evolutionary dynamics in frequency-dependent two-phenotype models

8/8/2019 Evolutionary Dynamics in Frequency-Dependent Two-phenotype Models

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THEORETICAL POPULATION BIOLOGY 25, 2 lo-234 (1 Y84)

Evolutionary Dynamics in Frequency-DependentTwo-Phenotype ModelsSABIN LESSARD *

Department of Mathematics, Stanford UniversityStanford, Calvornia 94305Received May 1983

General frequency-dependent selection models based on two phenotypic classesare analyzed with underlying one-locus multiallele phenotypic determinationsystems in diploid populations. It is proved that the mean phenotypic fitnesses tendto equality over discrete generations and genetic mutations if a phenotypicpolymorphism is to be maintained. The exact conditions are examined. The presentresults are valid for a wide class of models whenever random groupings or assor-tative patterns based on phenotype and affecting fitness, linearly or not, areindependent of sex, mating prcferenccs, or kinship. They can also be applied to two-sex haploid models.

1. INTRODUCTION

Theoretical population biology models based on randomly pairwiseinteractions may be an important source of ideas and principles that providesome insights on intraspecific selection. When the interactions are assumedto have additive effects (incremental or decremental) on individual fitness,such models lead to linear fitness functions which is a basic form offrequency-dependence with which to deal. In haploid models the recurrenceequations are usually analogous to those of standard (frequency-independent) diploid models but with fitness matrices allowed to be nonsym-metric. Such a generalization makes the dynamical analyses more delicate inmultidimensional situations. The analyses of Cockerham and Burrows(1971) on a discrete time model (cf. Schutz et al., 1968) and Zeemansstudies (1980, 198 1) on continuous time versions have revealed complexdynamics whose exhaustive classification might be prohibitive. The latterauthor was motivated by the notion of evolutionary stable strategy (ESS)introduced earlier in population biology (Maynard Smith and Price, 1973;Maynard Smith, 1974; Haigh, 1975; Bishop and Cannings, 1976; see alsoTaylor and Jonker, 1978; Hofbauer et al., 1979; and Hines, 1980a,b, 1982,

* Present address: Departement de Mathematiques et de Statistique. Universite deMontreal, Montreal, Quebec, Canada.210

0040-5809/84 $3.00Copyright 0 I984 hy Academic Press, Inc.All rights of reproduction in any form reserved.


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FREQUENCY-DEPENDENT TWO-PHENOTYPE MODELS 211for dynamical approaches). As a matter of fact, the hybrid concept of ESSmeshing game theory and ecology with perspectives originally dealing withanimal conflicts was initially defined in a framework of pairwise contests topredict the evolution and/or maintenance of behavioral traits. The notion ofevolutionary stability and its meaning in view of our analytical results willbe discussed in the last section of this paper. In diploid populations, furthermathematical difficulties come into play due to Mendelian segregation andrecombination. For models with two alleles at a single locus, directdynamical analyses are yet feasible assuming random mating (Schutz andUsanis, 1969; Cockerham et al., 1972; Maynard Smith, 1981). Note alsothat Matessi and Jayakar (1976a) generalizing Clarke (1972) investigatedsome multiallele models in an ecological context combining frequency- anddensity-dependent selection. In this framework, the assumption of linearity isequivalent to a first-order approximation leading to the Lotka-Volterraequations summarized in the community matrix introduced forinterspecific selection (see e.g., Levins, 1968; MacArthur, 1970;Roughgarden, 1972).In a more general perspective, individual selection is usually affectednonlinearly by population composition through competition for resources orcooperation for survival. Frequency-dependent selection has long beenrecognized to be entangled in complex relationships with viability selection(e.g., mimicry, parasitism) and/or reproduction with sexual competition andmating preferences (Fisher, 1930; Wright, 1955; Lewontin, 1958; Clarke andODonald, 1964; ODonald, 1980). More recently, the development ofgenetical models for behavioral traits that involve formation of basic groupswith/or without choice of partnership in series of encounters has led to evenmore intricate fitness networks (see, e.g., Boorman and Levitt, 1973, 1980;Cohen and Eshel, 1976; Matessi and Jayakar, 1976b; Axelrod andHamilton, 1981; Eshel and Cavalli-Sforza, 1982; Karlin and Matessi, 1983).In several of these models, the interdependence between phenotypes (usuallyrestricted to two, e.g., altruist and nonaltruist, cooperative andnoncooperative, etc.) is expressedby complex fitness functions.Population genetic models dealing with phenotypes (e.g., color and shapepatterns, sex) have traditionally been based on dominance-recessivityrelationships or homozygote-heterozygote schemes. With the emergence ofbehavioral traits, genetical zero-one determination mechanisms may not bealways appropriate for they are too limited in the expression of the traits.The introduction of probabilities genotypically determined describingtendencies among a given set of phenotypic possibilities is a response owardmore flexibility and generality that will be adopted in this paper. The conceptof partial penetrance though different in meaning is also more in agreementwith the game-theoretic approaches dealing with mixed strategies rather thanpure strategies.


