. Stability via Liapunov Functionals, I The material of this section is adapted from Leung (1979) and Leung

and Zhou (1988). Consider first the following predator-prey model:

where a , b , c , d , p , q , and r are positive constants and ki(O), 8 E [O,r], i = 1 , . . . , 4 , satisfy the following assumptions: ( H l ) ki(8) 2 0 and continuous for fl E [O,r],ki(r) = 0; (H2) $ ( 8 ) 5 0 and continuous for 8 E (O,r), lime+o+ ki(8) and limg+,- ki(8) both exist; (H3) kY(8) 2 0 and continuous for 8 E ( 0 , r ) . These assumptions amount to saying that delay effects diminish gradually in ever moderating pace as one moves backward in time, and the effects become negligible after a length of time r. Let Jlk;(8)d8 be 6, c, p, ij, respectively, for i = 1,2,3,4. We assume further that (H4) .(P p) > ( b 6)d;

+

+

6. Global Siabiliiy for Mulii-Species Models

207

so that (2.1) has a positive steady state ( z * , y * ) , where

As usual, we assume that initial data for (2.1) are taken from C([-r, O],R$). Theorem 2.1. Suppose ( z ( t ) , y ( t ) ) is a solution of system (2.1) with initial data satisfying z(0) > 0, y(0) > 0. T h e n

provided that f o r all 0 E ( 0 ,r ) , k;(8) (i = 1 , . . . ,4) satisfy (H5) p.-1(ki(8))2 < 2br-lk:(O) and cp-1(ki(fl))2 < 2qr-'k:(O). Proof. Since b and q are positive constants, it is easy to see that the solution ( z ( t ) , y ( t ) )of system (2.1) stays positive and is bounded. Let u ( t ) = z ( t ) - z*, v ( t ) = y ( t ) - y*; system (2.1) becomes

Let ($(s),$(s)) E C([-r,0],R2) besuch that 4 ( s ) + z * s E [-r, 01, 4(0) z* > 0, $ ( O ) y* > 0. Define

+

+

2 0, $(s)+Y* 2 0,

208

Delay Differeniial Equations

We have

Separating the last two integrals preceding into four terms and integrating them by parts leads to

Clearly, (H5) implies that V ( $ , $ ) = 0 if and only if 4(s),$(s) = 0 for s E [-r,O], and V(q5,$) < 0 otherwise. By applying Theorem 2.5.5, we have limt,,(u(t),v(t)) = ( O , O ) , and the conclusion of Theorem 2.1 follows. 0

all

209

6. Global Stability for Multi-Species Models

Note that Theorem 2.1 is valid even if (H2)-(H3) for k2 and k3 are replaced by somewhat weaker assumptions, such as k2(8), k3(8) are only continuously differentiable for 8 E (0, r ) . When k3(8), k4(8) do not satisfy (H2) and (H3), we have the following. Theorem 2.2. Assume (2.1) satisfies ( H l ) and (H2), k1(8), k z ( 0 ) ~ a t i s f y (H2)-(H3) with strict inequality in (H3), and k3(8),k 4 8 ) are continuously differentiable on (0, r ) . Assume further that, for 8 E (0, r ) , (H6) (kh(d))’/k!(@) < 2br-’,

Then limt+m(z(t), y(t)) = (z*,y*). Proof. The proof is almost the same as that of Theorem 2.1, except we define now

One can show that

The quadratic expression inside the first is clearly nonpositive. The expression inside the second {.} is also a quadratic in $(O), +(O), 4(-s) ds], and [J,”+(-s) ds]. By completing squares three times with the expression in the two {-}, we see that (H7) and k;’(O) > 0, k t ( 8 ) > 0 imply that V ( + , + ) = 0 if and only if 4(s),+(s) are identically zero on [-r,O]. The 0 rest of the proof follows that of Theorem 2.1. {a}

[Joe

210

Delay Differential Equations

As an example of applying the preceding approach to higher dimensional systems, we consider the following n dimensional system:

where, for i , j = 1,. . . ,n, e ; , p ; j , s;j are constants and (H3)and

ir

k ; j ( e ) d~

= 1.

kij(0)

satisfy

(2.5)

(H1)(2.6)

Furthermore, we assume that (2.5) has a unique positive steady state (xi,.. . ,x:). Let u i ( t ) = x;(t) - t;,i = 1,. . . , n . System (2.5) becomes

We have

6. Global Stabiliiy for Multi-Species Models

211

Following the proof of Theorem 2.1, we obtain the next theorem. Theorem 2.3. Suppose that (i) there are positive condankr a, > 0, i = 1,. . . ,n, such that n

C aipijwiwj 5 o i,j=l

for all

..

( ~ 1 , . ,w")

E R";

(ii) sii < 0, k::(O) > 0 almost everywhere o n [0,r ] , i = 1 , . . . , n ; (iii) sij = 0 when i # j. Then ( x i , . . . ,xi) is globally asymptotically stable with respect to positive solutions of system (2.5). We conclude this section by giving an interesting application of T h e orem 2.3 (or Theorem 2.1). Consider the following simple predator-prey svstem:

where 6 is a positive constant, k;; (i = 1,2) satisfy (Hl)-(H3), and J{ lcii(8)de = 1. If k::(8) > 0 almost everywhere on [O,r], i = 1,2, then Theorem 2.3 implies that, for 0 < 6 < 1, all positive solutions of (2.8) tend to the unique positive steady state. This is in contrast to the case that when 6 = 0, all positive solutions, except the steady state ( l . l ) , are nonconstant periodic solutions. In this case, we may say that delay stabilizes the system. For more results of this kind, see Leung and Zhou (1988). 6.3. Stability via Liapunov F'unctionals, I1 The results in the previous section depend on some very specific properties of lc;(O), which in the real world may not be easy to verify. In this section, we use a slightly different Liapunov functional to obtain a set of rather different conditions for the global stability of the positive steady state (assumed to be unique) of general Lotka-Volterra type systems. We consider i i i ( t ) = bi(ui(t))Gi(ut(.)), i = 1 , 2 , . .. , n , (3.1) where

+ j2 a i j u j ( t ) + 2 bij J u j ( t + e) d p i j ( 0 ) . =l j=1 -' 0

G i ( u t ( * )= ) Ti

Here, T i ,

aij

and

bij

are all real constants,

7

(3.2)

is a positive constant, and

0

J-, Idpij(e)I = 1,

i,j=

1 , e . e

,n;

(3-3)

212

Delay Differential Equations

..

hi(-), i = 1 , . ,n, are continuous and bi(0) = 0. Moreover, we assume that ,ti:). steady state u* = (ti!,

...

strictly increasing functions with (3.1)-(3.2) has a unique positive

Theorem 3.1. If there i s co?st$nt positive diagonal matriz C = diag (c1,. ,G ) such that C A ATC is negative definite, then u* i n (3.1)-(3.2) is globally asymptotically stable. Here, = (ajj), where

+

..

Remark 3.1. When b;j = 0, the theorem reduces to the well-known result of Goh (1977). When n = 1, bl(ul(t)) = ul(t), (3.1) reduces to (2.5.11).

Thus, Theorem 3.1 partially generalizes Theorem 2.5.6. It also generalizes Lemma 2.1 in Kuang and Smith (1991). Proof of Theorem 3.1. Assume that the constant positive diagonal matrix = diag (c1, cp, . . . ,cn) satisfies the assumptions specified in the theorem. Define V : C([--T,01, R;) + R as

c

+

Clearly, V ( $ ) > 0 if $;(O) uf > 0 for i = 1 , 2 , . . . ,n, V ( 0 )= 0. Assume that ut(.) = u* is a solution of (3.1)-(3.2). Let V ( $ t ) denote the derivative of V ( $ t )along the solution of (3.1)-(3.2). We have (see Lemma 2.5.1) $ t ( e )

+

6 . Global Stability for Multi-Species Models

213

+ aTc

The assumption of being negative definite implies that if $ ( t ) # 0, then V ( q 5 t ) c 0. By Theorem 2.5.5, we have 4(t) + 0 as t + 00 and hence the theorem. 0 Note that negative definiteness of ATC requires that aii < 0; that is, roughly speaking, the nondelayed intraspecific competition should be strong enough to dominate the effects of delayed interspecific and intraspecific interactions. In the following, we consider system (3.1) with

CA +

n

n

.t

where yij are real constants, and Fij : functions normalized by

1

00

F&)ds=

1,

[e,m)

+

R+ are continuous

. . = 1 )... , n .

2,]

We assume that the initial data are taken from BC((-w,O],R;) with Ui(0) > 0, i = 1,. . . ,n . We further assume that Fij in (3.4) are convex combinations of functions of the form

Delay Differeniial Equations

214

Using the linear chain trick described in Section 3.6, we introduce new variables u n + l ( t ) ,... ,up of the form t

1, F,(t -

S)Uj(S)dS,

p = 1,. . . ,m,

where F,, runs through all functions of the form (3.5) appearing in Flj,. . . , F,j for j = 1,. . . ,n. The new variables un+l,. .. ,up satisfy a system of linear constant coefficient differential equations in u1,. .. ,up. Together with the equations for u1, . . . ,un, we have

P

t q t )= C Ulj"j(t),

1=n

+ 1,. . . , p ,

(3.7)

j=1

for some well-defined coefficients a;,, i, j = 1,.. . , p , with initial conditions UI(0)

= uio

> 0,

2

= 1,2,. . . , n

(3.8)

To simplify the presentation, in the following we do not specify the coefficients aij. The following theorem slightly generalizes Theorem 2 in Wiirz-Busekros (1978). Theorem 3.2. Suppose s y s t e m (3.6)-(3.7) has a unique positive steady state u* = (ui, uf,.. . ,u;). A s u f i c i e n t condition for u* to be globally asymptotically stable with respect to initial values of the f o r m (3.8)-(3.9) i s the existence of positive real numbers d l , .. . ,dn, and a positive definite ( p - n ) x ( p - n ) - m a t r i x 0 4 such that D A + A * D is negative definite, where A = (aij)pxp, D1 = diag ( d l , . . . , d n ) , and D = diag ( D l , D 4 ) . Proof. The proof is almost identical to that of Theorem 2 in WorzBusekros (1978). We define

+

where D4 = ( d i j ) , i,j = n 1,. . . , p . It is easy to see that V 2 0, and V = 0 if and only if u ( t ) = u*. The derivative of V along a solution of

215

6. Global Stability for Multi-Species Models

(3.6)-(3.9) is

+

22

[(uk - u;)aikdij(uj - u;) i,j=n+l k = l (Ui - u:)dijajk(uk - ti;)]. Here, we have used the fact that u* satisfies (3.6)-(3.7). In matrix notation, we have

+

= ( U - u*)(DA

+ ATD)(u - u * ) ~ ,

where

A A = (A:

A A:)

A1 = (aij)nxn, i,j = n + l , ... ,p.

7

A4 = (aij),

Since DA+ATD is negative definite, V 0 along every solution u of (3.6)(3.7) and equals zero if and only if u = u*. This leads to the conclusion of the theorem. 0 For more results that relate to Theorem 3.2, see W&z-Busekros (1978). 6.4. Stability via Razumikhin-Type Theorems-Theory

In this section, we consider the following general nonautonomous Lotka-Volterra-type system with infinite delays:

i = 1, ... ,n, (44

216

Delay Differential Equations

where the integral variable on the right hand side is s. In system (4.1), we always assume that, for i = 1,. ,n, the following hold (Hl) bi(0) = 0, b:(ui) > 0; (H2) ri(t) and a i ( t ) are continuous for t 2 0. a i ( t ) > 0 , Ti 2 0; (H3) p i ( t , S ) , pj;(t,s),and vji(t,s) are nondecreasing with respect t o s. Let

..

A

pi(t)=pi(t,O)- pi(t,-Ti), A p j i ( t ) = p j i ( t , 0) - pji(t7 --oo), A vji(t)= vji(t,0) - vji(t, -w).

We assume that p i ( t ) , p j i ( t ) and vji(t) are continuous for all t 2 0. As usual, we choose the UC, space, where g : (-m,O] + [1,00) satisfies (g1)-(g3) in Section 2.7. We assume our initial conditions satisfy E BC, +(O) > 0. (4.2) (H4) u,,(s) = +(s) 2 0 for s 5 0; Here, +(O) > 0 means +,(O) > 0 for i = 1,. . . ,n. We call such a +(s) an admissible initial function. It is easy to see that if +i(O) = 0, then ui(t) = 0 for t 2 u. The following fundamental lemma is a simple combination of some well-known results. L e m m a 4.1. Assume (Hl)-(H4) hold. Assume also that (H5) there is a g(s), satisfying (g1)-(g3), such that

+

lri

A '

pig(t)=

A

i = 1, ... , n ,

g(s)+i(t,s) < 00,

'

vjig(t)= L r n g ( s )dvji(t,s) < 00,

. .

2,3 = 1,. . . ,n,

and pi,, p j i g , vj;, are continuous for t 2 u. Then (4.1) has a unique solution u(u,+)(t)such that u(u,+)(t)> 0, for t 2 u in its maximal interval of existence. If u(u,+)(t) i s a noncontinuable solution, then, for any M > 0 , there is a t* > u such that Cy=lui(o,+)(t*)> Proof. It is straightforward to verify that the function on the right-hand side of (4.1) is locally Lipschitz in R x UC,. A simple application of Theorem 2.2 in Hale and Kato (1978) yields the existence and uniqueness of u(u,+)(t).If, for some 1 E {1,2,. . . , n } , t > u is such that u,(u, +)(t) = 0, then there is an N > 0 such that Iui(u,+)(s)l< N for i = 1,. . . , n , s I?. Thus, from (Hl)-(H5), we see there is a Q > 0 such that u;(t)2

-Qw(~),

t It,

6. Global Stabiliiy for Multi-Species Models

217

which leads to

3 q+(0)e-Q('-a) > 0, a contradiction to our assumption. The last statement is an immediate result of Theorem 2.3 in Hale and Kato (1978). 0 For the sake of simplicity, in the rest of this paper, we denote, for i = 1,... , n ,

Gi(t,ut) = r i ( t )- ai(t)ui(t)-

ui(t

+

S) d p i ( t , s )

The results in this and the next sections are taken from Kuang and Smith (1992a). Our object in this section is to establish sufficient conditions for system (4.1) t o have a globally asymptotically stable solution with respect to admissible initial functions. The next lemma presents sufficient conditions for the solution u(o, $ ) ( t ) of (4.1) to exist for all t 2 6. Lemma 4.2. In addition t o (Hl)-(H5), a s s u m e f u r t h e r that (i) lim~~pl,,[r;(t)/~;(t)] < +OO, i = 1,2,. . . ,n; (ii) there i s a 7 E (O,I), sach that n

rai(t) >

C pji(t), j=1

tL

6-

T h e n u(o,$ ) ( t )esists a n d is bounded for t 2 6. Proof. From (i), we see there is an M > 0 such that

i = 1,2,. . . ,n,

r;(t)[(l - r ) a ; ( t ) ] - l < M , Let

t 2 6.

u = max{M, Il$illm : i = 1,. . . ,n}.

(4.4) We claim that U i ( Y , t ) ( t ) 5 ii for t 2 6. Otherwise, there is an 1 E {1,2,. . . ,n } and a t, t 2 o,such that ul(o,$)(t) = U , hi(o,$)(t) 2 0, and q ( a ,$)(s) 5 ii for s 5 2. From (4.1), we have

+ j=1 c J_ "

h i ( t > 5 b / ( u l ) [ r / ( t-) a i ( t ) u / ( t )

Thus.

0

uj(t 00

+

dpjl(t,s)~-

Delay Differeniial Equations

218

0

a contradiction. This proves the lemma. We consider first the following simpler system:

+ j=1 cJ "

ri(t) - ai(t)ui(t)

0

uj(t

--M

Assume there exists u* = ( u i ,u;, u; 2 0 and, for t 2 u,

For this u * , we define, for ur

cJ "

r t ( t ) = ai(t)uf -

j=1

3)

dpji(t, s)

... , u i ) such that, for i = 1,2,. .. ,n,

> 0,

0

-m

+

+c "

u; d p j i ( t , s )

j=1

1 0

00

u; dvji(t, s),

t 2 0; (4.7)

and if ur = 0, we denote rT(t) = ri(t). Clearly, r;(t) 5 r f ( t ) , t 2 u. Proposition 4.1. I n addition to (Hl)-(H5), assume further that (H6) there i s a 0 < 70 c 1 such that, for i = 1,2, ... ,n and t 2 u,

and

where

a ( t ) = min{a;(t) : i = 1 , 2 , . . . ,n}.

(4.10)

6. Global Stability for Multi-Species Models

219

corresponding to an admisJible function 4. T h e n lim u i ( t ) = u r ,

t++m

i = 1,2,. . . ,n.

Proof. By substituting (4.8) into (4.10) and letting iii(t) = ui(t) - u r , i = 1,. . . ,n, we obtain

(4.11)

If

ur

> 0, then

the equality holds in (4.11).

Let

V ( i i ) ( t= ) max{ii,2 ( t ) : i = 1,2,. . . ,n, t 2 a } , 114 - u*llm = max(ll4i - urIlm : i = 1,2,. .. ,n}. We prove first that, for t 2 a ,

V ( WI 114 - u*llk.

(4.12)

Otherwise, there is a 2 2 a such that

V(G)(S)= 114 - u*Iloo, and V(ii)(t)2 0, where V ( i i ) always denotes the upper right hand side derivative. We let N ( t ) = {i E ( 1 , ... , n ) : iiP(2) = V ( i i ) ( t ) } . Then, for any i E N ( t ) , we have (regardless of whether ur = 0 or ur

> 0)

220

Delay Diffemntial Equations

< 0.

(4.13)

Since (4.13) is true for all i E N ( j ) , we arrive at a contradiction. This proves (4.12). Denote w 2 = limsupV(ii)(t). t++w

> 0, then there is an 6 > 0 such that, for y = yo + c, 2, - € > y1'2(v + €). By the definition of v, we see there is a tl > such that, for t 2 t l , If v

(T

Clearly, there is a

t2

V(ii)(t) I(v > 0 such that, for s g-'(s)

*

+

€)2.

5 -t2, [Id- u*IIm < v

+

6.

+ t2 such that, for t 2 T , V ( i i ) ( t ) < 0. + t2, such that

We claim that there exists T 2 t l Otherwise, we see there is a t, 1> t l

V ( u ) ( t )2 0. V(ii)(t) 2 (v - q, Again, we define N ( 1 ) as before. Then, for any i E N @ ) , we have

6. Global Stability for Multi-Species Models

221

This is a contradiction. Thus, we have, for t V ( i i ) ( t )< 0,

and

Hence, there is TI > T such that, for t

(v

> T,

lim V (i i )(t )= v2.

t-+m

2 TI,

+ € ) 2 2 V(ii)(t)2 v2.

Define

N* = {i : for any t > 0, there is a 2 > t , such that i E N ( t ) } . Then N* is not empty. Let 1 E N * . If (i) u; = 0, then for any > TI such that 1 E N(Z), we have

+);.

bi(W

= bi(@))

2

bi(v);

(ii) u; > 0, we claim that there is a S ( 1 ) > 0 such that, for any t such that 1 E N(Z), bi(ui(1) u ; ) 2 b(1).

> TI

+

Otherwise, there is a t i , ti > T I , 1 E N ( t i ) , iii(t1) < 0. Since u(t1) = u; > 0, we see iir(t1) > -u;. Thus, for Z > t i such that 1 E N ( t ) , we have (since V ( i i ) ( t )< 0, for t > T ) lii,(t)l< liii(t,)l, which implies that iil(t1)

+

./(I)

+ u; > i i i ( t i ) +

21;

hence, bl(ui(t)

A + u ; ) > b i ( i i i ( t i ) + u;)=6(1) > 0.

The preceding argument indicates that there is a 6 > 0 and a T2 such that, for all t > T2 and all i E N ( t ) ,

> TI

+

b;(ii;(t) ur) > 6. Now, for any t 2 T2, i E N ( t ) , by repeating the previous argument, we have zd( G j2( t ) ) l , = i

< 2 b i ( ~ i ( t+) ur)(u + ~ ) ~ [ - + ( rE o) . i ( t )

<-24v

+ €)2.j(Z).

222

Delay Differential Equations

+

Therefore, we have proved that, for t 2 T2, V ( i i ) ( t ) < - 2 4 v ~)~g(t). By (H6), we see that V ( i i ) ( t )+ as t 4 +cq a contradiction. This 0 implies v must be zero, proving the proposition. In the following, we assume that there exist ii = ( i i l , . ,i i n ) such that, for i = 1,. .. , n , iii 2 0, and, for t 2 n, -KJ

..

(4.15)

Proposition 4.2. Assume the same assumptions a3 in Proposition 4.1 for (4.1). Let u ( t ) = u(n,#)(t)be the solution of (4.1). Then limsupt,, u ; ( t ) 5 iii. Proof. Let ii(t) be the solution of

with initial condition (4.2); then, from Proposition 4.1, we see lim i i i ( t ) = 4;.

(4.16)

t-.+oo

Assume i i i ( t ) = u i ( t ) and i i i ( s ) 2 obtain

U,(S)

for s I t. From (4.1), we

As the right hand side of the equation satisfied by ii(t) is a quasimonotone function, a standard comparison argument similar to Proposition 1.1 in Smith (1987) gives u ( n , $ ) ( t )I ii(n,+)(t) = ii(t)

for t 2 n.

This, together with (4.16), implies the proposition. 0 Now, we are ready to state and prove the main theorem in this section. The following assumptions are required: (A) There is a nonnegative vector ii = ( i i l , i i 2 , ... ,Gin) satisfying (4.15).

6. Global Siabiliiy for Multi-Species Models

223

(B) There is a 4 = (41, ... ,fin) such that, for i = l , . . . ,n, 4i

G;(t,C)5 0,

2 0,

and

QiGi(t,ir)= 0.

Theorem 4.1. In system (4.1), assume (Hl)-(H6), (A), and (B) hold, and n

limsup max t++oo

Tibi(ili)

j=l

llcln

I"

+ p i ( t ) + C ( p j i ( t ) + ~ j i ( t ) )= p

< 1. (4.17)

Then the solution u ( t ) = u(o,q5)(t) of (4.1) tends to ir as t + +w. Proof. From Proposition 4.2, we see that lim sup u , ( t ) I4,. t+m

Define ~ ( u ) ( t=> max{(u;

- f i ; 1 2 : i = 1 , 2 , . . . ,n},

v2 = limsup ~ ( u ) ( t ) . t-oo

From (4.17), we see that there is an

i=1,

Thus, for

~ ( € 1 )<

€1

> 0 such that, for 0 5 7 5 € 1 ,

...,n}) (

< 1, there exists

a TI = T I ( ( )> u such that, for

t 2 Tl,

Assuming v

> 0, then there is an 6 > 0 such that, for 7 = 70 + C, v - c > max{y'/2(v

+ c>,((v + c)}.

(4.18)

Clearly, assumption (B) implies the assumption (i) in Lemma 4.2, and (H6) implies the assumption (ii) in the same lemma. Therefore, we can define

224

D e l a y Differeniial E q u a t i o n s

U as in (4.4), which satisfies ii 2 Q i , i = 1 , 2,... ,n. Thus, i = 1,2, ...n and t 2 u. Let u1 > 0 be so large that

+

I

u/g(-a1)

and, for t 2 u1,

ui(t)

5 ii for

6

+

V(u)(tI ) (v €)2. Since U 2 i i i 2 0 , we have V ( u ) ( t )I U 2 for all t 2 u. Let u2 = m a x { 2 u ~ , u+ ~ T,T~ + T } , where T = maxl
- C i ) 2 = V(u)('i)2 (v - € ) 2 ,

then

8 E [-Ti,O]. - C i ) ( U i ( T + 8 ) - C i ) 2 0, Otherwise, there is a 8 E [ - q , O ] such that u i ( t + 8) = i i i . Thus, (Ui(2)

L I(. + E ) , where t* E (1,t+ 8). Thus, -6 I

I(v

+

€),

which contradicts (4.18). This proves our claim. Again, arguing as in the proof of Proposition 4.1, we claim that there exists T 2 2 u 2 such that, for t 2 T 2 , V ( u ) ( t ) < 0 . Otherwise, there must be a t 2 u2 such that

V(u)(T)2 (v - €) 2 ,

V ( u ) ( t ) 2 0.

6. Global Stability for Multi-Species Models

225

Then, by combining the conclusion we have just proved (to show that the new term J!ri ui(t s) d p ; ( t ,s) appearing in (4.14) is nonpositive) and a similar argument as the proof of the same claim in Proposition 4.1, we can easily obtain a contradiction, which proves the claim. The rest of the proof is very similar to the proof of Proposition 4.1. We omit the details here to avoid repetition. Our last theorem in this section is essentially the same as Theorem 4.1, although it appears to be more general. Theorem 4.2. I n system (4.1), i n addition to (Hl)-(H5), assume further that (i) there i s a 0 < 70 < 1 and 6; > 0, i = 1,2, ... ,n, such that

+

and

(ii) there is a nonnegative vector 4 = (41,.

a i = 1,2, ... , n } ) =

p

. . ,an) such that

< 1;

(iv) there is a nonnegative vector f = (fl,... , f n ) such that, for

i = 1,2,... , n ,

C; 1 0,

G;(t,6)

5 0,

and

f ; G ; ( t6) , = 0.

Then, the solution u ( t ) = u(0, $ ) ( t ) of (4.1) tends to f as t Proof. By letting w:(t) = S;u:(t),

4 00.

one can transform (4.1) and (4.2) to a system in terms of w(t) = (wl(t), . . . ,w n ( t ) ) . It is easy to see the resulting system satisfies all conditions of Theorem 4.1. Thus, the global stability of w(t) follows, which, in turn, implies the conclusion of our theorem. 0

Delay Diffemntial Equations

226

via R a z u m i k h i n - T y p e T h e o r e m s - A p p l i c a t i o n s In this section, we focus our attention on the autonomous system

6.5. S t a b i l i t y

iii(t) = bi(ui)[ri - a t' u t' -

J_, ui(t + 0

+urn uj(t

+

S)

S ) +i(S)

dpji(3) -

J=1

cJ "

j=1

0

uj(t

--M

+

8)

-

dvji(S)]

(54

For this system, we always assume that, for i = 1 , 2 , . .. ,n, ( A l ) bi(0) = 0, bi(ui) > 0 , and Ti 2 0; (A2) pi(S), P j i ( S ) and v j i ( S ) are nondecreasing for s 5 0, and A

< 00, vji = vji(0) - ~ j i ( - m < ) 00;

pji=pji(O) - p j i ( - m ) A

+

(A3) ai > Cin,l(pji vji). For convenience, we will denote p i = p i ( 0 ) - pi(-Ti). The following lemma is essentially well known. A similar result is given in Haddock e t al. (1989). The argument there can easily be adjusted to prove the present one. Therefore, we omit the proof. Lemma 5.1. I n system (5.1), there exists a 0 < 7 < 1 and a g ( s ) satisfying (g1)-(g3) in Section 2.7 such that, f o r i = 1 , 2 , . . . ,n,

Thus, the assumptions (H5) and (H6) for system (4.1) automatically hold for system (5.1). The next lemma can be found in Berman and Plemmons (1979), p. 274. Lemma 5.2. Suppose the matrix C = ( C i j ) n x n has positive diagonal element8 and is strictly diagonally dominant. Then, for any r E Rn, the linear complementarity problem of finding p E Rn such that

pLO,

C p - r 2 0,

and

[Cp-r]p=O

(5.2)

has a unique solution. We note that Hofbauer and Sigmund (1988) call p a saturated equilibrium for the Lotka-Volterra system n

6. Global Siabiliiy for Multi-Species Models

227

Therefore, Lemma 5.2 indicates that (5.1) always has a unique saturated equilibrium. We will adopt this name for the unique equilibrium of (5.1) that satisfies (5.2). The following theorem gives conditions for such an equilibrium to be globally asymptotically stable. T h e o r e m 5.1. Let U * = (ur,. . . ,u:) be the saturated equilibrium of (5.1) and p = ( P I , . .. ,pn) the saturated equilibrium of

In addition t o (A1)-(A3), a s ~ u m efurther that n

+ + C(/iji + ~ j i ):]i = 1,2,. ..

max{Tibi(pi)[ai pi

,TI}

< 1.

j=1

T h e n every solution of (5.1) corresponding t o a n admissible initial funct i o n tends t o u* a3 t + +m. Proof. This is an immediate consequence of Theorem 4.1 and Lemmas 5.1 and 5.2. 0 The autonomous version of Theorem 4.2 takes the following form. T h e o r e m 5.2. In system (5.1), assume (Al)-(A2), and (i) there esists 6; > 0, i = 1,2,. .. ,n, such that n

ai

> SF' C Gj(pji + Vji). j=1

(ii) For u* and p be defined as in Theorem 5.1,

T h e n , every solution u ( t ) of (5.1) corresponding t o admissible initial function tends to u* as t + +co. In the following, we restrict our attention to scalar equations. We consider

228

Delay Differenfial Equafions

where T O , a, ao, a l , a2, and TO are positive constants, 0 5 71, 72 5 +w. b(0) = 0, b'(u) > 0,p i ( s ) , i = 0,1,2, are nondecreasing and satisfy 0

Li

i = 0,1,2.

dp;(s) = 1,

For simplicity, we denote u* = r o / ( a

+ + a0

-a),

a2

uo = ro/(Q- Ql),

and T

= max{TO,q,T2}.

The initial condition for (5.3) is admissible, if u(8) = 4(8)

2 0,

4(0)

> 0,

8 E [-T,O],

and

4 is continuous.

The following result is a simple consequence of Theorem 5.1. Theorem 5.3. In (5.3) assume a > a1 a2, and

+

Then every solution u ( t ) of (5.3) corresponding to admissible initial function tends to u* as t + +w. Theorem 5.3 generalizes Theorem 1 in Miller (1966) in two ways: (i) we allow discrete delays; (ii) we don't require the undelayed term to dominate the other terms. Miller (1966) also.studied the scalar equation

k(t)= N ( t ) [ a- b N ( t ) -

/ f(t t

0

- s ) N ( s )d s ] ,

with respect to initial condition N ( 0 ) = NO > 0, where a > 0, 6 > 0, f(t) E C[O,w) n L1[0, w). Assuming

Miller (1966), Theorem 2, proved that lim N ( t ) = N * = a[b

t+m

+/

m

0

f(s) ds]-'.

6. Global Siabiliiy for Mulii-Species Models

229

In the following, we consider

(5-4) where the parameters and functions appearing in (5.4) satisfy (A1)-(A3). We assume

The initial condition of (5.4) takes the form u;(O) = u;o

> 0,

i = 1 , 2 , . . .n.

One can view (5.1) as the limiting equation, as t --t 00, of (5.4). The following theorem improves Theorem 2 in Miller (1966). Theorem 5.4. Let U* = (ti; ,... , u i ) and p = ( P I , ... ,pn) be the saturated equilibria of (5.1) and (5.4), respectively. Assume system (5.4) satisfies all the hypotheses of Theorem 5.1. Then, every solution u ( t ) of (5.4) corresponding to positive initial data tends to u* as t --t +00. Sketch of the Proof. We can show first that u ( t ) is bounded. In fact, we can choose 0 < y < 1 , g ( s ) satisfying (g1)-(g3) of Section 2.7 such that, for i = 1 , 2,... ,n,

then, we can show that

By a standard comparison argument, one can show that the solution of

Delay Differential Equations

230

dominates the solution of (5.4), where both have the same initial function. By essentially the same argument as given in the proof of Theorem 4.1, one can show the solution of (5.5) tends to p as t + +oo. The only tricky step needed here is to note that

= -ai(ui - pi)

Thus, we can show that

By applying the same trick as (5.6) to system (5.4) and making a similar argument as the proof of Theorem 5.3, one can easily arrive at the 0 conclusion of the theorem. The following theorem is obvious: Theorem 5.5. Assume all the hypotheses of Theorem 5.2 are satisfied. Then every solution u ( t ) of (5.4) corresponding to positive initial data tends to u* as t + +oo. It should be mentioned here that we can allow system (5.4) to have variable coefficients. In this case, a result similar to Theorem 4.2 can be established. By a lengthy argument, Gopalsamy (1980a) was able to obtain the following global stability result for a well-known Lotka-Volterra type two species competition model: Theorem A. Assume a l , ag, 71, 72, b l , bz are positive constants satisfying ala,' > 717;' > bib,'. (5-7) Let K 1 ( . ) and 1(2(-) be nonnegative continuous functions defined o n [-T,O] (T i s any positive constant) such that

6. Global Stability for Mulli-Species Models

231

T h e n the soIution [ u ( t ) , v ( t ) of ]

I

h ( t ) = u[rl - a l u - bl

1

1, 0

I(1(3)V(t

0

; ( t ) = 4 7 2 - u2

44 = h ( t ) L 0,

-T

I(2(S)U(t

+ S) dS],

+ s)ds - b w ] ,

t 2 0,

+ l ( O ) > 0 , v(t) = & ( t ) 1 0 , 42(0) > 0 , t E [-T, 01,

satisfies [ u ( t ) , v ( t ) ]+ (u*,v*)

as t + +w,

where

In our next theorem, we will see that this result can be easily extended to the following more general two species competition system:

.

.

where all the parameters are positive constants. It is assumed that p i , k i , i = 1,2, are nondecreasing and satisfy 0

0

Li

dp:(.) = 1 = L r n d k i ( S ) ,

2

= 1,2.

Moreover, we assume, for i = 1,2,

bl(0) = 0,

b’,(*)> 0.

Theorem 5.6. In (5.8), assume (5.7) and (9 max{Tlbl ( ~ ) ( Q . + 2 a 1 ) , 7 2 b z ( ~ ) ( P + b z + ~ b
as

t -+ +w,

(5.9)

232

Delay Differential Equations

Proof. Clearly, (Al)-(A2) hold in (5.8). Since ala,' > blb;', we see b2 > a z b l / a l . Thus, there is an € 1 > 0 such that, for 0 < 6 5 €1,

Let

61 = ( b i

+€)/ail

62

(5.10)

= 1.

Then,

and

(5.11) That is, the assumption (i) in Theorem 5.2 is satisfied. Clearly, we have limsupu(t) 5 y l / a l , t++w

From (5.9), we see there is a 0

and

limsupv(t)

5 72/b2.

t++w

< €2 5 € 1 such that, for 0 < 6 5 €2,

< 1. However, (5.11) is precisely the assumption (ii) of Theorem 5.2 if we choose 6 2 as in (5.10) with 0 < c 5 € 2 . It is easy to verify that (u*,v*) is the unique positive equilibrium of (5.8). Now, the conclusion follows Theorem 5.2. 0

61,

Theorem 5.6 improves Theorem A in three ways: (i) We do not require the delays to be bounded; (ii) We allow both discrete and distributed delays; (iii) Most importantly, we can tolerate large coefficients a and p, so long as 71 and 72 are small enough. We would like to mention that results similar to the preceding theorem can be established for general infinite delay Lotka-Volterra type population models involving mixed interactions, such as predator-prey, competition, and cooperation.

