Theory of Fitness in a Heterogeneous Environment

Theory of Fitness in a Heterogeneous Environment. I. The Fitness Set and Adaptive Function Author(s): Richard Levins Source: The American Naturalist, Vol. 96, No. 891 (Nov. - Dec., 1962), pp. 361-373 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2458725 . Accessed: 24/06/2011 01:07 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ucpress. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Vol. XCVI, No. 891

The American Naturalist

November-December, 1962

THEORY OF FITNESS IN A HETEROGENEOUS ENVIRONMENT. 1. THE FITNESS SET AND ADAPTIVE FUNCTION* RICHARD LEVINSt Department of Zoology, Columbia University, New York, N. Y.

In recent years there has been a gradual shift in interest in evolutionary theory fromthe quest for general principles to the interpretation of differences between genetic systems. Whereas previously the differentgenetic systems found in nature were interesting either as furtherexamples of universal phenomena or as especially favorable objects of research for studying some general principle, we are now in the first stages of comparative genetics, in which the differences are themselves of primaryconcern. A number of attempts have been made to account for differences among genetic systems in terms of differences in the kinds of environmentsfaced by the species. Thus Darlington (1946) suggested that the annual habit in plants required successful seed set every year and thereforeprecluded too much genetic "experimentation" whereas the perennials had a much greater probability of hitting at least one good year for seed production during the life span, and could thereforeaffordmore recombination than could annuals. Da Cunha and Dobzhansky (1954) proposed that chromosomal polymorphism in Drosophila is indicative of the exploitation of environmental heterogeneity. Lewontin went furtherand argued (1958) that balanced polymorphism will disappear in a uniformconstant environment. The present study is an attempt to explore systematically the relationship between environmental heterogeneity and the fitness of populations. For each pattern of environmentexamined, we will determine which population characteristics would provide the maximumfitness, where fitness is defined in such a way that interpopulation selection would be expected to change a species toward the optimum(maximum fitness) structure. That is, we are going to establish a correspondence between the optimal structures of populations and species and the pattern of environmentalheterogeneity in space and time. This will make it possible to test the theory against natural and experimental populations. The study is divided into two parts which use differentkinds of models and depend on differentmathematical techniques. In this and the next paper (Levins, 1963) we assume that populations are static, unchanging in time. This would be optimal for situations in which the environmentis constant in time or in which there is no correlation between the environments of successive generations. Subsequent papers will consider the changing population in a fluctuating environment. *This paper is dedicated to Professor L. C. Dunn. tPresent address: Department of Biology, University of Puerto Rico, Rio Piedras, Puerto Rico.

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Throughout the present paper we assume the following model: The environment exists in two discrete alternative phases or niches, in each of which a differentphenotype is optimal. The two niches occur in the proportions p, 1 - p within the territoryof a local population, where p may be constant or variable. However, if p is variable, we assume that there is no correlation between the environments of successive generations. In such situations, the optimumpopulation remains constant regardless of the environment of the previous generation. The fitness of an individual in a given niche is assumed to depend on the phenotype of the individual and the environmentof the niche but not on the composition of the species. In section 1-the Fitness Set-we introduce a multi-dimensional representation of the fitness of individuals and populations, in which each axis corresponds to the fitness of an individual (or a population) in a different niche. For convenience we will consider two niches and a two dimensional Fitness Set, but the extension to a more general situation can be made readily. The fitness of a population is not uniquely determined by the Fitness Set. It depends on the fitnesses in the niches taken separately, but also on the way in which the niches are related to each other, that is, whether they In section 2 the Adaptive Function occur simultaneously or successively. of the whole population in the as a of measure the fitness is introduced form depends on the pattern of that heteroheterogeneous environment. Its geneity. environmental In section 3-Examples of Adaptive Functions-several patterns are examined, the corresponding Adaptive Functions are defined, and the optimumpopulation characteristics found for each. The results are expressed in terms of the shape of the Fitness Set, especially whetherit is convex or concave. In section 4-the Shape of the Fitness Set-we consider the biological meaning of formas related to homeostasis and degree of difference between niches.. The results of the investigation are summarized, but discussion and conclusions are deferredto the following paper (see Levins, 1963), in which we consider modifications in the theory resulting froma relaxation of the assumptions that the phenotype is independent of the environment,that the fitness of an individual is independent of the population composition, and that individuals are distributed at random among the niches. This concluding section (Levins, 1963) examines the nature of the correspondence between the optimal populations predicted by the theory and actual populations in nature. Several approaches are suggested for testing the theory. In subsequent investigations we intend to analyze the problems of a population that is responding to selection, with or without advantage. The adaptive significance of mutation and migration will be considered. 1. THE FITNESS SET

Let there be n differentenvironmentsnumbered 1 to n and let wi be the average number of offspringthat would be produced by an individual of a

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given phenotype in the ith environment if no resource like food or space were in short supply. As defined, wi is independent of population density, and we will furtherassume for the present that wi is independent of population composition. Thus, wi is analogous to Andrewarthaand Birch's innate capacity to increase (1954). Then every phenotype can have its fitness represented by a point (w1, w2, . . . wn) in an n-dimensional fitness space. For convenience we will assume that the environmentconsists of two discrete phases or niches, so that the fitness space is two-dimensional and the two axes are w1 and w2, the fitnesses in niches 1 and 2 respectively. If we represent all phenotypes possible for a given species by points in the fitness space, we get a set of points which we will call F, the Fitness Set. F is assumed to be bounded and continuous. Phenotypes represented by the same point in the Fitness Set are indistinguishable to natural selection under our assumptions and will be considered identical in the discussions that follow. A population consists of one or more phenotypes gj each occurring with a frequency qj. Since wi is independent of population composition, the population is represented in the fitness space by the point (w1, w2) where wi -

qt wij

and wij is the fitness of phenotype in niche i. Thus, all populations composed of a mixture of individuals of two phenotypes will be represented in the fitness space by points lying on the straight line joining the representations of the component types. We define the Extended Fitness Set, F'. as the set of all points in the fitness space that represent possible populations, that is, all possible mixtures of the phenotypes represented in the set F. A convex set is a set of points such that the straight line joining any two points of the set lies entirely within that set. It follows fromthe definition of convexity and fromthe continuity of F that the set F' of all points representing populations is the smallest convex set enclosing F. If F is itself convex, then F = F' identically whereas if F is concave F' will contain points that are not in F., In other words, the convexity of F is equivalent to the statement that all possible fitness points can be attained by monomorphic populations, whereas if F is not convex there are fitness points that can only be attained by polymorphicpopulations. The values of w1 and w2, the average numbers of offspringin niches 1 and 2 in unlimited space, can be related to each other in a numberof ways. We will indicate three models and then proceed only with the third. Model I-Let w1 and w2 vary independently, with 0 A w1 Max(w1) and o ? w2 ? Max(w2). Then F is a rectangle, and identical with F'. Model II-The average number of offspring depends on some generalized propertywhich operates in all niches. For example, let w1 = Cw2 where C is a positive constant. Then F is a degenerate convex set, the line segment 0 < w 1 = Cw 2 < Max(w 1).

364 Model III-The

THE AMERICANNATURALIST OptimumDeviation Model.

If the fitness of a given phenotype is measured over a range of values of some environmental variable, for example, temperature,the result is generally a roughly bell-shaped curve. Such a curve can be described by the height of the peak, the location of the peak (that is, the environmentwhich is optimal for that phenotype), and the way in which fitness declines when the environmentdeviates fromthe optimal environment. We simplify the situation by assuming that all the peaks have the same height and that fitness in environmenti is given by wi=

w(i,y)=

w[CIS-i

yL

where Si is the value of the actual environmentin niche i, and y is the optimumenvironmentfor the phenotype in question. The variable y depends on many morphological and physiological properties of the organism. It is therefore a measure of the phenotype although expressed in the units of the environment. Therefore, Si - y is also the deviation of the actual phenotype fromthe optimumphenotype for niche i. W(z) is a non-negative, monotonically decreasing function of its argument. Therefore, the fitness wi has its maximumwhen y = Si, C is a positive constant which measures the intensity of selection against a given deviation fromthe optimum. We can regard C as measuring inversely the homeostasis of the organism, its ability to survive and reproduce in non-optimal environments. We assume that W(z) is the same function for all phenotypes, which differ only in y and C. C is assumed to be the same in each niche for a given phenotype and capable of varying independently of y. We exclude a zero value of C since that would result in all phenotypes being equally fit in all environments. 2. THE ADAPTIVE FUNCTION

In the previous section we defined the fitness wi in a single environment. With two niches the fitness is represented by a pair of numbers w1, w2. We would like to measure the fitness in a heterogeneous environmentby a single numberwhich depends on w1 and w2. Therefore, we define A(w1, w2) as the Adaptive Function. We make the reasonable assumption that A(w1, w2) is a monotonic increasing function of both arguments, that is, A(w1, w2) is always increased by increasing either w1 or w2. The Adaptive Function is chosen so as to describe as well as possible the selective value or fitness of a population relative to other populations in interpopulation selection. Its functional formwill depend on the type of selection and the pattern of environmentalheterogeneity in space and time. We define the optimumpopulation as that population whose representation in the Extended Fitness Set F' has the greatest value of A(w1, w2). We can now prove the following theorem: In the optimummodel, any population which maximizes Adaptive Function A(w1, w2) is represented by a point on the boundary of F' between the points [W(1, Sj), W(2, Sd] and [W(1, S2)7 W(2, S2)1*

THEORY OF FITNESS.

I.

365

Proof: The point (wl, w2) must lie on the boundary of F', because if it did not there would be some point (wl', w2') in the neighborhood of (wl, w2) for which wl' > w1 and w2' >w2. But then A(w1', w2') would be greater than A(wl,

w2).

If yo is some value of y < S1 < S2, and then PO is a population which includes a proportion q0 of yo, then the fitness of PO in niche i is W(i, PO) -

1=11

W(i, y'i) + q0 W(i, Yo)

But since W(i, y) = W(C ISi - y I ) increases monotonically with y for y < S1 <SK2 we can improve Wi for i = 1 and i = 2 by replacing yo individuals by individuals of phenotype y = S1. Thus, no optimumpopulation contains individuals with phenotypes y < Si or y > S2. All boundary populations composed of phenotypes y such that Si $ y ? S2 will be represented by points on the boundary of F between the points

[W(1, S1), W(2, Si)] and [W(1, S2), W(2, S2)], which proves the theorem.

This segment of the boundary of F will be designated the class of points representing admissible populations. If we agree to consider as equivalent all phenotypes represented by the same point in the fitness space, the following observations hold:

1. If F is curved and convex on the admissible boundary, all optimumpopulations are monotypic. The boundary of F' coincides with that of F. Any population of two or more phenotypes will be represented by a point on the straight line joining the representations of those types, and hence lying in the interior of F. 2. If the admissible boundary of F is a straight line, a polymorphicoptimum population is possible but not necessary since the same point also represents amonomorphic population. 3. If F is concave on the admissible boundary, the boundary of F' is the straight line joining [W(1, SI), W(2, S2)] and [W(1, S2)7 W(2, S2)]. A point

on this line a fractionq of the distance from[W(1, SI), W(2, SI)l to

[W(1, S2), W(2, S2)] represents a population containing a fraction q of individuals with the phenotype y = S 1 and a fraction 1 - q of y = S2. Except when part of the boundary is a straight line, every point on the admissible boundary represents only one population. Thus, when the representation of the optimumis known, the optimumpopulation is uniquely specified. It will be a monomorphicpopulation if F is convex, and can be monomorphic or polymorphicfor at most two phenotypes when F is concave. This and the preceding section constitute a translation of statements about populations into a geometric language. The investigation will now proceed with the geometric approach and the results will be retranslated into biology.

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3. EXAMPLES OF ADAPTIVE FUNCTIONS

interpopulation Whena species is expandingrapidlyinto new territory, selection mightbe expected to favorthose populationswhichincrease most patterns. rapidly. Wewill consider several environmental 1. Environment stable in time,heterogeneousin space consists of two niches, 1 and 2, in the We assume that the environment fixed proportionsp, 1 - p and that individuals are distributedamong the niches withoutregardto phenotype. The naturaladaptive functionis then the rate of expansion, A(wj,

w2) = pwl + (1-

p)w2

and is constant in time. The value of the adaptive function is constant for each of the straight lines of the family pw1 + (1 - p)w2 = k w2

as k varies.

=

or

k I i-p

p Ii-p

W

These are the parallel lines shown in figure 1.

For 0 < p < 1

a~~~~~ W2-

W2\

WI

WI

FIGURE 1. Linear adaptive function(brokenline) maximizedon (a) convex fitness set, and (b) concave fitness set. these lines always slope down toward the right. The maximumachievable value of A(wj, w2) will occur at that point of F' where the line corresponding to- the largest value of k touches the expanded fitness s et, the point M-in the figure. Regardless of the shape of F. the optimumis a monomorphic population. (Note: if the boundary of F' is a-straight-line with slope P.

I1-p

any boundary point is an equally good optimum.) If F is convex, any I

admissible boundary point can be an optimum depending on the value of p. For 0 < p < 1, the phenotypein the optimummonomorphic populationwill be optimally adapted to neither niche but will survive moderately well in

THEORY OF FITNESS.

367

I.

both. For p= 1 and p= 0, the optima are those which are specifically adapted to niches 1 or 2 respectively. As p varies along a geographic gradient, the slope of A(w1, w2) changes and the optimumadaptive system will show a dine in phenotype fromy = S to y = S2 with each local population optimally monomorphic. If F is pure concave on the admissible boundary, the admissible boundary of F' is the straight line segment with the end points given by y = S1 and y = S2. It is readily seen fromthe figure that the optimumpopulation is represented by the upper or lower end point depending on whether-p/(1 - p) is greater or less than the slope of the boundary of F'. Thus, the optimumis

W2

W

WI

FIGURE 2. Maximization of A(W1, W2) = WIPW2l-P and (b) concave fitness sets.

WI

on (a) convex,

monomorphic,of phenotype best adapted to one or the other niche. Even if p is allowed to fluctuate this will be the optimumprovided that -p/(1 - p) is always either less than or greater than the slope of the boundary of F'. Along a geographic gradient in p, the optimumwill remain uniformlyof the phenotype best adapted to niche 1 until p reaches some critical value, and then will be uniformlyof the type best adapted to niche 2. The optimum situation would be one in which there are discrete races with isolating mechanisms operating near the critical value of p, since crosses between individuals adapted to the two niches mightbe expected to segregate some types optimal in neither. 2. Environmentuniformin space, heterogeneous in time Let the environmentof the tth generation consist entirely of niche 1 or entirely of niche 2 with probabilities p, 1 - p respectively. The environment is uniformspatially for each generation. We must make the additional assumption that there is no correlation between the environmentsof successive generations, so that the optimumis always the same and fitness could not be improved by change. After n generations the population size has been multipled by W1nq W2n(1-q) where q is the proportion of generations during which the environmentwas niche 1. The rate of increase per generation is the nth root

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368

of this, W1q W2J-q. As n increases without bound, q approaches p, the probability that niche 1 occurs, and the adaptive function is A(W1, W2) W 1P W21-

P.