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212 SABIN LESSARDIn this study, general frequency-dependent selection schemes based on two

phenotypes segregating in large diploid populations will be analyzedassuming random mating and discrete nonoverlapping generations. Thepresent results will be valid for a wide class of models whenever randomgroupings or assortative patterns based on phenotype and affecting fitness,linearily or not, are independent of sex, mating perferences, or kinship.Mendelian multiallele one-locus systems will be assumed to determine thephenotypic expression. This will make possible an investigation of theevolutionary dynamics of the population composition under geneticmutations. We refer to Uyenoyama et al. (198 1) for a similar attempt in kinselection theory and Eshel and Feldman (1982) for a theoretical study on theevolution of the sex ratio in random mating populations. These authors dealtwith necessary and sufficient conditions for initial increase of mutant allelesusing heuristic covariance methods and/or standard local linear analyses. Incase of departure from a previously stable equilibrium in the sex ratioevolution models, Karlin and Lessard (1983) proved that the only attainablestable equilibria exhibit a sex ratio closer to 1: 1 than the originalequilibrium. Note also that Eshel (1982) considered a pure (linear) game-theoretic approach for the present class of models and showed that inpresence of a mixed ESS a mixture of strategies is protected.

For the class of models at hand with general fitness functions, two classesof equilibria can be distinguished: phenotypic equilibria characterized byequal phenotypic fitnesses (or phenotypic fixations) and genotypic equilibriaarising from the underlying genetic systems analogous to standard viabilityschemes. This observation formalizes a general principle that has been putforward in the recent literature (Lloyd, 1977; Slatkin, 1978, 1979; see, e.g.,Maynard Smith, 1982, for some examples of dimorphism in naturalpopulations). Moreover, the former class generally corresponds toequilibrium surfaces for which local stability analyses fail. With linearphenotypic fitnesses independent of sex and parental type, a globaldynamical analysis will be provided (Section 2). It will be shown that thedistance between the phenotypic fitness functions is a Lyapounov function,i.e., a function monotone over time with equality only at equilibrium. It canbe proved that the genetic composition of the population will alwaysconverge at a geometric rate in generic cases (Appendix). In the nonlinearcase, a stability analysis is possible in the vicinity of every equilibrium(Section 3). In all the cases considered, only phenotypic equilibria have beenfound to be evolutionary attractive (cf. Definition 1). More precisely, twophenotypic classes segregating in a population are expected to equalize theirmean titnesses or be driven toward either fixation over successive generationsthrough a series of random mutations if necessary. The exact conditions anddomains of attraction will be examined. Finally, our results for one-sexdiploid models will be applied to two-sex haploid versions (Section 4). In this


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FREQUENCY-DEPENDENTTWO-PHENOTYPE MODELS 213context, the above principles will hold based on the notion of reproductivevalue which assigns a weight to each sex inversely proportional to its meanfitness (i.e., its relative size at the mating phase). Note that such anaveraging leads to nonlinear cases when different linear fitness functions areoperating in the female and male populations. Our analytical results will becomplemented by a discussion on the concept of ESS. In this respect, thepresent study offers a rigorous treatment and an extension in support ofEshels claim (1982) that natural selection should lead to ESSs at least inone-locus random mating models.

2. BASIC MODEL OF ADDITIVE PAIRWISE INTERACTIONSOR LINEAR MODELConsider an infinite diploid population in which two phenotypic classes V,

and VI are segregating. Suppose that n alleles A, ,..., A,T ocated at a singlelocus are responsible for the phenotypic determination such that an offspringof genotype AiAi expresses phenotype R, with probability cii and PI with thecomplementary probability 1 - ui,i (i, j = I,..., n). The symmetric matrix V =~(u~~I\~,~~,ill be called the phenotypic determination matrix. In order toavoid unimportant technicalities, the following generic assumptions on V willbe imposed throughout:

(i) 0 < vii < 1 for i, j = l,..., n.(ii) Every principal submatrix of V is nonsingular.