6. Global Stability for Multi-Species Models

233

Generally speaking, the application of our main results (Theorems 4.1, 4.2, 5.1, and 5.2) may involve some complicated computations. In the case of Theorems 4.2 and 5.2, some additional guesswork may be required. Nevertheless, the application of Theorem 5.1 is quite routine. Consider the following autonomous system:

where (Al)-(A2) are assumed to hold. In order to apply Theorem 5.1, the following algorithm should be used: Step 1: Check whether or not n

n

j=1

j=1 j#i

If the answer is no, stop here. Otherwise, continue to Step 2. Step 2: First find the saturated equilibrium p = ( P I , . . . , p n ) of

then, find a set of positive numbers

i = 1,2,... , n ,

T,,

i = 1 , 2,... , n , such that, for

Step 3: Check whether or not

If the answer is no, stop here. Otherwise, the conclusion of Theorem 5.1 is valid for system (5.12). A “no” answer to Step 1 or Step 3 implies that Theorem 5.1 does not apply to system (5.12). In these cases, one may want to try Theorem 5.2. In applying Theorem 5.2, one needs to do the additional guesswork of finding positive constants S;, i = 1 , 2 , . . . , n . If one can find such S;,

234

Delay Differential Equations

such that (i) and (ii) of Theorem 5.2 are satisfied for system (5.12), then the conclusion of Theorem 5.2 applies. Otherwise, the task of applying Theorem 5.2 becomes very much involved. It should be mentioned here that the method can be extended to cover more general systems. For instance, we may consider

tii(t) = bi(Ui)Fi(t,ut),

.

u = ( ~ 1 , .. ,u"),

i = 1,. . . ,TL,

where Fi(t,ut) may not be linear but is globally Lipschitz. The bounded delay ri in Eq. (4.3) can be replaced by a positive continuous function q ( t ) . 6.6. When Nondelayed Diagonal Terms Do Not Exist Most of the convergence results appearing so far for delayed LotkaVolterra-type systems require that undelayed negative feedback dominate both delayed feedback and interspecific interactions. Such a requirement is rarely met in real systems. In this section, we present convergence criteria for systems without instantaneous feedback. Roughly, our results suggest that, in a Lotka-Volterra-type system, if some of the delays are small, and initial functions are small and smooth, then the convergence of its positive steady state follows that of the undelayed system or the corresponding system whose instantaneous negative feedback dominates. In particular, we establish explicit expressions for allowable delay lengths for such convergence to sustain. We study first a class of Lotka-Volterra-type infinite delay systems, where delays for the negative feedback are expected to be small. If the system is diagonally dominant (to be defined in what follows), and initial functions are relatively small and smooth in the short past, then we can show that the unique saturated equilibrium or steady state attracts such kind of neighboring solutions. Similar results are found to be true for delayed Volterra-Liapunov stable Lotka-Volterra-type systems. Our approach involves constructing suitable Liapunov-Razumikhin functions, carefully selecting initial function sets, and estimating the length of relevant delays. We consider first the following general autonomous Lotka-Volterratype infinite delay system:

where

6. Global Stability for Multi-Species Models

235

( u i ( t ) , u 2 ( t )., . . , u n ( t ) ) . Throughout the rest of this section, we u(t) assume that, for i = 1,. . . ,n, ( H l ) bi(0) = 0, b i ( . ) is continuously differentiable, and bi(.) > 0; (H2) r i ,a i , and Ti are constants; in particular, a, and T , are positive; (H3) p i ( @ )are nondecreasing, pi(0) - pi(m-7,) = 1; (H4) p i j ( 8 ) are bounded real valued Bore1 measures on (-oo,O] with total variation Ipijl. Note that (6.1)-(6.2) is a special case of (4.1). As usual, we assume the initial conditions satisfy (H5) uo(s) = +(s) 2 0 for s 5 0; E BC, +(O) > 0. In this section, we choose the norm I I in R" as

+

I+(s)I

= max{l4i(s)l : i = 1 , . . . ,n},

where +(s) = ( +1(s),. . . ,&(s)) in Section 2.7, we have

ll+ll,

E

R". Thus, for g ( s ) satisfying (g1)-(g3)

= supmax{ -: 2 = l ,

1

..., n ,

s
+

where E UC,. We say system (6.1) is diagonally dominant if

It is well known that (see Berman and Plemmons (1979)) system (6.1) has a unique saturated equilibrium if it is diagonally dominant. It is also easy to see that if (6.1) is diagonally dominant, then there is a g ( s ) satisfying (gl)-(g3) such that ai > C;=, l p i j g l , where l p i j g l .Pmg(s)Idpij(@)l, i = 1 , . . . , n . For simplicity, we denote T

= max{.ri : i = I , . . . ,n } .

(6.4)

Throughout this section, we assume that system (6.1) is diagonally dominant, and denote g = g ( s ) as a function that satisfies (g1)-(g3), a, > C;=1 I p i j g l , i = 1,..., n , and g(s) = 1 for s E [-T,O]. Clearly, such a g ( s ) always exists as long as (6.1) is diagonally dominant. Also, we always denote p as the unique saturated equilibrium of (6.1). The material of this section is adapted from Kuang and Smith (1992b). Lemma 6.1. In addition to (Hl)-(H5), assume that system (6.1) i s diagonally dominant. Assume further that

Delay Diffemntial Equations

236

=

+

+

u,(+)(t), u;(t 0) = ui($)(t 8), i = 1, ...,n. Clearly, Zi'implies that ui 5 K + p ; and Iui(t+O)-pi1/g(8) 5 K for i = 1,. . . , T I , 0 I 0. Thus, where u;

I l ~ t ( $ ) ( . > - p 1 15 ~

Note that we have used the assumption that g(8) = 1 for 8 E [-T,O]. This shows that Ihi(4)(t)l di(K)Zi' follows from Ilut(q5)(.)-pllg 5 Ii' fort 1 0. Assume that there is a t > 0 such that llul(+)(.) - p1Ig = Ii' and l l u t ( 4 ) ( - )- p1Ig < ( I for t E [O,i). Then, the definition of 11 [Ig together with the fact that 114 - pllg c Ii' implies that there is an i E (1,. . . ,n} such that I l u j ( + )(-)- pllg = Iui(t) - pi[. The previous arguments ensure that Ih;(t)l 5 d;(Ii')Ii' for t E [-.,I]. Denote

-

~ ( t=)(ui(t) - pi)2.

6. Global Stability for Multi-Species Models

237

(6.5)

It + e,?] is determined by mean value theorem.

where ( = ((0) E then for 4 E [-t,8],

Iu,(t

+ e ) - uj(t)l 5 d,(Zt')Zie

If

t < T,,

< di(Zt')KTi,

and, for 8 E [-q, -t], Iui(t

+ 0) - u;(t)l I Iui(t + 4)- ui(O)l+ Iui(O)- ui(f)l 5 di(Zi)Zt'lt

Hence, in all these cases,

Therefore,

+ 81 + di(Zi)Zt't

= d i ( K ) Z i e 5 di(Zt')KTi.

238

Delay Diffemniial Equaiions

which contradicts (6.5). This implies that no such t exists and, hence, for 0 t L 0, Il.t(4)(.) - Pllg I K. Now we are ready to state the main result of this section. Theorem 6.1. Assume that all the assumptions of Lemma 6.1 are satisfied. Then lim u ( + ) ( t ) = p. t++m

Proof. Let a E ( 0 , l ) be a constant such that n

ai(a - di(K)Ti) >

C lpijg1-

j=1

We claim that there is a TI > T such that, fort 2 2'1, ~ ~ ~ ~ ( q 5 ) ( ~aK. ) - p ~ ~ ~ ~ < There are two possibilities: (i) For any large T , there is a t > T , i E { 1 , . . . ,n}, such that

K2 2

K ( i ) > (Y21i2,

k(i)2 0.

(6.7)

However, a similar argument as the proof of Lemma 6.1 yields

which contradicts (6.7). (ii) There is an i E ( 1 , . . . , n } such that G ( t ) < 0, and Iim ~ ( t=) a 2 ~ { 2 . t++m

In this case, a similar argument as in case (i) implies that

e(t)5 2 a b i ( O l ~ ) [ - u i ( a- a i ( z i ) T i ) +

c n

Ipijsl]K2 < 0,

j=l

which leads t o limt++mK(t) = -00, a contradiction. This proves the claim. By Lemma 6.1, we know that, for t 0, i = 1 , . . . ,n, J u ; ( t )- pi1 5 11'. Since 4 E BC and lims+-mg(s) = +m, we see that there is a S1 > 0 such that (114 - pllm + 1 0 < al(. g(--S1)

6. Global Stability for Multi-Species Models

Thus, for t 21'7

239

+ S1 = ul,

Observe that d , ( K ) is strictly increasing with respect to K. If we replace K by a K in Lemma 6.1, we see that all its assumptions are satisfied. Clearly, n

ai(a

- di(aK)Ti) > a i ( a - d i ( K ) T i ) > C

Ipijgl.

j=l

Hence, we can repeat the previous argument and conclude that there is a u2 > u1 such that, for t 2 u2,

By repeating such an argument again and again, we obtain a sequence < u2 < . * . < u i < u;+l** * , limi++, a, = +oo, such that, for t 2 a;,

u1

This clearly implies that

limt++cc .(W)= P. 0 Equivalently, we can state the preceding theorem as follows: Theorem 6 . 2 . In addition to (Hl)-(H5), assume that system (6.1) is diagonally dominant. Denote, for i = 1 , . . . ,n,ZC > 0,

Assume further that, for some K > 0, 7, 5 T i ( K ) , i = 1,...,n, and satisfies the assumptions (ii) and (iii) of Lemma 6.1. Then

4

lim u(+)(t) = p.

t++m

Proof. Note that ~i 5 T , ( K )is equivalent to (i) in Lemma 6.1. The rest 0 is trivial. An immediate consequence of the preceding theorem is the following corollary. Corollary 6.1. In addition to the assumptions of Theorem 6.2, assume further that limsupt,+, \u(q5)(t)l < K. Then Ti I Ti(K) implies that limt-+, .($)(t) = p.

Delay Differeniiol Equaiions

240

The more general version of Theorem 6.2 takes the following form: Theorem 6.3. In addition to (Hl)-(H5), assume that in (6.1) there ezist 6 ; > O , i = 1 , 2 ,... ( 1 2 , s u c h t h a t

c n

a;

> 6;

6;yp;jl.

j=l

Assume further that there is a I(

> 0 such that r; 5 r;(K),where

i = 1 , 2 , . . . ,n, and 4 satisfies the assumptions (ii) and (iii) of Lemma 6.1. Then limt++oo u ( $ ) ( t ) = p. Proof. If we denote U;(t) = 6;u;(t), then U ( t ) = (U,(t), , . . , U n ( t ) ) satisfies i = 1,. . . ,TI, (6.10) ~ ; ‘ (= t ) a;b;(6;’Ui(t))~i(U~i(.)), where

then (6.10) and (6.11) are identical to (6.1) and (6.2) (except the bars). The condition (6.9) thus implies that (6.10) is diagonally dominant. The rest follows Theorem 6.2. 0 Remark 6.1. If we replace a;,!J u,(t 0) d p i ( 0 ) by aju;(t) in the system (6.1), then the assumption of diagonal dominance implies that all solutions of (6.1) tend to the unique saturated equilibrium p. Equivalently, this means that p is globally asymptotically stable with respect to admissible initial functions. Our results indicate that this global asymptotical stability of p is maintained provided that (i) initial functions are bounded (in BC norm), and the derivatives of their ith component are also bounded properly in the interval [-q, 01; and (ii) r; are small enough. In view of the fact that many real systems are studied with bounded initial functions with bounded initial derivatives, and the time delays ~i are usually regarded as small, our results suggest that when the system is diagonally dominant, instantaneous negative feedbacks ( a ; u ; ( t ) )are suitable approximations of delayed negative feedbacks (ai J!ri ui(t 0) d p i ( 0 ) ) . It should be pointed out that the infinite delays appearing in (6.2) do not create any real

+

+

6. Global Stability for Multi-Species Models

24 1

difficulties in our analysis; therefore, restricting them to a finite delay case will not provide any new results from our method. If a uniform bound can be found for solutions of (6.1), then Corollary 6.1 asserts that p is globally asymptotically stable as long as the Ti are small enough. If the initial functions are not differentiable, then one can replace q5 by u~(q5)for some T > max{r, : i = 1,. .. ,n}. This way the initial function becomes differentiable for s 2 -r,. The resulting estimate may change, though. Our estimate of the size of delay is not optimum even in scalar cases. We consider now a finite delay version of system (6.1): 0

[

hi(t>= bi(ui(t)) ri - ai

Li

ui(t

+ 6) dpi(e) + f:Jo u j ( t + 4) dpij(e)], j=1 -rij

(6.12) where rij > 0, i , j = 1,. .. ,n. (H4) thus reduces to (H4)' p ; j ( S ) are bounded real valued Bore1 measures on [-q,01 with total variation IpijI. Denote T = max{q,qj : i,j = 1,. . . , n } . Clearly, (H5) should be replaced by (H5)' ~ o ( s = ) 4(s), s E [-.,O], 4 E BC([-T, 01, R"),4(0) > 0. Our objective in the following is to establish convergence criteria for system (6.12) when it is not diagonally dominant. For notational convenience, we define dpij(O),

i , j = 1,. . . ,n,

D=diag(dl,dz ,...,d"),

d;>0,

i = l , ...,n.

For such a positive diagonal matrix D and a nonnegative steady state p = (PI,. . . ,pn) of (6.12), we define

where R; = ( ( ~ 1 , .. . ,un) : ui 2 0, i = 1,. . . , n } and Int R; denotes the interior of R;. It is easy to show that VD,,(U) > Vo,,(p)= 0 if u # p and u E Int R;. In the rest of this section, we assume that p is the unique positive steady state of system (6.12). For any number VO > 0, we denote

A(V0)= { u : u E R;,VD,,(U) = Vo}.

242

Delay Diffemniial Equaiions

In this section, we adopt the standard Rn norm; if u = ( u i ,. .. ,un) E R", then 1161 = (ELru?)''~.Since limlUI+mVD,~(U) = +oo, we see that A ( b ) is compact. We can thus define

It is easy to see that p(V0) is strictly increasing with respect to VOand 4 0. It should be pointed out here that p ( b ) also p ( b ) + 1 as depends on D and p. Denote A = ( a i j ) n x n , where (1.. II

- -a. I + pi,;

and a,, = p i ,

when i # j, i, j = 1,. . .,n.

We call system (6.12) VL-stable (Volterra-Liapunov stable [see Hofbauer and Sigmund (1988)]) if there is a positive diagonal matrix D = diag ( d l , . . . ,d,) such that D A + ATD is negative definite. Finally, we define, for any positive numbers M , N ,

F ( M , N ) = { d : d E c'([--7,ol,R;), IId(*)- PI1 c M , lldf(.)ll < N ) , where II-II is the uniform norm (with respect to 1.1 in R")in C([--7,O],R;). We are now ready to state and prove our main result of this section. Theorem 6.4. In addition to (Hl)-(H3) and (H4)', assume that (6.12) is VL-stable. Then for any positive number M , there i s an N = N ( M ) and T ( M ) such that -7 < T ( M ) and E F ( M , N ) implies that limt4+m u(r$)(t)= p, where p as the unique positive steady state of (6.12). Proof. Since (6.12) is VL-stable, there is a D = diag(d1,. . . , d n ) such that E = D A ATD is negative definite. Thus, there is a A > 0 such that, for any u = ( u , .. . ,u n ) ,

+

uEuT

< -Xu. uT = - A

n

C uz. i=l

Denote, for u ( t ) E Int R;,

W(U(t)= ) maxe€[-r,o] V ( 4 t + 6 ) ) . The derivative of V ( u ) ( t )along a solution u ( t )of (6.12) takes the form n

V ( u ) ( t )=

C di(ui(t)- pi)Qi(ut(*)),

i=l

243

6. Global Siabiliiy f or Mulli-Species Models

where

Note that

Denote Vo = % ( M ) = max{V(u) : Ju- pJ 5 M } , u = u ( M ) = max{lu - pI : V ( u )= Vo}, q = q ( M ) = max{bi(p;

+ ii)(ai + 2 Ipjjl)

: i = 1,2,.

.. , n } ,

j=l

then Iu($)(t) - pJ < ii, t 2 0. Otherwise, there is a t* > 0 such that v0 = V ( u ( t * )= ) W ( u ( t * )and ) I/i/(u(t*))2 0. We note from the definition of p(V0) that p(&)-’. 5 [up’)- p l 5 u.

244

Delay Diffemniial Equalions

i.e., k ( u ( t * ) )< 0, a desired contradiction. This proves the claim. Denote

Q

= Q($) = limsup Iu($)(t) - PI, t++w

$ E F(M,N).

If the theorem is false, then, for some 4 E F ( M ,N ) , C following that this is the case. Denote

> 0.

Assume in the

V* = limsup V ( u ( $ ) ( t ) ) , t++w

u* = max{lu

- PI : V ( u )= V * } ,

u* = min{lu

- PI : V ( u )= V * } .

The proof of the previous claim clearly implies that V* 5 V,. And it is trivial to see that u* 2 C 2 U, > 0. Since - i X p ( h ) - ' T q cE1 d i ( a i Cj"=,p i j ) < 0, there is an €0 = q(Q)> 0 such that 1

-'XP(VO 2

n

n

i=l

j=1

+ C O ) - ' ( Q - + [ T q C di(ai + C €0)

pij)](C

+

+

+

< 0. (6.13)

€0)

By the definition of Q, we see that there is a t l > 0 such that, for I min{ii,Q €0). Since u+ = p ( v * ) - l u * , we can choose a small positive € 1 such that

+

t 2 t i , I p ( $ ) ( t ) - pi u* - € 1

> p ( v * ) - ' ( u * - €0) 2 p ( &

Clearly, there is a t2 = t 2 ( ~ 1 > ) tl l u ( N t 2 )- PI

+

T

> u* - € 1 ,

+ € O ) - l ( C - €0).

such that

V ( u ( 4 ) ( t 2 )L ) 0.

(6.14)

245

6. Global Siability for Multi-Species Models

Note that, for t 2

tl

+ 7,

Now, the fact that Iu(q5)(t2)- p ( > u, - €1 > p ( h with (6.13) imply that W q 5 ) ( t 2 ) ) < 0,

+ to)-'(& +

€0)

together

which contradicts (6.14). This indicates that & must be zero, proving the theorem. 0 Similar comments to those included in Remark 6.1 can be made for the preceding theorem. It is well known that if a nondelayed Lotka-Volterratype system is VL-stable, then its unique positive steady state (if any) is globally asymptotically stable with respect to positive initial data. Roughly, Theorem 6.4 asserts that if (i) all involved delays are small and (ii) the initial (positive) functions are small and smooth, then this positive steady state continues to attract neighboring solutions. This partially justifies that in some real life systems if delays are expected to be small, one can approximate these systems by models consisting of only ordinary differential equations. 6.7, Remarks a n d O p e n Problems We have restricted our attention on systems of the form (1.2)-(1.3) in this chapter. Some of the results may be extended to more general systems such as (1.1); others may not. The linearity of G ; ( t , u t ( - )with ) respect to

246

Delay Differential Equations

u t ( . ) played an important role in almost all of our discussions. It is thus important and interesting to discuss cases when G,(t,t i t ( . ) ) is not linear with respect to u t ( - ) . As we can see, all the global stability results of Sections 2-5 require some kind of dominance of the nondelayed intraspecific competition. This restriction is very severe and may not be necessary at all. A reason for this is that, for retarded single species models, the global stability of the positive steady state usually persists, so long as the time delays are short. The results in Section 6 gave only estimates of regions of attraction of the positive steady states. We thus propose the following important research problem. Open Problem 6.1. Obtain sufficient conditions for the global asymptotical stability of saturated steady state in system (6.1)-(6.2). The following open problem related to Theorem 4.1. It may be less complicated than Open Problem 6.1. Open Problem 6.2. In system (4.1), assume that (Hl)-(H6), (A), and (4.17) hold. Let u ( ' ) ( t ) and ~ ( ~ )be( tany ) two solutions of (4.1)-(4.2). Is it true that

One can also find some global stability results of delayed multi-species models in Gopalsamy (1992). For local stability results, see Cushing (1977) and Gopalsamy (1992).

Periodic Solutions in Multi-Species Models 7.1. Introduction It is known from Chapter 5 that time delays have a tendency of producing oscillations, or periodic solutions in otherwise nonoscillatory models of single species growths. This is also true for multi-species systems. A standard way of illustrating this is applying the so-called bifurcation results (Theorem 2.9.1, or Theorem 6.1 in Cushing (1977), or theorems in Hassard e t al. (1981)). Examples of such efforts are well documented in Cushing (1977) and Gopalsamy (1992). Stability analysis of the bifurcating periodic solution is given in Gopalsamy and Aggarwala (1980) for the following competition model:

where k i , a i j , i , j = 1,2, a are positive constants. Oscillation results on delayed Lotka-Volterra-type systems can be found in Gopalsamy (1992), and Gyori and Ladas (1991). The main concern of this chapter is the global existence of periodic solutions in delayed multi-species models due to delays and/or periodicity of environments. We present first a global existence result of periodic solutions in a class of delayed autonomous Gause-type predator-prey systems in the following section. In Section 3, we present an existence and unique ness result on periodic solutions in a class of delayed periodic systems. We end this chapter with a brief discussion and some open problems. 7.2. Periodic Solutions in Delayed Gause-Type Predator-Prey Systems Our objective in this section is to establish sufficient conditions for the global existence of nonconstant periodic solutions in the following Gawe247

Delay Differeniial Equaiione

248

type predatorc-prey system:

$(t>= z(t)b(z(t)) -p(z(t))y(t)l, Y(t) = Y ( t ) [ - Y h ( 4 t - 4 ) 1 ,

(2.1)

+

where z ( t ) , y ( t ) stand for the population density of prey and predator at time t , respectively. We always assume that Y ( 0 ) > 0,

48)=

$(a 8 E [-.,OI,

$ E C"--7,Ol,R+),

40)

> 0.

(2.1)' Moreover, we awume the following hold: ( A l ) g ( z ) E C2([0,+oo),R);there exists 20 > 0 such that g ( z ) > 0 for z E [O,zo), g ( z 0 ) = 0 , and g " ( z ) 5 0 for z 2 0. (A2) p ( z ) E C'([O,+oo),R), and p ( z ) > 0 for z 1 0; p ( z ) is monotone nonincreasing for z 2 0. (A31 h ( z ) E W O , +W), R), and h'(z) > 0 for z 2 0; h ( 0 ) = 0. Thus, a solution of (2.1) and (2.1)' exists and is unique and stays positive. The material of this section is adapted from Zhao et al. (1992). Theorem 2.1. Let ( z ( t ) , y ( t ) ) be the solution of (2.1) and (2.1)'. T h e n there i s a constant M > 0, independent of initial data, such that max{limsupz(t),limsupy(t)}

5 M.

f+m

t-+w

Proof. We have g ( z ) = g'(zo)(z-zo)+g"(()(z-zo)2, where ( is between t o and z. Clearly, if z > 0, then ( > 0, which implies g''({) 5 0. So we have $ ( t ) 5 z(t)(-zOg'(zO) g ' ( z o ) z ( t ) ) , which leads to

+

limsup z ( t ) 5 zo. i++m

There are two cases to consider. (i) If v > h ( z o ) , then there is an From (2.2), there is a 21' > 0 such that z ( t ) < zo

+c

6

> 0 such that v > h ( z o + 6).

for t

1 2'1.

Since h ( z ) is increasing, we have

$(t)< y(t)[-v

+ h(z0 + c)] < 0

which implies that limsup y ( t ) = 0. t++W

for t

2 2'1 + 7 ,

7. Periodic Solutions in Multi-Species Models

(ii) If v

249

I h(zo), there is a T2 > 0 such that z ( t ) < zo + 1 for t 2 T2;

we thus have C(t) I y(t)[-v

Denote A = h(z0

+ h(zo + l ) ]

for t 2 T2

+ 1) - v . It follows that A > 0, $(t) I y(t) A for t 1 T2 +

+ 7.

(2.3)

T;

hence, we have

and thus

It is clear that solutions of

i ( t )= g‘(zo)z2(t), tend to zero uniformly as t -+ t o , to), such that

< Z ( t 0 ) < zo + 1 ,

(2.5)

+oo. There exists T3 > 0, independent of

~ ( t<) zo where zo satisfies h(zo) = v / 2 . Denote

If there is a T > 0 such that

0

for t 2 T3,

(2.6)

Delay Diffemniial Equaiions

250

which implies lim y(t) = 0,

t++w

and this clearly contradicts (2.8). Let

M = max{zo

+ 1,

(2.11)

It is clear M is a constant independent of initial data. It is obvious that limsupt++m z ( t ) < M. Now, we want to show that limsupt++m y(t) 5 Mi exp[A(T2 T3 T ) ] . Otherwise, there exists 0 < 11 < & such that

+ +

y(5i) = Mi,

y(Z2) = Mi exp[A(Tz

+ T3 + .)I

and y(t)

for t E (tl,tz];

> MI

y(t2)

2 0.

Then, from (2.4), we see that t2

- 51 2 T2 + T3 + T .

Then,

i ( t ) < g’(zo)z2(t)

for 51

for tl

+ T2 + 7’3

+ ~2

5 t 5 t2.

+T

5 t 5 52.

This leads to i(t)<0

In particular, y(52) < 0, contradicting

i(52)

2 0. This proves the theorem.

0

The proof of the following theorem can be found in Zhao, e t al. (1991). Theorem 2.2. Suppose system (2.1) has no positive steady state, that is, v 2 h ( s 0 ) . Then limt-,m(z(t),y(t)) = (z0,O). Since we want to establish the existence of periodic solutions in (2.1), it is thus necessary to assume that it has a positive steady state. Without loss of generality, we assume in the following that this positive steady state is (1,l). This is equivalent to (HI) = P(l), v = Also without loss of generality, we can assume (by time scaling) that (H2) T = 1. For convenience, we make the change of variables u ( t ) = z ( t ) - 1,

v(t) = y(t) - 1.

(2.12)

251

7. Periodic Soluiions in Multi-Species Modeb

This results in h ( t ) = (1 & ( t )= (1

+ u ( t ) ) [ g ( u ( t+) 1) - p ( u ( t )+ + v ( t ) ) [ - v+ h(u(t - 1) + l)].

4t))I,

(2.13)

u(o) > -1.

(2.13)’

The initial condition (2.1)’ becomes

v(o) > -1,

$(e) 2 -1,

uo = $,

The variational system of (2.13) at (0,O)takes the form h(t)= -au(t) - pv(t), & ( t )= wu(t - l ) ,

(2.14)

where a = p’(1) - g’(l), p = p(l), w = h‘(1). In what follows, we always assume that (A4) P‘W > g‘W* The characteristic equation of (2.14) is

x2 + a~ + @we-’

= 0.

(2.15)

Suppose (Hl), (H2) hold. Then 3ystem (2.1) ha3 a Hopf bifurcation at pw = a(un(a)/sinun(a)), n = O,1,2, ..., where u n ( a ) E (2n?r,2n?r (?r/2)) satisfies tanu,(a) = a/u,(a),n = O,1,. . . . Proof. Let X = X ( a , p w ) = p iu;(2.15) reduces to

Theorem 2.3.

+

+ p2 - u2 + a p + i3we-P cos u = 0, 2pu + au - pwe-” sin u = 0.

Let p = 0, u > 0; then, we have 02

cos u = -, Pw

CrU

sin u = -. Pw

It follows that (2.15) has a root X = iu, u > 0, if and only if (pw)’ = u4 a2u2and a (2.16) t a n u = -.

+

U

Equation (2.16) has countably many positive roots { u ~ }with ~ ! On ~ E (2n?r,2n?r (n/2)). In this case,

+

pw = a

sin

i? &(a),

(a)

n = 0, 1,2,. .

252

Delay Differeniial Equaiions

By straightforward differentiations of (2.15), we obtain

+ a2al(a)+ aal(a) > 0, + + a i ( a ) ( a 2+ 4)]

a:(&)

-

I(n(a)[(a

Q~((.Y))~

(2.17)

(2.18) It is clear that the pure imaginary roots of (2.15), Xn = ian7n = 0, 1 , 2 , . . . , are simple and, for any fixed n , the root X # A, satisfies X # mX, for every integer rn. Therefore, according to Theorem 2.9.1, we have that system (2.1) has a Hopf bifurcation at pw = Iiln(a) = a

an(a)

0 sin an(a)* The proof of the following theorem is obvious. Theorem 2.4. Assume that ( H l ) , (H2) hold. If pw > I(o(a), where I(o(a) = a(ao(a)/sinoo(cr)), and ao(a)E (O,?r/2) satisfies tanao(a) = (a/ao(a)),then the steady state ( 1 , l ) of system (2.1) i s unstable, and (2.15) always has a root X = p ia such that p > 0 and o E (0,n/2).

+

(2.19) We assume further that (A5) q(.) is concave, i.e., q"(.) 5 0. The following theorem is the main result of this section. Theorem 2.5. In system (2.13), assume that (Al)-(A5) and ( H l ) , (H2) hold, and (B) W P > Iio(a), where Ii'o(a) is described in Theorem 2.4. T h e n the system has at least one nonconstant positive periodic solution, with period T > 2. In the following, we always assume that (Al)-(A5), ( H l ) , (H2), and (B) hold. From ( A l ) and (Hl), we get q(0) = 1, which implies that 50

>1

and

v = h(1)

< h(z0).

From (A4), we see that

(2.20)

7. Periodic Soluiions in Mulii-Species Models

253

The relation (2.20) and (A5) imply that zo - 1 is the unique positive root of q ( - ) , and q(z) - 1 has at most one negative root. We need the following definitions: hl = min{h'(u) : u E [1,20]}, h2 = max{h'(u) : u E [l,zo]}, Pl = min{p(u) : u E [Lzo]}, ~1 = exp[hz(zo - 1)(2 h,' P:')], N = max{l,Nl - 1,zo - l},

+

+

'I = exP[-P(Wl+ N)1.

It is clear that all these numbers are positive. Define = {c01(41,42) : 4 E C([-l,O],R), i = 1,2; 41(-1) = 0, 4l(O) is nonincreasing, &(O) 1 2 9, while 42(-1) 2 0, 42(8)

+

is nondecreasing; max{l$,(8)1 : 8 E [-1,0], i = 1,2} 5 N } .

It is not difficult to see that It' is a closed, bounded and convex subset of the Banach space C([-1,0],R2) with the standard norm

This set is of decisive importance in the proof of our main theorem. The following notations are needed: Q1 = { ( u , v ) : u 2 O,V 2 0); Q3 = { ( u , o ) : u 5 O,V 5 0); RI = { ( u , ~:)21 I o , f ( u , v ) 2 0); R3 = {(u,v ) : u 2 0, f(u,v) 5 0);

= { ( u , ~:)21 Q4 = { ( u , v) : 21 Q2

R2 = { ( u , ~:)u R4 = { ( u , v) : u

I 0,v 2 01, 20 ,I ~ 01, I O,f(u,v) I O},

1 0, f ( u , v ) 1 O } ,

+

where f ( u , v ) = v 1 - q ( u ) , q(.) is defined in (2.19). The following lemma asserts that if @ E K\{O}, then ( u ( t ) , v ( t ) )= ( u ( t ,a), v ( t , @)),the solution of (2.13) with initial data a, eventually enters the fourth quadrant Q4. Figure 7.1 should be helpful for the subsequent arguments. Lemma 2.1. There ezists a continuous f u n c t i o n 01 : @ + al(@) f r o m K\{O} t o [ O , o o ) such that u(al(@)) = 0, v(al(@)) I0, and ( u ( t ,@), v ( t , @I) E Q2 U Q3 for 0 It Iai(@). Proof. For any @ E K\{O}, we notice that ;(t) = -[1 u(t)]p(u(t) l)f(u(t),v(t)), so ; ( t ) 1 0 ( 0) I if and only if f ( u ( t ) , v ( t ) )5 0 ( 2 0). This implies that as long as ( u ( t ) , v ( t ) )stays above (or on) the curve

+

+

Delay Differenfial Equalions

254

Fig. 7.1. The graph of map F : K

+

K.

r : f(u,v) = 0 in the uv-plane, then u ( t ) is decreasing, which indicates that ( u ( t ) , v ( t ) )cannot cross the v-axis from 9 2 into Q1 before crossing the curve Clearly, for small t > 0, ; ( t ) I 0. We assume in the following that

r.

+2(0)) < 0 can be dealt with similarly. Suppose first that $752(0) > 0. We have the following. Claim A. Them ezists a 71 T I ( @ ) , 0 5 71 < +oo, such that f ( u ( ~ ~ ) , v ( ~=l 0, ) ) ~ ( 7 1< ) 0, and ( u ( t ) , v ( t ) )c ~ o s s e s at 71.

The case that f(&(O),

=

r

7. Periodic Soluiions in Mulii-Species Modeb

If f(+l(O),r$2(0))

255

= 0, then we can take

71

= 0. Assume that

> 0.

f(41(0)142(0))

If Claim A is not true, ( u ( t ) ,v ( t ) ) will remain in R1.By the monotonicity of u ( t ) and v ( t ) in R1,thereis a (u1,vl)E R1 such that limr++oc(u(t),v(t)) = (u1,vl). From (2.13),we have 0 = (1 + u1)b(u1 0 = (1 v1)[-v

+

If q51(0)

+ 1) - P(W + l)(Vl + 111 = -(I + Ul)P(Ul +

+ h(u1 + l)].

< 0, then u1 5 &(O)

l)f(?41v1)1

< 0 since h ( t ) 5 0 for t 2 0. This yields

v1 = -1,

111

= -1,

a contradiction to the fact that (u1,vl)E R1 since f(-1, -1) = -q(-1) = -(9(0)/P(O))< 0. If41(0) = 0, f(41(0),42(0)) = 1+42(0)-1 = 42(0) > 0. (If &(O) = 0, then q52(8) 3 0 for 8 E [-1,0];also, &(O) = 0 implies &(8) 0 for 8 E [-1,0],a contradiction to (D = (&(8),&(8)) # 0.) This leads to h(0) < 0 and again implies that u1 = -1, 01 = -1, proving the

=

claim. If ~(71) 2 0, we have the following claim. Claim B. There ezistcr tl : 71 5 tl < +oo such that v(t1) = 0, u(t1) 5 0, and ( u ( t ) , v ( t )c~o89es ) the u-azis at tl. If v(q) = 0, let tl = 71. Then the claim is true. If ~ ( 7 1 ) > 0, it is clear that ( u ( t ) , v ( t ) )cannot cross the curve T from R2 into R1 before crossing the u-axis. If ( u ( t ) ,v ( t ) )remains in the interior of R1 n Q 2 forever, then we have h ( t ) < 0 for t > 71 and lim ( u ( t ) ,v ( t ) )= (0,O).

(2.21)

t*+w

Note that (B) of Theorem 2.5 implies that u p > a,which in turn implies that q'(0) h'(1) > 0. There exists 6 > 0 such that

+

d ( t )+ h'(q) > 0

for

It1 < 6,

17 - 11 < 6.

(2.22)

By (2.21),for preceding 6 > 0, there exists 7'1 > 0 such that lu(t)l Iv(t)l < 6 for t 2 7'1. From the second equation of (2.13),we get

v(T1

+ 1) - v(T1)

+

+

=L ?+l(l w(t))[-v h(u(t - 1)

I--v

+ h(u(T1) + 1).

+ l)]

dt

< 6,

256

Delay Differential Equaiions

Since f ( u ( T l ) , v ( T l ) ) = v(T1) q(u(T1)) - 1. Hence, we have

+ 1 - q(u(T1)) I

0, we have

v(Ti)

I

(2.23) here. Also,

The relation (2.22) combined with (2.23) implies v(T1 diction. Furthermore, we have

0

< tl - TI 5 1.