This gives the family of curves shown in figure 2. We see that if F is convex the optimumpopulation is monomorphicof phenotype y which varies continuously with p in space. In general, the optimum population is not optimal in either niche but moderately suited to both. If F is concave, F' has a straight line boundary. The optimal population is polymorphic, a mixture of types optimally adapted to the two niches occurring in proportions that vary with p. Thus, along a gradient in p we should observe a gradient in the proportions of the same two types. 3. Environmentheterogeneous both in time and space Here we assume that the two niches occur in the proportions p, 1 - p where p is now a random variable. This is the more general case including the models of 1 and 2 above, so we should expect the results to be like those of one or the other model depending on the relative predominance of spatial or temporal heterogeneity.

s.. \

W2-

WI

FIGURE 3. Adaptivefunctionformaximizingequilibriumpopulation density on a convex F. Aftern generations the population has increased at the rate in 1~ ~~~1 n

=1 t]t=1W

+n

-dW2

I

_ZI = log[PtWl+(1-pt)W2]

where Pt is the value of p in generation t, and the Pt's are independent of each other. The average rate of increase is 1n A(W1, W2) = E en t1 log

Ltwi+(1-pt)W2j

where E means expected value, taken over all Pt. proaches

As n increases this ap-

A(W1, W2) = elln "lPtW1+(-POW2]}

369

THEORY OF FITNESS. I.

with errorof order I/n. Since EI log [pW, + (1 - Pt)W2B1is the same for all t, the adaptive function becomes

A(W1, W2) _-e E log[PtWI1+ (I1-

P tW21

Since the exponential function is monotonic, ez > ex whenever z > x. also maximizes Thus, the point which maximizes e E logiPtW1 + (1-Pt)W21 Elog[ptW1 + (1 -Pt)W2]This can be rewrittenE log [W2 + P(W1 - W2)] It is a well known propertyof the logarithm which follows fromits concavity that E I log x I S log E I x }. The equality holds only when x is constant, and the inequality becomes greater as x becomes more variable about its mean. The x in this case is p(W1 - W2). Thus, the fitness of a population decreases as p is more variable and as the difference between its fitness in the two niches increases. Thus, if there are several populations for which the average fitness over the niches is the same, in a heterogeneous environmentthe most fit will be the one for which fitness varies least from niche to niche. After a species has occupied a region, the rate of increase declines and a steady state is approached. We might then ask what population structure permits the greatest mean population size in the steady state. The rates of reproduction of the various types in a population may respond differentlyto population density. However, in the absence of definite informationwhich would permit a general model we will make the simplest assumption, namely that the rates of reproduction of all types are reduced in the same proportion depending only on the niche and population size N. Let the rate of reproduction given by PW g1(N)

+ (1 - p)W2g2(N)

where g1(N) and g2(N) are monotonic decreasing functions of N bounded in the closed interval 0, 1. First, we suppose that p is constant. Then in the steady state pWjg1(N)

+ (i-p)W2g2(N)

= 1,

In order to maximize N, we have to find the intersection of the fitness set F' with the member of the family of straight lines described by the above equation which has the largest value of N. The family is shown in figure 3. For each N the intercepts are [

(1 - P)g2(N)]

ad

pgiN)'

]

Thus, as N increases both intercepts increase and the lines, although not necessarily parallel, do not intersect in the first quadrant. If there is no value of N for which the line touches F', there is no steady state and the population cannot survive. Otherwise, the optimum will be monomorphic, specialized to the more frequent niche if F is concave and intermediate if F is convex, and p neither zero nor one.

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THE AMERICANNATURALIST

If the environmentconsists entirely of niche 1 or entirely of niche 2 with probabilities p, 1 - p respectively, the mean log rate of increase is plogWI

+ (1 -p)logW2

+ ploggI(N)

+ (1- p)logg2

N)

which is zero in the steady state. Since g, (N) and g2 (N) are both less than The larger the value of p log WI one, their logarithms are negative. + (1 - P) log W2, the smaller is the value of p log g I(N) + (1 - p) log g2 (N) and the larger the value of N. Thus, the optimum population maximizes p log W1 + (1 - p) log W2 as in section 2. 4. THE SHAPE OF THE FITNESS SET So far we have considered the consequences of convex and concave fitness sets without indicating where each is likely to occur. Now, we will show the effect on the fitness set of the degree of difference between the optima in the two niches, and homeostasis (represented inversely by the constant C). The function W(C (S - y)) which gives the relation between fitness and deviation fromthe optimumis roughly bell-shaped, with a peak at zero deviation. Near the peak it is convex upward, and furtheraway it becomes concave upward. Therefore, we assume that our W(z) has the following properties: 1. W(z) 2.

?

6. W'(z) > 0 for z < 0

0

lim W(z) = 0

7. W '(- z) =W

'(z)

3. W(- Z) = W(z)

lirn W'(z) 8. I oo 0Iz-

= 0

4. W'(z) < 0 for z > 0

9. W X"(z)

< 0 for fz < zo

10. W"(z)

> 0 for j z > zo

I z I - 00

5. W'(0) = 0

When the second derivative of W2 with respect to W1 is positive along the admissible boundary, F is concave. pressions W=

We can evaluate d2 W2 from the exd W12

W(CISI-yj) Y =W(C152-yj)

and the relation d W2 d W1

dW2

dy

dy

dW

dW2

dW2/dW

or d W1

d y /d y

Since the first derivatives are non-zero within the interval of interest, and since W2 is a single valved function of W1, the second derivative can be

THEORY OF FITNESS.

I.

371

shown to be d2W2

dW 2

(dWi

d2W2

dW2

\dy

d Y2

dy

d2_W_2__

W_

3

d Y2 // \y

At the midpoint of the boundary, where y = S1SI+2 S2

CIS1 -

= CIS2 - Yl.

W and d 2 are of opposite sign, whereas the second The derivatives dd dy dy derivatives have the same sign. Thus, d2 W2

d2 W12\ //dW1\ 2

/d2 W2

dy2 //\dy/

d W12 \d Y2 which is

d2 W2= 2W"

IS2

d12

i - S1|)(W

)

Therefore, the second derivative will be positive at the midpoint of the

boundary whenever W''[C (S2 c (S2

-

SI)

is

positive,

that is,

whenever

Si)> Z0 2 Now consider a point near the end of the boundary, when y is very close to S1. Here W[C(S2 - y)3 can be approximated by -

W[C(S2-y)] = W[C(S2-S1)]

+ C(SI-y)W'[C(S2-SI)1 +

and W[C(S

C2 2 (S1 - y)2W"'[C(S2 --S1)1

y)] by

W[C(S 1-y)] = W(O) + C(S 1-y) W' ()

+

2

(S1 - y) W"(O).

Then at this point, the fitness set is concave whenever W(O) + W[C(S2 - S1)1 > W(0) + W[C(S2 - S1) + 2C C2 S-

+ (S1 -y)W'[C(S2

y)2[W"'(O)

- S1)

+ w"t[c(52 -

Sl)1

which holds when C(S-y)W'[C(52-Si)]

+ C2

-y)21W"-(O)

+ W"(C(52-

is negative. For any fixed y no matter how small, S2 can be taken sufficiently large so that W' [C(S2 - SI)! and W" [C(S2 - S1)i are as small as desired. Thus for any y, the fitness set is concave at w 1 = W[C(S1 -y)] W2 = W[C(S2 - y)] provided G(S2 - SI) is sufficiently large.

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Thus, when C is large (poor homeostasis) or the niches very different,the fitness set will be concave in the middle region, and as C(S2 - S1) increases the concavity extends towards the ends of the admissible boundary. Our result can be stated as follows: when the difference between the environments experienced is large compared to the tolerance for environmental diversity, the fitness set will be concave. SUMMARY

A method is presented for representing the fitness of populations-in a heterogeneous environmentin terms of their fitness in the various niches taken separately, and the distribution of the niches in space and time. The characteristics of the optimal population can be found for each environmental pattern, and conditions determined in which the optimumis specialized or generalized, mono- or polymorphic, differentiated into discrete races or gradually along lines. These results are shown in table 1. The next paper in this series will extend the model and discuss experimental approaches for testing the theory. TABLE 1 Niche differencelarge compared to tolerance

Niche difference small compared Environment stable to tolerance in time,variable in space in more Optimum commonniche and and Monomorphic Optimum population Monomorphic specialized unspecialized Optimum geographic Continuousdine Discrete races pattern

Optimum phenotype Intermediate

uniform Environment in space, variable in time foreither Optimum niche Polymorphic,with specialized types Cline in frequencies of specialized types

ACKNOWLEDGMENT

I am greatly indebted to Dr. Howard Levene for his patient criticism throughoutthe research and preparation of this study, and to my teachers and colleagues in the Zoology Department of Columbia University whose discussions stimulated parts of the investigation. The work was supported by National Science Foundation graduate fellowships for the years 1957-58,

1958-59, and 1959-60.

SUMMARY OF SYMBOLS

Wij average numberof offspringof an individual of type in niche 1. Wi average number of offspring of anpoindividual in the population in niche i.

C coefficient of sensitivity to the environment(inverse measure of homeostasis). Si optimumphenotype in niche i.

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373

F Fitness Set F' expanded Fitness Set (smallest convex set imbedding F). .A(W1,W2) Adaptive Function, a monotonic non decreasing function of W and W2. p proportionor probability of niche 1. qj frequency of individuals of type i in population a, b coefficients measuring the interaction of different types in section 6. r rate at which individuals find some niche. See section on Niche Selection (Levins, 1963). A rate of decline of niche selectivity. See section on Niche Selection (Levins, 1.963). B coefficient of developmental flexibility. See section on Developmental Flexibility (Levins, 1963). LITERATURE

CITED

Andrewartha, H. G., and L. C. Birch, 1954, Distribution and abundance of animals. Chicago University Press. daCunha, A. B., and Th. Dobzhansky, 1954, A furtherstudy of chromosomal polymorphism in Drosophila willistoni in relation to its environment. Evolution 8: 119-134. Levins, R., 1963, Theory of fitness in a heterogeneous environment. II. Developmental flexibility and niche selection. Am. Naturalist. (In press). Lewontin, R. C., 1958, Studies on heterozygosity and homeostasis, II. Evolution 12: 494-503.

Theory of Fitness in a Heterogeneous Environment. II. Developmental Flexibility and Niche Selection Author(s): Richard Levins Source: The American Naturalist, Vol. 97, No. 893 (Mar. - Apr., 1963), pp. 75-90 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2458643 . Accessed: 24/06/2011 01:10 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ucpress. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Vol. XCVII, No. 893

The American Naturalist

March-April, 1963

THEORY OF FITNESS IN A HETEROGENEOUS ENVIRONMENT II. DEVELOPMENTAL FLEXIBILITY AND NICHE SELECTION* RICHARD LEVINSt Department of Zoology, Columbia University, New York, N. Y.

The previous paper in this series (Levins, 1962) described a method for determining the optimal population structure of a species in terms of the statistical pattern of the environment. Three restrictive assumptions were made: 1. The phenotype is fixed independently of the environment; 2. Individuals are distributed at random among the niches; 3. The fitness of an individual depends on his own phenotype and the environment,but not the composition of the population. These restrictions will now be relaxed, and we will show the effects on our previous results of developmental flexibility, active niche selection, and interaction among differentphenotypes. The final section discusses the conclusions of both papers and several approaches fortesting the theory. 1. DEVELOPMENTAL FLEXIBILITY

The course of development of an individual is subject to long term, irreversible modification by the environmentas well as to transitoryfluctuations. The pattern of this long term modification may itself be subject to selection, so that the normof reaction with regard to particular aspects of the environmentmay become an "optimal" normof reaction. Three types of developmental flexibility will be considered: The phenotype may vary in a continuous way with the environment(Schmalhausen's dependent development, Schmalhausen, 1949). The phenotype may switch fromone discrete state to another at some threshold value of the environmentalfactors (Schmalhausen's regulative autonomous development). Or, there may be a stochastic switch mechanism in which a given genotype gives rise to two or more discrete classes of phenotype in proportions that vary with some environmental factors. An irreversible modification of development at any early stage will in general be advantageous only if the environmental factor evoking the modification is correlated with the environmentat later stages when selection is operating. This may occur in several ways. The environmental stimulus may be an indicator of the niche the individual already occupies. For example, shade or moisture might indicate a forest niche. Or the environmental factor which serves as a cue may be pervasive over the whole region *This paper is dedicated to Professor L. C. Dunn. tPresent address: Department of Biology, University of Puerto Rico, Rio Piedras, Puerto Rico.

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and yet be correlated with the proportions of various niches. For example, the photoperiod is an indicator of season and thereforeis an indirect cue as to the likely proportions of available fruits. The factors of the environment which modifydevelopment need not be the same factors for which the modifications are adaptive. The problem of findingthe optimal normof reaction has many complexities since there are many possible relations among the niche proportions, cue stimuli, and viabilities of phenotypes. Here we indicate an approach to these questions in some simplified situations. The same adaptive functions can be studied here as were considered in previous sections with the difference that the probability that a given individual will ultimately be in niche 1 is now some function of the early environment, e, of that individual and will be designated Pe. Similarly, the probability of ultimately being in niche 2 is 1 - Pe. If e has a certain distribution over the area of a population, the expected value or mean of Pe taken over all e will be denoted by p. The first model to be considered is as follows: let the mean proportion of niche 1, p, be constant in time but not in space and let the environmental cue e have the same distribution over the region in each generation. Then the fitness of the population is equal to the average of the fitnesses of individuals exposed to differentvalues of e. This will be a maximumwhen it is maximumfor each Pe. For each value of Pe the adaptive function is A(W1,W2) = PeWi + (1

-

pe)W2-

Following the argumentof 3.1, when the fitness set is convex the optimum phenotype for each Pe varies continuously with Pe so that the optimumnorm of reaction is dependent development. When F is completely concave, the optimumnorm is one which produces a single type of individual which is optimal for niche 1 or niche 2 depending on whether Pe is above or below some threshold value. Thus on a concave fitness set the optimumnormof reaction is a switch mechanism (regulative autonomous development). In the second model we suppose that the environmentconsists entirely of niche 1 or of niche 2 with probabilities Pe, 1- Pe respectively and that the environmentalcue e is uniformover the region. Then, as in 3.2, A(W1

W2)=

W1 Pew21-Pe.

If the fitness set is convex, the optimum norm of reaction is again dependent development, with the phenotype varying continuously with Pe. If F is concave, the optimumis a mixture of types in proportions that depend on Pe, so that the optimumnorm of reaction is a stochastic switch mechanism. Development can follow one of two channels, and the probability of taking a given channel depends on the environment. We now restrict ourselves to dependent development and examine the effect of developmental flexibility on the fitness set. It will be shown that the fitness set is made more convex.

FITNESS AND NICHE SELECTION

77

Let the probability of an individual being in niche 1 be Pe and in niche 2 be 1 - Pe where Pe depends on the environmental factor e. As before, the average value of Pe taken over all e is p, the proportionof niche 1 in the region. Since the optimumphenotypes in niches 1 and 2 are S1 and S2 respectively, we can define the average optimumphenotype, Se, by Se = peSi + (1 - pe)S2. The mean value of Se taken over all e is then Se = PS1 + (1

-

p)S2-

If the early environmentprovides a clue as to the later environment,then the average value of Se among those individuals that ultimately mature in niche 1 will be closer to S1 than the average of Se over the environmentsof individuals that are ultimately in niche 2. The mean value of Se given that the later environmentis niche 1 is El Se | niche 1}Se

-( 1-p)k

and for niche 2, ElSeIniche 21 = Se + pk where Si < S2 and k > 0. Hence the mean value of Se is ElSe} I= P[Se

-

(1- p)k] + (1- p)[Se + pk]

Se.

Since Se is always between S1 and S2, ElSelniche 1} is always between E(Se) and Si. A larger k indicates that Se is closer to Si and that e is a better early indicator of later environment. Let the phenotype y be influenced by the environmentas follows: y

yo + B(Se - yo)

where B is the coefficient of developmental flexibility and y0 is the phenotype when B = 0. The absolute value of the deviation of the actual phenotype fromthe optimumin niche i is -Si-y-

B(Se -Yo)

We always have Si < Se < S2. Whenyo = Si, the absolute value of the deviation is BISe - Sil which is greater than zero if B > 0. Thus when yo = Si the organism's fitness in niche i is reduced by developmental flexibility. Therefore, the end points of the admissible boundary when developmental flexibility is allowed are the same as in the case when developmental flexibility is excluded, namely y0 = Si, B = 0. On the other hand, when o =PS1 + (1 - p)S2

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THE AMERICANNATURALIST

there are values of B which improve fitness simultaneously in both niches. Let the fitness of an individual in niche i be Wi = W(Si - Y)

where W(Z) is a bell-shaped curve with center at Z= W1= EWIS2 - YO- B(Se e

= EWj(1 e

- yo)}

-

0. Then

p)(S1 -S2)

-

B(Se

- Yo)}

and W2 = EW[S2 e

-

YO - B(Se

-

yo)]} = E1W[p(S2 e

-

S1)

-

B(Se

-

Yo)]}

where the expectation is taken over all e. For small values of B, W1 and can be approximated by W = W[(1

-

p)(S1

-

S2)] - B(Se - yO)W[(1

-

p)(S1

W2

S2)]

-

and W2 =

W[p(S2

-S1)]

-

B(Se

- yo)WV[P(S2 - S1)].

We now take the expected value over all e given niches 1 and 2 respectively. W, = W[(1 - p)(S1 - S2)] - B[-(1 - p)k]W'[(l - p)(S1

-

S2)]

and W2 = W[p(S2 - S1)] - B p k W'[p(S2 - S1)].