(iii) If x = (x, ,..., x,) > 0 (i.e., xi > 0 for i = l,..., n) is such that(VX)~ = Ci-, v~~x,~= 1 where xi > 0, then (Vx), # 1 where xi = 0.Although these assumptions are not required for the main results of thispaper (namely, Propositions 1 and 2), they will simplify the exposition andafford easier understanding afterwards. Note that they are also standard forthe classical one-locus multiallele viability model with I as a viabilitymatrix. In this case, they ensure that the number of equilibria is finite andlocal linear analyses sufficient to investigate their stability properties.Consult, e.g., Kingman (1961a,b), Nagylaki (1977), or Karlin (1978) for areview of results on this model.The frequency of genotype A,Ai will be denoted by 2pii when i # j and piiwhen i = j. The frequency of allele A, is then given by pi = C, , pii.Assuming nonoverlapping discrete generations with random mating, theproportion of offspring in the phenotypic class H, (WI) in the next generationis

U = w(p) = T7i.7 I

uijPiPj (2.1)


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214 SABIN LESSARD(1 - W, respectively), where p = (p, ,..., p,). Then selection is assumed o actas follows: pairwise associations (one or several for each individual) orsequential encounters occur at random between the members of thepopulation and as a result of cooperation or confrontation affect theindividual expectations to participate to the next mating phase. Let &measure the resultant fitness of an individual of phenotype q havinginteracted only with individuals of phenotype E$ (i, j = 1, 2). Let us assumefij > 0. The fitness matrix F = jlfii/i~,j=, may be nonsymmetric. Assumingthat multiple random pairings have additive effects on individual fitness, themean fitness of the phenotypic class q is frequency dependent (unlessf;., = &) and given in the linear form

Fi = Fi(w) = ~fi, + (1 - w)&, i= 1, 2.These selective values transform the genotypic frequencies into

(2.2)

pt.= PiPjLvijFl + C1 - ulj> *IIJ wF,+(l-w)F, i,j= l,..., n, (2.3)

yielding the new allelic frequenciesp! = PilwiFl + (I - wi) F21

I wF, +(I - w)F2 i = l,..., n, (2.4)where wi = w,(p) = C,j=, vi,ipj. Then at equilibrium we must have therelation

Pi(W - Wi)(F, -F,) = 0, i = l,..., n. (2.5)Therefore two types of equilibrium can exist:

(I) Phenotypic equilibrium when F,(w) = F,(w), i.e., both phenotypicclasses have the same mean fitness. This condition only involves thephenotypic parameters of the model. Note that the equilibria w = 0 andw = 1 may also be included in this category.(II) Genotypic equilibrium when F,(w) # F*(w) and pi(w - wi) = 0 fori = l,..., n, i.e., (VP), = w(p), where pi > 0. In this case the equilibrium allelicfrequencies correspond exactly to those of the standard one-locus multialleleviability model with V as a viability matrix and w(p) as a mean fitnessfunction, i.e., for the transformation p,! = pi( Vp)Jw(p), i = l,..., n. Such an

equilibrium is independent of the phenotypic selection taking place. Notethat in case of coincidence of a genotypic equilibrium with a phenotypicequilibrium, the latter designation would prevail by definition. But this is adegeneratecase that will be ignored hereafter.Observe that the above conditions are necessary and sufficient for


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FREQUENCY-DEPENDENTTWO-PHENOTYPE MODELS 215equilibrium and do not depend on any special form of the phenotypic fitnessfunctions. When these are linear functions with respect to the phenotypicfrequencies as in the case at hand, the dynamics of the recurrence system iselucidated by the following result:

PROPOSITION 1. Let F; and F; be the next values of the mean phenotypicfitnesses F, and F, as defined in (2.2) following transformation (2.4).

(i> rf (f,, -f,* -f,, +fJ > 0, then IF; -F;I > IF, -F,/ withequality only at the equilibria of (2.4).09 V L -fiz- fi, + f&) < 0, then IFI - FSI G/F, - F,( withequality only at the equilibria of (2.4).