+ 1) < 0, a contra(2.24)

Thus, we have proved the claim. Claim C. There is a 01 e q ( @ )2 71 such that u ( q ) = c, v(a1) 5 0 and ) the v - a z i s from R2 into R3. at (TI,( u ( t ) , v ( t ) crosses If u(t1) = 0, let q = tl; then, the claim is true. If u ( t l ) < 0, then from Claim B, we know ( u ( t ) , v ( t )will ) go into R2 after t l . If the claim is not true, then the monotonicity of ( u ( t ) ,v ( t ) )again implies that there is a ( 2 1 2 , q ) with -1 < u2 5 0 , v2 < 0, and f(u2,vg) < 0 such that

By the first equation of (2.13), we see that

which implies limt++cs, u ( t ) = +oo, a contradiction. This proves the claim. If v ( q ) < 0 (this is possible if there exists uo < 0 with q(u0) = l), by the same method used in the preceding, we can also show the existence of 01. But in this case ( u ( t ) ,v ( t ) ) crosses the u-axis before 71. Assume now that 42(0) = 0 (this is impossible if uo does not exist). This implies 42(8) = 0, -1 5 8 5 0, which leads to &(O) < 0, since ($1,$2) $ (0,O). By the same method used in Claim A, we can show the ) T at 71. The conclusion of our existence of 71 such that ( u ( t ) , v ( t ) crosses lemma follows. implies that q ( @ ) is The continuity of (u(@,t),v(@,t))in t and 0 continuous in K\O. Lemma 2.2. There ezists a continuous function 02 : @ ---t Q(@) from K\O to [ O , + o o ) such that 4 0 ) > q ( @ ) + l , u ( u 2 ( @ ) )= 0, v(a2(@)) L 0,

257

7. Periodic Solutions in Multi-Species ModeLg

+

( u ( t ) , v ( t ) )E R3 for t E [ g i ( @ ) , g i ( @ ) 11, (u (t ),v(t ))E 9 4 U 91 for t E [01(@),g2(@)],and ( ~ ( tv)(,t ) ) E Ri n 9 2 for t E [02(@),~ 2 ( @ ) 11. Proof. We observe first that for t E [ q ( @ ) , q ( @ ) 11, (u(t),v(t))E R3, since 6 ( t ) 5 0 in this interval and ti(t)5 0 in R1 U &. Claim D. There is a j n i t e time 72 = 7 2 ( @ ) such that 4 7 2 ) > 0, v ( ~ 2 )< 0, and f(.(.2), v(72)) = 0. Otherwise, (u(t),v(l)) will remain in R3 after crossing from R2 into R3, because of the monotonicity of ( u ( t ) ,v ( t ) ) in R3 for t 1 cq 1. So there exists (u3,v3) E R3 with u3 > 0, f(u3, v3) 5 0 such that

+

+

+

By the second equation in (2.13), we know 0 = (1

+ D3)[-V + h(u3 + I)] = (1 + v3)h'(Ou3, +

+

where 1 5 { 5 1 u3. This implies v3 = -1, but 213 2 v ( q 1) > -1, a contradiction. Claim E. There is a finite time t 2 = t 2 ( @ ) > 7 2 such that ~ ( t 2 2 ) 0, v(t2) = 0, and ( u ( t ) , v ( t ) )crosses the u-azis at t2. If not, we have

) (0,O). lim ( u ( t ) , v ( t ) =

t++w

Since q'(0)

+ h'(1)

(2.25)

> 0, there exists €1 > 0 such that q'(0)

This implies that there exists

d ( t )+ (1 - +'(d

+ (1 - €l)h'(l) > 0. 61

>0

> 0 such that for

It1 < 61, IT - 11 < 61.

(2.26)

Let €2 = min{cl,61}. From (2.25), we can say that there exists T > 0 such that lu(t)l < €2, [v(t)l < €2 for t L T . From the second equation of (2.13), we have

v(T

T+2 + 2) - v(T + 1) 2 (1 - €2)/,+, [-v + h(u(t - 1) + l)]dt 2 (1 - €2)[-Y + h(u(T + 1) + 1))

(2.27)

258

Delay Diffemnfial Equations

+

Since f ( u ( T l ) , v ( T From (2.27), we get

v(T

+ 1)) 2 0, this leads t o v(T + 1) 2 q(u(T + 1)) - 1.

+ 2) 2 q(u(T + 1)) - 1 + (1 - €2)[-V + h(u(T + 1) + l)] = q'(t)u(T + 1) + (1 - + ' ( V ) U ( T + 1) 2 q'(t)u(T+ 1) + (1 - +'(q)u(T + 1) +

= [ d ( t ) (1 - 4h1(V)14T where

+ I),

(2.28)

+

0I (I u(T 1) < €2 I 6 1 0I q - 1s u ( T + 1) < €2 561.

Combining (2.26) and (2.28) we have v(T have thus proved the claim. Furthermore, we have

+ 2) > 0, a contradiction.

0 5 t2 - T I 2.

We

(2.29)

If u(t2) = 0, we let 42(@) = t2, and the lemma is proved. If u(t2) > 0, by the same method used in proving Claim C of Lemma 2.1, we also can show the existence of 4 2 . Also, because of the continuity of ( u ( @t,) ,v(@, t ) ) I3 in t, @, we have the continuity of 4 @ ) in K\{O}. Let

Define an operator F on I( as follows:

F @ = Uu(@)

for @ E K\{O},

and

FO = 0.

(2.31)

Lemma 2.3. F maps I( into I(, and F : K\{O} t I( is completely continuous with respect to the C topology. If F @ = @ E K\{O}, then ( u ( t ,a),v ( t , a)) is a nonconstant periodic solution of period c(@) > 2. Proof. Since f(zo - 1,-1) = 0, from the proof of Lemma 2.2 we know that 0 I u(t) < 10 - 1, t E [41,42]. And it is clear that Iuu,+i(e,@)l I 1 5 N, 8 E [-1,0]. Since col(uu,+l(@),vu,+l(@)) is a solution of (2.13), co1(uu,+~(@),vu2+~(@)) E C'([-1,0],R2). Also, uu2+1(-1, @) = 4 4 2 , @) = 0, %,+1(-1, a) = 4 4 2 , @) L 0, huz+l(e, @)= q O 2 1 e, a) 5 0, 6 u 2 + 1 ( ~ , a=) 6(a2 1 e , q 2 0,

+ + + +

e E [-1,01, e E [-1,01.

259

7. Periodic Solutions in Multi-Species Models

+

Now, we want to show that v(u2 l , @ ) 5 N . C l a i m F. u2 - t2 Ih;' p;' 1.

+

+

(2.32)

If a 2 # t 2 , then U(t2) > 0, V(t2) = 0, and 4 4 2 ) = 0, v(t2) > 0; this implies that there exists t3 with t2 < t3 < a 2 such that 4 t 3 ) = 4 t 3 ) # 0, u ( t ) 1u(t3) = v(t3) 2 v ( t ) , v(t) 2 v(t3) = u(t3) 2 u ( t ) , If t3

t E [t2,t3], t E [t3,62].

> t 2 + 1, integrating the second equation in (2.13) we get v(t3) - v(t2) =

J

t3

(1

t2

+ v(t))[-I/ + h(u(t - 1) + l)] dt

2 h'(<)J t 3

u(t

- 1)dt 2 h'(<)u(t3 - l)[t3 - t2]

12

<

2 hlu(t3)(t3 - t2)7 5 u ( t - 1) + 1. Since v(t2) =

where 1 5 t3 - t2 I h r ' , so we can say

t3 - t2

c,

v(t3) = u(t3), we have

I 1 + hT1.

Integrating the first equation in (2.13) from t3 to u2, we get

=

1:

1

+ u ( t ) [q (u ) - 1 - ~ ( t )dt.] +1)]4

[p(u(t)

(2.33)

Since q(0) = 1, q'(0) < 0, q ( - ) is concave, and u ( t ) L 0 for t E [t3,a2], we have for t E [t3,02]. q ( u ( t ) )- 1 5 0 From (2.33), we have

Hence, 0 2 - t3 5 p;', which implies (2.32), and proves the claim. From the second equation of (2.13), we have -6(t)

+

1 v(t)

- -I/

+ h(u(t - 1) + 1) = h'(t(u(t - l)))u(t - l ) ,

260

Delay Differeniial Equaiions

where ( ( u ( t - 1)) is between 1 and 1 a2 1, we get

+

ln(1

+ v(0g + 1))= / u z + l

+ u(t - 1).

Integrating from t 2 to

+

h'(()u(t - 1)dt 5 hz([o - l ) [ a 2 1 - t 2 ]

t2

5 h2(20 - 1)(2 + h,' This implies v(a2

+ P;').

+ 1) I N .

From the first equation of (2.13), we have

-Zi(t) 1

+ u ( t ) - [ g ( u ( t )+ 1) - p ( u ( t )+ l)(v(t) + l)] 2 -p(u(t)

Integrating from

02

to

a2

ln(1

+

+ 1, we have

+

+

+ l)(v(t) + 1).

4 0 2

+ 1)) 2 -p(l)(l + N ) .

This implies 1 u(a2 1) 2 9 . We thus have shown that F : K + K . Also, we have shown that max{ lu(t@)l,Iv(t, @)I} 5 N , t E [ O , Q 11. From system (2.13), we have

+

So F K is a bounded subset of C'([-l,O], R2)in C' topology, which leads to the equicontinuity and boundedness of F K in C topology, and hence the complete continuity of F . The second part of the lemma follows from the uniqueness of solutions 0 of system (2.13). In the following lemma, we will show that when the initial data is sufficiently small, the "first return time" a2(@)is uniformly bounded. Lemma 2.4. There ezists c > 0 (suficiently small) such that a2 : B, n ( K \ { O } ) + ( O , + c o ) i s completely continuous, where B, denotes the closed ball i n C with radius c. Proof. For b > 0 in (2.22), since u ( t ) and v(t) are monotone for t E [ O , T ~ ] and f ( u , v ) 2 0 in concave down, we can choose €3 > 0 such that

26 1

7. Periodic Solutions in Mulii-Species Modeb

maxt~[o,?(~)]{lU(t,@)l, I.(t,@>I) < 6 for @ E BC3 n (K\{OI). According to (2.24), we get for @ E BC3. t l ( @ )5 1 By the same method as that used in Lemma 2.2, we can show that q(@ -)t l ( @ ) is bounded for @ E BC3provided €3 is sufficiently small. By the continuity of the solution with respect to the initial data, for the € 2 in (ii) of Lemma 2.2, there exist 6 > 0 and 6 < €3 such that

From (2.29), we can say t 2 ( @ ) - a l ( @ ) 5 2, @ E B,, so t 2 ( @ ) is bounded for @ E B,. From (2.32), we know that Q(@)- t 2 ( @ ) is bounded. This implies q(@ is ) bounded for @ E B,, which leads to the complete continuity of a 2 in B, n (It'\{O}). In the following, we prove the ejectivity of (0,O) with respect to F, according to Theorem 2.9.4. Lemma 2.5. Let X = p + io be the simple root of (2.15) with p > 0, o E ( O , ( T / ~ ) ) . Then inf{ll{q@ll : @ E It',11@11 = 6 > 0) = v(6) > 0. Proof. From (2.14), we know

LQ where

+

0

= AQ(O) ~ q - 1 )=

A=(i2

i'),

J-1

d V ( qq e ) ,

B=(w 0 0o ) ,

(

A(X) = XI - A - Be-' = -we A!+

{) . -

It is clear that V = (A, -p) is a nontrivial solution of V A(X) = 0 and = is a nontrivial solution of A(X). u = 0. Since x is simple, PA is a one dimensional subset of C,and u

(4:)

is a basis of PA. So we have

262

Delay Differential Equations

where Q ( t ) = Ve-" = (Xe-", -(le-At), and ( Q , @) is a bilinear functional defined by

Suppose that liminf,-+,

This implies limn,+, [-1,0]. From limn,+,

$1 (n) (0)

e-''([+')

= 0 is true; then, we have

cos o(t

+ l ) $ p ) ( t ) = 0 uniformly for t E

R(@(")) = 0, we have lim

42(n) (0) = 0.

n-+m

Since 0 5

$p)(e)5 $p)(c),8 E [-1,0],

This implies limn,+,

we have

llq(n)(d)[[= 0, a contradiction. So we must have lim inf n-+m

+

~P'(o)< 0. <

Since o E (0,7r/2), we have o(c 1) E (0,7r/2) for E [-l,O]. sing([ 1 ) 2 0. So I(@("))5 o4?'(0), which leads to

+

This implies

7. Periodic Solutions in Multi-Species Modeb

263

lim inf I ( @ ( " ) )I a lim inf n-+w

n++w

+P'(o)< 0,

also a contradiction. 0 Finally, we are ready to state the proof of Theorem 2.5. Proof of Theorem 2.5. By the definition of K, we know that it is

a closed, bounded, convex set of infinite dimension in the Banach space C([-1,0],R2). F as defined in (2.31) is completely continuous by Lemma 2.3. By Lemma 2.4, Theorem 2.9.4, and Lemma 2.5, we see that 0 is an ejective fixed point of F. We conclude by Theorem 2.9.2 that F has a fixed point in K\{O), which by Lemma 2.3 corresponds to a nonconstant periodic solution of system (2.13) with period a(@) > 2. This completes n the proof. U Corollary 2.1. Let Y , a , b, and d be positive constant3 and 7 = a b ( l c)-', v = d(l c)-l in

+ +

+

(2.34)

Assume further that

where a0 E (O,n/2) is such that tanuo = u ; ' ( u -

-).bc (1

+ c)2

T h e n system (2.34) has at least one noncondant positive periodic solution with period larger than 2.

7.3. Periodic Solutions in Periodic Systems We consider first the following nonautonomous delay system

& ( t )= zi(t)Gi(t,zl(t),..., Z n ( t ) , Z l ( t - 7 ( t ) ) , . . .

,Zn(t

- ~ ( t ) ) ) , (3.1)

where i = 1,... ,n, z = (21,... , ~ n )E RT = { Z E Rn : 2 0). Denote Int R; = { Z E R; : Z , > 0). We assume that Gi is continuously differentiable and

Delay Digeredial Equations

264

(Hi) and

> 0 for j # i, > O , i , j , k = 1, ... ,n.

aci(t,z,,...,zn,yl,...,Yn) azj aCi(t,zl,..., Z n , Y 1 , . . . ,yn) aYk

is continuously differentiable, nonnegative, and bounded above by 7'. For x,y E R", z 2 y means x i 2 y;, i = 1,... ,n, and x > y means xi > yi, i = 1, ... ,n. If +,$ E C, we write $ 5 $ (+< 4) if the indicated inequality holds pointwise, with the preceding partial ordering on R". I n the rest of this section, we always assume that +(O) 2 0, 4(0) > 0. It is easy t o see that, for any given 4 E C with +(6) 2 0, +(O) > 0, there exists a E (0, co) and a unique solution x ( t ) = x ( t ;4) of Eq. (3.1) on [-T*, a), which remains positive for all t E [0, a). (Hl) guarantees that system (3.1) generates monotone flows. We also assume that (H2) there is a p = (p1, ... ,pn) E I n t R f such that, for t E R, i = 1, * 3 12, G;(t,P I , . . . 3 pn, P I * p n ) < 0; G i ( t , X x l , . . , X x n , X y l , . - YXyn) 2 XGi(t,xl,... , x n , Y l , * . . 7 (H3) yn) for X E (0,1], where i = 1,. . . ,n; (H4) G i ( t , x l , . . . ,xn, y 1 , . . . , y n ) is uniformly continuous with respect to ( x i , ... , x n , y i ,... ,yn) and Gi(t W , X I , . . . , x n , y 1 , . - . , y n ) = G i ( t , x l , . . . , z ~ , Y .~. ,,y n ) for some w > 0 and i = 1,. . . ,n. The material of this section is adopted from Tang and Kuang (1992b), where one can find the detailed proofs of the following lemmas and theorems. Lemma 3.1. Suppose that system (3.1) satisfies (Hl)-(H4). T h e n the following hold ( i ) For any q E Int RY, there ezists M ( q ) E Int R; such that, for any 4 E C with 0 5 4 5 7 o n [-7*,0], one has 0 5 x ( t , + ) 5 M ( 7 ) for all t 2 0; (ii) There ezists a A E Int R; such that f o r any a E R;, there is a constant T = T ( a ) > 0 such that, for any 4 E C with 0 5 5 a o n [ - T * , 01, one has 0 5 ~ ( 4) t , 5 A for all t 2 T ( a ) . If (H3) is replaced by the following assumption: (H3)* G i ( t , f i ( X ) x 1 , . . . , f n ( X ) x n , f i ( X ) Y 1 , - . ., f n ( X ) y n ) 2 gi(X) x G i ( t , x l , . . . ,xn, y1,. . . ,yn) for X E [0,1] and i = 1,. . . ,n, where fi,gi : [O,1] + [O,1] satisfy f i ( O ) = 0, gi(O) = 0, fj(1) = gi(1) = 1, and f i , g i are nondecreasing i = 1,. . , n ; then Lemma 3.1 is still true. We need the following result from Horn (1970) to establish the existence of an w-periodic solution. Lemma 3.2 (Horn's fixed point theorem). Let SOc S1 c 5'2 be convez subsets of the Banach space X , with So and S2 compact and S1 open relative to S2. Let P : Sz -t X be a continuous mapping such that, f o r T(t)

.-

-

)

+

.

+

.

7. Periodic Solutions

in Multi-Species Models

265

some integer m > 0, (a) Pi(S1) c S2, 1 Ij Im - 1, and (b) Pj(S1) G So, m I j 5 2m - 1. Then P has a fixed point in So. Consider now the periodic system

-.- ,zn(t),zl(t - T(t)), . . . ,zn(t - ~ ( t ) ) ) ,(3.2) where 2 = (xi,. . . ,Z n ) E R;. We assume that Fi(t, 21,. . . ,xn, y i , . . . ,Yn) k i ( t ) = zi(t)Fi(t,Z(t),

is continuously differentiable in its variables, and there is an w that F,(t

+ w,

XI

7 .

..

7

-

xn, ~ 1 *, *

7

Yn) =

Fi(t, $1 7 *

-

* 7

-

xn, ~ 1* ,

7

> 0 such

Yn)

for i = 1 , . . . ,n; ~ ( t is) also a continuously differentiable w-periodic - - ~ ( t )Assume . function, and ~ ( t2) 0 for t E R. We denote T * = maxo 0, liminft-_,+oox ( t , 4) 2 6, where liminft-r+w x ( t , 4) = (liminft++ooxl(t,q5),. . . ,liminft++w xn(t,$)); (A3) For any It' > 0, there exists an L(It') > 0 such that, for 11411 I It', IF(t,$)I= Ebl IJ'i(t,4)1 I L(It') for t E R ; (A4) There are G , ( t , q ( t ) , . . ,zn(t),xl(t - ~ ( t ) .).,. ,z n ( t - ~ ( t ) ) ) such that

.

F , ( t , x l ( t ) , * * ,xn(t),xI(t * - ~ ( t ) ) , . ,xn(t ** -~ ( t ) ) ) IGi(t,zl(t),. , x ~ ( t ) , x l ( t - ~ ( t ) ) , . .?. ~ n ( t - ~ ( t ) ) )

...

for i = 1 , . . . ,n , where Gi(t,xl(t), ,x n ( t ) , q ( t - ~ ( t ) .).,. ,zn(t - T(t))) (i = 1 , . . . , n ) satisfies the assumptions (Hl)-(H4).

Note that condition (A2) is in fact the definition of uniform persistence (Section 8.2). Lemma 3.3. Suppose that system (3.2) satisfies (Al)-(A4), and G; (i = 1 , . . . , n ) satisfies (Hl)-(H4). Then there are 8, A 0 E Int R; such that, for any 771,772 E Int R$ with 71 5 772 and any compact subset S of C that satisfies S c {4(0) E C : 771 I 4(0) 5 q2}, there exists a constant T = T(q1,772, S) > 0 such that, for any 4 E S , one has

A 2 x(t,$) 2 8

f o r all t 2 T.

266

Delay Differeniial Equaiions

With the help of the preceding lemmas, we can show the following.

Theorem 3.1. Suppose that the system (3.2) satisfies (Al)-(A4) and has n o positive steady states. T h e n the system (3.2) has a nonconstant positive w-periodic solution. The following assumption is less restrictive than (H2). (H2)* There exist positive w-periodic functions Bl(t),. . . ,Bn(t)and p = ( p i,... , p n ) E IntR; such that, for t E R a n d i = 1,... ,n,

The following theorem is slightly more general than Theorem 3.1. Suppose that system (3.2) satisfies (Al)-(A4), and G i ( t , x l ( t ),... , x n ( t ) , x i ( t - 7 ( t ) ) ,... , x n ( t - 7 ( t ) ) )(i = 1 ,... , n ) satisfies the assumptions ( H l ) , (H2)*, (H3)*, and (H4). T h e n (3.2) has a nonconstant positive w-periodic solution provided that it has no positive steady states. Let y ( t , + ) = ( y l ( t ) ,... , y n ( t ) ) > 0 be a solution of system (3.2), and, for i,j = 1 ,... ,n,

Theorem 3.2.

Assume that z ( t , 4) = ( z l ( t ) ., . . , x n ( t ) ) is another solution of system (3.1), and denote

n

We make the following assumptions on system (3.2):

7. Periodic Solutions in Multi-Species Models

267

( B l ) There exist positive constants p i , . . . ,pn, q1,.

C P i I b i j ( t ) J- qj(1 - 2 ( t ) )I0,

j = 1,.

.. ,qn such that

-

972.

i=l

(B2) For any r) E Int R;, there is an M ( p ) E Int R; such that, for any q5 E C with 0 Id(0) Ir) on [ - ~ * , 0 ] ,one has

z(t,+)IM ( q )

for t 2 0.

Theorem 3.3. If system (3.2) satisfies ( B l ) and (B2), then it has at most one positive w-periodic solution z ( t , 4 ) . If z ( t ,4) is the positive w periodic solution, then it is globally asymptotically stable with respect to c+= {d(e) E c : 4(e) 2 o , ~ ( o )> 01. If we replace ( B l ) by the following assumption: (Bl)* There exist positive constants p i , . . . ,p,, 91,. . . ,qn and positive w-periodic functions Bl(t),. . . ,B,(t) such that, for j = 1 , . .. ,n, n

+

+ qj c 0,

CpiBj(t)Iaij(t)I Pjajj(t)Bj(t) i=l n

C p i B j ( t - T(t))Ibij(t)l- qj(1 - T ' ( t ) ) I0; i=l

then we have the following. Theorem 3.4. If system (3.2) satisfies (Bl)*, (B2), then it has at most one positive w-periodic solution. If x ( t , 4) is the positive w-periodic solution of (3.2), then it is globally asymptotically stable with respect to

c+.

Now we apply the preceding results to some well-known systems. We consider first the following n dimensional delayed Lotka-Volterra system:

For system (3.3), we suppose that

(3.3)

268

Delay Differeniial Equaiions

(i) a i j ( t ) ,c i ( t ) ,bij(t) ( i , j = 1,. . . ,n ) , and T ( t ) are continuously differentiable w-periodic functions, c i ( t ) > 0, ai,(t) > 0, a ; j ( t ) 2 0, bij(t) 2 0 , and ~ ( t2) 0 for t E R; (ii) there exist positive differentiable w-periodic functions B l ( t ) ,. .., Bn(t) and p = ( p i , . . . , p n ) E Int R; such that, for t E R, i = 1,. . , ,n ,

i#i

Theorem 3.5.

If system (3.3) satisfies (i) and (ii), and it has no positive

steady states, then it has a nonconstant periodic solution. To prove Theorem 3.5, we need the following lemma. Lemma 3.4. If system (3.3) satisfies (i), then there ezists a y > 0 such that, for 9 E C with 9 2 y 1, we have f o r t 2 0. z(t,(b) 2 y * 1 Proof. By assumption (i), system (3.3) generates a monotone flow, and, for any 9 E C with 4(0) > 0 on [ - ~ * , 0 ] one , has z ( t , + ) > 0 for t 2 0. Thus, we have i ; ( t )2 Z i ( t ) [ C i ( t ) - a i i ( t ) ~ i ( t ) ] , i = I , . . . , n , and the conclusion follows. 0 Proof of Theorem 5.1. Let F i ( t , X l ( t ) , . . . , z l ( t ) , z l ( t - ~ ( t ) ) , - . ,. z n ( t - ~ ( t ) ) )

then it is easy to check that the assumptions (Al)-(A4) and (Hl),(H2)*, (H3)*, (H4)in Theorem 3.2 are all satisfied, and the conclusion thus follows.

0 For the uniqueness of the nonconstant periodic solution, we need the following condition. (iii) There exists p’ = ( p i , . . . ,pL) E Int R; and q = ( q i , . . . ,qn) E Int Rn such that, for j = 1,. . . ,n, n

+ qj < 0,

Cp:Bj(t)Iaij(t)I -p>ajj(t)Bj(t) i= 1 n

C p:Bj(t’ - T(t))lbij(t)l- qj(1 - ~ ‘ ( t 5) )0. i= 1

7. Periodic Solutions in Multi-Species Models

269

Theorem 3.6. A s ~ u m ethat system (3.3) satisfies (i)-(iii) and has no positive steady states. T h e n it has a unique and globally asymptotically stable nonconstant w-periodic solution with respect to C+. The following corollary improves the main result of Freedman and Wu (1992). Corollary 3.1. Consider the following equation:

where a ( t ) ,b(t),c ( t ) , and 7 ( t ) are continuowly diflerentiable, w-periodic functions and a ( t ) > 0, b(t) > 0, c ( t ) 2 0, T ( t ) 2 0 for t E R. Suppose that ( i ) the equation a ( t )- b ( t ) K ( t ) + c ( t ) K ( t - T ( t ) ) = 0 has a positive, w-periodic, continuously differentiable solution I((t); (ii) b ( t ) > c(t)(lt'(t - 7 ( t ) ) / K ( t )for ) all t E [O,w]. T h e n Eq. (3.4) has a positive w-periodic solution Q(t). If, in addition, rnin0 m a x o g ~ w [ c ( t ) l~ (~t ( t ) ) / (l ~ ' ( t ) ) then ] , Q ( t ) i s global&asymptotically stable with respect to C+. Proof. Let B ( t ) = K ( t ) ; then, condition (ii) implies that there is a pl > 0 such that pl > ( a ( t ) lic(t)l/lc(t))/(b(t)Zc(t) - c(t)K(t- ~ ( t ) ) ) ; i.e., the condition (ii) in Theorem 3.5 is true. Hence, the existence of periodic solutions follows from Theorem 3.5. A simple calculation yields the uniqueness and globally asymptotic stability of the periodic solution.

+

n U

The main results of this section can also be applied to the following delayed periodic predator-prey system:

where a i ( t ) ,b;(t),c;(t) (i = 1,2), and d ( t ) are nonnegative continuously differentiable w-periodic functions. This was done in Tang and Kuang (1992b). 7.4. Remarks a n d Open Problems

As far as the global existence of periodic solutions in delayed populations systems is concerned, numerous models remain to be studied. The following are three simple looking models for which such problems remain open.

270

Delay Differeniial Equations

Open Problem 7.1. Obtain sufficient conditions for the global existence of periodic solutions in

assuming that it satisfies (Al)-(A3) in Section 2, and q(6) is nondecreasing such that q( -T,') - q( - T - ) = 1. Open Problem 7.2. Obtain sufficient conditions for the global existence of periodic solutions in k(t)= rH(t)[l - ~ (- T)/K] t -c r ~ ( t ) ~ ( t ) , (4.2) b(t)= -bP(t) B P ( t ) H ( t ) . The system (4.2) was introduced in May (1974), p. 103 as a predator-prey model. Open Problem 7.3. Obtain sufficient conditions for the global existence of periodic solutions in the following delayed Lotka-Volterra competition system: k ( t ) = z(t)[rl - u z ( t - T ) - by(t)], (4.3) Y(t) = Y(t)[T2 - 4 t ) - dY(t)l, where all parameters are positive constants. It is of course very interesting to study these problems for higher dimensional (systems with more than two equations) delayed systems. The only existing work of this kind was given by Mahaffy (1980), who established sufficient conditions for the global existence of periodic solutions in certain protein synthesis models. Also, it is interesting to ask the preceding questions when delays are state dependent. Another effective method of obtaining sufficient conditions for the existence of periodic solutions in delayed periodic systems was described in Cushing (1977), p. 180. Roughly, one shows the existence of a positive w-periodic solution by proving that a bifurcation of such solutions takes place as the average of the w-periodic intrinsic growth rate of one of the species, say the ith, passes a critical value. Such bifurcation originates from a positive w-periodic solution of the subsystem obtained by deleting the ith species. It is also interesting to study some delayed almost periodic systems. An example of such effort was documented in Gopalsamy and He (1992) for the almost periodic delayed competition system

+

fil(t) = Nl(t"l(t) - U l l N l ( t - W )- a12(t)N2(t - 7(t))l, l;v,(t) = N2(t)[r2(t) - .2l(t)Nl(t - T ( t ) ) - .22(t)N2(t - T ( t ) > l ,

(4*4)

7. Periodic Solutions in Mulli-Species Models

271

where 7, r;, a;j ( i , j = 1,2) are continuous positive almost periodic functions. They were able to derive sufficient conditions for the existence of a globally attractive almost periodic solution in system (4.4). Even if 7, r i , aij ( i , j = 1,2) are all positive constants, it was shown by Shibata and Saito (1980) that system (4.4) can produce chaotic solutions.

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Permanence 8.1. Introduction Given a system of equations (ODEs, DDEs, PDEs, or difference equations) modeling some population growth, we often begin our analysis of it by studying the local stabilities of its steady states, followed by the global stability analysis, and then the existence and stability (often very difficult) of periodic solutions. These analyses can be very formidable for most systems of nonlinear nonautonomous DDEs with several discrete delays or distributed delays, or systems with more than two populations. While these qualitative properties of a given system are interesting and important mathematically as well as biologically, a more basic and important biological question t o ask is whether or not those involved populations will be alive and well in the long run. This turns out t o be a very interesting mathematical question in its own right, which is often referred to as permanence of the system (or populations). Other relevant and frequently used notions include persistence (slightly weaker than permanence), permanent coezistence, cooperativeneq ecological coezistence (see Butler et al. (1986), Hutson and Schmitt (1992)). Roughly speaking, we say a population z ( t ) is permanent if there exist two positive constants m and M , rn < M , such that, for large t (dependent on q),

rn 5 z ( t ) 5 M . We say a system is permanent if all its populations are permanent. Most of the existing permanence results are established for autonomous systems of ODEs and make use of the dynamical system theory of finite dimensional systems; see Butler et al. (1986), Butler and Waltman (1986), and Hofbauer and Sigmund (1988) for details. Recently, permanence theory of nonautonomous and infinite dimensional systems (such as ODEs and PDEs) has also received much attention. An excellent survey of such activities is given by Hutson and Schmitt (1990). A general permanence theory for autonomous infinite dimensional systems is documented in Hale and Waltman (1989). See also Thieme (1991) for some relevant results. Specifically, permanence results for autonomous delay differential population models are documented in Burton and Hutson (1989) for 273

Delay Differential Equations

274

Lotka-Volterra-type systems with infinite delays; in Cao et al. (1992) for two species Kolmogorov-type systems with a single discrete delay; in Cao and Gard (1992) for delayed Kolmogorov-type systems with a single discrete delay; in Freedman and Ruan (1992) for general delayed systems; in Kuang and Tang (1992a) for nonautonomous Kolmogorov-type systems with general delays; in Kuang and Tang (1992b) for Lotka-Volterra-type systems with distributed delays; and in Wang and Ma (1991) for two dimensional Lotka-Volterra type predator-prey systems with discrete delays. The results in the first four papers exploit the dynamical system properties of the solution maps of the considered systems, while the last three make use of some comparison arguments and/or permanence (or Liapunov-type) functions. In the following section, we present the general persistence theory of Hale and Waltman (1989), which is very useful when dealing with general autonomous DDEs. In Section 3, we describe a powerful permanence theory for delayed autonomous Lotka-Volterra systems due to Burton and Hutson (1989). Section 4 deals with permanence theory of general two dimensional nonautonomous Kolmogorov-type delayed population models. Section 5 gives sharp permanence results for nonautonomous delayed Lotka-Volterra-type systems. As usual, we devote the last section to a discussion that contains some remarks and open problems. 8.2. Persistence in Infinite Dimensional Systems In this section, we present a general persistence theory for infinite

dimensional systems due to Hale and Waltman (1989). This theory can be applied to both systems of DDEs and reaction and diffusion equations, which frequently appear in the literature. Let X be a complete metric space with metric d , and suppose that T ( t ) : X -+ X , t >_ 0, is a CO-semigroup on X; that is, T ( 0 ) = I, T ( t + s ) = T ( t ) T ( s )for t , s >_ 0, and T ( t ) x is continuous in t , x. Recall that the positive orbit r + ( x ) through x is defined as r + ( x ) = U,>o{T(t)x},and its w-limit set is w ( x ) = C1 U t > , { T ( t ) x } . Also, recallihat a set B in X is said to be invariant if F(t)B = B for t 2 0. This implies, in particular, that there is a negative orbit through each point of B that belongs to B. In the following, we restrict the negative orbits to those remaining in an invariant set, say B, and denote the negative orbit through x in B by a g ( x ) . Accordingly, we define the a-limit set a ( x ) of x and o-limit set a ( A ) , A c X, in a similar manner, taking into account the possibility of multiple backward orbits. Sometimes, it is convenient to have the alpha limit set of a specific full orbit, ~ ( x through ) a point x, to be denoted by a,(+

n,>o

275

8. Permanence

A nonempty invariant subset M of X is called an isolated invariant set if it is the maximal invariant set of a neighborhood of itself. Such a neighborhood is called an isolating neighborhood. The stable (or attracting) set of a compact invariant set A is denoted by W s and is defined as

W s ( A )= {z : E X , ~ ( z #) 4,

W(Z)

C A},

and the unstable (or repelling) set, W" is defined as

W " ( A )= {z : z E X, there exists a backward orbit 7-(z) such that a Y y ( z#) 4, cq(z) C A } . The weakly stable and unstable sets are defined as follows:

W 3 A ) = {. : z E x,4.) W$(A)= { z : z E X , a(.)

# 4, 4.) n A # 41, # 4, a ( z )n A # $}.

A set A c X is said to be a global attractor if it is compact, invariant, and, for any bounded set B c X, h ( T ( t ) B ,A ) -+ 0 as t -+ +w, where

&(B,A ) = sup inf d ( y , z). y E B zEA

A global attractor is always a maximal compact invariant set. The semigroup T ( t ) is said to be asymptotically smooth if, for any bounded set B c X for which T ( t ) B c B for t 2 0, there exists a compact set K = K ( B ) such that h ( T ( t ) B , K ) + 0 as t -+ +w. In particular, w ( B ) c I(. T ( t )is said to be point dissipative in X if there is a bounded nonempty set B c X such that, for any z E X, there is a t o = to(z,B) such that T ( t ) zE B for t 2 t o . The following basic results on the existence of global attractors can be found in Hale (1988). Theorem 2.1. There i s a nonempty global attractor A an X , i f (i) T ( t ) i s asymptotically smooth, and (ii) T ( t ) i s point dissipative in X , and (iii) r + ( U ) is bounded in X i f U is bounded in X . Theorem 2.2. There as a nonempty global attractor A an X , i f (i) there is a to 2 0 such that T ( t ) is compact for t > to, and (ii) T ( t ) is point dissipative in X . In the following, we consider a particular system motivated by biological considerations. We will assume that X is the closure of our open set X o ;that is, X = X o u axo,where axo,assumed to be nonempty, is

276

Delay Differeniial Equations

the boundary of Xo. Also, we assume that the Co-semigroup T ( t ) on X satisfies T ( t ) : xo--t xo, T ( t ) : axo+ axo. (2.1) We denote To(t>= T(t)lxo,

Ta(t>= W)laxo.