From the shape of W(Z) we know that W'(Z) < 0 when Z > 0 and W'(Z) > 0 when Z < 0. Thus, in both W1and W2the coefficient of B is positive. This means that a small positive value of B improves fitness over what it would be for B = 0. For all yo such that Se -(1-p)k
Se +pk

fitness is improved in both niches by developmental flexibility. Thus, the fitness set bulges out for these values of yo, and F becomes more convex. 2. NICHE SELECTION

It is, of course, unrealistic to assume a random distribution of individuals among the niches. In nature there is a wide range of selectivity, fromcomplete restriction to a single food plant or host throughgraded preferences to a widely polyphagous condition. Temperature and light preferences are widespread, perhaps universal. Therefore, the proportionof individuals in a given niche will be greater than the frequency of the niche if that niche is preferred,and less than the niche frequency if it is the less desired niche. It is clear that absolute preference for the niche in which the type does best would give maximal fitness if that niche is readily available, but could result in extinction of the species if that niche is sometimes absent. Thus it would seem that an optimal mode of behavior would be one in which the individual searches for the preferredniche for a given period of time, and if

FITNESS AND NICHE SELECTION

79

it has not found it then accepts some other mode. However, such an instantaneous change in behavior after some critical interval has elapsed is unrealistic. Therefore, we assume that niche preference is initially absolute and decreases exponentially with the passing of time. Let p be the frequency of the preferredniche and 1 - p the frequency of the less preferred niche. Let the probability of finding some niche in the interval (t, t + dt) be rdt. If it is the preferredniche, it will always be accepted. If it is the less desired niche, it will be accepted with the probability 1 - e-At where A is a measure of the rate at which selectivity deThen the probability of no niche being accepted in the interval creases. (t, t + dt) given that no choice has been made previously is

1

r[1 - (1

p)eXt]dt.

-

Let q, be the probabilitythat no choice has been made up to time t. Then fromthe above it follows that qt+dt = qt[1 - r(l - (1 - p)e At)dt] and qt+dt - qt = -rqt[l

-

(1 - p)e-kt]dt.

As dt approaches zero, we obtain the differential equation

dq

- = -rq[1 dt

(1

-

-

't

p)e-t]

which has the solution -Xt

r(l-p)

qt = q (e A) where q0 = 1. The probability that a niche is accepted in the interval (t, t + dt) is qtr[1 -(1

-p) et]

dt.

If a niche is chosen in that interval, the probability that it will be the preferredniche is p p +(1

-

p)Q-,t

Thus, the total probability that niche 1 (the preferred niche) is chosen in interval (t, t + dt) is prqtdt. The proportion of individuals in niche 1 is then the integral, p*, of prqtdtover all t. Hence, p* = prfe

0, r(,=p) (1-e-kt)-t

Withthe substitution rt= u, we have

X

dt.

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THE AMERICAN NATURALIST

p*

f

p

Au\

1 P) (1e

coo(

e

)U

du.

(

X

Thus, the effect of niche selection is to replace p in the adaptive function by p*. We see fromthe expression for p* that p* is greater than p since

T

00

eudu=

1

and rim lap)

(lye

)

e

>1.

Further, the variance of p* is less than the variance of p. This follows froman examination of the ratio of any two values of p, Pi and P2, and the corresponding ratio of the P*'s. P-P __

Pi

'?

Pi J

eJ

du

P2

{e P2.

- Au

ie P2*

Let Pi >

e

__)_U

(1-1PP2)

(

k

er r-u

r du.

A

Then

e

du <j

e

du.

Therefore, the ratios of the p*'s are closer to unity than the ratio of the p's; there is less spread in the distribution of p*, and the variance is smaller. Thus, niche selection reduces the effective variance of the environment,so that on a concave fitness set a population with niche selection is less likely to have a polymorphicoptimum. This result depends on the special assumption that all individuals prefer the same niche, and that within a niche fitness is independent of population density. If we relax these assumptions and suppose instead that fitness declines with population density, that niche preference is modified by density, and that each phenotype in the population selects preferentially that niche in which it would be more fit, we would be led to the opposite conclusion. For then phenotypes which are adapted to usually rare niches can be kept in the population without much loss of fitness, and the population will be more evenly spread over its environment.

FITNESS AND NICHE SELECTION

3.

81

INTERACTIONS BETWEEN INDIVIDUALS

A mixed population can be represented by a point on the straight line joining the representations of the separate types only if the differenttypes do not interact to effect each other's fitness. There is a growing body of evidence (for example, Levene, Pavlovsky and Dobzhansky, 1954; Lewontin, 1955; and Sakai, 1955) that such interactions may be important. In the absence of a general quantitative description of these mutual interaction effects, we will take for illustration the simplest model, quadratic interaction. Let the fitness of a given type in a given niche vary with the proportion of individuals of other phenotypes. If there are two types in the proportions q, 1 - q, the fitness of type 1 in niche i will be wil* = wil[I + a(1 - q)] and the fitness of type 2 in niche i will be wi2* = wi2[1 + bq] where wil and wi2 are the fitnesses of the two types when alone, wil* and wi2* are their fitnesses when both are present, and a and b are measures of the interaction effects. Then the fitness of the mixed population

in niche i is

qwil* + (1 - q)wi2* = qwil + (1 - q)wi2 + q(1

-

q)[awil

+ bwi2].

The fitness of a mixed population is represented by a point on a parabolic arc joining the fitness points of the separate types. In such a situation the expanded fitness set F' is not the smallest convex set that contains F. However, if we redefine "'straight line" to be the arc of the interaction curve, in this case parabolic, the previous theory holds. The well known result follows that, if interaction is positive, F' can more readily contain points outside of F, giving polymorphic optima, whereas negative interaction makes monomorphicoptima more likely. CONCLUSIONS AND DISCUSSION

The fitness of a species over its whole distribution can be resolved into the contributions of several adaptive systems. First, there is the fitness of individual members of the species. This in turn can be separated into "physiological" homeostasis and developmental flexibility. The formerincludes the tolerance of the organism for non-optimal conditions and its capacity for rapid reversible changes in internal state corresponding to transitory fluctuations in the environment. The latter refers to irreversible developmental modifications of an adaptive character. The distinction between physiological homeostasis and developmental flexibility is an arbitraryone which must be defined operationally for each experimental situation. Suppose that the life span of an organism is divided into two periods which we designate early and late. Let groups of individuals be subjected to various combinations of early and late environments at two intensities, arbitrarilydesignated +1 and -1, and let the fitness of each group be measured (for example, by the average numberof seed set per seed sown). Suppose the results are those shown in table 1.

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THE AMERICANNATURALIST TABLE 1 L ate -1

1.6

1.4

+1

Early In this table, fitness(W)

+1

1 + .5E

+ . 1L.

These data were obtained fromthe model which expresses fitness by W = 1 + .5E + 1L where E and L stand for early and late environmentsrespectively. The coefficients of E and L indicate how sensitive the organism is to environmental differences at the corresponding stages of development, and thereby measure inversely the physiological homeostasis of the organism in the two stages. Now consider the hypothetical data in table 2. TABLE 2 Late +1 Early

In this table, fitness(W)

+ 1

1.7

-1 5

1+ .1E+ .1L + .5E x L.

These data were obtained fromthe expression W = 1 + .1E + .1L + .5E x L where E and L are again the early and late environments. The coefficients of E and L inversely measure the physiological homeostasis in the two stages. The coefficient of E x L measures the increment in fitness when the early and late environments are the same. Therefore, it can be interpreted as a measure of the long lasting modification of development in the early stage in the direction of improved survival in that environmentat the expense of reduced survival in the opposite environment, that is as developmental flexibility. A second adaptive system is based on static polymorphism, the existence of a permanent and stable diversity within populations, in which each of the types is superior to all the others in some environmental phase or niche. This is a sort of insurance in the face of random fluctuations in the environment. However, it was shown that static polymorphismcan only be advantageous when the difference between the different niches exceeds some minimal value which measures individual tolerance or homeostasis. A third adaptive system is the geographic differentiationinto local races, subspecies, ecotypes, or lines.

FITNESS AND NICHE SELECTION

83

Finally, there is the genetic flexibility of the species - its ability to respond to the selection pressures of a changing environment. This component of fitness will be considered in a separate paper. The relative importance of these components of fitness depends on many factors-conflicting selection pressures, populations size, past history, available cytological mechanisms, etc. In the study just referredto we considered the selection pressures acting on the components of some of these adaptive systems. It was found that the optimumadaptive system depends on the pattern of heterogeneity of the environmentin space and time. The optimumwas defined in such a way that interpopulation selection would be directed toward the optimum. Intrapopulation selection may reinforce interpopulation selection or act against it. However, if the fitness of individual types does not depend on the frequencies within the population, we might expect that intrapopulation selection would generally also tend to the same optimum. The structure of actual populations may differfromthe theoretical optima for a number of reasons. If the assumptions of the model can be applied reasonably to a given population, departures from the predicted might be ascribed to the effects of other factors than those considered here -antagonistic selection, migration,population size, etc. Or it may be that the population is not yet in equilibrium with its environment,that its present properties are more determinedby past history than by present conditions. Thus, the correspondence between the predicted (optimal) population characteristics and those observed in nature will be only statistical. It For would be possible to summarize our results by listing "tendencies." example, species occupying more heterogeneous environments "tend" to be more polymorphic than species living in more uniformenvironments. Such an approach would result in a series of generalizations analogous to the familiar eco-geographic laws. However, we prefer a differentinterpretation. There are two kinds of about causes of evolugeneralizations in evolutionary theory-statements An identified' tion and statements about frequent results, or tendencies. causal factor is assumed to be present whether or not its effect is obvious, whereas a tendency is sometimes expressed and sometimes not. For example, we could talk about a tendency for unsupported objects to move toward the center of the earth, or we could discuss gravitational attraction. In the formercase, each bird, airplane, or rocket is an "exception" to the tendency. In the latter case we claim that gravity is just as relevant forobjects moving in the wrong direction, and that a knowledge of the magnitude and direction of the gravitational attraction helps identify and measure the opposing forces. The state of a species or population at any time is one of tenuous equilibrium among opposing factors or of transition from one equilibrium to another. Its evolution can be studied by identifying and measuring these factors, the vectors acting on a population. Then the correspondence be-

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THE AMERICAN NATURALIST

tween the predicted and observed situations becomes one of the ways of measuring the importance of a particular factor, and any correlations derived fromthe data must be subordinated to the dynamics of change. Otherwise statements about the results of evolution are too readily converted into statements about causes, as in the claim by one eminent biologist that the evolution of the brain in a given phylogeny was "guided by the law of increasing size." We claim that in this study the direction of a particular component of selection has been identified. The next step, the determinationof whether it is of sufficient magnitude to exert a noticeable influence on evolution, is an experimentalproblem. There are three principal experimental approaches for testing the theory: 1. The study of the responses of experimental populations to differentpatterns of environmentalheterogeneity; 2. The comparison of the structures of natural populations of the same or related species; 3. The statistical analysis of the adaptive systems for a large number of taxonomic groups. Experimental Populations If the fitness of each phenotype is independent of the composition of the population and if the density of the population does not greatly affect the relative fitnesses, it seems reasonable to expect that intrapopulation selection will move in the same direction as the theory predicts for interpopulation selection. Then it might be possible to observe the formationof optimal population structures. In order to set up such experiments, it is necessary to obtain the fitness set for a given population and a pair of defined niches. This could be done by dividing the progeny of single pairs into two batches raised under the two environments. The fitness of each group could be estimated using such informationas the percentage reaching maturity, average number of eggs produced, etc. The data for the two batches together define a single point in the fitness space, and the array of all such points for many single pairs would constitute the fitness set. Alternatively, one mightprefer to consider the fitness set for particular chromosomes extracted from the population and compared to some standard chromosome. A convex fitness set is readily attainable by taking the two environments close enough together. A concave fitness set might be more difficult to find. However, it is suggested that we could produce a concave fitness set as follows: An initial population is divided into two lines, designated High and Low. They are selected for survival in increasingly extreme conditions in opposite directions, such as high and low temperature. As the two niches diverge, the initially convex fitness set will become concave when each line can grow in an environmentlethal to the other, if not before then.

FITNESS AND NICHE SELECTION

85

At different stages in the divergence, experimental populations can be established using founders fromboth the High and Low lines, and exposing themto the appropriate environmentalpattern. In the early stages of divergence, when the fitness set is convex, we would expect the equilibrium population to have a unimodal distribution whose fitness in each niche will be intermediate between those of the founding populations. In the later stages of divergence, after a concave fitness set has been attained, the equilibrium population will depend on the environmentalpattern. If both environments are always available to the population in about constant proportions, the optimumpopulation will be adapted to one of the two niches and should soon be equal in fitness in this niche to the founding population raised in that environment. An alternative optimum would consist of the formationof two populations each adapted to one of the niches. That is, an isolating mechanism might arise giving in affect two populations. If the two environments alternate about once per generation, the expected result would be a polymorphic population which would show a bimodal distribution of fitnesses in each niche, and the sensitivity of the whole population to selection in either niche would gradually decline. The "equilibrium" we are discussing here is not the initial equilibrium reached after some 10 to 50 generations when differentkaryotypes of Drosophila of the same geographic origin are introduced into a population cage. It would involve a more radical reconstruction of the genetic system through the selection of many modifiers. Heterosis may arise where initially there was none, or heterosis may break down. Perhaps several hundred generations may be required before an "optimum" equilibrium is established, and even then its maintenance may be accompanied by genic and chromosomal turnover. Comparisons within species or groups of species Two theoretical models for the distribution of polymorphism within a species are of current interest. Da Cunha and Dobzhansky (1954) suggest that the amount of polymorphismis correlated with the number of niches available. This is based on the demonstration by Levene (1953) that when more than one niche is available, the necessary conditions for stable polymorphismare less stringent. Thus this theory is concerned with stability rather than optimality. The data forDrosophila robusta, D. willistoni, and a number of other species support the hypothesis, but the cosmopolitan species show very little polymorphism and in D. pseudoobscura the data are ambiguous. Carson (1955) seems to accept the argument that niche diversity makes polymorphismmore likely. However, since his interest is species formation he emphasizes the distinction between central (multi-niche) and marginal (few niche) populations. Both of these approaches and our own make similar predictions in expecting polymorphism and environmental heterogeneity to go together. They

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THE AMERICANNATURALIST

differ,however, in the measurements of polymorphismand of environmental heterogeneity, and also seem to disagree in several predictions. Carson measures polymorphismby the proportionof the genotype in which inversions block free recombination. His theory refers specifically to cytological polymorphism. Dobzhansky et al. measure polymorphismby heterothe numberof inversion heterozygotes per individual or the numzygosityber of distinct inversions in the population. We are concerned only with polymorphismfor niche adaptations. The only types we wish to recognize are those which are fitter than every other type in at least one niche and which therefore are represented on the admissible boundary of the fitness set. Thus the Lancetilla population of D. tropicalis, in which more than 70 per cent of the flies are heterozygous for an inversion, is highly polymorphic fromDobzhansky's viewpoint. However, it is by no means certain that the three karyotypes represent adaptations to different niches. The heterozygote may be superior to both homozygotes in all environments encountered by the population. On the other hand, populations which are cytologically homozygous may be polymorphic with respect to many genes of ecological significance. Thus, although we have spoken of polymorphism, our theory is concerned only with ecological polymorphism,the coexistence of two or more phenotypes each adapted in one or more niches. A proper test of the theory would require identification of the environmental factors to which the polymorphs are adapted. In the absence of specific informationon this point, the number of inversion types in a population (rather than the proportion of heterozygotes) can be taken as a very rough guess as to the numberof ecotypes. For Carson's purposes the location of a population or its relative abundance is sufficient indication of environmental heterogeneity. Da Cunha and Dobzhansky use a quasiquantitative scale devised by Dansereau (1952) which includes both spatial and temporal heterogeneity. However, in our theory, these two aspects of the environment have very different signifiEcological polymorphism is optimal only when the environment cance. undergoes changes of sufficient magnitude so that the fitness set of the population with respect to the two extreme environments is concave, or when two niches which are themselves sufficiently distinct occur in changing proportions in such a way that each niche is the predominant one part of the time. Spatial heterogeneity contributes to polymorphism only indirectly. If there are more niches, one might suppose that some of these are likely to be sufficiently far apart to give a concave fitness set. Further, an abundance of differentvegetation types, each with its own fruitingseason, gives rise to a great temporal heterogeneity in the available food for Drosophila even when the climate is rather uniform,as along the Amazon. I have recalculated the data of Da Cunha and Dobzhansky (1953) and Da Cunha, Burla and Dobzhansky (1950) in the following way: the total number of chromosome inversions reported for each locality was found. The localities were divided according to temporal heterogeneity (seasonal changes). The average numberof chromosome types for the uniformlocali-

FITNESS

AND NICHE SELECTION

87

ties was 19.2 and for the seasonal ones 27.2. Thus, it is apparent that much of the variation in chromosomal polymorphism among the localities can be ascribed to temporal heterogeneity of the environment although this appears as only a 0 or 1 on the scale that goes from5.5 to 12. Since Da Cunha et al. are interested in the stability rather than the optimality of polymorphism, the likelihood of polymorphism should not be increased in their view by a concave fitness set. On the contrary,it would seem as if a convex fitness set is more likely to give stability. For example, let the niche optima be s, -s and let the three genotypes (or inversion types) have phenotypes AA AA' A'A '

a 0 -a

Assume furtherthat the two niches are equally frequent. Then each homo1 zygote has a mean - [W(s + a) + W(s -a)] which is less than W(s), the fit2 ness of the heterozygote (when F is convex). When the fitness is concave, the mean fitness of the heterozygote is less than that of the homozygotes although the harmonic mean may still be more than that of the homozygotes and polymorphismmightbe stable. Thus, a comparison of successful species in the same region would (on the Da Cunha-Dobzhansky theory) reveal a position or zero correlation between individual homeostasis and polymorphism, whereas we would expect a negative correlation since good homeostasis is more likely to give a convex fitness set and therefore a monomorphic optimum. In particular, we expect that D. willistoni, since it is highly polymorphic, will prove to have relatively poor individual homeostasis compared to the cosmopolitan species and to its own siblings. A second discrepancy between our view and that of Carson as well as Da Cunha et al. concerns submarginal populations. Toward the ecological extreme of a species' distribution, the numberof niches utilized presumably But relatively minor environmental differences between niches decreases. may become more important when some pervasive factor is extreme and homeostasis possibly weakened. Thus, the fitness set may become more concave toward the margins, giving rise to submarginal polymorphismand race formation. A final prediction concerns race formation. Discrete races would be expected when the fitness set is concave and the local optimumis monomorphic. Thus, subdominant species with little polymorphism should show greater differences between geographic populations than more polymorphic dominants. A dominant species with low polymorphism (convex fitness set) should show continuous variation along an environmentalgradient.