Proposition 1 means that IF,(w) - F2(w)( is a global Lyapounov functionincreasing in the case of overall homogeneity advantage, i.e., f,, + fiz >f,, + f,, , and decreasing in the case of overall heterogeneity advantage, i.e.,f, i + fz2 < f,* + fi, . The proof is set forth in the Appendix. It will be usefulfor further analysis to highlight an equivalent statement.

PROPOSITION 1. Let w be the next frequency of the phenotypicclass g, as defined in (2.1) following transformation (2.4). If(f,, -f,* -fz, +f**) (s)O, then (w-w*l(z)iw-w*j where

fiZ -f,,w* =fil -f,* -f*, +f**

with equality only at the equilibria of (2.4).Remark. The function w(p) is actually monotone over successivegenerations except perhaps in the case f, , < f,, and fzz < f,*. (SeeAppendix.)The alternative version of Proposition 1 comes from the equality

F,(w)-F,(w)= (fiz-fiz)+ (f,, -f,z-fi, +fiz)w= (f,, -f,, -f*, +fz*)(w- w*)* (2.6)

In particular, F,(w) = F,(w) if and only if w = w* (ignoring the degeneratecase f,, - flz - fi, + f,, = 0). This equation defines a level curve of thequadratic form w(p), which can intersect the (n - 1)-dimensional simplex offrequency vectors

A = 1p = (p I,..., p,): pi > 0, +ie, Pi = I (2.7)


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216 SABIN LESSARDonly if the condition 0 ,< w* < 1 is satisfied. When it exists, let us denote byL* the equilibrium manifold in A corresponding to w*, i.e.,

L*=(pEA:w(p)=w*). (2.8)In general, the intersection of w(p) = w* with A may create several separatedequilibrium branches of dimension n - 2. On the other hand, with ourgeneric assumptions on V, the number of genotypic equilibria is necessarilyfinite. These are isolated points in A. The existence of a global Lyapounovfunction entails convergence of the allelic frequency vectors to theseequilibrium points or to the equilibrium manifold Lx. Here convergence toL* means convergence of w(p) to w*, i.e., convergence of the phenotypicfrequencies. For further discussion we shall concentrate on the case f, , < fi,and f12 c .flz, i.e., the case of overall heterogeneity advantage with 0


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FREQUENCY-DEPENDENT TWO-PHENOTYPE MODELS 217that w(p) is a quadratic form and consequently a local maximum (minimum)within A is actually a global maximum (minimum) over all A. Furthermore,the restriction of w(p) to any straight line is a quadratic form. Therefore, thelast statement of Corollary 1 is immediate in generic cases where thegenotypic equilibria (including those corresponding to the maxima andminima of w(p) in A) do not coincide with phenotypic equilibria.

It is worth noting that a locally stable genotypic equilibrium on theboundary of A cannot be a priori precluded by the existence of a stablephenotypic equilibrium, and a coexistence is actually possible. This occursfor instance when w(p) exhibits a local maximum smaller than w* on theboundary of A, while its global maximum and global minimum lie on eitherside of w*. In any case, an analysis of the evolutionary dynamics of thewhole system under genetic mutations can be made on the basis of ourresults.

Suppose that the conditions of Corollary 1 are in force and let p =(6 , ,..., lo,,) be a locally stable genotypic equilibrium involving the allelesA , ,..., A, (i.e., fii > 0 for i = l,..., n) such that 6 = w(p) < w*, or equivalentlyF,(G) -F,(G) > 0. Let us introduce a new allele A, +, to this originalequilibrium system such that the mutant marginal phenotypic frequency ofg, at p, namely,

wn+,(fi)= i.ntlPi, (2.9)i-lsatisfies w ,,+ i(p) > w(C). By continuity we must have w,+,(p) > w(p) andF,(w) - F,(w) > 0 in some neighborhood of 13 n which the following relationfor the frequency of A, + , holds:

P Pntlt1= \w+,(~)P,(w) F*(w)1 Fdw) /I w(p)[F,(w) - F2(W )] + F,(w) \ > pn i . (2*10)This local increasing property excludes convergence to 13 n the augmentedallelic system. To the contrary, the reverse condition w,, ,(p) < w(P) wouldhave preserved local stability for P. This is also precisely the condition for astable equilibrium fi with w(P) > w * to become unstable under geneticmutations. To sum up, a stable boundary equilibrium 13= ($i,..., 6,)becomes unstable following the introduction of a new allele A,, , if and onlyif w,+ ,(p) is in the direction of w* with respect to w(p). (Compare with theconditions used by Eshel and Feldman (1982) to define EGS (evolutionarygenetically stable) sex ratios.)