(2.2)

Clearly, axois also a complete metric space with metric d. It is easy to see that if T ( t ) satisfies the conditions of Theorem 2.1 in X, then so does Ta in axo,and there is a global attractor Aa in axo. However, when T ( t ) satisfies conditions of Theorem 2.1 or Theorem 2.2, To(t)may not even have a maximal compact invariant set in Xo, since there may exist z E Xo such that ~ ( zn) # 4. It can also happen that ~ ( z n) dXo = q5 for all z E X o and there still does not exist a maximal compact invariant set in Xo, for example, the solutions of the simple Lotka-Volterra predator-prey model,

axo

k = X(U - by),

j( = y(-c

+ dz),

~ ( 0 >) 0 ,

y(0) > 0,

(2.3)

(where a , b, c, d are positive constants) are all periodic. The semigroup T ( t ) is said to be persistent if, for any z E Xo, liminft,, d ( T ( t ) z a, x o ) > 0; it is said to be uniformly persistent if there is an 77 > 0 such that, for any z E Xo, liminft,,d(T(t)z,dXO) 2 7. If T ( t ) is persistent, it may not be uniformly persistent, which can be seen from (2.3). To obtain an equivalent formulation of persistence in terms of attractors, it is convenient to introduce two more definitions. We say that U c Xo is strongly bounded in Xo if it is bounded in X and there is an 77 > 0 such that d ( z , d X o ) 2 77 for z E U . When we restrict T ( t ) to Xo, we may think of d X o as being part of "infinity." The "bounded" set in Xo should then be the strongly bounded sets in X . We say that T ( t ) is strongly point dissipative in Xo if there exists a strongly bounded set B c Xo such that, for any z E Xo, there is a to = t o ( z , B ) such that T ( t ) z E B for t 2 to. The following proposition is obvious. Proposition 2.1. Suppose that T ( t ) i s point dissipative in X. Then the following statements are true: (i) T ( t ) is persistent if and only if ~ ( z is) strongly bounded for each 2

E

xo.

(ii) T ( t ) i s uniformly persistent af and only af T ( t ) is strongly point dissipative in Xo. A0 is said to be a global attractor for T ( t ) in X o relative to strongly bounded sets if A0 c Xo is compact, invariant and S(T(t)U,Ao)+ 0 as

8. Permanence

277

t + +ca for all strongly bounded sets U c X o . The proof of the next theorem is given in Hale and Waltman (1988). Theorem 2.3. Suppose T ( t ) satisfies (2.1) and (i) there i s a t o 20 such that T ( t ) i s compact f o r t > to; and (ii) T ( t ) i s point dissipative i n X ; and (iii) T ( t ) i s uniformly persistent. Then there are global attractors A i n X and Aa i n and a global attractor A0 i n X o relative to strongly bounded sets. Furthermore, A = Ao U W U ( A a ) where , W"(Aa)= {z E A : c r ~ ( z C ) Aa}. For retarded delay differential systems with finite delays, (i) is almost always satisfied because of the smoothing property (see Theorem 2.2.8). (ii) is also generally true, but may need nontrivial arguments. The most difficult hypothesis to verify is, of course, (iii). In the following, we present an effective method for obtaining uniform persistence. We need some additional definitions. Let M , N be isolated invariant sets (not necessarily distinct). M is said to be chained to N , written as M + N , if there exists an ) W s ( N ) . A finite element 2, x 4 M U N , such that x E W U ( M n sequence M I ,M2,. . . ,Mk of isolated invariant sets will be called a chain if MI + M2 + . - .+ Mk ( M I + M I , if k = 1). The chain is called a cycle if Mk = M I . The particular invariant sets of interest are

axo,

Aa = U

~(z).

z€Aa

& is isolated if there exists a covering M = Mk of by pairwise disjoint, compact, isolated invariant sets M I ,M2, . . . ,Mk for Ta such that each Mi is also an isolated invariant set for T . M is then called an isolated Aa is also called acyclic if there exists some isolated covering Mi of Aa such that no subset of the M,s forms a cycle. Such an isolated covering is called acyclic. The following theorem is the main result of this section. Theorem 2.4. Suppose that T ( t ) satisfies (2.1) and that (i) there is a t o 2 0 such that T ( t ) is compact for t > t o ; and (ii) T ( t ) is point dissipative i n X ; and (iii) Aa i s isolated and has an acyclic covering M . Then T ( t ) i s uniformly persistent if and only i i f o r each Mi E M ,

YYGL1

Ws(Mi)n X O = 4.

(2.4)

278

Delay Differential Equations

The preceding theorem can be viewed as a corollary of the following: Theorem 2.5. Suppose that T ( t ) satbfies (2.1) and that (i) T ( t ) i s asymptotically smooth; and (ii) T ( t ) i s point dissipative i n X ; and (iii) r + ( U ) i s bounded i n X if U i s bounded i n X ; and (iv) Aa i s isolated and has an acyclic covering. Then the conclz~sionsof Theorem 2.4 are valid. For the proof of Theorem 2.5, we need the following result. Lemma 2.1. Assume that T ( t ) satisfies (i)-(iii) of Theorem 2.5. Let 7 ; be a sequence of precompact semiorbits with w-limit sets w,. Suppose that M is a compact, isolated invariant set with wn n M = q5 f o r n large. If p , E w, i s such that d ( p , , M ) -+ 0, then there ezist ~ e q u e n c e s{q,}, { r , } , q,, r , E w,, and elements q E W S ( M ) \ M , r E W'(M)\M with limn-mqn = q and limn+oorn = r . The q and r can be found i n an arbitrarily small neighborhood of M . Proof. According to Theorem 2.1, T ( t ) has a global attractor A . The w-limit set of a semiorbit y+ consists of full orbits and belongs to A. Let Ii' be the set of nonempty compact subsets of R = C1 U z E ~ w ( x with ) Hausdorff metric p. Since A is compact, the subsets w, are uniformly bounded compact subsets of R. Therefore, there is a subsequence of w,, which we again label wn, such that p(w,,w) + 0, w E Ii'. Clearly, w is invariant. Let U be an isolating neighborhood of M and V an open set such that M C V C Cl(V) C U . Then, pn E V for large n. Since pn E W n , which is invariant, there is a full orbit through p,. Since p , E V, there is a corresponding Y n E dV, with T ( T n ) P , = Y n and T ( t ) p , E V for 0 > t > 7,. Since y , E R, we may select a convergent subsequence, say q,, such that limn+m q, = q E d V n w and, in particular, q 4 M . If (7,) were bounded, we could select a convergent subsequence T, + T , which, by continuity, makes T ( T ) q E M and hence T ( t ) q E M for t > T , and q E Ws(M)\M. We may thus assume that 7, --f -m. This has the consequence that r+(q)c V, which implies that q E W s ( M )by the definition of W S ( M ) . The proof for the r,s follows similarly, taking into account the possibility of multiple backward orbits. Note that, in this case, we need only one backward orbit to remain in V. This completes the proof of the 1emma.O Proof of Theorem 2.5. The necessity of (2.4) is clear. Suppose (2.4) holds and T ( t )is not uniformly persistent; then, there are two cases to be considered. There is a sequence of points p , with (i) either p , = r + ( t , ) for some orbit r+(z),x E X ; or (ii) p , E wn for some sequence of omega

279

8. Permanence

limit sets, such that d ( p , , b X o ) + 0 as n + 00. Choose a subsequence such that p, --+ qn and, if in case (ii), such that w, + w as in the proof of Lemma 2.1. Let R be W ( Z ) in the first case or w in the second case. Clearly, 7 + ( q ) c b X o and q E W t ( M ; )for some Mi, say for M i . By Lemma 2.1, there exists a point q1 E Ws(M1)n R and q1 4 M I . Since C? is invariant, there exists a full orbit 7(q1) through q1 that lies in R. ar(ql) exists, and, since a,(ql) is invariant, + ( p i ) n WG(Mj) # 4, for, say, j = 2. If ay(ql)C M2, then M2 is chained to Mi. We can then choose a new sequence of points pn either on the sets wn or on the orbit 7 + ( 5 ) whose distance from M2 tends to zero. By Lemma 2.1, we can choose q2 and repeat the previous argument. If a r ( p l ) is not a proper subset of M2, then we proceed essentially as in the proof of Butler and Waltman (1986), Theorem 3.1, to reach a contradiction of the no cycle condition. The proof then follows the general scheme of the proof of Theorem 3.1 in Butler and Waltman (1986), keeping in mind the two cases already mentioned, and taking care to use a full orbit when constructing the alpha limit sets needed 0 to chain sets. As an example, we consider the persistence question of the following simple delayed Lotka-Volterra type competition system:

i ( t )= r 1 2 ( t ) [ l - ~ ( t - 1 ) - p l Y ( ~ ) ] ,

i ( t ) = r2Y(t)[l-Y(t-l)-~22(t)],

(2.5) where r i , r 2 , p 1 , and p2 are positive constants. We will show in the following that if p1 and p2 are less than 1, then the system is uniformly persistent. Here, the initial function space X is the positive cone of C[O, 11 x C[O,I]. Clearly, X is invariant. We denote ( ~ ( t y(t)) ) , = ( z ( t ,4, $), y ( t , 4, $)) the solution of (2.5) with ( 2 0 , YO) = (4, $1. Hence, 0 E [-I, 01. T(t)(4,$ ) ( 6 ) = ( 4 t + 6 , 4, $1, Y ( t + 6 , 4, $I), Then, according to Theorem 2.2.8, T ( t )is completely continuous for t > 1.

Also, it is easy to show that limsupz(t) 5 e r l , t-oo

limsupy(t) 5 er2, t+oo

which implies that T ( t )is point dissipative. There are three steady states on the boundary of X:

Eo = (O,O),

El = (l,O),

E2 = ( 0 , l ) .

The origin is clearly unstable. We linearize around E2 and obtain the characteristic equation

+

[A - rl(1 - ~ I ) ] [rae-'] A = 0.

280

Delay Differential Equations

There is a unique positive eigenvalue X = rl(1 - PI), which corresponds to an eigenvector of the form (1, O)T. The eigenvalues of X r2e-A = 0 all have negative real parts, these corresponding to an eigenvector of the form (0, l)T,and hence corresponding to solutions that remain in the part of the boundary of X given by z ( t ) 0. Thus, the stable set of E2 does not intersect the interior of X. A similar conclusion applies to El for p2 < 1. Since Eo is unstable, & is just the union of the three steady states. Taking the Mi to be these steady states, there are no cycles in the boundary of X. Also, these steady states are isolated invariant sets by the linear theory in Section 2.8. Hence, (2.5) is uniformly persistent, and there is a global attractor in the interior of X. More interesting applications of Theorem 2.4 are given in Cao et al. (1991).

+

8.3. Permanence in Autonomous Loth-Volterra-Type Systems Consider the following system of autonomous equations with infinite delay:

for t 2 0, i = 1,... , n , t : R + R; with initial value of z given on (--00, 01. The system (3.1) includes the following Lotka-Volterra system as a special case:

where a;, bij are constants and I(;, are L1 functions. Since we are dealing with infinite delay systems, we tentatively adopt the UC, space as initial function space. Hence, we assume that there is a continuous function g : (-m,O] --t [ l , ~ )satisfying , (g1)-(g3) in Section 2.7 and such that if 'p : (-w,O] --t R" is continuous and bounded in UC,, then

Jo

(HI) Iqi(-e,'p(e)lg(e) de c -ca We denote in the following

00,

i = 1,.. . ,n.

W )= [g(e)11'2,

(3.3)

and 11 * Ilh the norm in UCh. Definition 3.1. Solutions of (3.1) are said to be h-uniformly bounded (h-U.B-1 in U C l e { V : CP E UCg, CP = ( P I , . . . , c ~ n ) , vi(e) 2 0,

28 1

8. Permanence

0 E (-w,O]} if, for each B1 > 0, there exists B2 > 0 such that cp E UC;, llcpllh I B1, t 2 0 imply that Iz(t,cp)( < 8 2 , where I I is a norm on R”. They are said to be h-uniformly ultimately bounded for bound B (h-U.U.B.) if, for each B3 > 0, there exists T > 0 such that cp E UC;, llcpllh I B2, t 2 T imply that Iz(4cp)l < B. It is known (Burton, 1985, p. 248) that U.B. and U.U.B. are independent properties. We assume in the following that (H2) for each cp E UC;, there is a unique solution z(cp) on [0, w), and solutions of (3.1) are h-U.B. and h-U.U.B. From the form of (3.1) and as a consequence of (H2), we see that for each B1 2 B there exists B4 0 (and, indeed, a smallest B4) such that Q E UC;, llcpllh I B1, t 2 0, imply that Ij.(t,cp)I I 8 4 . We may therefore define p : [0, w ) ---t [0, w) by p(B1) = B4 if B1 2 B and p(B1) = p(B) if B1 I B. We now formally define the initial function space Y for (3.1): y = {Q :Q E

ucl, llQllh <

- dS2)1

00, Iv(.l)

I P(llcpllh>h(-l~ll- IS2l)lSl - S211. The Lipschitz condition is required in order to prove certain compactness results; moreover, the Lipschitz “constant” must grow in order for zt(p) to enter a prescribed compact set for large t , independent of the magnitude Of

llPllh.

We will also assume that (H3) If cp E U C l , then any solution x(t,cp) defined for t 2 0 satisfies x i ( t , 9) 2 0 for O I t < 00, i = 1 , . . . ,n. Note that (H3) is automatically true for the system (3.2). Definition 3.2. For the B of h-U.U.B., let

Definition 3.3. System (3.1) is said to be permanently coezistent if there exist m, M with 0 < m 5 M < 00 such that, given any initial function Q E Y with cpi(0) > 0, i = 1 , . . . ,n, there exists t o = to(cp) such that m 5 x;(t,cp) 5 M ,

t

to,

1

5i In.

Note the slight difference between uniform persistence and permanent coexistence. The former requires only point dissipativity and is less restrictive on cp, and the latter depends on the specific initial function space Y.

282

Delay Diffemntial Equations

We also assume in the following that (H4) for each cp E Y , z(t,cp)exists on [O,m) and is continuous on [ 0 , 4 x y in II * 119. It is easy to see that (H4) holds automatically for (3.2). It is also easy to show that, under assumptions (Hl)-(H4), the set X is compact in the 11 norm. Define

-

S = { cp : cp E X , cpj(0) = 0 for at least one j E { 1,2,

Let p : R; ---t R+ be a $ : X\S -+ R by setting

C' function with

The function II, is continuous on X\S. be extended to X by putting $ ( u ) = lipjzf VEX\S

p-'(O)

. . . ,7 1 ) ) .

= aR;.

Define

If it is also bounded below, it may

$(v),

u

E S,

(3.5)

which yields a lower semicontinuous $ : X -+ R. The following result, due to Burton and Hutson (1989), is the main theorem in this section. Theorem 3.1. FOT(3.1), let (Hl)-(H4) hold, and assume that S and X\S are positively invariant. If there exists p with the preceding properties such that II, (defined b y (3.4)) is bounded on X\S, and if rt

f o r each u E w ( s ) , then (3.1) is permanently coesistent. In applying Theorem 3.1, the difficult part is that of finding function p (often referred to as the persistence function) such that (3.6) holds. Consider now the system (3.2). We define p : R; 4 R+ as n

p(2) =

nz y ,

2

= (21,.. . ,Zn),

(3.7)

i=l

where a; are strictly positive constants to be chosen. A straightforward calculation yields

8. Permanence

283

Kjjvj(0) do. Clearly, (3.8)in where F;(v)= a; - Cj"=, bijvj(0) - C;=, fact holds by continuity on the whole of X . Hence, we may define

and

n

~ ( z =)

C aiGi(z)i=l

(3.9)

The following result of Burton and Hutson (1989) is very helpful in establishing permanent coexistence in the system (3.2). Theorem 3.2. Assume ( H l ) and (H2) hold and there is a choice of a; such that p ( z * ) > 0 ( p is defined by (3.9)) at every steady state of (3.2) in S. Then (3.2) as permanently coesistent. In the rest of this section, we give two examples to illustrate the applications of the two theorems. Consider first the following predator-prey system with a possibly saturating predator:

f k l ( t ) = zl(q[a - bZl(t) [ i 2 ( 1 ) = ~ ( 1 ) [- d + /

- czz(t)],

1 --oo

+

K ( s - t ) X xl(s) vx1(s) d s ] ,

(3.10)

where a , b, c, d, X are positive constants, v is a nonnegative constant, and I { ( - ) is also nonnegative such that

L

K ( u )du E L1(-00,0].

Hence, there is a g satisfying (g1)-(g3) in Section 2.7 such that PO

/

rs

/---oo g(s)

~

-m

K ( u )du ds < 00.

Again, we denote h ( s ) = (g(s))'/2. The proof of the following lemma can be found in Burton and Hutson

(1989). Lemma 3.1. Suppose that I{ : (-00,Ol + [ O , o o ) is continuous, J!-oo K ( u )du < 00 and m !J m J: K ( u )duds c 00. Then solutions of(3.10) are h-U.B. and h-U.U.B. in UC:. If the conditions of Lemma 3.1 hold, then (3.1) is permanently coexistent whenever

Theorem 3.3.

a(bX

+ av)-'

J"

-m

K ( 0 )d0 > d.

(3.11)

Delay Diffeerential Equations

284

Proof. First, it is easy to check that (Hl)-(H4) hold for (3.10). The objective is to use Theorem 3.1. We note that it is sufficient for (3.6) if there is a p for which 1c, is such that 1c, > 0 for all u E w(S). If 4 0 ) = 0, then we have q ( t ) + a / b as t + 00; if q ( 0 ) = 0, then 22(t) + 0 as t 00. Thus, w ( S ) is {(0,O)}U{(a/b,O)}. Choose now p(z) = z;1lz;’, where z = (z1,zz). Then --$

Let a1 = 1; then, $((O,O)) > 0 for small a 2 < d-l. With (3.11), 1c,((u/b,O)) 0 is always positive. The conclusion now follows from Theorem 3.1. Consider next the following delayed three species competition system:

II-

= q ( t )[l - q ( t )-

$1

t

K ( s - t)52(S) ds -

-00

t

[ L

= I2(t) 1 -

$2

J

L

L ( s - t ) X l ( S ) ds - Z Z ( t ) -

t

1,

J

t

L ( s - t)23(S) ds] ,

K ( s - t)23(S) d s ] ,

--m

t

1

Lrn

L ( s - t)22(S) ds - 23(t) , (312) where Ii‘ and L are nonnegative functions in L1(-00,0]. Let k = J!OO K ( 0 )d0 and m = J!OO L ( 0 ) do. Theorem 3.4. If 0 < k < 1 < m and k + m < 2, then system (3.12) is permanently coezistent. Proof. We intend to apply Theorem 3.2. First, we verify (Hl)-(H2). Since k and m are finite, it is possible to find a function g : (-00,0] -, [l,m), satisfying (g1)-(g3) of Section 2.7 such that J!,g(e)K(0>d0 < 00 and J!OO g ( e ) L ( 0 )d0 < 00, and hence (Hl) is true. The question of h-U.B. and h-U.U.B. in UC: is trivial. Let V(21,22, 23) = 2; + 2; + 23. We have 23

[

= 23(t) 1 -

K ( s - t ) l l ( S ) ds -

dV/dt 5 - 2 ( 4

+ xi + 2:) + 2(2; + + 2:) 5 -1 2;

+ +

2; 2: 2 30. This implies (H2). In order to apply Theorem 3.2, we need to examine the steady states, which include (O,O,O), (l,O,O), (O,l,O), (O,O,l). From Theorem 3.2, it is enough to show that there exists a; > 0, i = 1,2,3, such that

if

2:

+ +

+

a2(1 - m ) Q3(1 - k) > 0; a2 ‘33 > 0; a1(1 - m ) a 2 ( 1 - k) ~ l (-l k) + a 3 ( 1 - m) > 0;

Q1

+

> 0.

(3.13)

8. Permanence

285

One can show that (3.13) is satisfied if (1 - k)3 > (rn - 1)3, which holds if k rn < 2. This completes the proof. 0

+

8.4. Permanence in Nonautonomous Systems In this and the next sections, we consider the permanence aspect of nonautonomous systems. This section deals with Kolmogorov-type two species models of the form

where f and g are continuously differentiable with respect to their arguments. The initial function space is C+ = C([-T,O],R:),where T is a positive constant. Since (4.1) is nonautonomous, general permanence t h e ories given in the previous sections no longer apply. To overcome these difficulties, we construct a set of proper autonomous ordinary differential systems whose solutions can serve as lower or upper bounds for the delayed system (4.1) in certain regions. Such comparison arguments may be extended to higher dimensional systems (multi-species interaction models). It is easy to show solutions of (4.1), with initial functions taken from C+, are nonnegative, and if s(0) > 0, y(0) > 0, then x ( t ) > 0 and y(t) > 0 in the maximum interval of existence. The material of this section is adapted from Kuang and Tang (1992a). Competition Assumptions: The following assumptions on f and g render (4.1) a competition model. ( C l f ) There exist positive constants 61 = 61(f), 62 = 62(f), 1i'i = Kl(f), Ii'2 = 1i'2(f), with Ii'1 < K 2 such that, for all t 2 0,

f(t,.t,O)

> 61,

+

[o, K1],

+

OO),

for ~ ( t0 ) E

8E

[-T,o],

(4.2)

and for ~ ( te) E

f ( t , x t , ~< ) -b2,

8E

[-4 (4.3)

( c 2 f ) f ( t , x t , O ) 2 f(t,xt,yt) for all t 2 0, xt, yt E C+, and there exist positive constants 63 = 63(f) and k = k ( f ) such that, for all t 2 0, xt E

c+,

f ( t , x t , ~ t< ) -63,

for y ( t

+e)

Also, for each pair (xo,yo), 20 > 0, yo f(t,.t,yt)

L --1(xo,yo),

for

E[ k , ~ ) , E

[-.$I.

(4.4)

> 0, there exists l(z0,yo) such that

114l 520, IlYtll 5 YO.

(4.5)

286

Delay Diffemniial Equations

(C3f) There exists a positive constant M = M ( f ) such that f ( t , 21, ~ t )I M , for t 2 0, zt, yt E C+. (4.6) (Clf) assumes that the growth rate for small population in the absence of competitors is positive, while there is a self-crowding effect creating a negative growth rate at high population levels, even in the absence of competition. (C2j) states that the existence of y is negative to the growth of z and, when the population of y is large, the growth rate of z becomes negative. The relation (4.5) assumes that the negative fluctuation effect on the growth rate of x is limited for limited population densities of species z and y, while (C3f) assumes that there is an upper bound for the growth rate of z. In (Clf)-(C3f), we replace f by g and denote the resulting assumptions as (C1g)-(C3g), respectively. When the system (4.1) satisfies (C1f)(C3j) and (Clg)-(C3g), we call it a competition system. Predator-Prey Assumptions: The following assumptions on f and g make (4.1\ a predatar-yey made\. (Plj The same as (CIjJ. (P2) The same as (C2f). (P3) The same as (C3j). (P4) g(t, zt, 0) 2 g(t, Z t , Yt), t > 0, z t , Yt E c+. (P5) There is a continuous function m ( z ) for z 2 0 with m(0) < 0 such that, for each zo > 0, t 2 0, 21, yt E C+,g(t,xt,yt) 5 m(zo) when 11~t115 xo. (P4) assumes the existence of a self-crowding effect for species y. (P5) says that the growth rate of y is uniformly limited by the prey density z. When prey is absent, predator density y decreases. Remark 4.1. Clearly, (C3f) implies that, for t 2 0, z ( t ) is bounded by z ( 0 ) e M t ,as long as y(t) exists. For the competition model, the same conclusion applies to y(t); therefore, (z(t),y(t)) exists for all t 2 0. For predator-prey models, (P5) implies that y(t) exists so long as z ( t ) exists, and hence ( ~ ( t y(t)) ) , exists for all t 2 0. A semidynamical system is said to be dissipative if there is a bounded globally attracting set. In particular, we say a delayed population model (with H ( t , ut)continuously differentiable with respect to (t, u t ) )

q t ) = H(t,ut),

E C([--7,O],R;), is dissipative if there is a positive constant M, independent of 210

(4.7) uo, and a

t o = to(u0) such that

0

5 u ( t ) = u(t,uo) 5 M ,

t 2 to.

(4.8)

8. Permanence

287

Following Hutson and Schmitt (1990), we define the following. Definition 4.1. The system (4.7) is called permanent if it is dissipative and uniformly persistent. Theorem 4.1. Let F ( t , z t ) = f ( t , q , O ) , and 2et f satisfy (Clf)-(C3f). Then k(t) = F(t,zr), zo E c+, s(0) > 0 i s permanent.

Proof. In other words, we need to show that there are two positive constants (independent of so) 91 and 92, 91 < 92, such that, for large t (depending on so), z ( t ) E [91,92]. We show first that we can choose q 2 = K2eM'. From (4.3) in ( C l f ) , we see that, for any t o > 0, there is a t l > to such that s(t1)5 K2; otherwise, z ( t ) tends to zero, contradicting (4.3). If, for large t , s ( t ) > 92, then there exist t l , t 2 , t 2 > t l > 0, such that Z(tl)

=K2,

k(t2) L 0,

4 2 ) = 92,

z ( t ) E [1-2,921,

for t E [ t l , t z ] .

Since ( C ~ J holds, ) we have, for t 2 t i ,

which implies that

t2

- t i 2 T. However, by (4.3), we then have f ( t z , s t , , O ) < -62

< 0,

and hence s ' ( t 2 ) < 0, a contradiction. Let 91 = K1 exp(-l(q2 l ) ~ )where , Z(s) G l ( z ,0) as defined in (C2f). We show now that, for large t , z ( t ) > 91. By (4.2), we see that, for any to > 0, there is a t i > t o such that z(t1) 2 11'1; otherwise, ~ ( twill ) tend to infinity, contradicting (4.2). Assume that, for t 2 t > T , z ( t ) 5 92 1. If, for large t , ~ ( t<) 71, then there are t l and t 2 , t 2 > t l > 1 T, such that

+

+

4tl)

= 1-17

k(t2) 50,

4 t 2 ) = 91,

fort E [tl,t2]. Since (4.5) holds, we have, for t E

z ( t )E [91,1-11,

[tl,t2],

z ( t >> s ( t l ) e - ~ ( ~ 2 + l ) ( t - ~ l ) . Hence, t2

- t l L 7,

+

288

Delay Difemntial Equations

and by (4.2), we must have i ( t 2 ) > 0, a contradiction. 0 We assume first that system (4.1) satisfies (Clf)-(C3j), (C1g)-(C3g). For convenience, we denote 61f = 61(f),and similarly for 62,, K l f , 61g, 62g, M f , M,, l f , I,, etc. We have the following dissipativity result for the system (4.1). Lemma 4.1. L e t vZ = K2 fex p (Mp ), qy = K2 g e x p (Mg ~ ),a n d ( s ( t ) ,y ( t ) ) be a solution of (4.1) s u c h t h a t z(0) > 0, y ( 0 ) > 0. T h e n limsups(t) t-+w

5 qz,

limsupy(t) t++w

5 qr.

(4.9)

Proof. We need only show that limsupt-r+w z ( t ) 5 qz; the case of 5 ly can be shown similarly. Indeed, the arguments limsupt,+,y(t) are very much like the first half of the proof of Theorem 4.1. If the conclusion is false, then there exist t l , t2, t 2 > t l > 0 , such that 4 t l ) = I{Zj, for t E

4 t 2 ) = 772,

[ t l , t 2 ] . From

qt2)

L 0,

Z(t)

E [1{2f,VZ],

(C3f), we have t2 - tl

> 7.

And (4.3) leads to i(t2) < 0, a contradiction. This proves the lemma. 0 Theorem 4.2. In s y s t e m (4.1), a s s u m e (Clj)-(C3,) a n d (C1g)-(C3g) hold. A s s u m e f u r t h e r t h a t (C) there is a positive constant 60 s u c h that, for all t 2 0, (i) f ( t , z t , y t ) > 60 for Ilstll 5 6 0 , llytll I V Y + 60; (ii) g(t,zt,yt) > 60 for 11xt11 I yz 60, llytll I 60, where 71, a n d qy are defined in L e m m a 4.1. T h e n s y s t e m (4.1) i s p e r m a n e n t . Proof. We adopt a similar approach as in the proof of Theorem 4.1. Let

+

iiz iiy

= 6oexP{-lf(Vz = 60exp{-lg(qz

+ 60,Vy + 6 0 ) T ) ,

+ 60,lly +

6O)T).

We prove here that liminf z ( t ) 2 t-+m

qz.

(4.10)

The proof of lirninft,+, y(t) 2 ijy is similar. From Lemma 4.1, we know that there exists t o > 0 such that, for t 2 t o ,

289

8. Permanence

If (4.10) is false, then there exist z(t1)

4 t 2 ) = qz,

= 60,

> tl > t o , such that

tl,t2, t2

L 0,

i(t2)

z ( t ) E [i?z,601,

for t E [ t l , t 2 ] . Clearly, (4.5) implies that t2

- tl 2

7.

However, by (i), we must have q t 2 ) = +2)f(t2,

z:a, YtJ

> 6 0 4 t 2 ) > 0,

0

a contradiction.

In the following, we apply the preceding theorem to the nonautonomous Lotka-Volterra-type competition system with distributed delays of the form

(4.12) where 0 5 r < +m, and p,, i = 1,2,3,4, are nondecreasing functions satisfying pi(O+) - / i i ( - T - ) = 1. a ( t ) ,b ( t ) ,c ( t ) ,k ( t ) ,h ( t ) , and f ( t ) are bounded positive continuous functions that are also bounded away from zero. For convenience, we assume that a ( t ) E [!!,a], k ( t ) E [Ic, LI,

b ( t ) E k761, h ( t ) E E,ill,

44 E [c,4, f(t) E [f,fl,

(4.13)

where a ,-b ,-c ,-k ,-h , and f are positive constants. It is easy to see that assumptions (Clf)-(C3,) and (C1g)-(c3g) are satisfied by the system (4.12). Applying Theorem 4.2 to (4.12), we have the following result. Corollary 4.1. Assume that the system (4.12) satisfie8 -

-k

a-

Er

cIe

>O

and

- a ar

I-h-e -b

>O.

(4.14)

Then (4.12) as permanent.

In the autonomous case, that is, when ii = a, b = b, ?i = c, = i, = &, and f = L, condition (4.14) reduces to the uniform persistence condition (4.4) in Cao et al. (1991). When, in addition, r = 0, i.e., when (4.12)

290

Delay Diffemniial Equaiions

reduces to an autonomous ordinary differential system, our permanence condition (4.14) becomes

which in fact is both necessary and sufficient for uniform persistence (Waltman, 1986). We assume now that the system (4.1) satisfies (Pl)-(P5). Note that we do not assume that g is strictly decreasing with respect to yt, which amounts to the so-called self-crowding effect. If such an effect exists, then boundedness of solutions of (4.1) is easy to obtain. This is the case for the work of Wang and Ma (1991). As we have mentioned earlier, it is easy to show that z ( t ) is bounded (see also the proof of Lemma 4.1). The next lemma shows that y(t) is also bounded. Lemma 4.2. Assume the system (4.1) satisfies (Pl)-(P5). Then there i s a positive constant q such that limsupy(t) 5 q t++w

for all solutions of (4.1).

Proof. Let q2 = K 2 f e x p ( M f ~ )then, ; a similar argument as in the proof of Lemma 4.1 shows that

limsupx(t) 5 9%

(4.15)

t++m

for all solutions of (4.1). Since ( C 2 f ) holds, there exist positive constants for all t 2 0, xt E C+, f(t,xt,yt) < -63,

for y(t

+ 8 ) 2 k,

63

and k such that,

8 E [-~,01.

From (P5), we see that there exists a positive constant p such that m(p) < 0, g(t,xt,yt)

t2

5 m(p) < 0,

O,Zt,Yt

E

c+, Ilxtll 5 P .

Let E E ( 0 , l ) be a constant. There exists a constant T solution z ( t ) of

i ( t )= - & x ( t ) , 0

I +o)

592

+

6,

satisfies z(t) 5 p

for t - t o 2 T .

> 0 such that the

291

8. Permanence

Denote

We claim that limsupy(t) 5 qy.

(4.17)

t++W

Assume in the following that (4.17)is false. Note that, for any t o > 0, there is a t > to such that y(t) < k. Otherwise, s ( t ) tends to zero and we must have, from (P5), Y(t) 5 Y(th(P), and, therefore, y(t) --t 0 as t -+ +oo, a contradiction. Since (4.15)holds, there is a t o = to(€)> T such that 11xt11 5 q2

+

6

for t 2 to.

The preceding arguments indicate that there exist Y(t1) = k

Y(t2) = 71g,

Y(t2)

2 0,

t2

Y(t) E

> ti 2 t o such that

Ik,Vy),

From (P5),we obtain

+

Y(t) Im(712 f)Y(t) = E Y ( t ) , Hence, we must have t2

t

L to.

- t l 2 T + T.

However, the solution z ( t ) of

q t ) = z(t)f(t, st, Yt),

11511

satisfies k(t)5 -6344 and, hence,

This implies that

5

1 1 ~ 1 ~ 1 1 p;

hence,

a contradiction. This proves the lemma.

II I712 + 6,

2

E

[W21.

Delay Differential Equations

292

In the following, we denote

f(t,2,Y)

f

f(t,2 , $1,

g ( t , 2,Y)

3

g ( t , %3),

where we denote ?,i E C+,?(O) = Z, $(e) = y, 8 E [-T,O]; that is, replacing in f ( t , + , + ) , g(t,q5,+) by constants z and y, respectively. Also, we assume that there exist continuously differentiable functions f ( ~ , y ) ,~ ( z , Y and ) , g ( i , y ) such that, for t 2 0,

+,+

I f(4 Y), &, Y) I g ( t , 9 ) I S(Z1 Y).

f(Z>Y)

$7

$1

Assume further that (P6) f is strictly decreasing with respect to both 2 and y, f(0,O) > 0, and lim f(x,O) = lim f(O,y) = -00; z-.+w

y++m

(P7) 9 and ij are increasing with respect to 2, but nonincreasing with respect to y; g(0,O) < 0. Let c E ( 0 , l ) be a constant. From the proof of Lemma 4.2, we see that there is a t o = t o ( ~ , c p l , c p 2 )such that, for t 2 t o , Z(t)

where

I 92 + 6 ,

Y(t) I

vy

+

€,

vy is defined as in (4.16). Clearly, -a

= -a(€)

f

f(qz

+

E, vy

+

€)

< 0.

(4.18)

And hence the solution ( q , yt) of (4.1) satisfies

k(t) 2

Z(t)f(vZ

-k

E,vy

+

f)

= --(YZ(t),

i 2 to

+ T,

which implies that z(t

+ e) 5 x(t)e-cre,

8E

[-T,o],

t 2 to

+ T.

t 2 to

+

Also, we have, from (C3f),

~ (+te) 2 x ( t ) e M e , Similarly, we have that, for t 2 t o

e E [-T, 01,

+ T,

T.

293

8. Permanence

and, hence, for t 2 t o

+ T,

For convenience, we denote

-

F(X,Y)

= f(ea'z,eP'y),

G(z, y ) = g(ea'z, e--"y),

G(z,y) = p(e-Mrz, eP'y).

In the following, we need to compare solutions of (4.1) with those of the following two nondelayed autonomous predator-prey systems:

qt)= u(t)F(u(t),v ( t ) ) , 6 ( t ) = v(t)G(u(t), v(t)),

and

4 0 ) > 0, v(0) > 0;

(4.19)

qt)= u ( t ) F ( u ( t )v, ( t ) ) ,

4 0 ) > 0, (4.20) v(0) > 0. 6 ( t )= v ( t ) q u ( t ) , v ( t ) ) , Further, we let z* > 0 be the unique solution of F ( s ,0) = 0. We now state

and prove the main result of this section. Theorem 4.3. Assume system (4.1) satisfies (Pl)-(P7), and (P8) G(z*,O) > 0. Then (4.1) i s permanent. Proof. We need only show that there is a 6 > 0, independent of initial data, such that liminf y(t)} > 6. min { lim inf t++w

&++a,

Denote 21 > 0 the unique solution of G(z,O) = 0. We claim that there is a Z 2 min{z*,z1} and a sequence {ti}Ellt i = t;(cpl,cp2) > T , t i + +OO i -+ +OO, such that z(t;) = Z. Otherwise, we have for all large t , y(t) < 0, and hence we must have limt,+,y(t) = 0, which leads to lim sup s ( t ) 2 z*, t-+a,

and (P8) thus implies that, for all large t , c ( t ) > 0, a contradiction. Recall that we have assumed that, for t 2 t o = t 0 ( ~ , ( ~ 1 , ( ~ 2 ) ,

z ( t >I 72

+

€1

Y(t) I7 y

+

6.