THE AMERICAN NATURALIST

88

Statistical Analysis of a Biota The plants and animals which live together in the same region do not necessarily experience the same kind of environmental heterogeneity. For example, the meadow environmentis more variable than the forest and less variable than disturbed ground. The average conditions over a longer life span have a smaller variance than over a short life span. The seed of climax forest trees usually fall near the parent and are more likely to grow under similar conditions than would the seed of weeds. Thus, fromthe general ecology of a species it is possible to derive a quasiquantitative measure of environmentalheterogeneity. For plants, an index of temporal heterogeneity can be established as follows, wherepositive scores indicate more variability: 1. Habitat: -1 for forest turbed ground.

0 for meadow or savanna

2. Successional position: - 1 for climax colonizers and weeds. 3. Seed dispersal: - 1 forpoor dispersal 4. Life span: -.1 for perennials

1 for dis-

0 for subclimax

1 for

1 for good dispersal.

0 for annuals

1 for ephemerals.

Similarly the effective spatial heterogeneity of a population's environment depends on the area covered by the panmictic unit, the occurrence of microniches large enough for an individual to grow in, and the number of niches occupied. Thus, an index can be constructed in which positive values indicate more spatial heterogeneity. 1. Occurrence: 2. Size:

-

1 for rare or specialized

- 1 for large trees lianas, and epiphytes.

1 for ubiquitous

0 for shrubs

1 for small herbs,

3. Pollination (indicating size of panmictic population): - 1 for selfers 0 for bee pollination 1 for pollination by wind or less specialized insects. The optimum structure also depends on individual homeostasis. This might be measured directly by transplantation experiments or could be inferredto some extent fromhabitat. Such a study might be most effective using pairs of related but ecologically distinct families such as Malvaceae-Bombacaceae, SolanaceaeScrophulariaceae, etc. Over a geographic gradient in the environmentof the region studied, we would anticipate the following sort of results: 1. The annual weed mode of adaptation: high individual homeostasis; random genetic differences between populations of the same area; little genetic differentiationalong the transect; each species occurs over a wide ecologi-

FITNESS

89

AND NICHE SELECTION

cal range; little response to selection for adaptation to particularmicroniches (oftenassociated withpolyploidy). Genera maybe large but without division into coenospecies.

weeds, secondarygrowths,subclimax:intermediate. 2. Forest herbs,shrubby 3. Climax vegetation, especially forest trees: lower individual homeostasis; adaptive genetic differentiationalong a transect; each species confinedto narrowrangeof ecological zones. Little adaptivepolymorphism within populations; allopatric races or coenospecies; freer response to selection. Caution must be observed in avoiding the circular reasoning of definingthe ecological zones by their unique climax vegetation and thendiscoveringthatthese trees are confinedto one zone. of the genSimilarindices could be workedout foranimals. Axsummary eral resultsis given in Table 3. TABLE

3

Difference large between niche optima Difference small beas compared to individual homeostasis tween niche optima as compared to indi- Environmenthetero- Environmentheterogeneous in time vidual homeostasis geneous in space Optimumphenotype Intermediate between Optimumphenotype Specialized to one or the other for more frequent optima in the two niche (specialized niche niches to one niche) Optimumpopulation structure

Monomorphicof moderate fitness in each niche

Pattern along geo- Continuous dine in phenotype graphic gradient in niche frequency

Monomorphic, specialized

Polymorphic mixture of specialized types

Discrete races separated at some critical value of niche frequency

Cline in proportions of same polymorphic types

SUMMARYOF SYMBOLS

A summary of symbolsmaybe foundin the precedingpaper of this series (Levins, 1962). ACKNOWLEDGMENT

I am greatly indebted to Dr. Howard Levene for his patient criticism the research and preparationof this study, and to myteachers throughout and colleagues in the Zoology Departmentof Columbia Universitywhose discussions stimulatedparts of the investigation. The workwas supported by National Science Foundationgraduatefellowshipsforthe years 1957-58, 1958-59, 1959-60.

90

THE AMERICAN NATURALIST LITERATURE

CITED

Carson, H. L., 1955, The genetic characteristics of marginal populations of Drosophila. Cold Spring Harbor Symp. Quant. Biol. 20: 276-286. Da Cunha, A. B., and Th. Dobzhansky, 1954, A furtherstudy of chromosomal polymorphism in Drosophila willistoni in relation to its environment. Evolution 8: 119-134. Dansereau, P., 1952, The varieties of evolutionary opportunity. Rev. Canad. Biol. 11: 305-388. Levene, H., 1953, Genetic equilibrium when more than one ecological niche is available. Am. Naturalist 87: 311-???. Levene, H., Th. Dobzhansky and 0. Pavlovsky, 1954, Interaction of adaptive values in polymorphic experimental populations of DrosophZla pseudoobscura. Evolution 8: 335-349. Levins, R., 1962, Theory of fitness in a heterozygous environment. I. The fitness set and adaptive function. Am. Naturalist 96: 361-378. Lewontin, R. C., 1955, The effects of population density and composition on viability in Drosophila melanogaster. Evolution 9: 27-41. Li, C. C., 1955, Population Genetics. Chicago Univ. Press. Sakai, K., 1955, Competition in plants and its relation to selection. Cold SpringHarbor Symp. Quant. Biol. 20: 137-157. Schmalhausen, I. I., 1949, Factors of evolution. Blakiston. Philadelphia.

J. Theoret. Biol. (1964) 7, 224-240

Theory of Fitness in a Heterogeneous Environment Part III. The Response to Selection RICHARD LEVINS Department

of Zoology, Columbia University,

New York, N. Y., U.S.A.-f

(Received 17 October 1963) A method is presented for estimating the average fitness of a population which is responding to selection in a fluctuating environment. The environment is assumed to be a stationary Markov random process of the type described by Uhlenbeck & Ornstein (1930), and the genotype is restricted to a single locus with two alleles. The fitness of each genotype depends on the deviation of its phenotype from the optimal phenotype in the given environment. Finally the fitness of the population is expressed as a function of the parameters of the genetic system (average phenotypic effect of a gene substitution, mutation rate) and of the environment (mean, variance, and autocorrelation). Conditions were found under which it is advantageous for a population to be polymorphic (that is, under which the average phenotypic effect of a gene substitution should differ from zero for maximum fitness). These conditions are of two very different kinds. First, regardless of whether the environment is autocorrelated or not, phenotypic diversity is advantageous if the environment is sufficiently heterogeneous compared to the tolerance of an individual for non-optimal conditions. When W(Cz) is the fitness of a genotype whose phenotype differs from the optimum phenotype by z, the tolerance is defined as the distance from the peak (at z = 0) to the point of inflection of W(Cz). Since W(z) has its inflection at some zO, the tolerance of each genotype is z,/C, and C is an inverse measure of homeostasis. If the environment consists of discrete niches or is bimodal with modes (or optimal phenotypes in the niches) further apart than z,,/C, it is sufficiently heterogeneous for polymorphism to be optimal. This kind of polymorphism is stable. The different genotypes should be retained in the same frequencies without responding to selection, so that the optimal genetic system will be one in which there is large genetic variance due to epistatic interactions but a minimal additive component. The second kind of optimal polymorphism occurs when there is a high autocorrelation in the environment. Here the advantage of genetic variance is only that it permits response to selection. Thus it will consist mostly of additive variance. t Present address: Department Puerto Rico.

of Biology, University of Puerto Rico, Rio Piedras, 224

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These two kinds of optimal polymorphism have very different consequences for speciation. In the first case, recombination between populations will result in F2 breakdown, so that natural selection will tend toward the establishment of isolating mechanisms. Therefore we expect that groups in which there is low individual tolerance of suboptimal environment will have high levels of static polymorphism based on epistatic complexes, a high degreeof F2 breakdown,and a high rate of speciation.In the second case, where additive variance predominates,Fz betweenpopulations is more or less intermediate and there is not any strong selection for isolation. Hencea high individual toleranceof non-optimal environments will be accompanied by genetic systems with more additive variance, fewer inversionsholding epistatic blocks, low F2 breakdown, and a low rate of speciation. It is suggestedthat the willistoni group of sibling speciesand Drosphila melanogaster are representativesof thesealternative systemsof adaptation.

Introduction In the previous papers (Levins 1962, 1963) it was assumed that a population consists of one or more types of individuals in fixed proportions which do not change under selection. The optimum population was specified by the proportionsf, of type i. Under that model it was possible to maximize the adaptedness of a population by finding the optimum frequencies for the component types. The present paper considers Mendelian populations which are changing under selection, mutation, and other influences. We will assume that the fitness of the population is some function of Wright’s V, the adaptive value, which is given by

w = CfiWi

(1.01)

where wi is the average number of offspring per zygote of type i, and fi is the frequency of type i. In particular, we will consider the fitness to be measured by Witself or its logarithm, m. The latter, which is the Malthusian parameter, gives the rate of growth of the population and is the appropriate measureof fitnessin a continuous model. When a new population is founded,

often by a single various accidents small. During this The maximization

pair, there is an immediate danger of extinction due to or the inability to find mates while the population is initial period the discrete model seems to be more realistic. of W minimizes the probability of extinction in the initial

phase of colonization. We assumeW to be greater than zero and bounded

away from zero and m bounded in probability to exclude extinction. The average fitness cannot be found without specifying how fi change. When this is done, the mean fitness E(w) or E(m) is some function of the parameters of the genetic system-the average phenotypic effect of a gene

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substitution, the coefficients of dominance and epistasis, linkage, the rate of mutation, etc. These parameters appear as constants in the equations describing the changes in gene frequency. They are themselves quantitative characters which depend on the genotypes in the populations and are therefore in principle subject to selection. In general one might expect that interpopulation selection would change the genetic system towards its optimum. The object of this study is to find the optimum genetic systems under a number of models. The present section will review briefly some of the relevant mathematical theory of gene frequency. In section 2 we investigate the correlation between gene frequency and the environment and find some approximations for the average correlation. Section 3 applies these results to different genetic models. The theoretical description of the change in gene frequency has been developed mainly by Haldane (1924) Fisher (1930) and Wright (1931). It has become increasingly complex in a number of directions as the effects of population size, inbreeding, linkage, assortive mating, polyploidy, aberrant segregation, and other factors have been added to the model. Wright (1939) summed up his results until that year and presented the differential equation for genetic change in a very general form. More technical discussions are found in Wright (1935) Feller (1950) and Kimura (1955a). In the most widely quoted early studies, the wij were constants. However, the u’~ have also been considered to depend in a deterministic way on the gene frequencies (Wright, 1939) or to be random variables depending on environmental fluctuations (Wright, 1931; Kimura, 1954), but successive environments were supposed to be statistically independent. Our own models will be the simplest possible in most respects, and will concentrate on the effect of random variation in selection when the environment of one generation is correlated with the environments of previous generations. The study of genetic change as a random process is for the most part based on the adaptation of Kolmogorov’s results (1931) for diffusion processes. A discussion of the elementary theory together with its application to a number of interesting cases is found in Kimura (19558). A number of investigators have used continuous time models (e.g., Moran, 1958, 1959) with finite populations. Starting with the probability of an individual of a given type being formed or dying in the time interval (t, I + dt), these authors pass directly to a differential equation in the probabilities in which population size N figures as a parameter which may then be allowed to become infinite (Kimura, 1955b). However, we are interested in the differential equation for change in gene frequency itself, and prefer a more elementary but direct approach.

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The population will be assumed infinite, that is sampling error can be ignored and the gene frequencies treated as continuous variables between zero and one. We will consider a locus with two alleles A, -4’ whose frequencies are X, 1 -X respectively. The frequencies are assumed to be equal in both sexes and mating to take place at random, so that the HardyWeinberg law holds and new zygotes are formed in the familiar proportions X2 : 2X(1 -X) : (1 -X)‘. It will be further assumed that the selective value of each genotype depends on its own phenotype and on the environment but not on the composition of the population. A continuous time model for genetic change must take into account the effect of age on the rate of reproduction. The assumption that the HardyWeinberg law holds is equivalent to the assumption either that the time between birth and sexual maturity is infinitesimal compared to the length of the generation (so that zygotes formed in the previous instant can be assumed to contribute to the reproductive gene pool) or that selection is very slow, so that the gene pool is about the same whether or not we include the newly formed zygotes. In the applications that follow, the latter assumption will justify the model. Consider a population at time t with gene frequency X(t) and the three genotypes in Hardy-Weinberg proportions. After a short time h, these have been multiplied by 1 fhm,,, 1 +hm,,, and 1 +hm,, respectively, and the whole population has been multiplied by 1 + hfi where 1%= x2m,,+2x(1-X)m,,+(l-X)*m,,. (1.02) In the next interval, the population is again multiplied by 1 + hCi so that after time t the population has been multiplied by (1 + hfi)'ih. As h + 0 this becomes erm SO that eZ is the number of offspring per generation, or Wright’s IV, and fi is its logarithm, the Malthusian parameter. The new gene frequency X(t +h) is given by X(t+h) = {X2(t)(l+hnz,,)+X(r)[l-X(t)](l+hm,,)}/(l+hm). (1.03) Hence the change in X is

AX(t) = hX(t)Cl-X(t)l{[m,,(t)+m,,(t)-2m,,(t)lX(t)(1.04) [m,,(t) - ~,2(01)/(~ + fJfi(O) which is AX(t)= hX(l-X)[(m,,+m,,-2m,,)X-(m,,-m,,)]+R (1.05) where the remainder is R = h’E(t)X(l-X)[(m,,+m 22-2m,,)X-(m,,-nz,,)]/(l+hm). (1.06) The mij and Si are random variables which depend on the environment. If they all have finite variances and higher moments, which will be the case for the models studied in the next sections, then for each m and for all k

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lim Pr{ hEI > k} = 0

(1.07)

h-0

that is lim(l+h@}

= 1

(1.08)

h-+0

with probability one. Hence the remainder vanishes as h -+ 0 and dx z = X(1 - Xl II<% 1(t) + m,z(t)-2m,,(t))X-(m,,(t)-m,,(t))l with probability