Whenever instability occurs in the model at hand, Proposition 1guarantees that w(p) will always move closer to w* until a new equilibriumis reached. Moreover, the phenotypic equilibrium manifold w(p) = w* itself


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218 SABIN LESSARDTABLE I

Evolutionary Attractive States for Model (2.4) in Terms of the Frequent)of the Phenotypic Class 6 and of its Initial Value M(,

w* ( 0 o


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FREQUENCY-DEPENDENTTWO-PHENOTYPE MODELS 2193. GENERAL CASE OF NONLINEAR FREQUENCY-DEPENDENT SELECTION

BASED ON Two PHENOTYPESWe propose to consider a generalization of the basic model allowing thephenotypic fitness functions F,(w) and F*(W) to be nonlinear positivefunctions of w. Such a situation may arise from a pairwise encounter modelif the assumption of additivity is dropped (and replaced, e.g., by amultiplicative model which is as much-if not more-relevant in the case ofsuccessivecontests eopardizing viability). But this is a very particular inter-pretation of a purely formal representation that can take into account severalforms of selection based on phenotype found in complex populationsinvolving, e.g., assortative behavior in partnership or any grouping patternsindependent of kinship followed by local individual interactions as well asfrequency-dependence arising from mimicry or parasitism widespread innatural populations. Without the linearity assumption, the scope ofapplication of our results will be substantially widened.With phenotypic determination matrix V and general positive fitnessfunctions F,(w) and F?(W) for the phenotypes F, and PI, respectively, wherew is the frequency of g,, the transformation equations assuming randommating are given by (2.4) which can be written in the form

\ (vP)i Q(w) + R(w) /" = Pi 1 we(w) +R(w) (' i = l,..., n, (3.1)where Q(w) = F,(w) -F,(w) and R(w) = F,(w) with w = w(p) =XI=1 PiCvP)iaFor simplicity, suppose that Q(w) has exactly N zeros in [0, 11 denoted byWT,..., w,: in this order, i.e.,

oq w: < wf < .*. < w,; < 1, (3.2)and these zeros are simple, i.e., the derivatives satisfy

2 (w?) 0, k = l,..., N.These genericy conditions on Q(w) will be assumed throughout this paper.Near any phenotypic equilibrium w*, i.e., any frequency w* such thatQ(w*) = 0, a linear approximation of F,(w) and F2(w) in the form

F,*(w) = wfi; + (1 - w)fi*, , i= 1,2, (3.4)


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FREQUENCY-DEPENDENTTWO-PHENOTYPE MODELS 221more, stable genotypic equilibria can coexist with stable phenotypicequilibria segregating the same set of alleles. As a general rule, stable andunstable equilibria alternate.

4. APPLICATION TO TWO-SEX TWO-PHENOTYPE HAPLOID MODELSAs mentioned in the previous section, multiplicative models are natural

alternatives to additive models. Also interesting is the peculiar fact that meresex differentiation may break linearity. In this respect, the generalized system(3.1) allowing nonlinear functions will prove to be particularly relevant fortwo-sex haploid models. Such models may have an interest of their own inaddition to enabling us to extend the principles established for one-sexdiploid models to two-sex populations in which fitness and sex ratio areindissociable.

Let A , ,..., A, be the haplo-types with frequencies p, ,..., p,, respectively, inan infinite two-sex population with 1:l sex ratio at conception. Assume thatan A,-type irrespective of sex exhibits a phenotype V, with probability ni andKZ with probability 1 - L~ (i = l,..., n). Suppose that the phenotypic deter-mination vector v = (v, ,..., v,) satisfies the genericy condition vi # ul for alli # j. Before selection takes place, the phenotypes P, and WI are present inthe haploid population with frequencies y = C;Y, pivi and 1 - y, respec-tively. The mean (positive) fitnesses of these two phenotypes in the male andfemale populations undergoing general frequency-dependent phenotypicselection will be denoted by the fitness functions M,(y), M,(y) and F,(y),F,(y), respectively. (The variable y will be omitted hereafter.) Followingrandom union of gametes for the formation of diplo-types, Mendeliansegregation and equal fertility in producing haplo-types, the next life cyclewill start with the allelic frequencies