We assume in the following that 6 < 1. We have two cases to consider: (i) there is a ti 2 t o T such that z(t1) = Z, F ( z ( tl) ,y ( tl) ) 5 0; (ii) there is a ti 2 t o T such that z(t1) = Z,F (z (tl),y (tl)) > 0.

+ +

Delay Differential Equations

294

We define

1

2 = - min{i*, 11). 2 We denote

v(t) = v(t,2, q ) ,

G ( t ) = u ( t , i,7]),

(4.21)

the solution of (4.19) with initial value ( 2 , q ) ,where q 2 m a x { ~ z + l , q ~ + l ) and satisfies F ( 0 , q ) < 0. Then, from a standard phase plane analysis of system (4.19), we know that there is a 71 > 0 such that

t(q)= 0 6(t)< o

and t ( t ) < 0 for t E [0,71].

for t E [O,q), and

Denote

u ( t ) = g ( t ,G(71),

.(t) = .(t7G(71)7

v(71)),

the solution of (4.19) with initial value ( G ( q ) , G ( q ) ) . 7 2 > 0 such that 4(72) = 0. Denote

5(71)),

Then, there is a

rl = {(ii(t),v(t)): o I t I r2 =

{(g(t),g(t)) :0 It

r3 = {(.,.(72))

r4 =

I 721, : g(72) Iz I 71,

{(7,Y) : d 7 2 ) L Y L 71,

rs = {(z,7) : i 5

5 7}.

ue1

r; constitutes the boundary of a closed bounded region R = R ( i ) in the zy-plane (see Fig. 8.1). We claim that, for t 2 t l = t l ( c , p l , p 2 ) , ( z ( t ) , y ( t ) )E R. Note that R is independent of initial value Then

(91792).

We consider first case (i). Observe that, for t 2 t l , ( z ( t ) , y ( t ) )can never leave R through r 1 , since if ( z ( t ) , y ( t ) ) = (ii(l),G(l)) E I'l, then 0 > k ( t ) 2 @), i ( t ) I f(l) < 0, which implies that dy/dz > du/dv. Similarly, we see that ( z ( t ) , y ( t ) ) cannot leave R through r 2 . Clearly, ( z ( t ) , y ( t ) ) cannot cross r3, r4, and r5. This proves the claim for case (i). Consider now case (ii). Clearly, we can replace i in (4.21) by a sufficiently small G(0) and construct a new region R(u(0)) accordingly to envelope (z(tl),y(tl)). However, as our notation suggests, the region R(ii(0)) now depends on (z(tl),y(t1)). Observe that, since (P6) holds, there is a constant PO > 0 such that if max{z(t),y(t)) < PO,

295

8. Permanence

'4

Fig. 8.1. Illustration of the proof of Theorem 4.3.

then k ( t ) > 0. By choosing a sufficiently small constant p i , we claim that, for any solution ( ~ ( t y) (,t ) ) of (4.1), there must be a t* > 0 such that min{z(t*), !At*))> p1. This is because, for a sufficiently small constant p > 0, y ( t ) cannot always stay in a1 = {(Z,Y) : 0 IY IPI, since otherwise liminfl,+,z(t) will be larger than or very close to the value Z* and hence forces y(t) > 0 for large t , because of (PS). Also, we ) always stay in knew earlier that ~ ( tcannot 0 2

= {(Z,Y) : 0

I

2

5 PI

296

Delay Differrnlial Equations

for sufficiently small p. Moreover, for p < po, ( z ( t ) , y ( t ) ) can only travel from 0 2 to 01, and not the other way around. We stress here that p1 is independent of (cpl ,cp2). Finally, we conclude that we can choose sufficiently small i i ( O ) , 0 < E ( 0 ) < it, such that { ( P l , Y ) : p1

IY

I du { ( Z , P l )

: p1

5 2 5 7 ) c fl(ii(0)).

For this O ( E ( 0 ) ) (independent of (cpl,'p2)), we have that, for any solution ( z ( t ) , y ( t ) ) of (4.1)with z(0) > 0, y(0) > 0, there is a t* = t*(cpl,cp2), such that for t 2 t * . ( z ( t ) , y ( t ) )E 0 This proves the theorem. 0 Clearly, when applying the preceding theorem, one can take c = 0 in selecting CY, p, 7 in F, G, G. In the following, we apply the proof of Theorem 4.3 to the nonautonomous Lotka-Volterra Michaelis-Menten type predator-prey system

again all the coefficients are positive continuous functions bounded both above and away from zero, and 7;2 0, p j ( 0 ) are nondecreasing, p;(O+) p j ( - ~ - ) = 1, i = 1,2,... ,6. We assume, in addition to (4.13),n ( t ) E

[n,4. Corollary 4.2. I n (4.22), let a be defined b y (4.18) with z* = g$-le-arl.

6

= 0 and

~f

-he-lr4z* >

+

x* 1 7

(4.23)

then (4.22) as permanent. When 7 1 = 74 = 7 5 = 0, (4.23)becomes

(4.24)

8. Permanence

297

Note that the condition (4.24) does not depend on T ~ , T ~ , T ~or, cf ,, If, in addition, (4.22) is autonomous (i.e., Q = si = a , 4 = b = b, k = i= k, = n = n, = h = h, 5 = z: = c, f = f = f ) , (4.24) reduces to

oh > E(b+ nu), which is exactly the necessary and sufficient condition for the uniform persistence of

8.5. Permanence in Nonautonomous Lotka-Volterra-Type

Competition Systems In previous sections, we obtained some general permanence results for two species nonautonomous competition systems, which depend on the lengths of delays. In this section, by a rather different approach, we present some sharper results on the permanence of nonautonomous Lotka-Volterratype competition systems of the form

(5.1)

where aij (t)( t ) (i) k = 1,. , . , n ; j = 1,. . , , m ) are nonnegative bounded uii ( i )( t ) and r;(t) (z = 1,. . . ,n ) are positive continuous functions; Cy==, bounded continuous functions that are also bounded away from zero. r > 0 and pij (k)( 6 ) are nondecreasing functions satisfying pij (k)(0+) - pij (k)(-r-) = l , i , k = l , . . . , n , j = l , ...,rn. Denote

= sup{aIk)(t): t E [O,oo)}, = inf{r;(t) : t E [O,m)},

i , k = 1,2,.. . , n .

Delay Differential Equaiions

298

and

.

= ( ~ 1 , . . ,rn)T.

We make the following assumptions: ( H l ) A is nonsingular, and Ax = r has a positive solution, i.e., zo = A - ' r > 0 , where xo = (xy,... ,x:) E R",zp > 0, i = 1 , 2,... ,n. (H2) k t A-' = ( P , k ) n x n . Assume that P;i > 0, P,k 5 0, i , k = 1 , . . . , n , i # k. The assumption (H2) says that A-' is an M-matrix (Berman and Plemmons, 1979). Lemma 5.1 (Berman and Plemmons, 1979). Let B = (bij),xn be a n n x n matrix with nondiagonal elements being nonpositive. T h e n the following statements are equivalent. (i) B is a n M-matrix. (ii) Every real eigenvalue of B is positive. (iii) There exist positive numbers d; (i = 1 , . . . ,n ) such that n

dibii >

C dklbiklk=l

k#i

Lemma 5.2. Suppose that A 2 0 and (H2) is true. T h e n there exist positive numbers d; (i = 1 , . . . , n ) such that n

k=l k#i

Proof. Let ( A - l ) T be the transpose matrix of A-*; then ( A - l ) T has the same real eigenvalues as A-'. Since A-' is an M-matrix, its every real eigenvalue is positive. Hence, every real eigenvalue of ( A - l ) T is also positive. Also, the nondiagonal elements of ( A - l ) T are nonpositive, and 0 the conclusion follows from Lemma 5.1. It is easy to prove (since Cy=laIj'(t) is bounded away from zero) the following lemma. Lemma 5.3. There exists a n M1 > 1 such that z(t,q?J)5 M i 1; i.e., every component ofz(t,q?J) i s less than or equal to Mi for all t 2 T(+), f o r some T ( $ )2 0, where 1 = (1,. . . , I ) . Obviously, we have the following: Lemma 5.4. Let f ; ( t , z , q >= r i ( t ) - C ; = , aii ( k )( t ) J0- r z k ( t

-

+

O)dp$)(B). T h e n there exists a n i = 1, ... ,n, then I f ; ( t , ~ , ~ t5 ) 1M2

> 0 such that if for i = 1 , . . . ,n.

11zitll

5

for

8. Permanence

299

Now we are in a position to state and prove the main result of this section. Note that the following permanence result is independent of delay size. Theorem 5.1. Suppose that the system (5.1) satisfie3 (Hl)-(H2). Then it is permanent. Proof. Let

1 = 1,. . . ,n,

x a!:)(s - e ) q ( s ) ds dp$’(S)],

then,

L ’

J

I=1

Since Cr==, plir; = (A-lr), = I:, k = 1. Hence, we have

C;=l & ; c $ ~ )

= 0 if

k # 1 and equals 1 if

1 = 1 , 2 , . . . ,n. l q t ) 2 &(t)[xY - 441, (5.2) Clearly, there exists 0 < 60 < 1 such that, if 0 < q ( t ) < 60, then i.(t)

I = I,. . . ,n.

2 ,4 ~(t),

-

-

We choose 61 < 60 such that f;(t,61 1,61 1) > we let 62 = 61e-~z‘. Denote

and

(5.3)

ir; for i = 1,. .. ,n, and

300

Delay Differeniial Equations

M3

= max Nl. lslsn

Claim 1. There exist positive numbers y1,. . . ,yn such that, for x = (21,. * * ,5,) E s 2 , i = 1, ... ,n. x i 2 yi, By induction, we can show that the n surfaces S!'), 1 = 1 , . . . ,n, intersect. In the following, we let y = (yl,. . . ,yn) be the intersection point of these n surfaces. We then have yi > 0 (i = 1,. .. , n ) . Let x E S2; then, xdlP11 . . .&Pln > dlP11 . . .y$P/n, 1 = 1,. .. ,n. 1 n - Y1 Hence (multiplying the preceding inequalities for all 1 = 1,. . . ,n ) ,

From Lemma 5.2, we have C;"=;dl,BI, > 0, i = 1,. . . ,n. It follows that there is a t least one 1 E { I , . . . , n } such that x1 2 yl; without loss of generality, we suppose XI 2 y1. Also, we have

Since x1 2 y1 and

C;"=2 dl,Bll < 0, we have

Lemma 5.2 implies that Eb2dl& > 0, k = 2,. . . ,n, so there is at least one 1 E (2,. . . ,n } such that x1 y ~ . Repeating this procedure proves Claim 1. Let x ( t , 4) be a solution of (5.1) with $i(fl) 0, + i ( O ) > 0,i = 1,. . . ,n. By Lemma 5.3, we may suppose that x ( t , 4 ) 5 A41 1 for t 2 0. a

30 1

8. Permanence

In the following, we prove that z ( t , 4) will enter S 2 and stay in S 2 for all large t ; i.e., the system (5.1) is permanent. (i) First, we prove that if there is some to > 0 such that z(to, 4) E S 1 , then x ( t , $ ) E S 2 for all t 2 t o . Suppose this is not true; then, by the definition of 5'2, there is a tl > to and some lo such that x(tl,~$)E S,!:'. Then, we can find t l L t 3 > t 2 2 to and some 1 E { 1,. . . ,n} such that 4t3,$)

and, for

t3

> t > t2,

E

sl"',

s(t,$) E S2\&

4t2,

$1 E Sfl'

and

= h(t2),

a contradiction that proves (i). Next, we prove that (ii) there is some t o > 0 such that . ( t o , $ ) E S1. Suppose this is not true; then, x ( t , $ ) E S \ S I for t 1 0, where S = {x = ( ~ 1 , .. . ,xn) E R; : 0 5 x 5 M 1 1). Similar to the argument @)(a) and (b) in the proof of Theorem 6.2 in the next section, we prove the following two claims. Claim 2. There esists 6, = 65($) > 0 and some to 2 0 such that

-

.(t,$) > 65 * 1

f o r t 2 to.

302

Delay Differential Equations

C l a i m 3. If there is some t l t o and Jome 1 E {I,. . . ,n} such that T l ( t l , 4 ) L 61, then zr(t,4) > 62 for t > t l . Since it is impossible that, for all t 2 t o and all 1 E ( 1 , . . . ,n}, x l ( t , 4) < 61, we have the following. C l a i m 4. There ezistcr some t l 2 to and some 1 E {I,. . . , n } ~ u c hthat xr(t1, 4) L 61. Similar to the argument of Claim 1, we can prove the next claim. C l a i m 5 . Let z = ( 2 1 , . . . ,r,) be the intersection point o f n surfaces S$’). Then 0

< r 5 6 2 ’ 1 < 61 - 1 .

Claim 2 yields that 65 1 < x ( t , + ) < M I 1, so we know that Q ( t ) is bounded. Now, x ( t , $ ) E S\S1. Claims 3, 4, and 5 imply that there is some 10 E { 1, . . . ,n} and some t 2 2 0 such that

Hence, it follows from (5.3) that Q ( t ) --f

00,

a contradiction.

0

8.6. Permanence in N o n a u t o n o m o u s Lotka-Volterra-Type

Predator-Prey Systems In the first part of this section, we apply the technique developed in the previous section to the following Lotka-Volterra-type predator-prey system:

where a i j ( t ) , b i j ( t ) (i = 1,2, j = 1,. . . , m )are nonnegative bounded continuous functions, and r i ( t ) (i = 1,2) are positive bounded continuous

303

8. Permanence

functions that are also bounded away from zero. We denote

F, = sup{ri(t) : t E [O,oo)}, ai= inf {cq;(t) : t E [0, O O ) } , = ~ inf { r i ( t ) : t E [0, OO)}, i, k = 1,2. ~ r k= , sup{crk,(t) : t E [O,OO)},

Theorem 6.1. I n s y d e m (6.1), if

then system (6.1) i s permanent.

Proof: Let

Similar to the argument of Theorem 5.1, we can use these two functions to construct the boundary of S1 and S2 and obtain the conclusion. 0 Corollary 6.1. Suppose system (6.1) i s autonomous and

j=1

Then it is permanent.

j=1

304

Delay Diffemniial Equaiions

In the rest of this section, we consider the Lotka-Volterra MichaelisMenten type prey-predator model:

-

c ( t )P r q ( t ) 22(t

+ 6 )d~4(t,e)

+ .PrZ(t) 51 ( t + 6 ) d ~ l 2 ( t0), h ( t )P r 1 , t ) zl(t + 6 ) +l(t, 0) k2 = x 2 ( k ( t ) m(t)+ L ( t ) Z l ( t + 0) dCl3(t,fl) -f(t) J0 2 2 ( t + 0) dp4(t,e)) m(t)

(6.3)

+

--r4(t)

9

where a ( t ) , b ( t ) , c ( t ) ,k ( t ) ,m ( t ) ,and h ( t ) are bounded positive continuous functions that are also bounded away from zero, and f ( t ) are bounded nonnegative continuous functions; r,(t) are nonnegative and bounded, r, = supiER ~ , ( t )and ; p, are nondecreasing functions satisfying p ; ( t , O + ) p i ( t , - ~ , ( t ) - ) = 1, i = 1 , 2 , 3 , 4 . Denote

a = inf { a ( t ) : t E R}, -b = inf { b ( t ) : t E R}, h = inf { h ( t ): t E R},

p1

= b(iih-lear1

ii = sup{a(t) : t E R}, -

b = sup{b(t) : t E R}, m = sup{m(t) : t E R},

+ m),

a1 = ah-

Ip1.

Note the slight differences between (4.22) and (6.3). In the following, we always assume that a1 > 0. By Lemma 4.2, we have the following: L e m m a 6.1. There ezs'st an M1 > 1 and a T ( 4 )2 0 such that

q ( t , $ ) 5 ab-lear',

2 2 ( t , 4 ) 5 MI

f o r t 2 T(4).

Let

F ( t , z i , ~ 2 , x i t , 2 2 tand ) G ( t , z l , ~ , q t , ~ gbe t ) the

left expressions

of the first and second equations in the system (6.3), respectively. It is

easy to verify the following: L e m m a 6.2. There ezists an M2 ((22t(( 5 M I for some t , then lf(t,Z1,22,211,22t)l

> 0 such that, if

5 M2 and

11qt11

(g(t,21,22,2it,Qt)l

I Zb-'ear1, 5 M2.

0

305

8. Permanence

Choose Po > 0 such that -

C

Pok- -M1 > 0, m -

where = inf{lc(t) : t E R},m = inf{m(t) : t E R},c = sup{c(t) : t E R}. Let &(z1,22) = z 1 z p ; then,

'

Choose 1 > 6 > 0 small enough so that if then

z l ( t +8)

5 6 for 8 E [ - T I , 01,

1

Lp i ,

V l ( t )= -

l(6.3)

+

+

and if z l ( t 8) 5 6, z 2 ( t 8) 5 6 for 8 E [ - q , O ] , then k l ( t ) > 0. Since al > 0, we see that if z l ( t 8) 5 Tih-'e"l for 8 E [--7,0], then

m(t)+ Pn(t)

P1 h ( t )

zl(t

+ 8) d P 3 ( 4 0)

+

- b(t)h

Let h ( z 1 , q )= (z1)hzf';then, if we have

zl(t+O)

5 Tib-'e"l

for 8 E [ - T , O ] ,

306

Delay Differential Equations

Choose 6 > 7 then

and

> 0 small enough so that if z2(t + 8 ) 5 7 for 8 E

11

5

[-T,

01,

5 MI}

2e8',22 b

and 5'2 = { ( 2 1 , 1 2 ) : ( 2 1 , 2 2 )

and

51

a -

< -ear b

and

E R:,ziziSo

22

> 62,(zi)'@

> 72,

< MI}.

Then, S1 C S2 C Int R.: The sets S1 and S2 are shown in Fig. 8.2. Theorem 6.2. Suppose, an the system (6.3), a 1 > 0. Then it i s permanent. Proof. Let ( q ( t , + ) , q ( t , + ) ) be a solution of (6.3) with 4;(0) 2 0, +;(O) > 0, i = 1,2. By Lemma 6.1, we can suppose that 7i-

Zl(tl4)

If q , -

22(t,+)

I Ml,

for t L 0.

In the following, we prove that ~ ( 4) t , will enter S2 and stay in S2 for all large t , i.e., the system is uniformly persistent. We will prove first that the following is true. (i) If there is some t o 2 0 such that ( q ( t o , + ) , ~ ( t o , + ) ) E S1, then ( Z l ( t , +),Z2(t, 4)) E s 2 for all t 2 t o . Suppose this is not true; then, by the definition of S2, there is a t l > t o such that ( 2 l ( t l , 4 ) , 2 2 ( t l , $ ) ) E L26 or ( 2 1 ( t 1 7 4 ) , 2 2 ( t l , 4 ) ) E L2q. we the other case is just consider the case that ( q ( t 1 , + ) , ~ ( t 1 , + ) ) E similar. Without loss of generality, we suppose that ( q ( t , +), 2 2 ( t , 4)) E S 2 for t E [ t o , t l ) ; then, Vl(t1) 0. Now, we can find t 2 E [ t o , t l ) such

<

8. Permanence

307

ge-iir

a

6

-e

h

Fig. 8.2.

and so

aT,

308

Delay Differential Equations

a contradiction. Claim 1 is thus proved. Hence, by (6.4), we must have Vl(t1) > 0, a contradiction. This proves that (i) is true.

Next we prove that the following holds: (ii) There is some t o 2 0 such that (21(t0,4),z2(t0,4))E

s1.

Suppose this is not true; then, z ( t , + ) E S\S1 for t 2 0, where

s = {(z1,z2): (z1,z2)E R:,O I z1 5 iih-'ear1,0 I z2 I M ~ } . We divide our proof of (ii) into two parts. (a) Ftepeating the constructions for S1 and 5'2, we can find another two i such that Si c Sk C Int R: and (zi(to,4),22(to, 4)) E S; sets S: and S for some to 2 0. Now, we apply the conclusion of (i) to S i and Si and conclude that the solution (zl(t,$ ) , ~ ( 4)) t , is bounded away from zero for t 2 t o ; i.e., there is a 6 3 > 0 such that z ; ( t , 4 ) 2 6 3 for t 2 t o , i = 1,2. Hence, & ( t ) and b ( t )are bounded. (b) Claim 2. If there is some t o > 0 such that z1(to,4) 2 6, and (> 61) f o r (z1(+?5),z2(t,4)) E s\s1 fo. t L t o , then Z l ( t 7 4 ) 1

t 2 to.

Without If not, then there is some tl > t o such that zl(tl,+)= loss of generality, assume that 5 z l ( t , $ ) 5 6 for t E [to,tl] and k i ( t i , 4 ) L 0. Now,

L C(tl - to), so tl - t o

2 r. Notice that (see Fig. 8.2)

s\sl= [{(z1,z2)E ~ 2 :+z1 2 6e-ar,(zl)%pc ql} u {(z1,z2)E R : : z15 6e-ar,(z1)%$ < 71) u {(z1,z2)E ~ 2 :+z15 6e-ar,z1z;'0 < b1}] n S, and so the facts that q ( t , 4 ) 2 6e-ar and (zl(t,4),~2(t,4)) E S\S1 for t E [to,tl] yield that (zl(t,+),z2(t,+))E {(zl,z2) E R$ : z1 L be-", ( z ~ ) ~ z 0, a contradiction. Claim 2 is thus proved.

309

8. Permanence

Now, we have the following cases to consider. (I) z l ( t , + )< 6 for t 2 0. This is impossible; otherwise, by (6.4), it follows that & ( t ) + +oo as t + +oo, which contradicts the boundedness of V1. (11) There is some t o 2 0 with s l ( t o , 4) 1 6. In this case, Claim 2 implies that z l ( t , 4 ) 2 61 for t 2 t o . Since (z&,d),z2(t,4)) E S \ S l , (zl(t,4),32(t,4)) E { ( 2 1 , 2 2 ) E R : : z1 L 61, ( q ) l z $ < 71) n S for t 2 t o , which implies that z2(t, 4) 5 for t 2 t o . Then, (6.6) yields that V4t) + +m as t -i +oo, a contradiction to the boundedness of V2. 0 The theorem is thus proved. The following corollary is an immediate consequence of the preceding Theorem. Corollary 6.2. Suppose that t h e s y s t e m (6.3) i s autonomous, i e . , a ( t ) = a , b ( t ) = b, c ( t ) = c, k ( t ) = k, h ( t ) = h, and m ( t )= m. If

ah - k(aearl

+ mb)> 0,

(6.7)

t h e n s y s t e m (6.3) i s p e r m a n e n t . 8.7. Remarks a n d O p e n Problems In this chapter, we have presented several approaches for the study of permanence of delayed population models. Each has its advantages and disadvantages. Most of the results are delay dependent, with the exception of those for Lotka-Volterra-type systems with linear per capita growth rates. However, all the results become sharp when applied to related autonomous ODE systems. Nevertheless, we strongly feel that, for most delayed Lotka-Volterra-type systems, sharper permanence results may be independent of the lengths and types of delays involved. A plausible reason for this is that in order for a solution to be very close to the boundary, it must take a very long time. The closer it gets to the boundary, the longer it takes, and the longer it takes to go away from the boundary, and such time approaches infinity when the distance to the boundary tends to zero. When this happens, the lengths of delays compared with the long time needed for a solution to be very close to boundary becomes rather small and seems to be irrelevant. Such an argument is also valid for general delayed population models. In particular, we propose the following two open problems:

310

Delay Differenfial Equations

Open P r o b l e m 8.1. Assume that Pij are nondecreasing and Pij(0') = 1. Then

Pij(-r-)

is permanent if and only if

is permanent. O p e n P r o b l e m 8.2. The system (6.3) is permanent if

-a h - kb(ab-1

+

7%)

> 0.

Generally speaking, permanence results for n (> 2) species models are just as hard (if not more so) to obtain as the stability results for ( n - 1) species models. This is the reason that most of the models considered in this chapter are of the two or three species kind.

Neutral Delay Models 9.1. Models and Preliminaries The simplest and most widely adopted single species growth model is the well-known logistic equation

i ( t )= rx(t)[l - x(t)/K],

(1.1)

where r is called the intrinsic growth rate of the species z,I( is interpreted as the environment capacity for z,and r[l-z(t)/K] is the per capita growth rate of x at time t. Based on his investigation on laboratory populations of Daphnia magna, F. E. Smith (1963) argued that the per capita growth rate in (1.1) should be replaced by r[1 - ( z ( t )+p?(t))/K], since a growing population is likely to consume more ( p > 0) or less ( p < 0) food than a matured one, depending on individual species (for details, see Pielou (1977)). This leads to the equation

i ( t )= rx(t)[l - (x(t) + pi(t))/K].

(1.2)

We may think of x as a species grazing upon vegetation, which takes time T to recover. In this case it will be even more realistic to incorporate a single discrete delay 7 in the per capita growth rate, which results in the neutral delay logistic equation

i ( t )= rx(t)[l - (x(t - T ) + p?(t - T ) ) / K ] .

(1.3)

This equation was first introduced and investigated by Gopalsamy and Zhang (1988). Subsequently, it was studied by Freedman and Kuang (1991). The topics covered in these two papers are local asymptotic stability and oscillation of solutions. Later on, Kuang and Feldstein (1991) considered the problem of boundedness of solutions in the following more general nonautonomous neutral delay equation:

i ( t ) = r(t)x(t)(a(t)- x(t - 1) - c(t)&(t- 1)). 311

(1.4)

Delay Differential Equations

312

Gopalsamy et al. (1991) then studied a periodic version of (1.3) that takes the form z(t

- mr)+ c ( t ) k ( t - mr) K(t)

where ~ ( t )c ,( t ) , and K ( t ) are positive continuous periodic functions of period r , and m is a positive integer. They were able to obtain sufficient conditions for the local asymptotic stability in the C' space of periodic solutions of (1.5). Motivated by some of these works, Kuang (1992a) introduced and studied the following general neutral delay system, which may model a two species interaction in a closed environment:

where r;, i = 1,. . . , 5 , p and c are nonnegative constants, and a and b are real numbers. It is assumed that 0 d p ; ( s ) = 1, i = 1,. . . , 5 ; (Hl) p ; ( s ) is nondecreasing and J-, (H2) g(z) is continuously differentiable such that g ( 0 ) > 0, g'(z) < 0 for z 2 0, and g( 1) = 0; (H3) p ( z ) = zq(z) is continuously differentiable, p ( 0 ) = 0, p ' ( z ) 2 0, for z 2 0. Notice that an environment's carrying capacity can always be scaled to 1. Clearly, system (1.6) includes the following Gause-type neutral delay predator-prey system as a special case:

Here, a, p, u are all positive constants, and p ( z ) is the predator response function for the predator species y with respect to the prey species z. A slightly more general version of (1.7) was introduced and studied in Kuang (1991a), where the focus of the study was the local stability and oscillation analysis of the system (1.7).

9. Neutral Delay Models

313

The system (1.6) also contains the following neutral delay competition model as a special case:

{

k ( t ) = rlz(t)[l - k l z ( t ) - az(t - 7 1 ) ) - P i ( t - T O ) - q y ( t - Q)], Y(t) = r2y(t)[l - c2z(t - 7 3 ) - h ~ ( -t 74)I.

(1.8) Here, all parameters except p are assumed to be positive constants. We have included k l z ( t ) into the per capita growth rate of z ( t ) , which may reflect the possible instantaneous interference within species 2. The system (1.8) was first introduced and studied in Kuang (1991b). Again, the focus of that work was the local stability and oscillatory analysis of system (1.8). Sufficient conditions for solutions of (1.8) to be bounded can be found in Kuang (1992a). In the most recent paper, Kuang (1992f) obtained sufficient conditions for solutions of (1.6) tend to its unique positive steady state. In the following, we collect some known results for the system (1.6) that will be needed in the subsequent sections. Let 70 = max{T, : i = 1,2,. .. ,5} and R+ = { r : r 2 0). We always assume that the initial conditions for (1.6) are of the type z(5)

= 4l(S)

Y(S)

=h

( S )

E [-70,01,41(0) > 0, and 41 E C1([--70,0],R+), s E [-70,01,42(0) > 0, 2 0, and 4 2 E C1([--70,0],R+).

2 0,

(1.9)

Wesay (z(t),y(t)) is asolutionof (1.6) on [ - T o , w ) if both z ( t ) andy(t) are positive continuously differentiable functions and satisfy both the preceding initial conditions and the system (1.6). We note that the first equation of the system (1.6) can be rewritten as (provided z ( t ) > 0)

By letting u ( t ) = In z ( t ) ,we can see that (1.6) falls into the class of neutral systems considered in Hale (1977), Chapter 12. Thus, local existence, uniqueness, and continuous dependence of solutions are guaranteed by Hale (1977), Section 12.2. The following result, given in Kuang (1991d), ensures that the solution ( z ( t ) ,y(t)) of (1.6) and (1.9) is positive and exists for all t 2 0.

314

Delay Differential Equations

Proposition 1.1. Assume (Hl)-(H3) i n (1.6). Then the so2ution ( z ( t ) ,y(t)) of(l.6) and (1.9) is positive and esists for all t > 0 . Moreover, there ezist positive constants A and B (depending o n initial functions $1 and $2) such that z ( t )< AeBt. Proof. Suppose the maximal interval of existence for z ( t )and y ( t ) is [O,w). The positivity of z ( t ) and y(t) follows from the fact that the system is of Kolmogorov form. Assume first w < +oo. Then, Eq. (1.10) implies that, for t E [O,w),

which leads to

Thus, for t E [O,w), In z ( t ) < In z(0) Let A = z(0)exp (PJ!~ implies that, for t < w ,

+pJ

0

+i(s)dC12(3) -7-2

+i(s)dp2(s)),

z(t)<

+ g(O)t.

(1.11)

B = g(0); then, we see that (1.11)

AeBt.

(1.12)

Hence, limt,,z(t) 5 AeBw < +oo. Clearly, (1.12) holds for all t E [O,w). The second equation of (1.6) implies that, for t E [O,w), d -(lny(t)) dt

I

I4 + Iblp(AeBW),

which leads to lim y(t) 5 y(0) exp(w[laI

t+w

+ I ~ I P ( A ~ <~ ~+m. )I)

(1.13)

By the well-known continuation theorem (Theorem 12.2.4 in Hale (1977)), we conclude that w = +oo, proving the proposition. 0 As one may already be aware, many real systems are quite sensitive to sudden changes. This fact may suggest that proper mathematical models of the systems should consist of some neutral delay equations. Even though the delay lengths may be short, and the neutral terms relatively small, it is still necessary, for the sake of rigorousness, to justify that the neutral term effects are not important. Indeed, occasionally we may find that

315

9. Neuirul Delay Models

neutral term effects can be quite significant. This is largely due to the fact that neutral delay equations are not structurally stable in the sense that the introduction of neutral delay terms may destabilize an asymptotically stable equilibrium. For example, 3k = -x has a globally asymptotically stable trivial solution, while the same solution in x 2k(t - T ) = -z becomes unstable for any T > 0. This is because the corresponding characteristic equation of the neutral equation may have roots bifurcating from infinity, a phenomenon that cannot occur in retarded equations. This indicates that it is important to deal with neutral delay equations in real mat hematical models.

+

9.2. Boundedness of z ( t ) in the System (1.6) In the rest of this chapter, we denote 11$(s)11 = max{c$(s) : s E [-TO, 01) for any continuous function $(s) defined on [-70,0].The material of this section is adopted from Kuang (1992a). Our main object in this section is to obtain conditions under which z ( t ) will be bounded. To this end, we will analyze the system independent of y. We need the following lemma. Lemma 2.1. If 0 < cr 5 e - l , then there ezists a X(cr), 1 < X(a) I e, such that exp(aX(cr)) = X(a) and exp(crz) > z for z < X ( c r ) . Proof. If cr = e - l , then one easily sees that eaz 3 z for all z E R, and eaz = z if and only if z = e . Clearly, for z > 0, eaz is strictly increasing with respect to a. Thus, we see that, if 0 < cr < e-', euz will intersect with z at exactly two distinct points, say z1(cr) and z2(a),and z ~ ( Q<) 22(cr). Then, we must have 1 < q ( c r ) < e < z2(cr). Let X(a) = z l ( a ) ; then the 0 conclusion of the lemma holds. The following theorem is the main result of this section. Roughly speaking, it states that if the maximum value of initial function $1 is strictly less than 1 and TI and p are small, then z ( t ) is bounded above by a fixed constant (between 1 and e ) depending only on the values of TI and P.

Theorem 2.1. Assume cr = g ( O ) ~ 1 +p 5 e - l , and let X(a) be defined as i n Lemma 2.1. Assume further that the initial function $1 for satisfies 11$111 < 1. Then z ( t ) < X(cr) f o r t 2 -TO. Proof. Since z ( t ) > 0, y ( t ) > 0, we have

If z ( t ) is not bounded by X(cr), then there must exist t* > t o > 0 such that z ( t * )= X(cr), z(t0) = 1, 1 < z ( t ) < X(a) for t E ( t o , t * ) , and z ( t ) < X(a)

316

Delay Differential Equations

for t E TO,^*). It is easy to see that (2.1) implies that, for t 2 t o ,

Since ~ ( t>)0 for t

Hence, we have

2

-TO,

we have

9. Neutral Delay Models

317

Therefore, (2.2) yields

Clearly, this is a contradiction to the definition of X(a). Therefore, z ( t * ) must be less than X(a),and the theorem is proved. 0 Recall that a function z ( t ) (defined on [0,+m)) is said to be oscillatory about z* if there exists a sequence {in} + +oo as n + +oo such that z(tn) = z*, n = 1,. . . ; otherwise, it is said to be n o n o d l a t o r y about z*. If z ( t ) is unbounded, then the following theorem may roughly characterize its behavior. Theorem 2 . 2 . In the system (1.6), if z ( t ) i s unbounded, then z ( t ) is oscillatory about 1. Proof. Assume that ~ ( tis)unbounded and not oscillatory about 1; then, there must exist a t o > 0 such that, for t 2 t o , z ( t ) > 1. Let t l = to 71; then, we have, for t > t l ,

+

and

Hence, from (2.2), we see that, for t

>tl,

which implies that ~ ( tis) bounded. This is a contradiction to our assumption, and the theorem is thus proved. 0 9.3. Boundedness R e s u l t s for the System (1.6) We consider first the case when c > 0 in the system (1.6). This will include the neutral competition system (1.8) as a special case. When a < 0, b > 0, (1.6) can be used to model the predator-prey interaction with selfcrowding effect on the predator. The material of this section is taken from Kuang (1992a).

318

Delay Differential Equations

+

Theorem 3.1. Assume c > 0 , a = g ( 0 ) q p 5 e-', X(a) as defined as in Lemma 2.1, and 11q5111 < 1. Then solutions of (1.6) are bounded. Moreover, i f z ( t ) < X(a) for t 2 -TO, and (i) if a < 0, b > 0, and p = max(0, bp(X(cr)) a } , then

+

(ii) i f a > 0, b < 0, then

Proof. The assertion on ~ ( tfollows ) from Theorem 2.1. Thus, in case (i) we have

+

It is thus clear that if b p ( X ( a ) ) a I 0, then limt++,y(t) = 0. Assume b p ( X ( a ) ) a = /? > 0; then, (3.1)implies that C (t) I Py(t). Hence, for t 2 to, y(t) I y(to)eP(t-to), which leads to

+

Y(t0) L Y(+-

P(t-to).