(1.09

one. Furthermore, lim,lj’

X(t+h)-X(t) --___~

_ dx” -~

= o

(1.10) (: dt i) so that dx/dt is the mean square differential coefficient of X(r). Finally, the X process is continuous almost everywhere since it is mean square differentiable (Bartlett, 1955). Wright (1938) showed that subject to the qualifications introduced by Kimura’s discussion of quasifixation (see below) the frequency distribution of gene frequency approaches a stable limit distribution h

h-0

(1.11) 1

where K is a constant that makes

4(x)dx = 1, and M(x) and V(x) are the s mean and variance of dx/dt conditional on X. It is also known that the maxima and minima of 4(x) occur at the roots of the equation 0

M(x)

iav(x)

= i 7’

(1.12)

The most important question about $(x) is whether one or another allele will necessarily be fixed, or polymorphism will be maintained indefinitely. In finite populations the rate of fixation has been studied by Wright (1931), Fisher (1930), Kimura (1955a, b), Moran (1961) and others. In infinite populations there is not true fixation of genes whose initial frequencies differ from zero. (The fixation of genes that start with effectively zero frequency, e.g. a single individual in an infinite popmation, has been studied by Fisher (1930) and others.) Kimura (1954) has studied the distributions for infinite populations and has introduced the concept of quasifixation. This refers to a situation in which, for t sufficiently large, ~$(x,t) is bimodal with peaks that approach the boundaries 0 and 1 and probability mass increasingly concentrated near

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the peaks. When such a process occurs, the limit distribution obtained from (1.11) is U-shaped and refers to the distribution of unfixed classes. A bellshaped curve decreasing to zero at 0 and 1 indicates a stable distribution. In that case, the ergodic properties hold. That is, the expected value of x at time t is equal to the average of x(t) taken over all t. When we encounter situations in which quasifixation occurs in the infinite population model, we have two alternative interpretations for what will happen in actual finite populations. We could assume that fixation actually occurs, or that it is prevented by mutation at an extremely low rate which does not affect the shape of (p(x) except near boundaries. 2. The Correlation Between Gene Frequency and Environment The adaptive value W depends on the gene frequencies and on the adaptive values of the different genotypes, which in turn depend on the environment. Therefore W (and also m) is a function of gene frequency, and the environmental variables, the calculation of the expected value of W is impossible without knowing something about the joint distribution of gene frequency and the environment. In our applications it is sufficient to find the covariance. In the application of this paper wij (and rn& will depend on the environmental variable s. The value of s at time t, s(t), will be a random variable, and the assemblage for all times t, s{(t)}, is called a stochastic process. If the environment changes very rapidly, its value in one generation can be regarded as independent of its value in the previous one. This situation is the easiest to handle mathematicahy but is also the least interesting biologically since it implies that a population which is changing in response to the environment does not thereby keep up with it or improve its fitness. We are more interested in determining whether, and in what circumstances, genetic change enables a population to follow the environment and adapt successfully to changing conditions. This will depend on the covariance of the gene frequency and the environment, which can be found from the equation for genetic change with time. From (1.09) we have the differential equation for x(t) dx

z = xWC1-x(01 (x(t)[m,,(s(t>)+m,,(s(t))-2m,Z(s(t))l+ EM)m22W))l.

For the immediate becomes

applications,

(2.01)

the m,(s) will be of such a form that (2.01)

dx 2; = d(x)

+ bB(x)s(t)

(2.02)

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RICHARD

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where a and b are constants and A(x) and B(x) are analytic functions bounded on the closed interval [0, 11. In our applications they will be polynomials up to the third degree. Further, the aA term may be replaced by a sum of such terms without altering the results. If the stochastic process {s(t)} is such that the s(ti), s(tz) . . . s(t,) are mutually independent for every set of distinct times ti, then the theory of section 1 can be applied. If there is a limit distribution it will be given by 4(x) of (1.08) with M(x) = aA and V(x) = b’a~B*(x), and it follows that the extreme points of 4(x) occur at the roots of

&4(x)= b2B(x)$. If the s(t) are not independent the limit distribution is not known. The simplest model after independent s(t) assumes that s(t) is an OrnsteinUhlenbeck random process (Doob, 1942) which has the following properties : (1) the {s(t)> process is completely stationary, that is s(t) has the same distribution as ~(2’) for all t, t’ and their joint distribution depends on It- t’l but not on t or t’ separately; (2) s(t) is Markovian. This means that if we know s(t), and if h and k are positive numbers, knowledge of s(r- h) does not increase our information about s(t + k) ; (3) E(s(t)} = 0. Th is is no restriction since any non-zero term can be included in the A(x) of equation (2.02); (4) s(t) has a finite variance rrz and COV (s(t+h), s(t)) = e-h’aa:. Here l/cr is a damping constant which measures the rate at which correlation decreases with h. A large value of CI indicates strong correlation. It can be shown (Doob, 1942) that the s(t) have a multivariate joint distribution, so that E{s(t+h)ls(t)}

= evh”s(t)

Gaussian ww

and Vur{s(t+h)ls(t)}

= (l-e-2h’“)cr,2.

(2.05)

Thus as h + 0 the variance of s(t) given s(t -h) vanishes and the s(t) process is continuous in mean square. Therefore dx/dt is a mean square differential coefficient of x, and the x process is continuous almost everywhere (Bartlett, 1955). We have (2.06) E(s(r + Wlx(O} = E{E{s(r + W)s(O, x(O}lx(t)).

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By its construction .x(t) does not depend directly on any s(z) with z > t. From the Markov property and stationarity we have the Markov property for time going backwards also, so that none of the s(z) with r < t depend on s(t + h) given s(t). Hence x(l) depends only on s(z) for T < t. Furthermore s(t+h) depends on x(t) only through the information x(t) gives about the s(r), and hence by the Markov property the conditional distribution of s(t+h) given s(t) is independent of x(t). Thus (2.06) can be written (2.07) JqsO+h)lx(O} = E{E{s(t+h)ls(t)}Ix(t)}, and finally (2.08) E{s(t+h)lx(t)) = E{e-h’“s(t)lx(t)}. Likewise (2.09) E{?(t+h)jx(t)} = E(E(~~(t+h)js(t))jx(f)} which is E{s2(t+h)lx(t)}

= E(Var[s(t+h)ls(t)]lx(t)}+ E{E{s(t+

h)ls(t>}21x(t)}.

(2.10)

This gives E(s2(t+h)lx(t)) = E((l -e-2h’a)a~~x(f))+E(e-2h’“s2(t)~x(t)}, (2.11) but since the first term is independent of x(t) the E may be dropped, giving (2.12) E{s2(t+h)lx(t)} = (1-e-2h’a)~~+E{e-2h’a~(f)l~(f)). We can use the above results to find the expected value of s(r)x(t) by means of a recurrence relation. Replacing x(r + h) by its value in (1.09) and multiplying both sides by s(t + h) we have s(t+h)x(t+h) = s(t+h)x(t)+had(x)s(t+h)+hbB(x)s(t)s(t+h)+O(h2). (2.13) Now define (2.14) ~(7, t; x(O) = E{s(M+(O} and (2.15) K(t, t; x(t)) = E(s~(T)x~(T)[x(?)). We are interested in Y(T,~; x(t)) and K(r,r; x(t)) for r = t. The expected value of (2.13) is taken conditional on x(t) and z is allowed to approach r to give y(t+h, t;x(t)) = e-h~ay(t,t;~(t))+he-h~ay(t,t;x(t))aA(x)+ +he-“‘“K(t, t;x(f))bB(x)/x2+O(h2). (2.16) Since y(t, t; x(t))/x(t) = E(s(t)/x(t)) the terms in (2.16) remain bounded when x = 0. Now subtracting y(t, t; x(t)) from both sides of (2.16) dividing by h and allowing h to approach zero, we obtain the differential equation for y:

(2.17)

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RICHARD

LEVINS

The same procedure can be used for K(t, I; x(t)): s2(t+ h)2(t+

h) = syt+ h)x2(t) +2h?(t

+ h)x(t)aA(x)+ 2hs*(t + h)s(t)xbB(x).

As T + t this has the expected value conditional K(t+h,t;x(t))

(2.18)

on x(t)

= (1-ee-Zhiu)~~x2+e-zhiaK(t,t;x(t))+ ‘cx) 2hxaA(x) [l - e-2h’a]0~ +2ha x-

2hbB(x)E{s*(t

+ h)s(t)jx(t)}

e -2h/rrK+ (2.19)

+O(h*)

and as h + 0 this gives d$ = acr:x* - ~K+2a;K+2bB(x)xE{?(t)lx(~)}.

(2.20)

The equations (2.17) and (2.20) can be rearranged for solution neighborhood of a = 0, b = 0

in the (2.21)

K = afx* +aa y

K+ccbxB(x)E{s3(t)lx(t)}

- id;.

(2.22)

We consider two cases, depending on the relation of a to b. Case I: Q smaller than b (or same order). Then a solution for y up to order b requires K only up to order 1, allowing us to ignore the u and b terms. We note that dK _ -aK + .-.-aKdx-(2.23) ds- aT ax dr and similarly dy ay ay dx -- = az + ai 2;. (2.24) dr Further, dx z = d(x)+ bB(x) (2.25) is of order b. Then it can be verified by direct substitution

that

K = afx* +0(b)

(2.26)

where O(b) means some quantity of order b, is a solution since dK - = 2x&b). dr

(2.27)

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This gives 4x1

y = abB(x)a,2 + au ,y-a%

(2.28)

+ O(b’),

for which y = abB(x)az

(2.29)

is a solution near b = 0. Case II: b = O(a). Here a solution can be obtained up to order ab by finding K up to order a K = o,‘x2+O(a2)+O(b).

This is verifiable by direct substitution

(2.30)

with

dK - = 2o;xaA(x)+O(b).

(2.31)

y=abB(x)c~z+au~?y-ag+O(ul,b).

(2.32)

dr

Hence (2.18) becomes

A solution is + a’ubofA(x)‘$

+ O(a2b)

(2.33)

which is verified by substituting

dy dr =

abo, 2aBG aA(x)+O(a2b).

(2.34)

We have of course not shown that the solutions (2.26), (2.29), (2.30) and (2.33) are unique. Suppose that for a given K there are several solutions of (2.21). Since that equation is linear in y and dy/dr, the difference between any two solutions, G(t, t; x(t)), satisfies G = aaA(x)G-adG X

(2.35)

dr’

We can obtain formal solutions of (2.35) of the form G = F(x) e” by substitution into (2.21). Then

44

F(x) ear = aa x

F(x) e”‘- alF(x)

(2.36)

aF Irdx e” - a z e z.

(2.37)

Under the assumptions of case II this becomes F(x) eLt = au q

F(x) e”-

c&‘(x)

e”‘- a E aA

e”‘,

(2.38)

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RICHARD

LEVINS

from which the solution is (2.39)

G(~,~;x(~))=~xexp[-(~)~~~e~‘. 0

In order for C to be different from zero, G(t, t; x(t)) must vanish for CI = 0. Therefore C is not zero only when

x dx __ is positive. But if the integral is s A(x)

positive then G = Cxe -r/ae’l vanishes for a near zero more rapidly than any power of a, and can be omitted from our approximation. A similar argument applies to K and to case I. TABLE

I

Formulae If m = log wis the Malthusian parameter for a population at time t, then the change in gene frequency under selection is g = fx(l - x) 2.

(1)

For the equation $ = aA(x) + bB(x)s(f) where E(s) = 0 and s(r) is a stochastic process with no autocorrelation, the limit distribution is d(x) = &

exp [2 j $$j?&]

(2)

dx.

0

The extreme points of ((x) satisfy the relation aA

= b?$B(x)

as

ax.

(3)

Let y(t, t; x(t)) = E{s(r)x(t)lx(r)} and K(t, t; x(t)) = E{sa(t)x2(r)/x(t)1. Then in the neighborhood of 5 = t we have yqy.b~K+aaA$y-a$

(4)

and K = o,axa + aa ‘+

K + axbB(x)E{s”(t)~x(t)}

- ; “df

where a = - l/lnp and p is the correlation between s(t) and s(t - 1). When the terms of order ab can be ignored, y = abc$B(x). When the approximation is taken to order ab. y = abuzB(x)

1 + era $$

+ a%ba,2A(x) ‘E.

(3

(6) (7)

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3. Exponential Quadratic Deviation Model In addition to the general assumptions of sections 1 and 2, the exponential quadratic deviation model specifies that at any given time there is a single environment, uniform in space but changing in time, for which there is an optimum phenotype s(t); that the stochastic process s(t) satisfies the assumptions of section 2: that the phenotype depends on a single locus with two alleles and no dominance on the phenotypic scale; and that the adaptive value WV has the form pfTj = ,-cw)-qrjP (3.01) where qij is the phenotype of genotype ij and C is a positive constant which measures the sensitivity of fitness to deviations from the optimum. It has the same meaning as the C in the previous paper, and inversely measures homeostasis. The quadratic deviation model, in which fitness declines with the square of the deviation of the actual phenotype from the optimum, was introduced by Wright (1935) and studied further by Kojima (1959) under the designation “optimum model”. These authors considered a constant environment, and were concerned with problems of equilibrium values. Besides being interesting in its own right, the optimum model is the first approximation to any model where fitness declines symmetrically with the deviation. However it has two disadvantages for our purposes. First, there is some threshold value of the deviation at which fitness becomes negative. This would require either that we restrict the environment in such a way that such values are never attained, or specify that fitness is taken to be zero whenever the quadratic deviation would give negative fitness. The second difficulty is that the equation for genetic change derived from this model cannot be put in the form of (2.02) and therefore is not amenable to solution by our methods. The exponential quadratic deviation model is approximately the same as the quadratic deviation model for small deviations, gives a value of Wij which is always positive, and gives a quadratic deviation model for the Malthusian parameter, from which a manageable equation for genetic change is derived. The model is described in Table 2. TABLE

____

T.B.

Genotype

Frequency

AA AA A’A’

2x(1 - x) (1 - x)”

2

Phenotype

X2

i -a

Adaptive value

Malthusian parameter

e-c(s-n)* e-C@) e--C(S+O)’

- C(s - a)” - C(s”) - C(s + a)” 16

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From the table we find the mean Malthusian parameter, iii = - C{x2(s - a)” + 2x( 1 - x)s2 + (1 - x)Z(s + u)‘}. Now from (1.09) we have dx z = Ca2X(1-X)(l-22x)+2aCx(l-x)s(t). Thus when s(t) is not autocorrelated

(3.02)

(3.03)

we have

= Ca’x(l-x)(1-2x)+2&x(1-x)E{s(t)}

(3.04)

and $x = 4a2C2a5x2(1 -x)‘. 1 II Then by formula 2 the limit distribution is given by

(3.05)

Var

(3.06) The shape of b(x) depends on the exponent.