Therefore,

since y(t)

> 0 for t 2 0. A substitution of (3.2) into (3.1) yields C(t) 5 Y(t)(P - ce-P"Y(t)).

(3.3)

Clearly, solutions of

C(t) = py(t)(l - cp-'e-PQ

Y(4)

(3.4)

319

9 . Neutral Delay Models

Therefore, solutions of (3.3) must satisfy lim sup y ( t )

5 c-lpepla

t++m

This proves case (i). For case (ii), we have

Thus, y(t) Therefore,

I

a y ( t ) , which implies

that y ( t

+ s)

2

y(t)e"" for s

I 0.

Y(t) I y ( t ) ( a - ce-"-Y(t)). Hence, by repeating the preceding argument, we can show that lim sup y ( t )

I c-lueaq.

t-+m

This completes the proof. In the rest of this section, we assume c = 0, u = -6 system (1.6) thus reduces to

0 < 0, b > 0. The

When delays are absent from the preceding system, it reduces to the socalled Gause-type predator-prey system. For (3.7), we have the following boundedness result. Theorem 3.2. I n (3.7), assume (Y = g ( 0 ) q p 5 e-', X(a) is defined as i n Lemma 2.1, and 11q5111 < 1. Let z > 0 be the unique solution of b p ( z ) = 6, 7 = min{q(z) : z E [o,X((Y)]}. Denote y = 7-"g(0) p X ( a ) ln(X(a)/z)], P = max{y(O),y), A = P ex~{[bp(X(a))- 6](73 74 1)). Then, for t 1 0, z ( t ) < X((Y), y ( t ) < A. Moreover, if X((Y) < 5, then limsupt+, y ( t ) = 0; and if X((Y) 2 5, then

+

+ + + +

) from Theorem 2.1. In the Proof. Again, the assertion on ~ ( tfollows following we assume y ( t ) is not bounded by A. Clearly, in this case 5 must

320

Delay Differential Equations

be less than A(a), since if 3 > X(a), then y(t) < 0; which implies that Y(t) I Y(0) < A. The first equation in (3.7) gives us, for t 2 t o ,

which leads to

The second equation of (3.7) implies that

Y(t) < [WW) - 6lY(t), which leads to

t L to. (3.9) Y(t) < Y(to)exP{[bP(X(a)) - 6l(t - t o ) ) , Since y(t) is not bounded by A, there must be t 2 > t l 2 0 such that y(t1) = i j , y(t2) = A, and y(t) L ij for t E [ t l , t 2 ] . From (3.9), we see that t 5 t2; then, t2 - tl > 73 74 1. In (3.8), we let t o = t l 73 and t o 1 I y ( ~ s) 2 ij for r E [to,t], s E [-r3,0]. Thus,

+

4t)

+ +

+

+

I X(a) exp{g(O)(t - t o ) + P X ( a ) l exp{-7ij(t - t o ) ) 5 X(a) exp{g(O)(t - to) + P+)) exp{-[g(O) + P V O ) + ln(A(a)/5)l(t - t o ) ) < A(a)exp{- ln(X(a)/?)) exp{-[pX(a) h ( A ( a ) / ~ ) ] ( t t o - 1)) = %exp{-[pX(a) + ln(X(a)/%)](t - t o - 1)).

+

(3.10)

Hence, for t E [ t o have

+ l,t2], ~ ( t5) 2. From the second equation of (3.7), we

(3.11)

9. Neuiml Delay Models

321

Clearly, -6+bJ!!, p(z(t+s))dp4(s) < bp(X(a))-6. For t1+73+74+1 5 t 5 t 2 , - s + b l ~ , p ( s ( t + s ) ) d ~ r ( s ) 5 0. Therefore, fortl+73+74+1 5 t 5 t 2 ,

+ 74 + 1)) = A,

~ ( t <) Gexp{[bp(X(a))- b ] ( ~ i

a contradiction to the assumption that y(t2) = A. This proves that Y(t) < A. Next we prove the second part of the theorem. Clearly, if X(a) < 5 , then y(t) < 0, and limsupt++m y(t) = 0. Assume > now that X(a) 2 5, and there is a y(t) such that limsup,,+,y{t) yexp{[bp(X(a)) - 6](~3 74 l)]}. We claim that there exists a t > 0 such that y(i) = j j . Otherwise, for large t , y(t) > 5. From (3.10), we see z ( t ) < 5 for large t ; indeed, in this case limt-r+ooz(t) = 0. This clearly implies that limt-r+m y(t) = 0, a contradiction. Obviously, for any T > 0, there is a 2 2 T such that y(2) 2 jjexp{lbp(X(a)) - 6](73 74 1)). Thus, for such a large 2, we can find a 2, 2 < t , such that y(t) = ij and y(_t) 2 ij for t E [i 21.,Now, we can repeat the previous argument (by letting t = t l , 2 = t 2 ) to derive a contradiction. The proof is thus complete. 0

+ +

+ +

9.4. Convergence in Single Population Models, I The material of this section is taken from Kuang (1992a). In this section, we restrict attention to the following neutral delay single population model, which results from (1.6) by taking y(t) G 0;

We always assume that the initial function of (4.1) satisfies the requirements stated in the first part of (1.9). Clearly, (4.1) has exactly two steady states. They are s ( t ) 0 and ~ ( tE) 1. The variational equation of (4.1) about s ( t ) = 0 is k ( t ) = g(O)s(t). Thus, we see s ( t ) = 0 is always unstable. In fact, we have the following stronger result: Proposition 4.1. Let ~ ( tbe) the solution of (4.1). Then limsupz(t) 2 1. t++m

Proof. Otherwise, there are 0 < 6 < 1, T > 0 such that, for t 2 T , z ( t ) 5 1 - 6. Then, for t > t o 2 T + 71 + 72,

Delay Differential Equations

322

and

Therefore,

2 ~ (to )ex P{ g (l- E ) ( t - t o ) - (1 - E

M ,

which implies that lim x(t) = +m,

t++m

a contradiction to the assumption that x(t) 5 1 - E for t 2 T. This proves the proposition. 0 In order to prove our global stability result, we need the following simple lemma. L e m m a 4.1. For 0 < r < 1, there is a strictly decreasing function h ( r ) such that h ( r ) = exp{r(h(r)-l)}, lim,,o+ h ( r ) = +m, limr+l- h ( r ) = 1, and 2 > exp{r(z - I)} for x E (l,h(r)). Pr oof . Clearly, for 0 < r < 1, er(’-l) always intersects with x at 2 = 1. As (d/dz)(er(z-l))Iz,l = r < (dx/dx)l,=l = 1, we see that er(‘-l) will intersect with x at another point, say h(r). Clearly, this h ( r ) has all those properties described in the lemma. 0 As in Theorem 2.1, we denote = 9(0)71

+ P,

and define X(a) as in Lemma 2.1. Assume

Q

(4.2)

5 e-’;

we define

G ( a )= max{ [g’(z)I : x E [0, X((Y)]}.

(4.3)

We denote T = max{r1,72}.

Now, we are ready to state and prove the main result of this section. T h e o r e m 4.1. In (4.1), assume 0 < (Y < e-l, and (i) X((Y)P < 1;

323

9. Neulml Delay Models

+

(ii) G(a)n 2p Ih-'(X(a)), where h-'(.) is the inverse function o f h ( r ) defined i n Lemma 4.1. Then limt-.r+ooz ( t , 4) = 1, provided that its initial function +(s) satisfies +(s) E C'([-.i.,O],R+), d(0) > 0, and 11411 = max{l+(s)l : s E [-T,O]} < 1. Proof. By Theorem 2.1, we know that z ( t ) < X(a),where 1 < X(a) Ie , and a G g(0)q p 5 e-l. Denote

+

v = limsup Iz(t)- 11; t-++oo

(4.4)

then, 0 5 v 5 e - 1. In the following we assume v > 0. We claim that z ( t ) must be oscillatory about 1. Otherwise, there is a T > 0 such that, for t 2 T , z ( t ) > 1 or z ( t ) < 1. We assume fii-st that z ( t ) > 1 for t 2 T. Let 1 2 T + 71 + 7 2 be such that

I&(?)[

= max{ Ik(t

+ s)l : s E [ - ~ 2 , 0 ] } .

If ;(I) 2 0, we have

and

which implies that (since X ( a ) p < 1)

If k(2) < 0, we have k(t)

+ pz(2)

and

which imdies that

J0

k(t

-T!

+ s) d p 2 ( s ) 5 (1 - X(a)p)k(t),

324

Delay Differeniial Equations

Hence, for all t L T

+71 +

72,

This, together with the assumptions v that

lo t

t++w lim

ma,{

g(z(7

+ s))

> 0 and z ( t ) > 1 for t 2 T , implies :s E

[-Ti,

01)

= -m.

Thus,

Therefore, we have, for any t o 2 T ,

since

This clearly contradicts the assumption that z ( t ) > 1 for t 2 T . The case of z ( t ) < 1 for t 2 T can be dealt with similarly. This proves the claim. Now, since z ( t ) is oscillatory about 1, we see that there is a sequence {Ti},i = 1, ..., such that 0 < Ti < < T; < T;+1 < * . * , lim;++,T; = +m, and z ( q ) = 1. Denote u = limsupz(t), !-+W

w = liminfz(t). !-+w

+

+

Then, either u = 1 v or w = 1 - v. Assume first that u = 1 v. Then, for any 0 < E < v, there is an i ( ~ > ) 1 such that, for t 2 Ti, 1 - v - E < z ( t ) < u E . Clearly, there is a t* = t * ( ~ 2 ) T; 7 such that z ( t * ) > u - E , and z ( t * )is the maximum in [Tj,Tj+l]for some j 2 i.

+

+

9. Neutral Delay Models

325

and

Thus, in both cases we have

< exp{ [ G ( a ) n + 2p](u + 6 - 1)). Since (4.6) is true for all 0

< 6 < v , by

I eXP{[G(a)Ti

(4.6)

letting c + 0 we obtain

+ 2p](u- 1)).

(4.7)

326

Delay Diffemniial Equalions

This is a contradiction to assumption (ii), since if G ( a ) q +2p then, for 1 < z 5 X(a), z

5 h-l(X(a)),

> exp{[G(a)q + 2p](z - 1)).

+

This indicates that u # 1 v if v > 0. Now we assume w = 1 - v. Then, a similar argument as before yields that, for any 0 < c < v, 1- v

+

6

> exp{-[G(a)q

+ 2 p ] ( v + c)},

which implies that 1 - v 2 exp{-[G(a)q

Thus,

+

+ 2pIv).

w L eXP{[G(+l 2pl(w - 1)). This is impossible for w < 1 and G(a)q 2 p 5 h -' (A (a )), since h-l(X(cr)) < 1 by Lemma 4.1. This proves that w # 1 - v if v > 0. Hence, v must be zero, proving the theorem. 0 In particular, for the equation

+

where T > 0, we have the following result. Corollary 4.1. In (4.8), let a = T T ~ p. Assume a < e-', A ( a ) p < 1, and T T ~ 2p 5 h-'(A(a)). Then, the conclusion of Theorem 4.1 as valid for (4.8). Proof. We note in this case g(x) = ~ (-x). 1 Thus, g(0) = T and G ( a ) = T . The rest follows from Theorem 4.1. 0 Remark 4.1. It should be mentioned here that assumption (ii) in Theorem 4.1 can be replaced by (ii)' G ( a ) q 2p < (e - l)-', which is more restrictive, but easy to verify. This is because 1 < X(a) < e for 0 < a < e-'; thus, by Lemma 4.1, we have

+

+

+

h - ' ( ~ ( a )>) h-'(e) = ( e - I)-'. Remark 4.2. As mentioned in the introduction, our results are rather limited and primitive. This can be seen by taking p = 0, i.e., when (4.1) is reduced to (4.9)

9. Neutral Delay Models

327

For (4.9), it is known that if Gq 5 1, where G is any upper bound of lg'(z)l, z 2 0, then z 1 is globally asymptotically stable with respect to initial function 4, 4 E C([-?,O],R+), +(O) > 0. This is clearly much sharper than the conclusion implied by Theorem 4.1. 9.5. Convergence in the System (1.6) The material of this and the next two sections is taken from Kuang (1992f). In order to state our main results in this section, we need some notation. For system (1.6), denote

9(A) = max{Is'(f)l : 2 E [O,AI}, = min{ls'(4l : 2 E [O, All, q(A) = max{ lq'(z)I : I E [O, A]}.

&v

(5.1) (5.2) (5.3)

We note that g(A(cr)) = G(a),where G(a) is defined in (4.3). In addition to (Hl)-(H3), we assume, throughout the rest of this section, that (H4) there is a unique positive steady state E(z*,y*)in (1.6). Clearly, (H4) implies that 9(I*)- q(z*)y*= 0,

a

+ bz*q(z*)- cy* = 0.

Denote further that ~ ( z , y ) ( t )= max{(z(s) - I * ) ~(y(s) ,

max{t - 270, -70)

- y*l2 : 5 s 5 t } , t 2 0,

X ( t ) = max{li(s)l : -70 5 s 5 t } , Y ( t )= max{ly(s)l : -70 5 s 5 t } .

(5.6) (5.7) (5.8)

The following lemmas are needed to establish our main theorems. Lemma 5.1. Assume that the solution (x(t),y(t)) of (1.6) with initial function 4 = (q5l(t),&(t)),t E [-70,0]satisfies (i) there are positive constants u,w such that Iz(t) - z*I 5 u, Iy(t) - y*l 5 w, t 2 -70; and p(u z*) < 1; (ii) X ( 0 ) 5 D ( u z*)Q, where Q = max{u, w}, and

+

+

D ( 2 ) = (1 - pz)-'.k(.) Then, f o r t 2 0, lk(t)l 5 D ( u

+ q(0) + q(z)(. + Y*)].

+ I*)&, and

328

Delay Differential Equations

Proof. From (i), we have Iz(t)l Iz* have

0

=

4, -

(Y(t

+

+ u, I y ( t ) l Iy* + v, t 2

0

3) - Y*) dCl5(3)

+ b /-

P’(@ 74

We

-TO.

+ s ) ) ( z ( t+ 4

z*) dCl4(3),

where p ( z ) = x q ( x ) , and ( ( t + s ) is the value determined by the mean value theorem. It is easy to see that max{p’(s) : o I2 Ix*

+ u} Iq ( 0 ) + 2( + u)q(z* + u ) . 2*

Hence,

Ii(t)l I(Y* +

+ Ibl(q(0) + 2(2* + .)+*

+ 4lI/VC.,

Y>(t).

This proves the second part of the theorem. Assume there is a t o > 0 such that

Ik(to)l > O(u and Ik(t)l < Ij.(to)l, for t

< t o . Clearly,

+ z*)Q

(5.9)

9. Neutral Delay Models

329

Hence,

Clearly, (5.10) and (5.11) together imply that li(t0)l

I [ l - p(u

+ 2 * ) ] - ' ( u + t*)

0 This proves the lemma. Lemma 5.2. Assume that (z(t),y(t))is the solution o f ( 1 . 6 ) with initial = ( + l ( t ) , + 2 ( t ) ) , t E [-70,0], such that v(z,~)(o) < v2 and function p(x* v) < 1 . Assume further that (i) I(v) g(x* + v) - [ q ( O ) + ~ ( z * v ) ( t * y* v)] > rn(v), where m(v) b(2* v)Ti p]D(2* V); (ii) c > Ibl[q(O) 2q(z* + v)(z* v)] 2c75n(v), where n ( v ) = (Y* .)[c Ibl(q(0) 2q(z* v)(2* 4)l; (iii) X ( 0 ) I D ( v z*)v. Then ~(z,y)(t) < v2 for t 2 0. Proof. If the lemma is not true, then there is a t o > 0 such that V(z,y)(to)= v2, V ( x , y ) ( t o ) 2 0, and V(~,y)(t)< v2 for t < t o . Thus,

+

+

+

+

+

+

+ + +

+

+

+ +

+ +

+

Assume first that

V(2,y)(to) Then, clearly

= ( X ( t 0 ) - 2 * ) 2 = v2.

330

for

Delay Differential Equations

-70

5 s I 0.

Also,

where we have used the fact that Ig(z(t0

+ s)) - g(z(t0))l I s(z* + v)lz(to +

3)

- z(t0)1,

+ s) - z(t0)I = lk(()ls I D(z* + v ) v q , since s < 71, and Ii(t)l I D(z* + v)v from Lemma 5.1. By assumption (i), we arrive at

and Iz(t0

I 2z(to){-+4v2

V(z,y)(to)

+ vm(v)v} < 0,

since ( ~ ( t o ) - z * = ) ~v2. This contradicts our assumption that V ( z , y ) ( t ~ 2 ) 0. Hence, V(z,y)(to) # (z (t 0 )- x * ) ~ . Assume now that V(z, y)(to) = (y(t0) - Y * ) ~ .We have V ( z ,Y)(tO) = 2Y(tO)(Y(tO) - Y*){ -c(y(to) - Y*)

+

to) - to +

+ b J O-

[z(to

d~ds)

+ s)q(z(to + 4)- z*q(z*)I

dPP(S)}.

r4

(5.13)

This leads to

V ( z ,y)(to) I 2y(to){ -[c-Ibl(q(O)+2~(.*+.)(.*+v))]v2+2c75.0 where we used the fact that Y(t0) Idto) - y(to

+ .)I

< n(v)v from Lemma 5.1, and I 2 v Y ( t o ) I2v74v)v.

< 0,

33 1

9. Neulrul Delay Models

Again, this is a contradiction to V ( z ,y)(to) 2 0. This completes the proof.

0

We observe that (i) D ( z ) is strictly increasing, as long as pz < 1; (ii) m(v) and n ( v )are strictly increasing; and (iii) Z(v) is strictly decreasing. These observations will be used in the proof of the following main theorem. Theorem 5.1. We consider the system (1.6). In addition to all the assumption^ of Lemma 5.2, assume further that p(x* v) < Then

+

3.

Proof. Denote

It is easy to see that 6(w) is a strictly increasing function of w. Hence, for 5 v, 6(w) 5 6(v) < 1. Of course, we have V(z,y)(t) < v for t 2 0 from Lemma 5.2. < r1 < 7-2 < 1 be two constants. We make the following Let claim. Claim 1. There is a 7'1 > 0 such that, for t 2 7'1, V(z,y)(t) < r$V2. We first prove. that if U(z,y)(t) = max{(z(t) - ~ * ) ~ , ( y ( t ) Y+)~} > rTU2, t 2 0, then U ( z ,y ) ( t ) < 0. Assume first that I z ( t ) - z* I > rlv; then, using Eq. (5.12) (change t o to t ) and the fact that V(z,y)(t) < w2, we have w

Similarly, we can show that if I y ( t ) - y * l > 7-12), then $(y(t) - Y * ) ~< 0. These imply that limsupt++w U ( z , y ) ( t )5 r;v2. Hence, there is a ti > 0 such that U ( z ,y)(t) < rgv2 for t 2 t l . Let T'1 = t l 270; then, we have ~ ( z , y ) ( t<) r$v2 for t 2 7'1, proving Claim 1. Since p(z* + v) < we can choose a constant 8 such that r2 < O3 < 1 and o(e)= e2(i - p(z+ .)) - e p(z* V ) < 0.

+

3,

+

+

+

This is possible because 0 ( 1 ) = 0, and O'(e)le=1 = 2 ( 1 - p ( z * + v ) ) - l Thus, e2(i - p(z* v ) ) < 8 - &* v), which implies that

+

> 1.

+

(5.15)

332

Delay Differeniial Equalions

Clearly, e > (r2)lI3 > r2. Next, we prove the following. Claim 2. For t 2 TI - TO, Ii(t)l 5 8D(v z*)v. We note that if the claim is not true, then there is a t o 2 that Ii(to)l > BD(v z*)v.

+

2'1

- 70 such

+

Thus,

ii(to)

+ p.(to) J--rzo 5(to +

4 4 4 2 (e - p(v + z * ) ) ~ ( + v 2*)v.

8)

A similar argument as that leading t o (5.11) leads to

+ q(v f 2*)(v + 2* + y*)lr2v e3

v(v + 2*)[g(v+ 2*) + q(0) e - p(v - . + x*)

+ q(v f + z*+ Y*)] (+ ~ z*)]-l(v + z*)[s(v+ z*)+ q ( o ) + q(v + + + y*)]v 2*)(v

< e[i - p

2*)(v

=

e q v + C*lv,

2*

i.e., 0 > 1. This is the desired contradiction, proving Claim 2. Up to now we have shown that (1) V ~ ( z , y ) ( t )< f12v2, for t 2 0, where

Vq(z,y)(t) = max{(z(s) - x*)~, (y(s) - Y * ) ~: TI

[s(z*+

+ +

+ t - 270 L s 5 TI + t } ;

+

(2) r p v ) > q v ) > m ( v ) > ev)T1 p p ( 2 * v); (3) c > pl[q(o) 2q(s* ev)l 2 C 7 5 n ( 8 V ) ; (4) X,,(O) 5 D(v+z*)Ov,whereXT,(t) = max{li(s)l : TI - t o

+

Tl

+t};

( 5 ) p(2*

+ ev)(2*+

LsL

+ ev) < 3.

By similar arguments as presented in the preceding,,,we can show that there is a T2 > TI such that (1)-(5) are satisfied by replacing TI with T2 and v with Ov. Continuing in this manner, we can show that, for any positive integer i, there is a positive constant Ti such that (1;) Vx(s,y)(t) < 02iv2,for t 2 0, where V ~ ~ ( z , y ) (= t ) max{(z(s) - x*)~, (Y(s) - Y * ) ~: Ti

+ t - 270 5 s 5 Ti + t } ;

333

9. Neutral Delay Models

+ + + +

+

(2;) i ( e i v ) > [g(z* eiv)T1 p ] ~ ( z * v ) ; (3;) c > Ibl[q(O) 2q(z* eiv)(z* elv)] 2c.rsn(eiv); (4;) XT~(O)5 D ( v z*)Oiv,where X T ~ (=~ max{li(s)l: ) Ti - t o

+

s

5 Ti

+t};

+

+

+

5

3.

(5;) p(z* 6%) c It is easy to see that, for t 2 0, i 2 1,

VT~(Z,y)(t) = V(z,y)(Ti+ t). Therefore, we have lim V(z,y)(t)= 0,

t++m

which is equivalent to the conclusion of the theorem,

9.6. Convergence in Lotka-Volterra Systems

We notice that the sufficient conditions for the global asymptotical stability of the positive steady state of (1.6) are rather complicated. Since many systems or equations appearing in the literature are of Lotka-Volterra type, it is thus desirable to have conditions specifically designed for these cases. This will be an objective of this section. We will continue to use the notations defined in the previous sections. We consider first the following general neutral delay Lotka-Volterra system, which satisfies all the assumptions made for the system (1.6):

In addition, we assume that r and c are positive constants. Note that (6.1) is obtained by replacing g(z)with r(1 -z),and q ( z ) with 1. If one replaces q ( x ) with a positive constant q, then by letting y = q y , the resulting system will have the form of (6.1). The positive steady state E(z*,y*)(if it exists) satisfies

{

r(1 - z*) - y* = 0, a bz* - cy* = 0.

+

Throughout the rest of this section, we assume that

334

Delay Differeniial Eguaiions

(A) r c + b > 0, rc - a > 0, and b + a > 0. Then, z* = r c - a / ( r c b), y* = r ( b + a)/(rc+b). Notice that if TO = p = 0, rc b < 0, r c - a < 0, and b a < 0, then ( z * , y * ) is unstable. Also note that if (A) is satisfied, then 0 < z* < 1. For the system (4.1), we have g ( X ) = g(X) = r , ij(X) = 0. Thus, (1) ~ ( z =) (1 r ) z ( l - pz)-l; (2) l (v ) = r - 1; (3) m(v) = [ r q p ] ( l + r ) ( l - p(z* v))-'(z* v); (4) 4.) = (Y* lbl). A direct application of Theorem 5.1 to system (6.1) yields the following theorem. Theorem 0.1. I n the system (6.1), assume that there is a positive constant v > 0 such that the following conditions are satisfied: (i) p ( z * v) < (6.2) (ii) r - 1 > ( r ~ 1 p)(l r ) ( l - p(z* v))-'; (6.3) (iii) c > Ibl ~ C T ~ (lbl)(y* C v). (6.4) Then the solution ( z ( t ) ,y ( t ) ) with initial function = ($l(t), 4 2 ( t ) ) , t E [-To,O], satisfies

+

+

+

+

+

+

+

+

3; +

+

+

+

+

+

+

+

provided that rnax{l$1(t) - z*I, ( 4 2 ( t ) - y * ( : t E [-70,0]} < v and rnax{l&(t)l: t E [ - 7 0 , 0 ] ) 5 (1 r)(z* v ) ( l - p(z* v))-lv. The following example may illustrate the preceding result. Example 6.1. Consider the following neutral delay Lotka-Volterra competition model:

+

{

+

+

i ( t )= z(t)[2(1 - z ( t - T ) ) - p i ( t - T ) - y(t)], i ( t ) = y ( t ) [ i - z ( t ) - 2 y ( t - 41.

4,

(6.5)

It is easy to see that r = 2, a = b = -1, c = 2 , q = 72 = T , T ~= 7 4 = 0, 3 3 and b a = 3. Hence, and 7 5 = 6. Clearly, r c b = 3, rc - a = 3, (z*, y * ) = ( 3 , l ) . Thus, conditions (i)-(iii) reduce to (1') p ( i v) < (2') 1 > (27 p)3(1 - p ( 3 v))-'; (3') 1 > 1 2 4 1 v). We can show that if p 5 Q, T 5 and u < then v can be any positive constant less than 1. Therefore, the solution ( z ( t )y, ( t ) ) with initial function = ( 4 1 ( t ) , 4 2 ( t ) )t, E [-To,O] satisfies

+

+

+

3;

+

+

+

&,

&,

9. Neutml Delay Models

335

provided that 0 5 +l(t) < $, 0 < & ( t ) < 2, t 5 0, and max{l&(t)l : t 5 0) 5 6. 0 R e m a r k 6.1. Clearly, the conditions imposed on the initial functions are very reasonable in the preceding example. We would like to comment here that other conditions are also very natural. Note that if a > 0, 6 < 0, then the system (6.1) can be viewed as a competition model. When TO = p = 0, it is well known that E ( z * ,)'y exists and is globally asymptotically stable in the positive cone if and only if r c + 6 > 0. It is easy to see that there is 0 < c c 1 such that rc > lbl c. By letting 5 = &z, j = y, the > 1, 151 = &c < c; i.e., the resulting resulting system will have F = system satisfies the so-called "diagonal dominance" condition. Clearly, as long as the system is "diagonally dominated" (r - 1 > 0, c > lbl), then we can always choose small p , TI,and 75 such that (6.2)-(6.4) are satisfied with reasonable size of v. In other words, we have shown that, as long as the delays 71 and 75 are short and the neutral coefficient is small, positive steady state will remain as the attractor for solutions with initial functions reasonably bounded in C' space. Roughly speaking, the convergence is structurally stable with respect to small delays as well as neutral terms, provided they are introduced as in the system (6.1).

+

6

9.7. Convergence in Single Population Models, I1 In this section, we again consider the convergence aspect of the following neutral delay scalar population models:

Alternatively, we can set q(z) = 0 in (1.6) to decouple the system. Thus, (7.1) becomes the first equation of the decoupled system. We denote 70

= max{71,72),

- 1)2 : max{t - 270, -TO) 5 s 5 t ) , X ( t ) = max{lk(s)l : -TO 5 s 5 t ) ,

~ ( z ) ( t=) max{(z(s)

and g(X), g(X) are as defined in (5.1)-(5.2). Since in the decoupled system z has nothing to do with y, an immediate result from Theorem 5.1 is the following global stability theorem for the scalar equation (7.1): T h e o r e m 7.1. Assume that there is a positive constant v such that

+

+

g(1+ v ) > g(1 v)(g(1+ v)71 p ) ( l - p(1+ w ) ) - l ( l + v). (*) Then the solution z ( t ) of (7.1) with initial function +(t),t E [-70,0], satisfies lim z ( t ) = 1, t++w

336

Delay Differential Equations

provided that max{Iq5(t) - 11 : t E [-70,0]} < v and max{l&t)l : t E [-70,0]} Ig(1 v ) ( l v ) ( l - p(1 v))-'v. Proof. In applying Theorem 5.1, we note that z* = 1, q ( 0 ) = ij = 0; D(1 v) = g(1 v ) ( l v ) ( l - p(1 v))-l; l(v) = g(1 v); and m(v) = 3 ( 1 + v ) ( g ( l + v)q p ) ( l v ) ( l - p ( 1 + v))-'. Clearly, condition (i) of Lemma 5.2 reduces to

+

+

+

+

+

+ +

+

+ v)[(g(l +

+

+

+

+ p)(l + v ) ( l - p(1 +.))-'I. Notice that f(z) = z/(l - z) is increasing, and f(3) = 1. g_(l v) > g(1

v)71

(7.2)

We see that (7.2) holds only if p(1 + v) < a condition required in Theorem 5.1. The 0 rest of the proof follows that of Theorem 5.1. In comparison with the result stated in Theorem 4.1, we see that Theorem 7.1 is easy to use. One need only evaluate g(X) and g(X); the condition (*) then provides bounds for p and 71, while in order to apply Theorem 4.1 one needs to estimate the two nontrivial functions X(a) and h-l(a). However, one cannot claim which result is more general, since Theorem 4.1 has no conditions imposed on q5 other than 11q511 < 1; while in order to satisfy Theorem 7.1, v may have to be small to satisfy (*); moreover, 141 has to be uniformly bounded. Let g ( z ) = r ( l - 3); then, (7.1) reduces to the following logistic-type neutral delay scalar equation:

3,

0

0

i ( t ) = z ( t ) [ r /-71 (1 - z ( t + S ) ) d p 1 ( s ) - p J

--r2

i ( t + s ) d p s ( s ) ] . (7.3)

For this equation, we have the following result. Corollary 7.1. Let v be a positive constant such that (r71

+ p ) ( l + v)[l - p(l + 41-l

< 1.

(7.4)

Then the conclwion of Theorem 7.1 hold3 for Eq. (7.3). Proof. It is easy to see that g ( 1 + v) = g ( 1 + v) = r , which reduces (*) of Theorem 7.1 to (r7

t p ) p + v)[l

n

+ vll-1. < 1. e fol owing more general scalar

U

- p-2

Finally, we would like o consider t neutral delay equation:

337

9. Neutral Delay Models

where pi(s), i = 1,2,3, satisfy (Hl) and g1(z) and -g2(z) satisfy (H2). In this case, (7.5) does not have the secalled negative feedback property enjoyed by Eq. (7.1). We assume that (7.5) has a unique positive steady state z ( t ) 1. Denote 70 = max{71,~2,~3}; i = 1,2; g;(A) = max{Ig’(z)I : x E [o,A]}, i = 1,2. $(A) = min{lg’(x)l : z E [o,~]}, By similar arguments to those presented in the proof of Theorem 5.1, we can establish the following global stability result for system (7.5). Theorem 7.2. Assume that there is a positive constant v m c h that

where the involved functions are evaluated at 1 z ( t ) of (7.5) with initial function qh(t),

+ v.

Then the solution

t E [-70,0],satisfies

lim z ( t ) = 1,

t-+w

provided that max{Iqh(t) - 11 : t E [-70,0]}< v and.max{l&t)l : t E [--7o, 01) I *(91(1+ ). 92(1 .)I.. Sketch of the proof. First of all, we show that if I z ( t ) - 11 is bounded by v, then Ij.(t)l is bounded by *)(g1(1 v) g2(l v))v. Second, we show that indeed Ix(t) - 11 is bounded by v. Then, we show that there are TI > 270 and 0 c 8 c 1 such that, for 2 2 T I , I z ( t ) - 11 c Bv and li(t)l < Bd(v), where d(v) = *j(ijl(l +v)+g2(1 +v))v. By repeating these arguments, we can show that, for any positive integer i, there is a Ti > 0 such that, for t 2 Ti, [ ~ ( t )11 < B’v and l i ( t ) l < Bid(v). This clearly proves the theorem. 0 Of course, the third term involving g2(z) can also be added in the first equation of the system (1.6) if needed. Our arguments will remain valid in spirit. The logistic version of (7.5) takes the form

+

+

+ +

+

338

Delay Diffemniial Equations

We assume that Q > p > 0 and a - p = r. This makes z ( t ) 1 the unique positive steady state of (7.6). In this case, Sl(A) = gl(X) = Q, 92(X) = g2(X) = p. The condition (*) of Theorem 7.2 thus reduces to a -P

> (Q71 + p73 + p ) ( a + p)

l+v 1 - p(l v ) '

+

(7.7)

Example 7.1. Consider

i ( t )= z(t)[l-z(t--)-

p

1

24

1

1 12

1

1 10

-4t--)-pqt-1)+-z(t--)+-z(t--)I.

2

4

1

1

4

(78.8) 3 Equation (7.8) can be converted to (7.6) by letting r = 1, Q = 3, 1 1 = 71 = Tz, 7 2 = 1, 73 = g, and

4,

1

q.

=

-.-1

12 ' 1, - 1 < s < o , 0, s = -1; 1 lo < s 5 0 , 1, -1

1

0,

s=

1 10'

--.1 8

It is easy to see that (7.7) reduces to

(A

1 > - + - +1'p6

) 1 !(i;?v)*

(7.9)

&.

&,

If we choose v = 1, then (7.9) is equivalent to p < That is, if p < 0 < $(t) < 2, l$(t)l 5 4/(1 - 2p), for t E [-1,0], then the solution z ( t ) of (7.8) with initial function $ ( t ) tends to 1 as t + $00. 0 9.8. Boundedness in a Nonautonomous Neutral Logistic Equation

The material of this section is taken from Kuang and Feldstein (1991). If all the coefficients of (1.3) are time dependent, then it becomes

i ( t )= r ( t ) z ( t ) [ a ( t ) b(t)z(t- 7)- c ( t ) k ( t - 41,

(8.1)

339

9. Neutral Delay Models

where r > 0, and r ( t ) , a ( t ) , b ( t ) ,and c ( t ) are continuous. Also, r ( t ) > 0, a ( t ) > 0, and b ( t ) > 0. Obviously, it is easier to work with equations that have less variables and parameters. Thus, we take

t = 75,

z(s) = z(7s),

T(s) = r r ( r s ) b ( T ( s ) ,

a(s) =

b( r s ) '

~(s= )

1 c(rs) 7 b(r9)'

--

With these scalings, (8.1) becomes d z(s) -

a(s)

ds

- z(s - 1) - c(s)

dZ(s

- 1)

ds

Therefore, through the rest of this section, we are going to discuss the equation i ( t ) = r ( t ) z ( t ) ( a ( t) z(t - 1) - c ( t ) i ( t - 1))) (8.3) rather than (8.1). For Eq. (8.3) we always assume that r ( t ) is positive, that a ( t ) c ( t ) # 0 , and that all of these three functions are continuous and bounded both f r o m below and above. The approach throughout this section is mainly motivated by the intrinsic interest of the equation and the ecological interpretation of the results. We present first conditions under which the positive solutions of (8.3) tend to infinity as t + 00. Before stating the related theorems, the following result will be needed. For its proof, see Kuang and Feldstein (1991). Lemma 8.1. If z(0) > 0 , then z ( t ) > 0 for all t 2 0 . If z(0) = 0 , then z ( t ) = 0 f o r all t 2 0 . We note (8.3) can be written as

-

i ( t ) - r ( t ) a ( t ) z ( t )= --r(t)z(t)(c(t)i(t - 1) + z(t - 1)).