If 8Ccri < 15 2%

U

the

distribution is bell-shaped and polymorphism will be stable. If only one of the inequalities holds, b(x) is J-shaped and fixation of the favored allele is suggested. If neither inequality holds, &x) is U-shaped and quasifixation of both alleles in different populations is indicated. Now as in section 2 we consider the case where s(t) is an OrnsteinUhlenbeck random variable. For mathematical convenience we assume that E(s) = 0. Then E(x) = l/2 and the mean Malthusian parameter is from (3.03), E(E) = -C{2u2(~+o;)-4uE{sx} +a;>. (3.07) We can find E(sx) by substituting the terms from equation (3.03) into formula 6 of Table 1. The coefficient of s(t) in (3.04) corresponds to bB(x) in formula 6, and the term without s(t) corresponds to uA(x). Then E(sx) is E(sx) = 2ctuCc7,2E{x(l-x)}. (3.08) Thus (3.09) E(m) = C{ - 2u2(a + c~f)+ 8u2aCof(f - of) - CJ;}. In order for a small value of a to give a higher mean rate of increase than that obtained for a equal to zero, that is for polymorphism to be desirable, the coefficient of u2 must be positive: (3.10) We must now consider whether (3.10) can be satisfied in nature. When polymorphism is stable and 9(x) is bell-shaped the variance flz is less than

THEORY

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IN

A HETEROGENEOUS

237

ENVIRONMENT

what it would be for the uniformly distributed x, which is l/12. The variance l/12 is approached as S&f + 1. Thus, for the largest permissible value of 8Ca,? which gives a non U-shaped 4(x) this becomes (3.11) However, when a = 4, l/a = 4 and the correlation between the environments of successive generations is e-+ or about 0% Brooks SCCarruthers (1953) report that the correlation between the temperatures or rainfall of successive days is about O-7, and for successive years virtually zero. Therefore it is not likely that (3.11) will be satisfied for environmental factors of a meteorological origin. For other aspects of the environment such as biotic succession the correlation between conditions of successive generations can be that high. For values of SC’af which are much smaller than 1, the variance is smaller than l/12, and in order for polymorphism to be desirable must exceed approximately 4/8&z, which is greater than 4. Thus it is less likely that the optimum value of a will be different from zero. These conclusions are only slightly altered when we allow mutation. We note that in (3.09) the coefficient of c: is always negative for any a different from zero. Therefore any process which reduces ts;fwithout changing the mean of x or reducing E(sx) improves the fitness of the population. Mutation occurring with equal frequency in both directions meets these requirements. For then the rate of change of x becomes dx - = Ca2x(1-x)(1-2x)+u(l-2x)+2aCs(t)x(l-x). (3.12) dt where u is the mutation rate. For non-autocorrelated s(t) the limit distribution is given, from formula 2, by +(x)

=

qx(l/&--wCb.~)

,-u/cc.*&f~-~)1~

(3.13)

The factor exp [ - u/Ca20~x(l -x)] approaches zero near the ends of the interval [0, I] and has a peak at x = l/2. Therefore mutation has the effect of concentrating the probability density more closely around the peak and reducing the variance. When s(t) is autocorrelated, we can find E(sx) from formula 7 of Table 1, Then E(sx) = 2aCaz[f - a: + 6auai]. (3.14) When this result is substituted into (3.07), the condition for optimum becomes 8Cafa[~-a~+6aua~] > t+af.

a> 0

(3.15) 16-2

238

RICHARD

LEVINS

The left side of the inequality has been increased both by the reduction of u; and by the term including u explicitly, while the right side has been decreased. Thus the inequality can now be satisfied with smaller values of CL The adaptive significance of mutation will be discussed in greater detail in the next paper of this series. The question arises, to what extent does the quadratic deviation model give a generally applicable result? To investigate this, we let the adaptive value of phenotype a be given by (3.16) W = W(s-a) where W(z) is a non-negative function of its argument with peak at z=O and is symmetric about its peak so that W(z) = W( - z), W’(z) = - W’( - z) Since for large deviations from the optimum the phenotype becomes inviable, we take W(z) to be asymptotically zero as Iz[ increases in either direction. Finally, W”(z) is negative for jz[ < z0 and positive for Izj > z,,. Thus z0 is the point of inflection of W(z). As in the previous model we assume .r(t) to be symmetrically distributed about a zero mean, so that the mean gene frequency will be l/2. Then W = x”[ W(s - a) + W(s + a) - 2 W(s)] +2x[ W(s) - W(s + a)] + W(s + a). (3.17) Since we are interested in what happens in the neighborhood can expand Wabout s to get, to order a2,

of a = 0, we (3.18)

W = W(s)+aW’(s)-2axW’(s)+a2W”(s)[*-x(1-x)]. Since W’(s) is an odd valued function of a symmetrical Therefore when s(t) has no autocorrelation E(w) = E{W(s)}+a2E{+-x(1-x)}E{W”(s)}.

s, its mean is zero. (3.19)

The term 1/2-x(1 -x) is never negative, so that the coefficient of a2 has the same sign as EW”(s). We now show that in order for EW”(s) to be positive, s must have at least a bimodal distribution with modes sufficiently far apart. Consider E( W”(s)) = j W”(s)P(s) ds -cc where P(s) is the distribution of s. If s had a uniform -K, K, so that

(3.20) distribution

over

(3.21)

THEORY

OF

FITNESS

IN

A

HETEROGENEOUS

then we would have

ENVIRONMENT

239

K E{W”(s)) = & j W”(s)ds

(3.22)

-K

which is

&v’fK)-

W’(-K)] < 0.

(3.23)

In order for E{ W”(s)) to be positive the integral in (3.20) must give greater weight to values of s for which W”(s) is positive, that is it must have a greater probability density for 1x1> z0 than for IsI < z,,. Hence P(s) must be at least bimodal. Therefore in the absence of an autocorrelated s, polymorphism will be desirable only when s has a bimodal distribution with modes sufficiently far apart (beyond + zO). The two modes of this distribution correspond to the two discrete niches of the previous papers (Levins, 1962, 1963). When EW”(s) is negative, polymorphism will only be advantageous if the autocorrelation of s is sufficiently high. This autocorrelation gives rise to a positive correlation between x and s, and a negative correlation between W’(s) and x which increases the value of R? E(W) = E{W(s))-2aCOV{x, JV’(s)}+a2E{W”(s)[+x(l-x)]}. (3.24) How high the autocorrelation must be depends on the form of W(s). Thus the quadratic model gives a qualitatively general result for all W’s for which E( W”(s)} is negative. I am greatly indebted to Dr. Howard Levene for his patient criticism throughout the research and preparation of this study. The work was supported by National Science Foundation graduate fellowships for the years 1947-58, 1958-59, and 1959-60. REFERENCES BARTLE~, M. S. (1955). “An Introduction to Stochastic Processes”. London: Cambridge University Press. BROOKS, C. E. P. & CARRUTHE~~, N. (1953). “Handbook of Statistical Methods in Meteorology”. London: Her Majesty’s Stationery Office. DOOB, J. L. (1942). Ann. Math. 43,2. FELLER, W. (1950). Proc. Second Berkeley Symp. Mathematical Statistics Probability. University of California Press. FLPHER.R. A. (1930). “The Genetical Theory of Natural Selection”. Clarendon Press. FISHER, R. A. 81,FORD, E. B. (1951). Amer. &ient. 39, 452. HALDANE. J. B. S. (1924). Trans. Cumb. ohi/. Sot. 23. 19. JEPSON, d. L., S&P&, G. G. & M~YR, E. (Ed.j (1949). Adaptation and selection. Chapter 20. In : “ Genetics, Paleontology, and Evolution”. KIMURA, M. (1954). Genetics, 39, 280.

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KIMURA, M. (1955a). Proc. MI. Aoad. Sci., Wash. 41, 144. m, M. (19556). Cold Spr. Harb. Symp. quant. Biol. 20, 33. KOJIMA, K. (1959). Proc. nat. Acad. Sci. 45,989. KOLMOGOROV, A. (1931). Math. Ann. 104, 415. LEVINS, R. (1961). General Systems, 6, 33. LEVINS; R. (1962). Amer. Nat. 96, (891), 361. LEVINS. R. (1963). Amer. Nat. 97. (893). 15. MORAN, P. k. P:(1958). Hum. Giiet. 53, 1. MORAN, P. A. P. (1959). J. Aust. math. Sot. 1, 121. MORAN, P. A. P. (1960). Trans. Camb. phiI. Sot. 57, 304. MORAN, P. A. P. (1961). UHLENBECK, G. E. & ORNSTEIN, L. S. WRIGHT, S. (1931). Genetics, 16, 97. WRIGHT, S. (1935). J. Genet. 30, 257.

(1930), Physical Review, 36, 823

WRIGHT, S. (1938). Proc. nat. Acad. Sci., Wash. 24,253. WRIGHT, S. (1939). Genetics, 24, 538.

The Theory of Fitness in a Heterogeneous Environment. IV. The Adaptive Significance of Gene Flow Author(s): Richard Levins Source: Evolution, Vol. 18, No. 4 (Dec., 1964), pp. 635-638 Published by: Society for the Study of Evolution Stable URL: http://www.jstor.org/stable/2406216 . Accessed: 24/06/2011 01:11 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ssevol. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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THE THEORY OF FITNESS IN A HETEROGENEOUS ENVIRONMENT. THE ADAPTIVE SIGNIFICANCE OF GENE FLOW RICHARD

IV.

LEVINS

Department of Biology, Universityof Puerto Rico Accepted July 9, 1964

Gene flowhas generallybeen regarded values for gene flowthat depend on the to a population,itsprincipal statisticalstructureof the environment, as detrimental effectbeing to swamp a local population and that natural selectioncan establish and preventits adaptationto local condi- these optimal levels (or more precisely, tions. The swampingeffecthas been con- that the actual levels of gene flowamong sideredso strongthat priorisolationwas populationsof different species differin thoughtto be a necessaryfirststep in thesame directionas theiroptimalvalues). speciation. More recentlyit has been The modelused hereis a continuation of recognizedthat strongselectioncan pro- that used in the previouspapers of this even in the face of a high series (Levins, 1962, 1963, 1964). It is duce divergence rate of migration(Thoday and Gibson, assumed that corresponding to each en1962; Streamsand Pimentel,1961), while vironment thereis an optimumphenotype the experiments of Koopman (1950) show S, whichmayvaryin timeand space; that that sexual isolationcan arise in popula- whentheactualphenotype is theoptimum, tionsin contact. the adaptivevalue is maximized;and that This has raised the oppositequestion: fitnessdeclinestowardzero as the deviawhydo we not have local speciationon a tion of actual fromoptimumphenotype much greaterscale, why are there any increases. (The phenotypeis genetically widespreadspeciesat all, whyis not each determined.Anydevelopmental or physiolocal ecotypea distinctspecies? The ob- logical modification of phenotypeby the vious firstanswerrefersto populationsize. environment has the effectof reducingthe sub- environmental becomesinfinitely As theenvironment variance,as explainedin the divided, local population size decreases, second paper of this series.) We further and the probabilityof randomextinction specializethe modelby assumingthat the increases. Suppose that the averagenum- fitnessof each phenotypedeclineswiththe ber of offspring per pair is r, and each has square of the deviationof that phenotype a probability1/r of survival. Then the fromthe optimum,so that probabilitythata populationof size 2n beWtj(t)= 1- [St(t) - ptj(t) ]2, comes extinctby chance is [1 - (1/r) ] rn (1) whichapproachesel-nforlarger. However, whereWij(t) is the fitnessof phenotypej this probabilitydecreasesrapidlywith n, in populationi at timet, Si(t) is theoptiand would not be an importantfactorfor mumphenotypeforpopulationi at timet, populationsgreaterthan severalhundred. and Pij(t) is the jth phenotypein populaA seconddisadvantageto small populationi. The fitnessof thewholepopulation tion size is the loss of geneticvariance is foundby averagingover all phenotypes throughrandomdrift,whichmightbe imto get portantin verysmallpopulations,but the same objectionapplies as in the previous (2) WJ.(t)= 1- [St(t) - P.(t)]2argument.Drift cannot account for the VAPHE paucityof abundantbut local species. It will be argued below that gene flow wherePi.(t) is themeanphenotypein popamongpopulationsis part of the adaptive ulation i at time t and VAPHE is the variancewithinthepopulation. systemof a species,thatthereare optimum phenotypic EVOLUTION

18: 635-638. December, 1964

635

RICHARD LEVINS

636

Averagingover all timest we obtain the so that mean fitnessof the population d=(S-P) (6) (3) E(Wi ) = 1 - (S - p)2 - var (S) var (Pi.)-E(VAPHE)

Xi

+

The summationtermis the total additive so thatour geneticvarianceforphenotype, theorem Thus fitnessis reducedby thedeviationof particularcase of thefundamental the mean phenotypefromthe mean opti- of naturalselectionbecomes mum,by the variancesof the environment dP and of the mean phenotype,and by the (7) d = V(S-P), variance,and is increased meanphenotypic by the covariance of environmentand whereV is theadditivegeneticvariancefor mean phenotype. phenotype. For conveniencewe assume In thepreviouspapersit was shownthat that V remainsconstant,a reasonableasin P are not the greaterthe additivegeneticvariance, sumptionif the fluctuations the greateralso will be VAPHE and the too great. constantin time, varianceof the mean phenotypein a flucCase I: Environment Thus, additivevari- variablein space, so that the populations tuatingenvironment. environments. different ance reducesfitness.However,if thereis have permanently in the Then thechangein meanphenotypein two a sufficiently high autocorrelation (correlationbetweenthe en- populationsis givenby thepairof equations environment of the of successivegenerations vironments samepopulation)thenthemeanphenotype (8) dPi =V[Si(t)-1] +m(P2- Pi), dt and therefore can followthe environment the covarianceof S and Pi. can increase and to compensatefor the fitnesssufficiently lossesdue to var (P) and VAPHE. In that (9) dP2 + M(Pl-N2, d 2= V[S2(t)-P2 dt case, the optimumamountof geneticvariance will be greaterthanzero. wherem is the fractionof each population Migrationbetweenpopulationscan affect exchangedwiththe other. Since the Si(t) each of the componentsof fitness. The are constants,an equilibriumvalue is natureof theseeffectscan best be seen by reachedat consideringthreespecial cases. First we - S2)/(2m + V) note that the change of mean phenotype (10) Pl = Sl -(Sl underselectionat n loci is and 2 cov (S,Pi.).

(4)

dP ~dt =

aP dxi axi dt'

( l l)

P2 = S2 -Mw(S2, - Sl)1(2m + V).

Thus migrationdisplacesthe mean phenoat the ith locus. typefromits optimalvalue by an amount wherexi is the frequency between Further,forcontinuousmodelswithover- whichincreaseswiththedifference and the amountof gene the environments lappinggenerations, flow,and decreaseswiththe additivevaridxi ance. This is the familiarswampingeffect. (5) =12Xi Xi) aW dt Oxi VAPHE is also increasedby migration since each populationis a mixtureof two In our model, means. Hence, populationswith different Ow withdifferbetween populations flow gene 2 2(S -P)OaP/Oxi reducesfitness. ent average environments Ox1

ADAPTIVE SIGNIFICANCE

OF GENE FLOW

63 7

Case II: Environments of bothpopula- mentalchangeswitha highautocorrelation tions fluctuateindependentlyabout the (long-termchanges), we can now assert same mean,and thereis no autocorrelation thatthe adaptivesignificance of gene flow betweenenvironments ofsuccessivegenera- is thatit permitsthepopulationto respond tions. The pairofsimultaneous differentialgeneticallyto long-term, widespreadenviequations(8) and (9) has the solution ronmentalchangeswhile dampingthe responseto local,ephemeralfluctuations.Or + 1/2e-2m(t-,)] Pi(t) = Vfte-(t-){[/2 in information terms,migrationis a filter (12) Si (y) + [ e/-l/e-2m(t-y)]. wherebyspatial information is used for S2(y)} dy, temporalprediction. wherey is a dummyvariableof integration. The above argumentleads to the conThus Pi(t) dependson the environmentsclusion that the optimalamountof gene of both populations,weightedin favorof flowbetweenpopulationsis increasedby recentenvironments. The relativeweight thetemporalvarianceof theenvironmental of the environment of its own habitat is variable and decreased by the spatial greatestfor recentenvironments and ap- gradient. From this several predictions proachesone-halfforthe environments of follow: the remotepast. The average P1 is the 1. In regionsof temporalstability,there same as P2, and is not affectedby migra- will be greatergeographicdifferentiation tion. The varianceof Pi is and speciation per unit spatial heterogeneity. (13) var(Pi) = % var(S)* 2. Stable habitats (large lakes in con[1 + 1/(1 + 2m/V)]. trastto streams,forestsin contrastto open When m is zero, var (Pi) 1/2 var (Sl) fields,subsoilas contrastedwithsurface) whileform/Vlargevar(Pi) /4var(S1). favorreducedgene flowand local differThus migrationbetweentwo populations entiation. can reducethevarianceof meanphenotype 3. Alonga giventransect,witha given by almosthalf (and formanypopulations, spatial heterogeneity, herbs, weeds, and theeffectis greater).Therewillbe a slight animalshave a morevariablelifetimeenincreasein VAPHE, but the overalleffect vironmentthan climax trees. Therefore, of migrationis to damp the responseto whereas the climax vegetationwill be ephemeralfluctuations of the environmentsharplyzoned,thesecondaryspecies,herbs, and thusto increasefitness. and weedswillcrossseveraltreezones and Case III: Environments vary in time, have moregeneflowalonga transect.This but are the same forthe two populations. willbe associatedwithsimultaneous flowerThen ing along the transectforthesespeciesas contrastedwith waves of flowering along (14) P(t) = V te-v(t-Y)S(y) dy thealtitudinalgradientforclimaxtrees. independently of m. 4. Species may be describedas having Comparingcases II and III, we see that fine-grained environments iftheindividuals migrationdamps the responseof popula- wander over many of the microclimatic tions to local fluctuations whilepermiting patches during their lives, and coarseresponseto widespreadchangesof environ- grainedif the individuallands at random ment. It remainsto be noted that long- or by habitat selection within a single termor major fluctuations tendalso to be patch and remainsthereforits wholelife. whileshort-term widespread, oscillationsof The environment ofa coarse-grained species environment are more localized (see, for withouthabitatselectionhas the greatest example,Hariharan,1956). Since it has temporalvariancefromgenerationto genbeen shown previouslythat responseto eration,the fine-grained species are next, selectionis advantageousonlyforenviron- and the coarse-grained withstronghabitat

RICHARD LEVINS

638

selectionhave the lowest temporalvariance. Plants are in the firstcategorywith respectto soil types; mostvertebrates are in the second, while the smallerinvertebratesare in thelast. The expectedamount of gene flowdecreases,and the degreeof diversification increasesfromfirstto last. The questionarises,to what extentare the resultsdependenton the detailsof the model? Considera more generalcase in which,insteadof equation (1), (13)

V IS1(t)- Pi1lI . WTj(t) W[

For a fixedSi(t) we can expand in Taylor's seriesabout S -Pi: (14) WVj(t)=W[Si(t)-Pi(t)] [Pi(t) -pj]

-

+

W'[Si(t) - Pi(t)] + 1/:, [S.&(t)_p,(t)] 2 W"[Si(t) - P-.(t.)]+

Now taking the expectedvalue over all phenotypeswe have (15) Wiij'(t)= W [ S(t.) - Pi(t) + ?/VAPHE W"[Si(t) - P.i(t)]

whichhas the averagevalue (17) W(Si-Pi) =W(S -P) + ?/[var(S) + var (P) - 2 cov (S, P)]X W"(S-P)

+

Once again,if the averageP is not too far fromthe averageoptimum,thenthe coefficientof var (S) + var (P) - 2 cov (S, P) is negativeand the qualitativeeffectsare unaltered. Therefore,the conclusionsof thispaperholdbeyondthequadraticmodel is not too variprovidedthe environment able or migrationtoo extreme. SUMMARY

of geneflowis The adaptivesignificance thatit permitspopulationsto respondunwideder natural selectionto long-term, spread fluctuationsin the environment while damping the response to local, ephemeraloscillations.The optimumlevel of gene flowincreaseswith the temporal and decreases varianceof the environment gradient. spatial the average with LITERATURE

CITED

1956. A

in spatial and

study P. S. of VAPHE is negative, HARIHARAN, If the coefficient frequency distribution of daily rainfall in the qualitative effectsare in agreement relation to a network of rain recordingstations. Indian J. Meteorol. Geophysics, 7 withthe previousspecial model. This will (3,3). alwaysbe thecase if Pi is nottoo farfrom K. F. 1950. Natural selection for KOOPMAN, unimodal is a non-negaSi, because W(z) reproductive isolation between Drosophila tive functionof its argument.Only when pseudoobscura and D. persimilis. EVOLUTION, z exceeds z0, the point of inflectionof 4: 135-148. W(z), does W"(z) become positiveand LEVINS, R. 1962. Theory of fitnessin a heteroI. The fitness set and fitnessincreasewith VAPHE. Similarly, geneous environment adaptive function. Am. Nat., 96: 361-373. expanding W(S - P) about the mean . 1963. II. Developmental flexibilityand values of Si and Pi, we have niche selection. Am. Nat., 97: 75-90.

(16) W(Si - Pi) = W(S - P) +

[ (Si - S)=- (Pi - P) ]

W'(S - P) + ?/2[ (Si - S) + (P- P) ]2. W"(S - P) +...

. 1964. III. The response to selection. J. Theor. Biol. (in press). 1962. IsolaTHODAY, J. M., AND J. B. GIBSON. tion by disruptive selection. Nature, 193: 201-210. 1961. F. A., AND DAVID PIMENTEL. STREAMS, Effects of immigrationon the evolution of populations. Am. Nat., 95: 201-210.

THEORY OF FITNESS IN A HETEROGENEOUS ENVIRONMENT. V. OPTIMAL GENETIC SYSTEMS RICHARD LEVINS Department of Biology, University of Puerto Rico, Rio Piedras Received May 31, 1965

F O R populations living in more or less constant environments there is a close connection between the frequencies of alleles at different loci and the direction of natural selection operating on these genes. Alleles at very low frequencies will, by and large, be those selected against but maintained by recurrent mutation. Alleles in intermediate frequencies are for the most part maintained by some form of balancing selection. This correlation between allelic frequencies and selective forces is disrupted in small populations where genetic drift will cause gene frequencies to differ from their equilibrium values, but for large populations the frequencies of alleles will, in general, be such as to maximize or nearly maximize population fitness. When we turn to a consideration of populations in fluctuating environments, the problem is much more complex. Alleles may have low frequencies because natural selection has reduced them in past generations, but these low frequencies may be very far from the equilibrium values dictated by current selective forces. Thus gene frequencies at various loci and in various populations are correlated with past selective forces, but not necessarily with recent forces, or, worse yet, they may be negatively correlated with current selection if successive environments are themselves periodic or correlated to the proper degree. Unfortunately, from the standpoint of the population, the fitness of the population is determined in any generation by the correspondence between the gene frequencies and selective forces at that instant. If the pattern of environmental change is such as to make past environments poor predictors of present environments, populations that have responded adaptively to past environments will be ill-adapted to present ones. Thus, it is clear that the average fitness of a population is a function of the pattern of fluctuation of the environment, on the one hand, and the way in which a population’s genetic composition changes in response to the environment. While the pattern of the environment is not generally under the control of the population (it may be if the organisms’ behavior allows them to modify their immediate environment adaptively) ,the way in which the population responds to selective forces is a function of the constants of the genetic system. Additive effects of genes, dominance, epistatic interaction, linkage, mutation rates, migration rates all are critical in determining whether the population changes quickly or slowly, whether different genotypes are differently fit, and This reaearch was supported by Atomic Energy Commission Grant AT(30 1)2820, principal investigator R C. LEWONTIN Genetics 52 : 891-904 November 1965

892

R . LEVINS

thus to what extent the population has a high fitness at any time. The hypothesis has been proposed (LEVINS1962) that the genetic parameters can be interpreted as adaptations to the statistical pattern of the environmental heterogeneity. For each such pattern of environment, there will be optimum genetic systems, that is, some values of the constants which maximize average fitness regardless of how fitness is defined. It is further argued that selection acting on the genetic system itself will change the constants toward their optimum values, so that existing populations might be expected to differ in the same directions as their optima differ. The present study is concerned with the optimal values and adaptive significance of additive phenotypic effect and to some extent linkage, under several models of natural selection. We are asking how different patterns of environmental variation change the average fitness of a population when the genes concerned have different additive effects and different linkage relationships.

Fitness and Phenotype The fitness of each genotype in each environment has to be specified by way of the phenotype. This can be done in two ways which are equivalent except for the units used. 1. Corresponding to each genotype there is an optimal environment. As the actual environment departs from the optimum in either direction, fitness decreases toward zero. The rate at which fitness declines, and the fitness at optimum can, of course, be different for each genotype. For instance, it is well known from the work of DOBZHANSKY and his collaborators that chromosomal heterozygotes are often less sensitive to environmental differences than are homozygotes, while the super-vital” homozygous types are often narrow specialists. However, since our primary concern here is with adaptations to different environments rather than the genetics of homeostasis, we will assume that all genotypes have equal peak fitness and equal rates of loss of fitness with departure of the environment from the optimum but that the particular environment that is optimal differs from genotype to genotype. Since the different phenotypes differ only in the value of the environmental variable that is optimal, the fitness of a given phenotype in a range of envrionments can be expressed as W ( S - y ) where W is a unimodel symmetric, nonnegative function of its argument (S-y). S is the optimal environment for the genotype in question and y is the particular environment for which fitness is being measured. Then, in Figure 1, the two curves show the fitness of two different phenotypes, I and 11, over a range of environments, y . Here both S and y are measured in units of environment. 2. A dual representation of this system is one in which the environment is considered fixed and S represents the phenotype that is optimal in that environment. I n such a case y represents the phenotype value of a genotype and the deviations (S-y) are measured in phenotype units. For this representation, the abscissa in Figure 1 is a measure of phenotype and the curves I and I1 are the fitnesses in two environments of a continuous range of phenotypes. 66

THEORY O F FITNESS

893

FIGURE1.-Relation between fitness, on the ordinate, and environment or phenotype, on the abscissa. In one model, cuwes I and I1 represent two different phenotypes and the abscissa is i n environmental units. In the dual representation, I and I1 are different environments and the abscissa is in phenotypic units.

Among the possible models of W (S-y) the quadratic deviation model is the simplest with the essential qualities. Here [l-(S-y)'/H 2 0 W(S--y) = (0 otherwise. where H is a measure of homeostasis, a large H reducing the sensitivity of the genotype to environmental change and therefore the effective environmental variance. For most of this paper we are not concerned with homeostasis and set H = 1 for convenience. Any fitness model which is symmetric about its peak will have a quadratic deviation model as its first approximation. In the previous paper (1964), it was shown that for a unimodally distributed S the quadratic deviation model has the same qualitative results as a more general W , while a bimodally distributed S is essentially the 2-niche model of the first paper (1962). The mathematical convenience of the quadratic deviation model is of course that the expected value of the fitness depends only on the means, variances, and covariances of the environmental variable and the gene frequency. Therefore, it has been used by a number 1930; WRIGHT1935; LEWONTIN 1964) in analyzing popuof authors (FISHER lations. More complicated functions for W ( S - y ) introduce higher moments of the variables, but the qualitative results remain. Methods and Approach What we wish to do in this paper is to determine the effect of different values of genetic parameters such as gene effect and recombination fraction on the average fitness of a population through time. The average fitness of a population segregating at a single locus is defined by = W,, 2, 2W,, x( 1-x) W,, (1-2)2 (1) where W,,, W,, and W,, are the fitnesses (relative probabilities of leaving offspring) of the three genotypes AA, Aa, and aa,and where x is the frequency of

w

+

+

894

R. LEVINS

the gene A . Thus, if we know the gene frequency in each generation and the and average these values through time. fitnesses we can evaluate (1) for When environment is constant (Wij’s are constant) we can make use of the equation for gene frequency change

to solve for the gene frequency x in every generation starting with any arbitrary gene frequency xo.These values, x, when substituted into (1) give the required value of W. Our problem is more complex, however. The fitnesses are varying each generation with some secular as well as random component to the variation. For any particular sequence of fitnesses through time we can determine Ax in each generation and find the average fitness over time. But we are concerned not with particular sequences of environments, but with general descriptions of environmental change. Thus, we would like to be able to specify the distribution of gene frequencies over time, the probability that a gene frequency will take a particular value, when the statistical characteristics of the environment are given. Such characteristics are the mean selection coefficient, its variance through time and the serial autocorrelation between successive environments, among other things. If we can specify the gene frequency distribution, it is then a simple matter to calculate the average fitness from equation 1. If the environment is a “pure noise” random variable with no autocorrelation (implying no secular trend), the methods of diffusion equations give the distribution of gene frequencies through time (KIMURA1955),and, using such distribucan be found. The most tions of x together with equation 1, average values of interesting problems arise, however, when successive environments are correlated, but no simple solutions are possible in such cases, and so I have resorted to a Monte Carlo simulation scheme. There is no question that, in general, environments are serially correlated to a greater or lesser degree. It is well known, for example, that in continental temperate regions a weather predictor can be correct more than 70% of the time by simply predicting that tomorrow’s weather will be the same as today’s. The serial autocorrelation grows weaker as the time interval between points grows greater so that the correlation between days one month apart is very low and that between days a year apart is essentially zero. The correlation between successive two-day periods would be .49, and for successive weeks or months would decline further. Thus short generations would permit environments with higher autocorrelation. This autocorrelation is completely independent of any cyclic changes in the environment. A processof this kind can be generated in the following way: Let S, = value of some environmental variable at some moment or during some time interval Stdl = value of the same variable at the equivalent moment or interval in the previous generation, year, month, etc. = average value of the variable over all serially equivalent moments or intervals E = a random variable with zero mean and variance, u2 and K,, K , = arbitrary weights

s

8

895

THEORY O F FITNESS

s

Then the deviation of S t from $ can be related to the deviation of St-, from by the relation St - S= K , (St-, - S) K , E (3) Provided that K , is less than unity, it can be shown that after a sufficiently long time, t, St has a stable distribution with a finite variance

+

and that the ordinary product-moment correlation between St and St-,, p, is, in fact, K,. In this paper I have investigated the effect of the autocorrelated process given is in expression 3 with K , = dl - K,' so that the environmental variance, qS2, equal to U:, and the correlation between successive generations is K,. By varying K , and 2,it is then possible to control the pattern of environment over time. I have examined the effect of this variation on the fitness of a population segregating at two loci each with two alleles. The loci are considered identical in their phenotypic effect. Table 1 shows the relation between fitness and phenotype at each locus according to the quadratic optimum model discussed above. The course of events in such a population subject to the correlated fluctuations in environment is followed by simulating the process with a digital computer as follows. The computer is started with some arbitrary gene frequency (usually .5) at each locus, and some arbitrary initial value of S (usually zero). The computer generates a random value of E by generating a uniformly distributed pseudorandom variable between 0 and 1 (BOFINGER and 'BOFINGER1958) and then using that variable as the argument in a uniformly spaced table of the inverse normal distribution. The value from the table is multiplied by a constant to adjust the variance of E . From the current value of equation 3 yields St which, when substituted in the relations in Table 1, gives the fitnesses, W,j. These in turn when substituted into (1) give the current value of and when used in (2) give the next generation gene frequency. This procedure is repeated sequentially, beginning with the current value of St and the gene frequencies, for 100 generations and the average is calculated. Such a sequence of 100 generations constitutes one replication and each parameter set is replicated ten times. Computations were performed on the IBM 7070/7074 Computer at the University of Rochester. An alternative model for the environment is based on seasonal change. Here the environment is allowed to alternate every n generations between S and -S.

w

w

TABLE 1 Quadratic deviation model for one locus Genotype

AA AA' A'A'

Frequency

Phenotype U .

2 2

0

2x(l-x) (1-x)Z

--a

Fitness

l-(S--a)Z>O, zero otherwise 1-S2 >0, zero otherwise 1- ( S f n ) 2 >0, zero otherwise

896

R. LEVINS

This may seem highly simplified compared to the familiar sinusoidal graphs of monthly temperature or rainfall. However these graphs are based on averages over many years data. In any one year the change over from winter to spring weather patterns is rather abrupt although the time of change varies. Thus a sinusoidal model is not necessarily more realistic. In this model, when the environment changes every n generations the autocorrelation is (n-2)/n so that there will be a positive autocorrelation only when the period exceeds four generations.

Optimum Phenotypic Effect

*

In the two locus model we have used, each locus contributes a to the phenotype (Table l ) . Thus, a large value of a represents a major gene while if a = 0 the locus has effectively disappeared. The first question is, what is the optimum value of a? In previous analytical investigations (LEVINS1964), it was shown that the optimal value of a is different from zero when the autocorrelation of the environment exceeds about 0.8. Therefore we calculated the mean for environments with different autocorrelations, p, and mean environment = 0. Figure 2 shows the relation between the average fitness and the additive phenotypic effect a for several values of p. We see that for the smaller values of p there is no intermediate value of a that is optimum. For populations living in such environments the highest average fitness is maintained by modifying the phenotypic effect until there is no effect of segregation at all. For p > .8, however, there is an

w

s

.10

.20

.30

.40

.50

.60

0

w

FIGURE 2.-The relation between and gene effect, a,for different degrees of environmental is compressed below .90. autocorrelation.The scale of

w

897

THEORY OF FITNESS

op::mum a different from zero. The value of this o p t h u m increases as p increases. Finally, whatever the phenotypic effect of the locus, an increase in autocorrelation results in an increase in mean fitness. For a more detailed examination of the relation between W and the genetic parameters, fitness can be broken down into several components: Wt = 1 - (St-Mt)' - VAPHEt (4) where M t is the mean phenotype of the population and VAPHE is the phenotypic variance in generation t. In our particular case the phenotypic variance is entirely genetic. Taking the expectation of over time we get: E ( W ) = l-u~'-(S--IM)'-u~' - E(VAPHE) f 2 C O V ( S , M ) ( 5 ) For a given set of environmental parameters, us2 and S are fixed whereas the other terms depend on the genetic parameters. In our program, a 2 , COV(S,M) VAPHE, SQDIF (defined as (S--M)2 for each 100 reps) and the correlation between S and M were calculated. Since the scale used in the model is arbitrary, the biologically meaningful comparisons are optimum a and VAPHE compared to variance of the environment and the proportion of the total fitness lost due to fluctuation that is restored by responding to selection. When there is no fluctuation, the optimum (I is zero and fitness equals 1. As the fluctuations increase but with a held at zero, fitness declines roughly by us2(but not exactly, since for extreme environments fitness ) the mean fitness reaches zero and no negative values are allowed). Let E ( W O be for a = 0. Then 1 - E(W,) is the loss of fitness due to fluctuation. For each environmental pattern there is some optimum a, with mean fitness E . Thus -E(WO) and the proportion the fitness restored by the optimal system is E ( of fitness restored is [E(W,) - E ( W , ) ] / [ l- E(W,)].

w

(w,)

w,)

TABLE 2 Optimum phenotypic effects

Model

P

Normal random environment 1 locus 0.8 1 locus 0.8 1 locus .9 1 locus 0.9 1 locus 0.99 1 locus 0.9 Periodic environment S = -+ .9 every 20 generations 0.9 S = 2 .6every 1 0 0 generations 0.98 S = f .9every 100 generations .98

Optimum a

0.1 0.2

0 .06 .06 .12 .20 .20

.081

any 0.1 0.05 0.1

VAPHE at optimum a

0

.002

Proportion fitness loss restored

0 .02

.002

.005 .012 ,007

.07

.9

.04.