(8.4)

If we assume c ( t ) # 0 for all t 2 0, then (8.4) is equivalent to

;(t) - r ( t ) a ( t ) z ( t = ) - c ( t ) r ( t ) z ( t ) ( i (t 1) + c - ' ( t ) z ( t - 1)).

(8.5)

This expression leads us to the conclusion of the following theorem. Theorem 8.1. In Eq. (8.3), assume r ( t ) > 0 , a ( t ) > 0, - c ( t ) > 0, and c - ' ( t ) = -r(t - l)a(t - l ) , f o r t 2 1 and z(0) > 0 . Then the following hold. (1) If i ( t ) c - ' ( t ) z ( t ) 2 0 for -1 5 t 5 0, then z ( t ) is unbounded. I n fact, z ( t ) 2 Z(O) exp($ r ( s ) a ( s )d s ) .

+

340

Delay Differential Equations

+

(2) If i ( t ) c-'(t)z(t) 5 0 for -1 z(0) exp($ r ( s ) a ( s )ds) for t 2 0.

5 t 5 0, then

z(t)

5

Proof. By Lemma 8.1, z ( t ) > 0; hence, -c(t)r(t)z(t) > 0 for t 2 0. Denote F ( t ) = i ( t ) - r(t)a(t)z(t) for t 2 0. (8.6) Equation (8.5) indicates that F ( t ) will not change sign for t 2 0, if i ( t ) + c-'(t)z(t) does not change sign in the interval [-1,0]. In other words, if i ( t ) + c-l(t>z(t) 2 0 for -1 5 t 5 0, then F ( t ) 2 0 for all t 2 0; therefore, for t 2 0, i ( t )- r ( t ) a ( t ) z ( t2) 0 which implies 4t)

2 4 0 ) exp

(/01

r ( s ) a ( s ) ds)

'

This proves (1). In the case that

i ( t )- r ( t ) a ( t ) z ( t 5 ) 0, we have

0

< z ( t )5 4 0 ) exp

(Jo'

t 2 0,

r ( s ) a ( s )ds)

7

0 which implies the conclusion in (2). Motivated by Theorem 8.1, we can prove the following more general result. Theorem 8.2. In Eq. (8.3), assume r ( t ) > 0, a ( t ) > 0, c(t) < 0, and c-'(t) = -r(t-l)a(t-l)+c(t), z ( t ) > 0 , f o r t E [0,1]. Then thefollowing hold. (1) If c ( t ) 2 0 and i ( t ) - r ( t ) a ( t ) z ( t )2 0, for -1 5 t 5 0 , then z ( t ) 2 4 0 ) exp(Ji r ( s ) a ( s )ds). (2) If c ( t ) 5 0 and k ( t ) - r ( t ) a ( t ) z ( t )5 0, for -1 5 t 5 0, then z ( t ) 5 z(0) exp(& r ( s ) a ( s )ds) for t 2 0. Proof. From Eq. (8.5),we arrive at i ( t )- r ( t ) a ( t ) z ( t= ) (€(t) r ( t - l)a(t - 1))-1r(t)z(t)(i(t - 1) - r(t - l)a(t - l ) z ( t - 1) + c (t)z(t - 1)). Hence, if c ( t ) 2 0, then

i ( t )- r(t)a(t)z(t)2 ( € ( t )- r(t - l)a(t - 1))-lr(t)z(t) x ( i ( t - 1) - r ( t - l)a(t - l ) z ( t - l)),

(8.7)

34 1

9. Neutral Delay Models

since we assumed that c ( t ) < 0, and, by Lemma 8.1, z ( t ) > 0. Obviously, k ( t ) - r ( t ) a ( t ) z ( t2 ) 0, for -1 5 t 5 0 ensures that i ( t ) - r(t)a(t)z(t)2 0 for 0 5 t 5 1. Induction immediately yields the conclusion that z ( t ) 2 z(0) exp(Jir(s)a(s)ds) for t 2 0 . In case of ~ ( t5) 0, (8.7) implies that

k ( t ) - r ( t ) a ( t ) z ( t )5 ( ~ ( t )r(t - l)a(t - l ) ) - l r ( t ) z ( t ) x ( k ( t - 1) - r ( t - l)a(t - l)z(t - 1)).

A similar argument as in the case of (1) yields

Remark 8.1. We observe that the assumption of c ( t ) < 0 in this section is in opposition to the assumption made in F. E. Smith (1963). However, we feel this may well be a reasonable hypothesis for some species. For example, if an immature organism consumes less food than a mature one, then c ( t ) should be negative rather than positive. Our results in this section indicate that the neutral term in Eq. (8.3) is destabilizing, since, if c ( t ) = 0, then all solutions of (8.3) are eventually bounded, as long as both r ( t ) and a ( t ) are positive and bounded from below and above. In the following, we try to obtain conditions for Eq. (8.3) under which solutions of it will be bounded. Throughout this section, it is assumed that there are finite constants T I , r2, c1, and c2 such that 0 < rl 5 r ( t ) 5 r2 and 0 < c1 5 c ( t ) 5 c2. Equation (8.3) may be rewritten as

&(t)+ c-l(t

+ l)z(t) = r ( t ) t ( t ) { a ( t+) r-'(t)c-'(t + 1) - c ( t ) x [&(t- 1) + c-'(t)z(t - l)]}.

Denote y(t) = k ( t )

+ c-'(t + l)z(t);

(8.8)

(8.9)

then, (8.8) becomes y(t) = r(t)c(t)z(t)(c-'(t)(a(t)

+ r-'(t)c-'(t + 1)) - ~

( -t 1)).

(8.10)

Denote

P ( t )= r - ' ( t ) ~ - ' ( t ) ,

+

Q ( t ) = c-'(t)(a(t) r-'(t)c-'(t

+ 1)).

(8.11)

Assume there are positive constants P , Q1, and Q2, such that P ( t ) 2 P , Q1 5 Q ( t ) 5 Q2. We have the following lemma.

342

Delay Differeniial Equaiions

Lemma 8.2. Let T 2 1. For t I T , a ~ ~ u m that e the ~ o l u t i o nz ( t ) of Eq. (8.3) satisfie3 0 I ~ ( t5) P(t)Ql/Q2 and 0 I y(t) I Q1, for -1 I t 5 0. Then, (1) 0 I y ( t ) I Q1, for all 0 I t I T ; (2) Jrfiin(O,t-2) Y(S) ds I 9 2 for 0 I t I T * Proof. We first assume 0 I t I 1. The assumption ~ ( tI) P(t)Q1/Q2 implies r ( t ) c ( t ) z ( t )I Q1/Q2. Since 0 I y(t) I Q1 for -1 5 t 5 0, we see y(t) 2 0 by the assumption Q(t) 2 Q I , and ~ ( t I) ~(t)P-’(t)Q(t) I (Qi/Q2)Q(t) 5 Q i , since Q(t) I Q 2 . Clearly, by induction, we can conclude that 0 5 y(t) I &1 for all 0 I t 5 T . Therefore, we have

Hence, 1

6

2

y(s)ds

I9 2

0

for 1 5 t I T.

From Eq. (8.9), we have (8.12)

Lemma 8.3. Assume, for -1 5 t I T , y(t) 2 0 and Jrfiinlo,t-2)y(s) ds

I

Then, for t E [O,T],~ ( t<) z ( O ) ~ - J ~ ~ - ~ ( + ~ +Q2e2cy1/(e2c;’ ~ ) ~ ~ - 1). Proof. Let us estimate Q2.

Since

343

9. Neutral Delay Models

we will concentrate temporarily on the estimation of the integral on the interval [t - 2(n l ) , t - 2nI. Since we are assuming c-'(t) 2 q1> 0, we have - J,'c-l(s 1)ds I -$(t - a).

+

+

Hence,

Therefore,

From Eq. (8.12), we immediately arrive at the conclusion of the 1emma.O Now, we are ready to state our boundedness result. Theorem 8.3. Assume, in Eq. (8.3),

(i) z(0)e-JiC-'(s+l)d8 + Q2e2C;1/(e2C;1 - 1) I P(t)Q1/Q2; and (3) 0 5 y(t) 5 Q1, f o r -1 I t 5 0, and z(0) > 0. Then

o < z ( t ) < z ( ~ ) e - J ~ c - 1 ( s + l ) d+3 ~2e2c;l/(e2c;I

- 1)

f o r t 2 0.

Proof. By Lemma 8.1, we know z ( t ) > 0 for all t 2 0. If the conclusion of the theorem is not true, then, there is a first time, say t*, t* > 0, such that (8.13) and 0

< z ( t ) < z(0)e--fiC-'(s+l)ds+ Q2e2ca1/(e2ca1- 1)

By assumption (i), we have 0 < z(tjr(t)c(t) I Q 1 / Q 2

for t E [ O , t * ] .

for 0

I t < t*.

344

Delay Differential Equations

Combining this with assumption (ii) and then applying Lemma 8.1, we see

0 I?At)5 91, and

.t

Lin{0,t-21

y(s) d s

for t E [0, t'].

I 92

From Lemma 8.2, we know this implies

+

z ( t ) < z ( 0 ) e - J ~ C - l ( s + l ) d s 9 2 e 2 ~ ; ' / ( e 2 ~-~ 1) '

for t E [0,t*1.

This inequality obviously contradicts Eq.(8.13), which proves the theorem.

0

This theorem indicates that as long as the two assumptions (i) and (ii) are met, the solution of Eq. (8.3) is bounded above by z(0) Q2e2'Y1/ (e2'Y1 - l), and eventually, the solution is bounded above by Q2e2';'/ (e2C;' - 1). In the case that all the coefficient functions are constants, we have the following counterpart of Theorem 8.3. Theorem 8.4. Consider the equation

+

i ( t ) = r z ( t ) ( a- z(t - 1) - c i ( t - l)),

(8.14)

where T , a , and c are real constants, and where r > 0 , c > 0 . Assume, i n Eq. (8.14), (i) z(O)e-'/' $(u + &)e2/'/(e2/' - 1) 5 $; and (ii) o 5 ? ( t ) c - l z ( t ) 5 i ( u $1 f o r -1 5 t 5 0. Then, for t 2 0,

+ +

o < z ( t ) < z(O)e-'/'

+

+

e2Ic/(e2lc- 1)

for t 2 0.

C

Proof. In this situation, c ( t ) = c1 = c2, p ( t ) = p = r-lc-', Q ( t ) = Q i = Q2 = $(u + and y(t) = i ( t ) c - ' z ( t ) . The conclusion follows from Theorem 8.3. 0 Corollary 8.1. In Eq. (8.14), if (i) z(0) > 0, z(0) + ;e1/'(a + &) < and (ii) o 5 ;(t) + c - ' z ( t ) 5 $ ( a + for -1 5 t 5 0,

A),

+

6)

A;

345

9. Neutral Delay Models

Hence, the conditions (i) and (ii) in Theorem 4.1 are met, which implies

0 for t 2 0. For example, in Eq. (8.14), r = 0.05, c = 4, a = 1, z(0) 5 1.1, and z ( t ) = z(0) for -1 5 t 5 0. Then, z(0) +e1Ic(a < 4.96 < = 5, and i ( t ) c - l z ( t ) = :z(O) < :(a = for -1 5 t 5 0. Therefore, by Corollary 8.1,

+ + 6) 4

+

0

< z ( t ) < 4.96

+ 6)

6

for all t 2 0.

If a < 0 in Eq. (8.14), then the corresponding boundedness result is very simple. Theorem 8.5. Ifa < 0, z(0) 5 -a, a + > 0, and 0 5 k ( t ) + c - ' z ( t ) 5 1 ,1( a E ) , then 1 for all t 2 0. 0 < z ( t ) < -, rc

+

Proof. By Lemma 8.1, we know z ( t ) > 0 for all t 2 0. Assume, at t = t*, x ( t * ) = ( r c ) - l , and 0 < z ( t ) 5 (rc)-' for t E [ O , t * ] . By Lemma 8.2, 0 5 y(t) 5 ;(a and Eq. (8.12) implies

+ A),

~ ( t=)z(O)e-('/c)t + e-(l/c)t

1 + ( a + -)(I - e --(l/c) t) rc 1 1 for t E [ O , t * ] . < z(0) + a + 5 -, rc rc

= z(0)

This obviously contradicts the assumption that z ( t * ) = ( rc) -', which proves the theorem. 0

Delay Differential Equations

346

9.9. A Periodic Neutral Logistic Equation We continue to consider Eq. (8.1). In addition to the assumptions made in Section 8, we assume now that r ( t ) , a ( t ) , b ( t ) , and c ( t ) are periodic of period w and T = 77w for some positive integer m. The following results are taken from Gopalsamy, et al. (1991). For convenience, we rewrite Eq. (8.1) as

k ( t ) = r(t)z(t){1 - [ ~ (-t mw)

+ c(t)k(t- m w ) ] / K ( t ) } ,

(9.1)

where r(t), c ( t ) , and K ( t ) are positive continuous periodic functions of period w. We define constants ro, ro as

similarly, we define a,co,KO,and K O . We shall first consider the periodic ordinary differential equation

which can be written as

One can see that positive solutions of (9.3) satisfy

and hence

We can conclude from (9.5) that

9. Neutral Delay Models

347

A positive solution p ( t ) of (9.3) is said to be globally asymptotically stable if every other positive solution, say ~ ( t )satisfies , t-00 lim

Iz(t) - p(t)l = 0.

(9.8)

Lemma 9.1. Suppose r, I(, and c are strictly positive continuous periodic functions of period r. Then (9.3) has a globally aJymptotically stable periodic solution of period r . Proof. The existence of at least one positive periodic solution of (9.3) is a consequence of (9.7) and Theorem 15.3 on p. 164 of Yoshizawa (1975). We shall show that if u and w are any two positive solutions of (9.3), then t+aY lim

Iu(t) - w(t)l = 0.

(9.9)

The uniqueness and the global asymptotic stability of the periodic solution will then follow from (9.9). We define w such that w ( t ) = ln[u(t)] - ln[w(t)]

(9.10)

and derive that (9.11) We observe

(eff)) ,

w ( t ) = ln[u(t)] - ln[w(t)] = [ u ( t )- v ( t ) ] -

(9.12)

where e ( t ) lies between u ( t ) and v ( t ) ,and hence O ( t ) is strictly positive and is bounded away from zero. Equations (9.11) and (9.12) together imply

from which one can derive that lim w ( t ) = 0,

t+m

and so (9.9) follows. This completes the proof. 0 Let p ( t ) denote the unique positive periodic solution of (9.3). It can be verified that p satisfies

348

Delay Differential Equations

and hence p also satisfies

Thus, the existence of a periodic solution of (9.1) is easily resolved. Furthermore, we note that

KO Ip ( t ) I KO and

-8 Ii ( t ) I 8 , where

do=

rOKO KO(KO K O con

+

-K ~ ) .

We let

4 t ) = 1 4 4 t ) l - Wwl, so that z(t) =p(t)ev(').

We derive from (9.1) that

2

is governed by

where

b ( t ) = -rc( t ))p ( t I((t)

- w).

The linear variational system corresponding to the solution p of (9.1) can be obtained from (9.13) by neglecting the nonlinear terms in (9.13). This leads to the linear variational equation y(t) = - a ( t ) y ( t - mu)- b(t)Y(t - mu).

(9.14)

Definition 9.1. The positive periodic solution p of (9.1) is said to be locally asymptotically stable in the C' metric if every solution of (9.14) satisfies and lim y(t) = 0. (9.15) lim y ( t ) = 0 t+w

t-oo

349

9. Neutral Delay Models

If the convergence in (9.14) is exponential, then the solution p of (9.1) is said to be ezponentially asymptotically stable in the C' metric. We remark that there are no routine techniques developed in the literature for the verification of asymptotic stability in the C' metric for neutral differential equations. Theorem 9.1. Suppose the delay mw and the neutral coefficient c ( t ) are small enough t o satisfy

+ bo < 1;

cod" < K O ;

+ bo) < ao; 1 - (sow + 60) uo(u'mw

where

+ cOdO];

uo = - TO "KO

a0

= -TO [KO

(9.16)

- cOdO];

KO

KO

T h e n the positive periodic solution p of (9.1) is (locally) exponentially asymptotically stable in the C' metric. Proof. It is enough to show that the trivial solution of (9.14) is exponentially asymptotically stable in the C' metric. We first rewrite (9.14) in the form

By the variation of constants formula, we have, from (9.17),

xe

-h:a(u)du

s," a(u)du ds.

e o

(9.18) Supplying y from (9.18) in (9.17),

Y ( t ) = -a(t)y(to)e

-

4 U ) du

(9.19)

Delay Differential Equations

350

and, estimating the terms of (9.19),

+ (a'mw + 6') sup ljl(s)l. alt

(9.20)

Rearranging the terms in (9.20), we obtain

[l - (a'rnw

+ bO)] sup(l$(s)leS;

44 du)

s5t

+ aO(aOmw+ 6') 1 (sup Ii(u)le.ft:

(9.21)

1

IaOly(to)l

10

du)

ds.

u<s

By the Gronwall-Bellman inequality, we derive, from (9.21),

(9.22) It is easily seen from (9.22) that

which implies by (9.16) that lim y(t) = 0,

t-00

(9.23)

the convergence in (9.23) being exponential. The fact that lirn y(t) = 0 i?-00

follows from (9.23) and (9.14). This completes the proof.

0

9.10. Remarks and Open Problems In this chapter we considered one or two species neutral delay population models motivated by the work of F. E. Smith (1963) in his investigations on laboratory populations of Daphnia magna. We have discussed here the boundedness (or nonboundedness) and convergence of solutions and existence of periodic solutions in a very special periodic neutral delay logistic equation. Local stability and oscillation analysis of (1.6) (or its

9. Neutral Delay Models

35 1

simplified version) can be found in Kuang (1991a,b). Similar results for the scalar equation (1.3) can be found in Gopalsamy and Zhang (1988). Discussion of numerical simulations of Eq. (1.3) and the system (1.6) is given in Kuang (1991d). The existence of Hopf bifurcation for Eq. (1.3) and the system (1.6) can be established by applying Theorem 2.9.1. It should be pointed out here that there is obvious overlapping among results presented in Sections 2 and 9, and those in Sections 4 and 7. However, they are largely independent of each other. This is clearly reflected by the differences in their restrictions on initial functions. Our analysis of (1.6) (or even (1.3)) is far from complete. Many interesting and important problems remain open. The following are two of the most relevant ones. O p e n Problem 9.1 (Gopalsamy, et al. 1991). Is it true that if Ic(t)l is sufficiently small for t 2 0, then Eq. (9.1) will have a globally attracting periodic solution? Note that when c ( t ) = 0 and T is small enough, this problem has an affirmative answer (Zhang and Gopalsamy, 1990). O p e n P r o b l e m 9.2. Obtain sufficient conditions for the existence of positive periodic solutions in

q t ) = r ( t ) z ( t ) [ a ( t-) @(t).(t) - b(t)z(t- T ( t ) ) - c ( t ) i ( t- T ( t ) ) ] , (10.1) b ( t ) , ~ ( t c) (,t ) are nonnegative continuous periodic where r ( t ) ,a @ ) ,@(t), functions. When r ( t ) , a ( t ) , @ ( t ) , b ( tand ) , ~ ( tare ) positive and c ( t ) = 0, such a problem was considered in Freedman and Wu (1991), and Tang and Kuang (1991b) (for systems). We stress here that ~ ( tis) not a multiple value of the period of these functions, as required in Section 9. It is also interesting to systematically study other types of neutral delay models. Such examples can be found in Freedman and Kuang (19921, Gyori and Wu (1991), Brayton (1966), and Driver (1984a,b). The works of Bartsch and Mawhin (1991) and Serra (1991) are also relevant, where conditions for the existence of periodic solutions in certain periodic neutral equations are established. Hopf bifurcation theory for neutral delay differential equations are considered in De Oliveira (1980), Staffans (1987), and Krawcewicz et al. (1992).

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APPENDIX Algorithm

375

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Appendix

377

.....................................................................

C * C The organization of this documentation is described below. C The algorithm SNDDELM for the numerical solution of systems of C neutral delay differential equations with state-dependent delays C (SNDDE) by Adams predictor-corrector methods is outlined in C section A. The definition of SNDDE that this code solves is C presented in section B. Three examples to show the user how to C code them are given in section C. Three optional output disk files’ C in which the first one contains numerical statistics; the second C one contains the order,stepsize, and numerical solution at the C grid points; and third one contains the interpolated numerical C solution at user-specified equal time interval are described in * C section D. An explanation of user-specified input parameters, C functions, and subroutines is presented in section E. C .1 C This code was written in VAX FORTRAN I 1 in double precision by:* C Edisanter Lo C Associate Research Technologist * C Arizona State University C Tempe, AZ 85281 1 C C Zdzislaw Jackiewicz C Professor of Mathematics C Arizona State University C Tempe, AZ 85281 C C June 30,1992 c-------------------------------------------------------------------* C C A. Description of the General-Purpose algorithm SNDDELM : C

C C C C C C C C C C

C C C C

C C C C

C C C

C C

C C C

This variable-step and variable-order code for the numerical * integration of systems of neutral delay differential equations with state-dependent delays (SNDDE) is based on Adams methods * methods implemented in PEICEI (Predict-Evaluate-InterpolateCorrect-Evaluate-Interpolate) mode with order p ranging from 1 to * 12. The Adams-Bashforth method of strong order p is used as the the Predictor, and the Adams-Moulton method of strong order p is used as Corrector and Interpolator.

*

The Adams methods are represented in divided difference form * for efficient implementation.The automatic detection of derivative* discontinuities at each step relies on the estimates of the local * discretization errors for orders p and p+l where p is the order of* the algorithm at that step. The restarting of the integration at each derivative discontinuity point relies on the step and order * changing mechanism based on the estimates of the local discretization errors for orders p-2,p-l,p,andp+l. These errors are estimated from the predictor-corrector formulas by the analogue of* Milne’s device. No extra cost is needed to obtain asymptotically correct estimates of the local errors for lower adjacent orders p-2 and p-1. However, it is necessary to use one extra function * evaluation and to use local extrapolation (i.e. accepting the * numerical solution corresponding to order p+l instead of order p at each step) to obtain asymptotically correct estimate of local error for higher adjacent order p+l. t

c-------------------------------------------------------------------* C C B. Definition of SNDDE : C C A system of neutral delay differential equations with stateC dependent delays of dimension n is defined as C C Y‘(t)=F(t,Y(t),Y(ALPHA(t,Y(t))), Y’(BETA(t,Y(t))) ),a<=t<=b,

* *

Appendix

c Y (t)=G(t) ,

*

tau<=t<=a, tau<=t<=a,

C Y'(t)=G'(t), C

c where the vectors are defined as the following:

... ... . . ...

* *

yl(t) y2(t) yn(t) 1' yl'(t) y2'(t) yn'(t)l; gl(t) g2(t) ... gn(t) 1 gl' (t) 92' (t) . gn' I' 1 F : = [ fl f2 fn I Y(ALPHA(t,Y(t))):=[ yl(alphal(t.Y(t))) yn(alphan(t,Y(t))) I ' Y'(BETA(t,Y(t))):=[ yl'(betal(t,Y(t))) yn'(betan(t,Y(t))) 1 ' in which the apostrophy on the brackets indicate TRANSPOSE and alpha's = delay functions, beta's = neutral delay functions, b = final point, g ' s = initial functions , * y's = solutions, f's = derivatives of the solutions' tau = value of the delay function at the initial point t=a * Y(t) :=[ Y'(t):=[ G(t) : = [ G' (t): = [

C

C C C C

c c C

c

C C

C

(5)

... ...

*

C

c-------------------------------------------------------------------* C

c c . Examples

:

C Three examples are illustrated below, but only exampleXl is * C C actually coded as a sample. ** C c ExampleX1: C The following system(dim=2) of delay differential equations c with state-dependent delay which has a periodic solution is coded C in this program as an example. C c Yl'(t) = - 2 yl(t-y2(t)) ,0<=t<=5, c y2'(t) = ( lyl(t-y2(t))I-lyl(t)l ) / ( l+lyl(t-y2(t))I ,o<=t<=5,* c yl (t) = 1.0 ,-o.s<=t<=o,* c y2 (t) = 0.5 , -o.s<=t<=o. P L

C Reference solution obtained with TOL=10**(-15) is C yl(5) = 0.6138 5211 7892 0276 C y2(5) = 0.7730 8554 3769 7122 C

* * *

C ~xample~2: * c The following system(dim=S) of delay differential equations c with multiple delays in Yl ** C ,O<=t<=l,* c yl'(t) = y5(t-1) + y3(t-1) c y2'(t) = yl(t-1) + ~ 2 ( t - 0 . 5 ) , O<=t<=l, ,O<=t<=l,* c y3'(t) = y3(t-1) + yl(t-0.5) , O<=t<=l,* c y4'(t) = y5(t-1) yd(t-1) ,O<=t<=l,* c Y5'(t) = yl(t-1) c yi (t) = y4(t) = y5(t) = exp(t+l) ,-1<=t<=o, c y 2 (t) = exp(t+0.5) ,-1<=t<=o, , -1<=t<=o, c y3 (t) = sin(t+l) C c has to be rewritten in the following equivalent form with single c delay SO that it can be coded in this Program C ,O<=t<=l, c yl'(t) = y5(t-1) + ~ 3 ( t - 1 ) , O<=t<=l,* c y2'(t) = y6(t-1) + ~ 2 ( t - 0 . 5 ) ,O<=t<=l, c y3'(t) = y3(t-1) + yl(t-0.5) , O<=t<=l, c y4'(t) = y5(t-1) yd(t-1) , O<=t<=l,* C y5'(t) = y6(t-1) , O<=t<=l,* C y6'(t) = y5(t-1) + ~ 3 ( t - 1 ) c yi (t) = y4(t) = y5(t) = y6(t) = exp(t+l) ,-1<=t<=o,* , -1<=t<=o,* c y2 (t) = exp(t+O.5) ,-1<=t<=o. c y3 (t) = sin(t+l)

C

c c

~xampleX3: The following second-order delay differential equation

Appendix

C

c c

379

Y"(t) + y'(t-1) + y(t) = 1 Y (t) = Y'(t) = 0

C C has to

Ne rewritten as a system(dim=

,0<=t<=2,

,-1<=t<=o, )

by defining

C

yl(t) = y(t) and y2(t) = y'(t). C so that it can be coded in this program C c yl'(t) = y2(t) c Y2'(t) = 1 yl(t) yZ(t-1) c yl (t) = 0 c y 2 (t) = 0

-

-

C

t

,0<=t<=2,* ,0<=t<=2,* ,-1<=t<=o. ,-1<=t<=o.*

c-------------------------------------------------------------------* C

C D. Output : 1 C 1 C There are 3 output files whose names are specified by the C variables OUTSTAT, OUTTVHY, and OUTSTEP. C C (i). OUTSTAT is a sequential text file of variable length and * C contains some numerical statistics such as C AERR=absolute error, NRS = Y of rejected steps, C RERR=relative error, NFE = # of function evaluations,* C NSD = # of significant digits, HMIN=minimum step size, C NS =# of successful steps, HMAx=maximum step size, C and numerical solution at endpoint t=b. The 1st table corresponds C to 1st component of solution, the 2nd to the second component,etc.* c c (ii). ou?TvHy is a sequential text file which has the following C characteristics: fixed length, direct access, formatted. This file' C contains the order, stepsize, and numerical solution(s) at each * C of the integration grid points and has the following structure: * C recordY1--> tl vl hl yl-1 y2-1 yn-1 C record#2--> t2 v2 h2 y1-2 y2-2 yn-2 * * C C where tj = the jth (nonuniform)grid point (real) C vj = the order at time tj (integer), C hj = the stepsize at time tj (real), * C yi-j = the ith solution at time tj (real), * C n = the number of equations in the system. C * C Each record/line in OUTTVHY has been written with format statement* C FORMAT(E13.7,1X,I2,(lX,E13.7)) for single precision, * C FORMAT(E22.16.1X.I2.cLEN>(1X.E22.16) 1 for double urecision. C where LEN = n+l. The following sample program could be-used to C read single-precison data in the output file whose name is given * C in OUTWHY in subroutine PARAM : C * C PROGRAM SAMPLE1 C INTEGER*4 NE,SIZE, I,J * C PARAMETER (NE=Z,SIZE=3000) C REAL*4 T(SIZE),H(SIZE),Y(NE,SIZE) !for double,use REAL*B * C INTEGER"4 V(S1ZE) C I=1 C OPEN(UNIT=10,FILE='EXAMPLE1.TVHY',STATUS='OLD') ClOO READ(10, *,END=200)T(I),V(I),H(I),(Y(J,I),J=l,NE) C I=I+1 * C GO TO 100 C200 CLOSE(10) c STOP

-

.....

.

C

E

N

... ...

D

*

L

C (iii). OUTSTEP is a sequential text file, which is usually used

C for plotting the solutions, with the following characteristics: C fixed length, direct access, formatted. This file contains the

*

A p p e n diz

380

C C C C C C C C C C C C C C C C C C C C C

interpolated numerical solution(s) at user-specified equal time increment, TINC, and has the following structure: record#l--> tl yl-1 y2-1 ... yn-1 record#2--> t2 y1-2 y2-2 ... yn-2

.....

* *

where tj = the jth time in which time increment is constant(real),* yi-j = the i th solution at time tj (real), n = the number of equations in the system.

*

Each record/line in OUTTVHY has been written with format statement* FORMAT(E13.7,(lX,E22.16)) for double precision, The following sample program could be used to read single* precison data in the output file whose name is given in OUTSTEP * in subroutine PARAM : t

PROGRAM SAMPLE2 INTEGER'4 NE,SIZE,I,J PARAMETER (NE=2,SIZE=3000) REAL*4 T(SIZE),Y(NE,SIZE) !for double,use REAL'8 I=l

C

*

OPEN(UNIT=11,FILE='EXAMPLE1.STEP',STATUS='OLD')

C10 C

READ(ll.*,END=20)T(I),(Y(J,I),J=l,NE) I=I+1 C GO TO 10 C20 CLOSE(l.1) C STOP C E N D C c-------------------------------------------------------------------*

* t

C

C C C C C C

E. Input (User-SuppliedValues):

*

The user needs to supply the values of some parameters through * (1) subroutine PARAM, (5) function BETA , (2) function Y, (6) function F , (3) function Z, (7) function YTHEOR, C (4) function ALPHA, ( 8 ) program MAIN. * C ..................................................................... C------ (1). The subroutine p m ----------------------------------* C User-specified variables are NE, TAU, INIV, ENDV, TOL, FORCE, C STAT, TVHY, STEP, DOUBLE, TABLE, TINC, OUTSTAT, OUTTVHY, and C OUTSTEP. The valid values to logical variables FORCE, STAT, TVHY, C STEP,and DOUBLE are either .TRUE. or .FALSE. * c-------------------------------------------------------------------* SUBROUTINE PARAM(NE,TAU,INIV,ENDV,TOL,FORCE,STAT,TVHY,STEP, & DOUBLE,TABLE,TINC,OUTSTAT,OUTTVHY,OUTSTEP) REAL*8 TAU,INIV,ENDV,TOL,TINC, PI LOGICAL*4 FORCE,WUBLE,STAT,WHY,STEP CHARACTER*30 OUTSTAT,OUTI'VHY,OUTSTEP CHARACTER'40 TABLE INTEGER NE PI=4.0DO*DATAN(l.ODO) NE=2 !number of equations in the system(=n) TAU=-O.SDO !minimum delay time at initial point(=tau),i.e. !min( alpha1 (t,Y (t), . . ,alphan (t,Y (t)) , ! betal(t,Y(t),..,betan(t,Y(t)) ) at t=INIV INIV=O.ODO !initial point of integration(=a) ENDV=S.ODO !final point of integration(=b) TOL=lO.ODO**(-I.2) !absolute error tolerance FORCE=.FALSE. !.TRUE. -->force endpoint t=b to be a grid STAT=.TRUE. !.TRUE. -->print a table of numerical statistics ! in OUTSTAT W H Y = .TRUE. !.TRUE. -->print order,stepsize,and numerical sol. I at grid points in OUTTVHY

38 1

Appendiz

STEP=.TRUE.

!.TRUE. -->print numerical sol. at equal time ! increment (TINC) in OUTSTEP DOUBLE=.FALSE. !.TRUE. -->store results in double precison in I OU'ITVHY and OUTSTEP TABLE='Table 1: Example 1' !title for the table in OUTSTAT TINC=O .2DO !equal time increment to print sol in OUTSTEP OUTSTAT='EXAMPLEl.STAT' !output file name for numerical statitics OU'ITVHY='EXAMPLEl.TVHY' !output file name for time,order,stepsize,sol OUTSTEP='EXAMPLEl.STEP' !output file name for sol. at equal time inc. RETURN

END

C------ (2). The function y ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t C User-specified function Y(IDX,T) computes the value of the C initial function g at time T for component IDX, that is, C Y := gl if IDX=l C Y := 92 if IDX=2, etc.

t

c-------------------------------------------------------------------*

FUNCTION Y (IDX,T) REAL*8 Y,T INTEGER IDX IF(IDX.EQ.1)THEN Y=l.OD0 ELSE Y=O .5DO ENDIF RETURN END C------ ( 3 ) . The function 2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t C User-specified function Z(IDX,T) computes the value of the C initial function g' at time T for component IDX, that is, * C Z : = gl' if IDX=1 C 2 : = 92' if IDX=2, etc. c-------------------------------------------------------------------t FUNCTION Z(IDX,T) REAL'8 2.T INTEGER IDX IF(IDX.EQ.1)THEN Z=O.OD0 ELSE Z=O .OD0 ENDIF RETURN

END

C------ ( 4 ) . The function ALPHA

C C

C C C C C C

.....................................

User-specified function ALPHA(IDX,T,YT) computes the value of the nonneutral state-dependent delay function alpha at time T for t component IDX, that is, ALPHA : = alphal(t,Y(t)) if IDX=l, t ALPHA : = alphaZ(t,Y(t)) if IDX=2, etc. * YT is the solution vector at time T where YT(1) := yl(t), t YT(2) := yl(t), etc.

c-------------------------------------------------------------------* €'UNCTION ALPHA(IDX,T,YT) INTEGER IDX REAL*8 ALPHA,T,YT(*) IF (IDX.EO .1\THEN ALPHA=T-YT (2) ELSE ALPHA=T ENDIF RETURN

END

C------ ( 5 ) . The function BETA ...................................... C User-specified function BETA(IDX,T,YT) computes the value of C the neutral state-dependent delay function beta at time T for

382

Appendix

C component IDX, that is, C BETA := betal(t,Y(t)) C BETA : = betaZ(t.Y(t))

if IDX=1, if IDX=Z. etc.

*

REAL'8 F,T,YT(*),YA(*),ZB(*) IF(IDX.EO.1)THEN F=-2.060 *YA ( 1) ELSE F=( DABS(YA(l))-DABS(YT(l)) )/(l.ODO+DABS(YA(l))) ENDIF RETURN END C------ ( 7 ) . The function YTHEOR ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~* C User-specified function YTHEOR(IDX,T) computes the theoretical C solution,if available,at time T. If not available, then YTHEOR has* C the reference solution, usually at end point t=b, for comparison C with the numerical approximation. The function YTHEOR is usually * C used for testing purpose in which the true/reference solution is c known. *

c-------------------------------------------------------------------* FUNCTION YTHEOR(IDX,T) INTEGER IDX REAL.8 YTHE0R.T IF(IDX.EQ.1)THEN YTHEOR=0.61385211789ZO276DO

____ ELSE

YTHEOR=O.7730855437697122DO ENDIF RETURN

END

C------ ( 8 ) . The program MAIN --------------------------------------* C User-specified variables are IP and MAX which are located at C at the declaration statement PARAMETER. * C IP := number of steps/grids allocated(memory allocation) C MAX:= maximum number of equations in the system allowed C (memory allocation) in which MAX is some finite number * C depending on computer memory and must be at least equal C to number of equations in the system. C-------------------------------------------------------------------~ PROGRAM MAIN INTEGER IP,MAX PARAMETER (IP=5000,MAX=6)

383

A p p e n diz

c-------------------------------------------------------------------* REAL'8 T(-2:IP),H(-2:IP),ZOUT(MAX,-2:IP),YOUT(MAX,O:IP), & PHIS(MAX,O:IP,15),PHIO(MAX,l5),PHIV(MAX,15),PHIl(MAX,lS), & PHI(MAX,15),CPl(MAX),GPl(MAX),GR1(15),ZHO(MAX),YHOV(MAX), & ZHOOV(MAX),YHOAV(MAX),ZHOOBV(MAX),ZHOV(MAX),YHlV(MAX), & YHlAV(MAX),ZHOBV(MAX),ZHlV(MAX),YHO (MAX),ZHOO (MAX), & YHOA(MAX),ZHOOB(MAX),EMAX(-2:1),E(MAX),TO,HO,TAU,INIV, & ENDV,TOL,SORTMAX,ALP,BET,Y,Z,ALPHA,BETA,F,PSI,YTHEOR,HN, B,C,SUM,TEMA,TEMB,ZMAX,ZTEN(MAX),ZH(O:15),TT(O:15),TINC, AERR(MAX),RERR(MAX),NSD(MAX),SOL(MAX),PPHI V(O:IP),LL,I,M,MM,VO,NUMF,FAIL,VN,NRS,K,

& &

INTEGER'4 &

NE, IDX,UNITl,UNIT2,UNIT3 CHARACTER'30 OUTSTAT,OUTTJHY,OUTSTEP CHARACTER'QO TABLE LCGICAL*4 FORCE,DOUBLE,NEXT,STAT,TVHY,STEP DATA UNITl/l/,UNIT2/2/,UNIT3/3/ CALL PARAM (NE,TAU,INIV,ENDV,TOL,FORCE,STAT,TVHY,STEP,

&

WUBLE,TABLE,TINC,OUTSTAT,OUTTVHY,OUTSTEP)

IF (NE.GT .MAX)THEN WRITE(6,*)'ERROR: Number of equations exceeds maximum allowed STOP ENDIF NRs=o NUMF=O

V(0)=1 V(l)=l I=O NEXT=.TRUE. C---- COMPUTING THE INITIAL VALUES: --------------------------c C---YOUT(IDX,O),ZOUT(IDX,-2),ZOUT(IDX,-1),ZOUT(IDX,O), --C C---PHIS(IDX,0,1),PHIS(IDX,0,2),PHIS(IDX,0,3), - -C C---PHI(IDX,l),PHI(IDX,2),PHI(IDX,3), - -C C---T ( - 2 ) , T ( - 1 ) , T ( 0 ) . T ( 1 ) , H ( - 2 ) , H ( - 1 ) . H o , H ~ 0 ~ ~ H ~ 1 ~ - -C W IDX=l,NE YHO (IDX)=Y (IDX,INIV) ENDW

W IDX=l,NE TEMA=ALPHA(IDX,INIV,YHO) TEMB=BETA(IDX,INIV,YHO) YHOA(IDX)=Y(IDX,TEMA) ZHOOB (IDX)=Z(IDX,TEMB) ENDW

DO IDX=l.NE

YOUT(IDX, 0)=YHO(IDX) ZOUT(IDX,O)=F(IDX,INIV,YHO,YHOA,ZHOOB) ZTEM ( IDX)=DABS (ZOUT( IDX,0 ) ) ENDDO

NuMF=NUMF+NE IF(NE.EQ.1)THEN ZMAX=ZTEM(l) ELSE ZMAX=SORTMAX(NE,ZTEM) ENDIF TEMA=O.lDO*(ENDV-INIV) IF(ZMAX.EQ.O.ODO)THEN H(l)=TEMA ELSE TEMB=0.5DO*TOL/ZMAX H(l)=DMINl(TEMA.TEMB) ENDIF TEMA= (INIV-TAU)/3.ODO H (0)=DMINl (TEMA,H (1)) H (-1)=H (0) H ( - 2 ) =H (0) T(O)=INIV T(1)=T(0)+H (1)

!'

Appendix

384 T(-l)=T(O)-H(O) T( -2)=T(-l)-H(-l) DO IDX=l,NE ZOUT(IDX,-1)=Z(IDX,T(-l)) ZOUT(IDX,-2)=Z(IDX,T(-2))

DO K=O, 2 ZH (K)=ZOUT(IDX,-K) T T ( K ) =T( -K) ENDDO DO M=1,3 PHI(IDX,M)=PPHI(O,M, ZH,TT,H) PHIS (IDX,0,M)=PHI (IDX,M)*B(l,M,H) ENDDO

ENDDO DO WILE(NEXT)

FAIL=O C----- CHECK IF WE WANT TO FORCE THE ENDPOINT TO BE A GRID ---C 2000 IF(FORCE.AND.(T(I+l).GT.ENDV))THEN T ( I+1) =ENDV H(I+l)=T(I+1)-T(I) ENDIF c-------------------------------------------------------------c HO=H(I+1) TO=T(I+l) vo=v( I+1) IF( (I+1).GT.IP)THEN WRITE(6,*)'ERROR: Insufficient storage(# of steps > IP) STOP ENDIF C----- UPDATE PHIS(I,M):M=1,2,..,P+2-------------------------C IF(FAIL.GT.O)THEN DO IDX=l,NE DO MM=l,VO+2 PHIS (IDX,1,MM)=PHI(IDX,MM)*B (I+l,MM,H) ENDDO

!'

ENDDO

ENDIF C----- PREDICTOR: YHO(T(I+l)) AT ORDER P----------------------C CALL GG( I,V0+2,1.ODO,H,GR1) DO IDX=l,NE SUM=O .OD0 DO M=l,VO SUM=SUM+GRl(M)*PHIS(IDX,1,M) ENDW YHO(IDX)=YOUT(IDX,I)+H(I+1)*SUM

C----- PREDICTOR: ZHOO(T(I+l)) AT ORDER P---------------------C ZHOO(IDX)=O.ODO DO M=l,VO+l ZHOO(IDX)=ZHOO(IDX)+PHIS(IDX, 1.M) ENDDO ENDDO C----- PREDICTOR: YHO(ALPHA(T(I+l))) AT ORDER P---------------C CALL YYHOA(MAX,IP,NE,I,T,H,V,YOUT,ZOUT,PHIS,YHO,GP1,YHOA) C----- PREDICTOR: ZHOO(BETA(T(I+l))) AT ORDER P---------------C CALL ZZHOOB (MAX, IP,NE,I,T,H,V,ZOUT,PHIS,YHO,CP1,ZHOOB) C----- PREDICTOR: ZHO(T(I+l)) AT ORDER P----------------------C DO IDX=l,NE ZHO (IDX)=F (IDX,TO,YHO,YHOA,ZHOOB) C----- PREDICTOR: PHIO(T(I+1),M);M=1,2,..,P+2 ----------------C PHIO(IDX,l)=ZHO(IDX) DO M=2,V0+2 PHI0 (1DX.M)=PHI0 (IDX,M-l) -PHIS(IDX,1,M-1) ENDDO C----- ERROR ESTIMATE AT ORDER p - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C

385

Appendix

IF EMAX(0) IS ACCEPTED, GO TO 1000 --------------------C IF(EMAX(O).LE.TOL)GO TO 1000 FAIL=FAIL+l IF(FAIL.EQ.1)THEN H(I+1)=0.5DO*HO ELSEIF( (FAIL.EQ.2).OR.(FAIL.EQ.3))THEN H(I+1)=0.5DO'HO V(I+1)=MAx0(1,VO-1) ELSE H(I+l)=O.SDO'HO V(I+l)=1 ENDIF T(I+l)=T(I)+H(I+l) GO TO 2000 IF((VO.GE.2) .AND.(VO.LE.l2))THEN ERROR ESTIMATE AT ORDER p-1 ---------------------------C DO IDX=l,NE E(IDX)=H(I+1)*DABS((GR1(VO)-GR1(VO-l))'PHIO(IDX,VO))

ENDDO EMAX(-l)=SORTMAX(NE,E) ENDIF

Do IDX=l,NE PREDICTOR: YHO(T(I+l)) AT ORDER P+1 -------------------C YHOV(1DX)=YHO (IDX)+H (I+1)'GR1 (VO+l)'PHIS (IDX,I,VO+l) PREDICTOR: ZHOOLT(I+l)) AT ORDER P+1 ------------------C ZHOOV(IDX)=ZHOO ( IDXj+PHIS(IDX,I,V0+2) PREDICTOR: YHOA(ALPHA(T(I+l))) AT ORDER P+1 -----------C ALP=ALPHA(IDX,TO,YHOV) IF (ALP.LE.T (I))THEN YHOAV(IDX)=YHOA(IDX) ELSE YHOAV(IDX)=YHOA(IDX)+H(I+1)'GP1(IDX)*PHIS(IDX,I,VO+1) ENDIF PREDICTOR: ZHOOB(BETA(T(I+l))) AT ORDER P+1 -----------C BET=BETA(IDX,TO,YHOV) IF (BET.LE .T( I))THEN ZHOOBV(IDX)=ZHOOB(IDX) ELSE ZHOOBV(IDX)=ZHOOB(IDX)+CP1(IDX)'PHIS(IDX,I,V0+2) ENDIF

ENDW

DO IDX=l,NE

PREDICTOR: ZHOV(T(I+l)) AT ORDER P+1 ------------------C ZHOV(IDX)=F(IDX,TO,YHOV,YHOAV,ZHOOBV) PREDICTOR: PHI(T(I+l),M);M=1,2,..,P+3 -----------------C PHIV(IDX,l)=ZHOV(IDX) DO M=2,V0+3 PHIV( IDX,M)=PHIV( IDX,M-1)-PHIS (IDX,1,M-1) ENDDO

ERROR ESTIMATE AT ORDER p+1 ---------------------------C E(IDX)=H(I+1)*DABS((GR1(V0+2)-GR1(VO+l~)*PHIV(IDX,V0+2)) ENDDO NuMF=NuMF+NE EMAX(l)=SORTMAX(NE,E)

A p p e n diz

CHECK IF THE SOLUTION IS ACCEPTABLE AT ORDER P+1 ------C IF(EMAX(l).GT.(2.0DO*TOL/DFLOAT(V0+2)))GO TO 3000 DO IDX=l,NE CORRECTOR: YHl(T(I+l)) AT ORDER P+l -------------------C YH1V(IDX)=YHOV(IDX)+H(I+1)*GR1(VO+l)*PHIV(IDX,V0+2) CORRECTOR: YHlA(ALPHA(T(I+l))) AT ORDER P+1 -----------C ALP=ALPHA(IDX,TO,YHlV) IF(ALP.LE.T(I))THEN YHlAV(IDX)=YHOAV(IDX) ELSE YHlAV(IDX)=YHOAV(IDX)+H(I+1~*GP1(IDX~*PHIV(IDX,V0+2) ENDIF CORRECTOR: ZHOB(BETA(T(I+l))) AT ORDER P+1 ------------C BET=BETA(IDX,TO,YHlV) IF(BET.LE.T(I))THEN ZHOBV(IDX)=ZHOOBV(IDX) ELSE ZHOBV(IDX)=ZHOOBV(IDX)+CP1(IDX)*PHIV(IDX,V0+3) ENDIF ENDDO

DO IDX=l,NE CORRECTOR: ZHl(T(I+l)) AT ORDER P+1 -------------------C ZHlV(IDX)=F(IDX,TO,YH1V,YH1AV,ZHOBV) CORRECTOR: PHIl(T(I+l),M);M=1,2, ..,P+2 ----------------C PHIl(IDX,l)=ZHlV(IDX) DO M=2,V0+2 PHI1(IDX,M)=PHI1(IDX,M-1)-PHIS(IDX,I,M-l) ENDDO CORRECTOR: ACCEPTED SOLUTION AT ORDER P+1 -------------C YOUT(IDX.I+l)=YHlV(IDX) ZOUT(IDX,I+l)=ZHlV(IDX) ENDDO NuMF=NuMF+NE COMPUTE THE NEW ORDER:=VN AND NEW STEP SIZE:=HN -------C CALL NEWPH (EMAX,FAIL,VO , HO ,TOL,VN,HN) H (I+2)=HN V(I+2) =VN T(I+2)=T(I+l)+H(I+2) DO IDX=l,NE COMPUTE PHIS(T(I+l).M);M=1,2,..,P+2 ------------>------C DO M=l,V(I+1)+2 PHI(IDX,M)=PHIl(IDX,M) PHIS(IDX,I+1,M)=PHI(IDX,M)*B(I+2,M,H) ENDDO ENDDO

CHECK IF T END 1s REACHED -----------------------------C IF (T( I+1).GE.ENDV)THEN NEXT=.FALSE. ELSE I=I+1 NRS=NRS+FAIL ENDIF

ENDDO

C----- OUTPUT STEP SIZES,ORDERS,SOLUTION,AND NUMERICAL STATS -C NRS=NRS+FAIL CALL OUTPUT (MAX,IP,NE,I,INIV,ENDV,T,H,V,PHIS,YOUT, &TOL,NRS,NUMF,TABLE,OUTSTAT,OUTTVHY,OUTSTEP, STAT,TVHY,STEP, &UNITl,UNITZ,UNIT3,DOUBLE,TINC,AERR, RERR,NSD,SOL) STOP END

c-------------------------------------------------------------c C---- FUNCTION PSI computes sum of step sizes. FUNCTION PSI (J,M,H)

- -C

387

Appendix

REAL'S H(-2:*),PSI INTEGER J.M.L PSI=O .OD0 Do LZ0.M-1 PSI=PSI+H(J-L) ENDW r ---

RETURN END

....................

C---- FUNCTION B computes ratio of two sums of step sizes. FUNCTION B(J,M,H) REAL'S H (-2: ') ,B , PSI INTEGER M,J,K IF (M.EQ .1)THEN B = l . OD0 ELSE B=l. OD0 Do K=l,M-l

--C

B=B'PSI(J,K,H)/PSI(J-1,K.H) ENDW

ENDIF RETURN END

c-------------------------------------------------------------c C---- FUNCTION C computes the coefficients associated with C---- the interpolating polynomials. FUNCTION C (J,M,R,H) REAL'S H(-2:* ) ,C,PSI,RH, R INTEGER M,J,K IF (M.EQ.1)THEN C=l.OD0 ELSEIF(M.EQ.2)THEN C=R ELSE RH=R*H(J+l) C=RH/PSI(J+l,1,H) DO K=2,M-1

--C

- -C

C=C*(RH+PSI(J,K-l,H))/PSI(J+l,K,H) ENDDO

ENDIF RETURN END ~

C------------------------------------------------------------- C C---- SUBROUTINE GG computes the coefficients of the Adams --C C---- methods recursively. --C

SUBROUTINE GG(J,N,R,H,G) REAL'S H( -2:* ) ,GM(15,15) ,G(15), PSI,ALP,R INTEGER M,N.J.Q.K W Q=l,N GM ( 1,Q = ( R ' ' Q ) /DFLOAT(Q) ENDW

W M=2,N W Q=l,N-M+l ALP=H (J+1)/PSI(J+1,M-1,H) IF(R.EQ.1.ODO)THEN GM (M,Q ) =GM (M-1,Q ) -ALP*GM (M-1,Q+1) ELSE &

GM(M,Q)=(ALP*R+(PSI(J.M-2,H)/PSI(J+l,M-l.H)))' GM(M-1,Q)-ALP*GM(M-1,Q+1) ENDIF ENDW

ENDW W M=l,N G(M)=GM(M,1) ENDW

RETURN

A p p e n d iz

FUNCTION PPHI ( J,M, ZH,TT, H) REAL*8 H(-2:*),ZH(O:15),TT(O:15),ZZ(0:15),PPHI,PSI INTEGER J . M.. K.. L

.

DO K=O,M-1 ZZ(K)=ZH(K) ENDDO IF(M.EQ.1)THEN PPHI=ZZ(O) ELSE Do Ikl.M-1 DO L=M-l,K,-l ZZ(L)=(ZZ(L)-ZZ(L-l))/(TT(L)-TT(L-K)) ENDDO ENDDO

PPHI=ZZ(M-l) DO K=l.M-l

PPHI=PPHI'PSI(J,K,H) ENDW

ENDIF RETURN END c-------------------------------------------------------------c C---- Function SORTMAX computes maximum value of the array C---- of dimension NE. FUNCTION SORTMAX(NE,ARRAY) INTEGER NE,COUNT,LOOP REAL*8 ARRAY(*),SORTMAX,TEM Do LOOP=l,NE-l Do COUNT=l.NE-l IF(ARRAY(COUNT+l).GE.ARRAY(COUNT)) GO TO 10 TEM=ARRAY(COUNT) ARRAY (COUNT)=ARRAY (COUNT+1) ARRAY (COUNT+1) =TEM 10 CONTINUE ENDDO ENDDO SORTMAX=ARRAY(NE) RETURN END

--C --C

c-------------------------------------------------------------c

C---- Subroutine YYHOA --C C---- computes the y (alpha(t ) ) for order P . --C C---h i+l i+l - -C C---- output: YHOA(IDX),GPl(IDX) for IDX=1,2,...,NE. --C SUBROUTINE YYHOA(MAX,IP,NE,I,T,H,V,YOUT,ZOUT,PHIS,YT,GP1,YHOA) INTEGER MAX,1P.M. I,J,V(0 : ) ,NE,DI REAL*8 T(-2:*),H(-2:*),ZOUT(MAX,-2:*),YOUT(MAX,O:'),YHOA(*), & PHIS(MAX,O:IP,*),YT('),GPl(*) ,R,ALP,ALPHA,Y,SUM,G(15),B DO IDX=l,NE ALP=ALPHA(IDX,T(I+l),YT) IF(ALP.LE.TLO))THEN YHOA( IDX)=YiIDX,ALP) ELSEIF( (ALP.GT.T(I)).AND.(ALP.LE.T(I+l)) )THEN C---- use method of order P --C C---i+l - -C R= (ALP-T(I)) /H(I+1) CALL GG(I,V(I+l)+l,R,H,G) SUM=O .OD0 W M=l,V(I+l) SUM=SUM+G(M)*PHIS (IDX,I ,M) ENDDo

YHOA(IDX)=YOUT(IDX,I)+H(I+~)+SUM

G P (IDX) ~ =G(V(I+1)+1)

389

Appendix

ELSE C---- use method of order (P + 1) C---j+l J=O DO WHILE((ALP.LE.T(J)).OR.( J=J+1 ENDDO R=(ALP-T(J))/H(J+l) CALL GG (J,V (J+1)+1,R,H,G) SUM=O .OD0 DO M=l,V(J+l)

--C --C ,P.GT.T (J+

SUM=SUM+G(M)*PHIS(IDX,J,M)

ENDW GPl(IDX)=G(V(J+l)+l) YHOA(IDX)=YOUT(IDX,J)+H(J+1)*(SUM+GP1(IDX)' PHIS(IDX,J+1,V(J+1)+1)/B(J+2,V(J+1)+1,H))

&

ENDIF ENDDO

RETURN END c-------------------------------------------------------------c C---- Subroutine ZZHOOB - -C C---- computes the z (beta(t ) ) for order P - -C C---h i+l i+l --C C---- output: ZHOOB(IDX),CPl(IDX) f o r IDX=1,2,...,NE. --C SUBROUTINE ZZHOOB(MAX,IP,NE,I,T,H,V,ZOUT,PHIS,YT,CPl, ZHOOB) INTEGER MAX, IP,M,I,J,V( 0 : ) ,NE, IDX REAL*8 T ( -2: ) ,H ( -2 : * ) , ZOUT(MAX, -2: ) , PHIS (MAX, 0 : IP, , ZHOOB ( * ) ,CP1(*) ,YT ( ) , R,BET,BETA,2 , B,C & DO IDX=l,NE BET=BETA(IDX,T(I+l),YT) IF (BET.LT .T(0) )THEN ZHOOB(IDX)=Z(IDX,BET) ELSEIF(BET.EQ.T(O))THEN ZHOOB(IDX)=ZOUT(IDX,O) ELSEIF((BET.GT.T(I)).AND.(BET.LE.T(I+l)))THEN C---- use method of order P - -C C---i+l - -C R= (BET-T(I)) /H(I+1) ZHOOB(IDX)=O.ODO DO M=l,V(I+l)+l ZHOOB(IDX)=ZHOOB(IDX)+C(I,M,R,H) *PHIS(IDX,1.M) ENDDO

ELSE --C C---- use method of order (P + 1) --C C---i+l J=O DO WHILE((BET.LE.T(J)).OR.(BET.GT.T(J+l))) J=J+1 ENDDO R=(BET-T(J))/H(J+l) ZHOOB(IDX)=O.ODO DO M=l,V(J+l)+l ZHOOB(IDX)=ZHOOB(IDX)+C(J,M,R,H)*PHIS(IDX,J,M) ENDDO CP1 (IDX)=C (J,V(J+l)+2,R,H) ZHOOB(IDX)=ZHOOB (IDX)+CP1 (IDX)'PHIS (IDX,J+l,V(J+l)+2) /B (J+2,V(J+1)+2,H) & ENDIF ENDW

RETURN END

c-------------------------------------------------------------c C---- Subroutine NEWPH computes new order and new step size.--C c----output: vN,HN --C

Appendiz

390 SUBROUTINE NEWPH(E,FAIL,VO,HO,TOL,VN,HN) REAL'8 E ( -2: 1) ,TEMA,TEMB ,HO ,HN,ERR,Q,TOL INTEGER VO,VN,FAIL,VMAX

IF (FAIL.EQ o )THEN IF((VO.EQ.l).AND.(2.ODO'E(l).LT.E(O)))THEN VN=VO+l ELSEIF((VO.EQ.2).AND. (E(O).GT.2.0DO*E(l)).AND.(E(O).LT.E(-l)))THEN

VN=vo+1 ELSEIF((VO.GT.2).AND.(VO.LT.VMAX).AND. (E(O).GT.2.0DO*E(l)).AND.(E(O).LT.TEMB))THEN

VN=VO+l ELSEIF((VO.GT.l).AND.(VO.LT.VMAX).AND.

(E(-1).LE.TEMA))THEN VN=vo- 1 ELSEIF((VO.EQ.VMAX).AND.(TEMB.LE.E(O)))THEN VN=vo- 1 ELSE

VN=vo ENDIF

ELSEIF((VO.GT.2).AND.(TEMB.LT.E(O)))THEN vN=vo- 1

ELSE VN=vo ENDIF ENDIF IF(VN.EQ.(VO-1))THEN ERR=E(-l) ELSEIF (VN.EQ .VO) THEN ERR=E ( 0) ELSE ERR=E( 1) ENDIF TEMA=ERR*(2.ODO*'(VN+l)) TEMB=O.SDO*TOL IF (TEMA.LE .TEMB)THEN HN=Z.ODO*HO ELSEIF(ERR.LE.TEMB)THEN HN=HO ELSE TEMB=l.ODO/DFLOAT(VN+l) TEMA=(0.5DO*ML/ERR)**TEMB TEMB=DMAXl(O.SDO,TEMA) Q=DMIN1(0.9DO,TEMB) HN=Q*HO ENDIF RETURN END c-------------------------------------------------------------c C---- Subroutine NUMSOL computes numerical solution --C C---- output: AERR(IDX),RERR(IDX),NSD(IDX),HMIN,HMAX,YT(IDX)--C SUBROUTINE NUMSOL(MAX,IP,NE,I,T,H,V,PHIS,YOUT,INIV,ENDV,TINC, & DOUBLE,STEP,OUTSTEP,UNIT3,AERR,RERR,NSD,HMIN,HMAX,YT) INTEGER IP,MAX,V(O:*),M,J,L,1.C0UNT.11, IDX,NE,RLEN,UNIT3 REAL*8 T(-2:'),H(-2:'),YOUT(MAX,O:'),PHIS(MAX,O:IP,*), & AERR(MAX) ,RERR(MAX),NSD(MAX),YT(MAX),G(15),Tl,R, SUM, HH,HMIN,HMAX, B,YTHEOR,YTRUE,TINC,INIV,ENDV & LOGICAL'I WUBLE,STEP CHARACTER*30 OUTSTEP IF (STEP)THEN

391

Appendix

IF(DOUBLE)THEN RLEN=(NE+l)*23-1 ELSE RLEN=(NE+l)'14-1 ENDIF OPEN(UNIT=UNIT3,FILE=OUTSTEP,STATUS='NEW',RECORDTYPE='FIXED', FORM='FORMATTED',ACCESS='DIRECT',RECL=RLEN) COUNT=JIDINT((ENDV-INIV)/TINC) IF(DFLOAT(COUNT)*TINC.LT.ENDV)THEN

&

couNT=couNT+1 ENDIF ELSE

couNT=o ENDIF DO II=l,COuNT+1 IF(II.EQ.(COUNT+l))THEN Tl=ENDV ELSE T1=INIV+DFLOAT(II-1)*TINC

ENDIF J=O &

DO WHILE(((II.EQ.(COUNT+1)).OR.(II.GT.l)).AND. ((Tl.LE.T(J)).OR.(Tl.GT.T(J+l))))

J=J+1 ENDDO

R=(Tl-T(J))/H(J+l) DO IDX=l,NE CALL GG (J,V (J+1)+1,R,H,G ) SUM=O .OD0 DO L=l.V(J+l) SUM=SUM+G (L)'PHIS ( IDX,J,L) ENDDO

'PHIS (IDX,J+1,V(J+1)+1)/B(J+2,V(J+1)+1,H) IF(II.EQ. (COuNT+l))THEN YTRUE=YTHEOR(IDX.Tl) AERR(IDX)=DABS( (YTRUE-YT(IDX))) IF(YTRUE.EQ.O.ODO)THEN RERR ( IDX)=l.ODO ELSE RERR(IDX)=AERR(IDXI/DABS(YTRUE) ENDIF C----- NSD=O<=>solution=O, and NSD=99.99<=>absolute error=O --C IF(RERR(1DX).EQ.O.ODO)THEN NSD(IDX)=99.99DO ELSE NSD(IDX)=-DLOG10(RERR(IDX) ENDIF ENDIF ENDDO IF (STEP)THEN IF (DOUBLE)THEN WRITE(UNIT=UNIT3,REC=II,F'MT=1O)T1,(YT(IDX),IDX=1,NE) &

FORMAT(E22.16,(lX,E22.16))

10

ELSE WRITE(UNIT=UNIT3,REC=II,FM"=20)T1,(YT(IDX),IDX=1,NE) 20

FORMAT( El3 .7, ( lX,El3 .7) ) ENDIF IF(II.EQ.(COUNT+l))THEN CLOSE(UNIT3) ENDIF ENDIF ENDDO

DO L=l,I Do M=l,I-l IF(H(M+l).LT.H(M) )THEN

Appendix

392

HH=H (M+1) H(M+1)=H (M) H(M)=HH ENDIF ENDDO ENDDO

HMIN=H( 1) HMAX=H(I) RETURN END

c-------------------------------------------------------------c C---- Subroutine OUTPUT prints the step sizes, order, and --C C---- numerical solution and statistics to the output files.--C SUBROUTINE OUTPUT (MAX,IP,NE,I,INIV,ENDV,T,H,V,PHIS,YOUT, &TOL,NRS,"MF,TABLE,OUTSTAT,OUTTVHY,OUTSTEP,STAT,TVHY,STEP, &UNITl,UNITZ,UNIT3,DOUBLE,TINC,AERR,RERR,NSD,SOL) INTEGER MAX,IP,NE,I,V( 0 :* ) ,IDX,NRS,NUMF, 11,UNITl,UNIT2,UNIT3, & LINE,LEN,RLEN REAL'8 T(-2:'),H(-2:'),PHIS(MAX,O:IP,'),YOUT(MAX,O:*),TOL, & AERR(MAX) ,RERR(MAX),NSD(MAX),HMIN,HMAX,SOL(MAX), & INIV,ENDV,TINC LOGICAL DoUBLE,STAT,TVHY,STEP CHARACTER'30 OUTSTAT,OUTTVHY,OUTSTEP CHARACTER'40 TABLE IF ( TVHY ) THEN LEN=NE+1 IF (DOUBLE)THEN RLEN=(NE+2)*23+2 ELSE RLEN=(NE+2)'14+2 ENDIF OPEN(UNIT=UNIT1,FILE=OU?TVHY,STATUS='NEW',RECORDTYPE='FIXED', & FORM='FORMAlTED'.ACCESS='DIRECT'.RECL=RLEN) Do II=O,I+l LINELII+l IF(WUBLE)THEN WRITE(UNIT=UNIT1,REC=LINE,FMT=100)T(II),V(II),H(II), . . .. . . . .. & (YOUT(IDX,11);IDx=i,mj 100 FORMAT(E22.16,1X,I2,(1X,E22.16)) ELSE WRITE(UNIT=UNIT1,REC=LINE,FMT=200)T(II),V(II),H(II), & (YOUT(IDX,II),IDX=l,NE) FORMAT(E13.7,1X,12,(lX,E13.7)) 200 ENDIF ENDDO

CLOSE (UNIT1) ENDIF C-- - - CONMPUTE THE NUMERICAL SOLUTION AT T=ENDV -------------I2 CALL NUMSOL(MAX,IP,NE,I,T,H,V,PHIS,YOUT,INIV,ENDV,TINC, DOUBLE,STEP,OUTSTEP,UNIT3,AERR, RERR,NSD,HMIN,HMAX,SOL) & IF (STAT)THEN OPEN(UNIT=UNITZ,FILE=OUTSTAT,STATUS="EW') C----- WRITING THE RESULT INTO THE OUTPUT DATA FILE ----------C WRITE (UNIT2,lO) TABLE FORMAT ( / ,25X.A40,/ ) 10 Do IDX=l,NE WRITE(UNIT2,20) FORMAT(9X,'TOL',SX, 'AERR',SX,'RERR',SX, 'NSD',4X, 20 & 'NS',4X,'NRS',2X,"FE',3X,'HMIN',SX,'HMAX',/) WRITE(UNIT2,30)TOL,AERR(IDX),RERR(IDX),NSD(IDX) & I+l,NRS,"MF, HMIN,HMAX FORMAT(7X,E7.l,lX,2 (E8.2.1X),FS.Z.lX,3 (15,lX),2(E8.2,lX) 30 WRITE(UNIT2.SO)IDX.ENDV.SOL(IDX) 50 FORMAT(' ~ R I C A SOLUTION: L ;,ix;'Y' ,12,' ( ' ,~23.17,') = & E23.17) WRITE(UNIT2, * ) ' ' 8 ,

Appendix

393

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Index A

competition assumptions 285 completely continuous 19 continuation 19 continuous dependence 19, 20 critical point 22 cycle 277

A.G.A.S. 149 absolutely globally asymptotically . . stable 149 acyclic 277 Allee effect 143 a-limit set 22 asymptotically smooth 275 asymptotically stable 25 asymptotically stable in the C'-metric 348 atomic 16 attracting set 275 autonomous 16, 17

D DDEs 3 decomposition theorem 55 delayed logistic equation 4 difference operator 17 discrete dynamical system 22 dissipative 286 dynamic diseases 7 dynamical system 22

B

E

Bellman's inequality 52 bilinear form 54

eigenfunction 54 eigenvalue 24, 54 eigenvector 54 ejective point 60 epidemic model 6 equilibrium 22 existence 19, 20 exponentially asymptotically

C Co-semigroup 274 chain 277 chaos 195 characteristic equation 24 characteristic root 24

395

396

Index

stable in the C’-metric 348

F FDEs 3 fading memory space 46 formal adjoint 56 formal adjoint equation 56 G G.A.S. 149 Gause-type predator-prey system 247 general linear system 51 generalized eigenspace 54 global attractor 275 global existence 51 global solution 134 globally asymptotically stable 39, 149 Gronwall’s inequality 52

isolated covering 277 isolated invariant set 275 isolating neighborhood 275

L Liapunov functional 28, 33 Liapunov-LaSalle type theorem 30 Liapunov-Razumikhin pair 48 linear chain 97 linear homogeneous 16, 17 linear nonhomogeneous 16, 17 linearized equation 26 Lipschitz 19 lossless transmission 8

M maximal interval of existence 19 Michaelis-Menton growth equation 132 Michailov criterion 106

H

N h - U.B. 281 h - U.U.B. 281 h-uniformly bounded 281 h-uniformly ultimately bounded 281 hematopoiesis 8 Hopf bifurcation 58 Horn’s fixed point theorem 264 Hutchinson’s equation 4

I ideal predictor 11 infinite delay 46 infinitesimal generator 53 invariance principle 48 invariant set 23

NFDEs 3 negatively invariant set 23 neutral delay competition model 313 neutral delay predator-prey system 312 neutral delay differential equation 17 neutral delay logistic equation 6 nonautonomous 16, 17 noncontinuable 20 nonoscillatory 120 null space 54

Index

0 ODES 3 w-limit set 22 orbit 22 oscillatory 120

P PDEs 3 periodic neutral logistic equation 346 permanence 273 permanently coexistent 273, 281 persistence 273, 276 point dissipative 275 positively invariant set 23 precompact 47 predator-prey assumptions 286 predator-prey model 5 principle of causality 3 process 21

R RDDE 3 RFDEs 3 Razumikhin type theorem 38 repelling set 275 resolvent set 53 retarded delay differential equation 15 retarded functional differential equation 15

S saturated equilibrium 226 smoothing property 20 solution operator 21

397

spectrum 53 stability of equation 72 stable 23, 25 stable set 275 stage structure 187 steady state 22 strongly bounded 276 strongly continuous semigroup 53 strongly point dissipative 276 structured population 191

T threshold delay 191 trajectory 22

U UC, 46 uniformly asymptotically stable 25 uniformly bounded 27 uniformly persistent 276 uniformly stable 25 uniformly ultimately bounded 45 uniqueness 19, 20 unstable 25 unstable set 275

V VL-stable 242 variation equation 26

W weakly stable set 275

Index

398

weakly unstable set 275 Wright’s conjecture 125 Wright’s equation 4

Y Yorke condition 141

Mathematics in Science and Engineering Edited by William F. Ames, Georgia Institute of Technology Recent titles

T. A. Burton, Volterra Integral and Differential Equations C. J. Harris and J. M. E. Valenca, The Stability of Input-Output Dynamical Systems George Adomian, Stochastic Systems John O'Reilly, Observers for Linear Systems Ram P. Kanwal, Generalized Functions: Theory and Technique Marc Mangel, Decision and Control in Uncertain Resource Systems K . L. Teo and Z. S. Wu, Computational Methods for Optimizing Distributed Systems Yoshimasa Matsuno, Bilinear Transportation Method John L. Casti, Nonlinear System Theory Yoshikazu Sawaragi, Hirotaka Nakayama, and Tetsuzo Tanino, Theory of Multiobjective Optimization Edward J. Haug, Kyung K. Choi, and Vadim Komkov, Design Sensitivity Analysis of Structural Systems T . A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations Yaakov Bar-Shalom and Thomas E. Fortmann, Tracking and Data Association V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Applications to Numerical Analysis B. D. Vujanovic and S. E. Jones, Variational Methods in Nonconsewative Phenomena C. Rogers and W. F. Ames, Nonlinear Boundary Value Problems in Science and Engineering Dragoslav D. Siljak, Decentralized Control of Complex Systems W. F Ames and C. Rogers, Nonlinear Equations in the Applied Sciences Christer Bennewitz, Differential Equations and Mathematical Physics Josip E. Pekari'c, Frank Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems E. Adams and U. Kulisch, Scientific Computing with Automatic Result Verification Viorel Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems Yang Kuang, Delay Differential Equations: With Applications in Population Dynamics W. F. Ames, E. M. Harrell 11, and J. V. Herod, Differential Equations with Applications to Mathematical Physics