.87

.36

.6

,007

.94

.81

.9

.34 .10

.96

898

R. LEVINS

Table 2 shows the phenotypic variance and the fitness loss restored by responding to selection for a number of models. In the first group of models the environment, s, is generated by the method already described involving the normal distribution. For the second set, S is not a random but a strictly periodic function. If S varies cyclically with the value f K for n generations followed by -K for n generations, it is simple to show that the serial autocorrelation p = (n-2) /n. Despite differences among the models, the optimum a is roughly of the same order of magnitude as the variance of the environment and VAPHE is 2 to 10% of that variance. The fitness restored by the optimal a is small compared to the fitness differences among genotypes, so that natural selection on a toward the establishment of the optimum will be considerably slower than the fluctuations in gene frequency except when the autocorrelation is very strong. However, even selective advantages on the order of .02 can still result in changes in a over periods which are short compared to the life of the species. Expression 5 allows us to examine in some detail the components determining the average fitness under different conditions and so to understand the existence of an optimum value of a different from zero. As a increases, the population becomes more sensitive to selection SO that the variance of the gene frequency increases. Since the mean phenotype is M = a(2.7-1) (6) its variance is 4a2ux2which increases more rapidly than the second power of a VAPHE increases also, since a given degree of genetic heterogeneity produces more phenotypic variance. However, its increase is less rapid than that of u 2 . VAPHE is given by VAPHE = 2a2 x ( 1-z) (7) so that its average value is E (VAPHE) = 2a21(1-1) - 2 2 uz2 (8) Hence the greater the variance in gene frequency, the more of the time the population is relatively homogeneous (displaced from the frequency of .5 at which VAPHE is greatest). The combined effect of VAPHE and uM2is to reduce fitness by 2a2Z( 1-1) 2a2ux2. SQDIF has the expected value over all possible series of 100 generations. E(SQD1F) = us2 uar2 2 COV(S,@) [E(S)--E(@)12 (9) The difference between the means of the replicate means will be very small. However, since successive values of S and of M are correlated, the variances of the replicate means are still large. For example, with an autocorrelation of 0.9, the variance of the mean of 100 generations would be about 0.19 times the variance of a single value. The covariance of 3 and i@ measures the effectiveness of selection over the 100 generations and would be positive even when there is no autocorrelation. When there is autocorrelation, part of it appears in COV(S,M) and part enters SDQIF through the COV (S,M).Over an indefinitely long period, SQDIF should vanish. It is included only as a bookkeeping device for finite replications. The overall effect of these three components is to reduce the fitness of the

+

+ +

+

899

THEORY O F FITNESS COR

(%MI-

.90

-

.80 -

.70 .60 -

....*.

..... *....+.... _*...-*

.10

.20

.30

.40

.50

.60

o_ FIGURE 3.-Relation between environment-phenotype correlation and gene effect, a, for a 2: variety of environmental situations. 1 : periodic environment, period 100, amplitude .9; 4: random environment, p = 9, a s z = . I ; 3: random environment, p = 90, asz = .I; random environment, p = .80, asz = .I.

population as a increases. However, this may be offset by an increased covariance term provided the autocorrelation is sufficiently great. In Figure 3 we show the relation between a and correlation of S and M . Table 3 shows the components of fitness for several models and values of a. The response to selection is not the only way in which genetic diversity may be advantageous. If the environment has a bimodal distribution with the modes sufficiently far apart, the optimal a will be equal to the mode even in the absence of autocorrelation. In this case it is advantageous to have a fixed genetic composition which does not change with selection. In the periodic environments in which the environment alternated between S and -S, this occurred when S exceeded .5 so that a genotype which had fitness 1 in one environment was lethal in the other. This aspect of the problem was studied analytically in LEVINS (1962) and will not be discussed further here. The Role of Linkage When we study two loci, each with two alleles, the genetic composition is defined by the frequencies of the gametic types AB,Ab, aBab. The equation for change is given by LEWONTIN and KOJIMA(1960) as AXAB= [ X A B( M F A B - W ) - R D W A ~ B ~ I / W (10)

900

R. LEVINS

TABLE 3 Components of fitness

-.

Model

Normal random environment us2 = .1, p = .9

usZ'= .1, p

= .99

Periodic environment S = t .33 every 20 generations

U

W

VAPHE

VAR(M)

SQDIF

COV

.06 .12 .24 .60 .06 .12 .20 .06 .I2

.9265 .9280 .9194

,0014 .0046 .0150

.0088 "7

--.0001

,0015 .0050 ,0124

.0002 .0017 .0112 .0976 ,0020 .0013 ,0047

.0010

.0006

,0030 .0071

"9 .01B

2314.0

.om

+.ooo5

f.0047 .0078 + . M O .ow9 +.0006 .0151 +.mi9 .00G f.0044 .0155 +.0002 .0018 .0079 .0069 .W30

.U)

.9603 .9672 9679 3575 ,8579 3590

.02 .06

.a901 ,8889

.0002 .0018

0 .oO01

.oO01

.io

.3201 3124 .8933

.0009 .0187

,0084 .3258 .7358

,0003 .o003 .o003

0

0 -.0002

S = f .9 every 20 generations

.60 .90

.Ma

+.0696 .4sH)5

.6624

where D is the measure of linkage disequilibrium, defined by

D =fARfab -fAbfaB.

(11)

Here, f i j is the frequency of the corresponding gamete, R is the frequency of recombination, and M F A S is the marginal fitness of the AB gamete. Thus, a negative D implies excess of repulsion gametes (with total phenotypic effect small, or zero when the two loci have equal effects). In all cases there was an excess of repulsion (negative D).This is what we would expect when we consider that a pair of repulsion gametes have an average fitness 1 - ~ ~ ~ - 0 ~ ~COV(S,M) /4 while a pair of coupling gametes have an */4 - a2.The excess of repulsion has average fitness of 1 - ~ ~ ~ - - ~ ~COV(S,M) the effect of reducing the genetic variance, so that tighter linkage reduces both VAPHE and U$. In Table 4 and Figure 4 we show the relationship among components of fitness and linkage for some typical cases. When there is no autocomlation in the environment, so that any response to selection is harmful, closer linkage is favorable at all values of a. The linkage disequilibrium reduces the effect of increasing a so that the decline of W with D is slower than for a single locus. However, when the environment is highly autocorrelated a certain response to selection is advantageous. Therefore, when a is small a maximum of recombination is desirable, releasing more genetic variance. But when a is large the total genetic variance may be excessive and tighter linkage is beneficial. The same total additive phenotypic effect may be divided among several loci. In Table 5 we show the effects of different divisions of the same total a between

+

+

901

THEORY O F F I T N E S S

TABLE 4 Eflect of recombination on components of fitness,

P

a

R

0

.06

0 .I

W

.90717 ,90674 ,90667 .go666 ,89898 35453 ,83056 .82380 .92749 ,92754 ,92755 ,92756 ,92646 .92568 ,92548 ,92542 ,92063 ,90165 ,89055 ,88728

.3

.5 .30

.9

.06

.12

0 .1 .3 .5 0 .I .3 .5 0 .I .3

.30

.5 0 .I .3 .5

us2

= .I for normal random environment

VAPIIE

M

,00301 ,00340 ,00346 .00347 ,01077 ,05100 ,07126 ,07690 ,00275 .00293 .00295 ,00296 ,00841 ,01033

,01060 ,01066 ,01991 ,05423 ,06572 ,06877

.00011 .OOO14 ,00015 .0oO15 ,00143 .00620 .01068

,01208 .OOO62 ,00070 ,03072 ,00072 ,00315 ,00476 .00506 ,00514 ,00991 ,03038 ,03927 ,04197

COV

--.WO39 -.00041 -.0m2 -.m2 $-.00007 -.OW87 -BO125 -BO134 -.oooo9 -.m3 --.00003 -.oooo3 +BO137 +.00244) f,00255 +.00256 +.00878 ,02452 ,02912 .03037

D

--.0340 -.0064 -.0023 -.0014 -.2189 -.lo35 -.0439 -.0272 -.0186 -.0038 --.0014 --.0009 -.0559 -.0119 -.0045 -.0028 -.1816 --.0686 --.0288 -.01797

R =recombination, p zautocorrelation of the environment, a = phenotypic effect of a single locus.

.93

.92

.9i

w .90

.89

-88

.20

.lo

.30

a_

v,

FIGURE 4.-Relation between mean fitness, and gene effect, a, for various values of recombination and environmental autocorrelation. 1: R = .5 p = 0; 2: R = 0, p = 0; 3: R = .5, p = .9; 4: R = .3, p = .9; 5 : R = 0, p .9.

902

R . LEVINS

TABLE 5 Components of fitness for two loci and normal random environment a, f a2 = .12, VAR (s) = .I, p = .9 a1

a2

W

VAPIIE

.I2 .09 .07 .06

0

.03 .05 .06

.9279 .9278 .9276 .9275

,004.0 ,0033 .0030 .0026

M

,0014 .om9 .o007 ,0006

SQDIF

.w ,0053

,0058 ,0060

cov +.0003 +.OOOl 0 -.0001

two loci. It is apparent that the optimal arrangement is one in which the effect is concentrated at one locus. The reason for this is that in these models VAPHE. the total genetic variance, is increased by the introduction of some epistatic variance as well as additive so that the response to selection is smaller compared to the VAPHE. The price paid for each unit response to selection is greater than in a single locus. DISCUSSION

It has long been known in qualitative terms that a response to selection increases the fitness of populations. This is not universally true however. Whether or not the response is beneficial depends on the other components of fitness (especially homeostasis) and the pattern of the environment (especially its variability and autocorrelation). The effect of good individual homeostasis is to reduce the effective variance of the environment, aS2/H,and therefore to make it less likely that selection will be advantageous. A highly variable and autocorrelated environment is necessary for the response to selection to be beneficial. Thus organisms with short generations will have more highly autocorrelated environments and will be more likely to depend on the response to selection for their adaptation. For both reasons fluctuating polymorphism is more likely to be important to insects than to mammals, and for the second reason more likely in Diptera than in the longer lived Lepidoptera or Orthoptera. The brief investigation of multiple loci indicated that a system with all its phenotypic variability concentrated at one locus could respond more effectively to selection than one in which the additive variance is spread over several loci. The degree of response to selection in a population was controlled in our model by the average phenotypic effect a. But although a is treated as a constant in the equations of changing gene frequency, it depends on the rest of the genotype and is subject to evolutionary change through the seletcion of modifiers. If the autocorrelation of the environment is too low or the level of individual homeostasis too high, the optimum value of a will be 0. The selection will work to modify the developmental pathways which link up with the locus in question in such a way that all genotypes at this locus will have the same phenotype. At this point the locus has disappeared at the level of gross phenotype although it may persist as a segregating isoallelic system at the enzymatic level.

THEORY O F FITNESS

903

The response to selection in a fluctuating environment is not the only adaptive advantage of genetic variance within populations. In the first paper of this series, it was shown that if the environmental heterogeneity is large compared to the individual homeostasis, and independently of any autocorrelation, a stable polymorphism is advantageous as “mixed strategy”. These two kinds of adaptive polymorphism can be distinguished in several ways: In mixed strategy polymorphism the optimal genetic variance for fitness is about equal to the variance of the environment on the same scale, whereas in response-to-selection polymorphism it is at least an order of magnitude lower. I n mixed strategy polymorphism the genetic variance is largely epistatic and stable whereas in response-to-selection polymorphism it is largely additive and easily altered. Therefore, in species with a predominantly mixed strategy type of polymorphism local populations are likely to differ in epistatic blocks of genes. Therefore, crosses between populations would be expected to produce F, breakdown. I n response-to-selection polymorphism the differences between local populations will be more additive and the recombinants will be intermediate on the average. Finally, the reduced fitness resulting from the F, breakdown in mixed strategy polymorphism creates a selective advantage to anything which reduces crossing. Isolating mechanisms would appear frequently in such species, and we would expect to see large clusters of similar species. In contrast, response-toselection polymorphism would lead to large, widespread, and taxonomically distinct species with little tendency to speciate. With respect to Drosophila, this permits us to recognize three modes of adaptation: ( 1 ) The D. melanogaster mode-broad niche, high individual homeostasis, low levels of polymorphism, genetic variance mostly additive, little F, breakdown, and little tendency to speciate. (2) The D. willistoni mode-broad niche, low individual homeostasis, high levels of polymorphism, variance more epistatic, higher F, breakdown, and strong tendencies toward speciation. ( 3 ) The D. prosaltans mode-narrow niche, poor individual homeostasis, low polymorphism. The predicted correlates within the adaptive system permit experimental testing of the theory. SUMMARY

In the optimum quadratic deviation model in an unstable environment, an increase in the phenotypic effect of a locus reduces fitness by increasing the variance of the mean phenotype and the average variance within the population. It also can increase fitness by increasing the correlation between the mean phenotype and the optimum, which is an environmental variable, provided the auto. correlation of the environment exceeds about 0.8. There is an optimal phenotypic effect a, different from zero, if the environment is sufficiently predictable. This optimum increases with the autocorrelation of the environment and with its variance but decreases with the homeostasis (environmental tolerance) of the individual. Thus species which are very sensitive to environmental change will depend more on genetic change to adapt. In any case, the average phenotypic variance of an optimal population is roughly 2 to 10% of the environmental variance.

904

R. LEVINS

The response to selection may restore a significant proportion of the fitness lost due to environmental fluctuation. For a normal random environment, the proportion of fitness restored is smaller than for environments that alternate periodically between discrete alternatives. Even a restoration of 2% would create enough selection pressure to move the population toward optimum on a time scale which is still short compared to the life of the species. Linkage reduces the response to selection and the average variance because there is an excess of repulsion over coupling gametes. Thus tight linkage is advantageous when the total phenotypic effects of the loci are above optimum, and disadvantageouswhen they are below optimum. Testable predictions are made relating the response to selection to other aspects of the adaptive system and species structure. LITERATURE CITED

BOFINGER, E., and V. J. BOFINGER,1958 On a periodic property of pseudo-random sequences. J. Assoc. Computing Machinery 5: 261-265. FISHER,R. A., 1930 The Genetical Theory of Natural Selection. Clarendon Press, Oxford.

KIMURA,M., 1955 Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harbor Symp. Quant. Biol. 20: 33-53. LEVINS,R., 1962 Theory of fitness in a heterogeneous environment. I. The fitness set and a d a p tive function. Am. Naturalist %: 361-378. - 1964. Theory of fitness in a heterogeneous environment. 111. The response to selection. J. Theoret. Biol. 7: 224-240. LEWONTIN,R. C., 196.F The interaction of selection and linkage. 11. Optimum models. Genetics 50: 757-782. R. C., and K. KOJIMA, 1960 The evolutionary dynamics of complex polymorphisms. LEWONTIN, Evolution 14: 458472.

WRIGHT, S., 1935 Evolution in populations in approximate equilibrium. J. Genet. 30: 257-266.