(p! = l \ PilviFl + C1 vi) F2 I 1I 7 ) yF,+(l-Y)F, r

\ PiIviMl + C1 i> M*l (I yM,+(l-Y)M, ! i=

that is,1 ..., 4 (4.1)

wherePf = (Pi12)lvis(Y) + T(Y)ly i = I,..., n, (4.2)

F, --F, M, -M2S(Y) = p + M and F2 M2T(Y) = -jy + z (4.3)


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222 SABIN LESSARDwith F=yF,+(l-y)F,, Q=yM,+(l-y)M,, and .~=y(p)=,Yy=, pivi. This recurrence system of equations is formally equivalent to(3.1) with matrix I= )Iu~u~~~;,~,, nd functions Q(w)= S(fi)/& andR(w) = T($), where w = y*. The genotypic equilibria are the geneticfixation states (i.e., pi = 1 for some ), and the dynamical propertiespreviously highlighted hold based on the functions y=y(p) and S=S(y). Ifwe assume without loss of generality ui < t, < ... < c,, then y(p) which is alinear function achieves its unique maximum at AH-fixation and its uniqueminimum at A r-fixation. According to Proposition 3, the former equilibriumis stable when S(v,) > 0 and the latter when S(u,) < 0. Similarly, thephenotypic fixation states y = 0 and y = 1 are evolutionary attractive if theyare locally more advantageous. Moreover, the polymorphic phenotypicequilibria are the zeros of S(y) whose stability properties are determined asfollows:

PROPOSITION 4. The evolutionary attractive polymorphic equilibria ofthe two-sex two-phenotype haploid model (4.2) are the zeros from plus tominus of

s(y)= F,(Y)--F,(Y) + M,(y)-M2(y)F(Y) WY) where Fi( y) and Mi( y) are the Jtnesses of 5 (i = 1, 2) in the female andmale populations, respectively, whose mean jitnesses are F(y) and a(y),respectively, when the frequencies of g, and U;; are y and 1 - y, respective&.

5. DISCUSSION

Maynard Smith (1982) made a clear distinction between an ESS orevolutionary stable strategy and a population in an evolutionary stablestate. An ESS is defined as an uninvadable (possibly mixed) strategy onceadopted by all members of a population: the population is monomorphic andany deviant strategy is selected against at least when rare. When apopulation composition (monomorphic or polymorphic) is stable against anysmall perturbations then the population is said to be in an evolutionarystable state.Let us recall that if r pure strategies are possible, then a mixed strategys = (sl )...,s,) is a frequency vector which assigns a probability sj to the purestrategy (0 < s j < 1, XI= i sj = 1). In a game theoretic approach of pairwisecontests allowing r pure strategies, an individual adopting a strategy 5 =(s; ,.**,S;) against an opponent adopting s = (s, ,..., s,.) receives a payoff in


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FREQUENCY-DEPENDENT TWO-PHENOTYPE MODELS 223fitness that will be denoted by E(S, s). In that context, a strategy s* =(ST,..., r*) is called an ESS if for every alternative strategy s # s* either

E(s*, s*> > E(s, s*)or (5.1)

E(s*, s*> = E(s, s*> and E(s*, s) > E(s, s)(Maynard Smith and Price, 1973). With linear games, .e., payoff function inthe form

E(i, s) = 2 fiaijsj (5.2)i,j= Ifor some coefficients aij, i, j = l,..., r. condition (5.1) is equivalent to

E(s*, (1 -E) s* + ES)= (1 - e)E(s*, s*) + cE(s*, s)> (1 - E) E(s, s) + EE(S, )= E(s, (1 - E) s* + ES) (5.3)

for every E > 0 small enough, or equivalently,E(s,(l -&E)s* $&S)

= (1 - c)E(s*, (1 -E) s* + ES) t cE(s*, (1 -E) s* + ES)> (1 -E) E(s*, (1 -E) s* t ES)+ EE(s, 1 -E) s* + ES)= E((1 - E) s* + ES, 1 - E) s* + es) (5.4)

for every E > 0 small enough (cf., e.g., Hofbauer et al., 1979). In a moregeneral context of frequency-dependent selection (and after Taylor andJonker (1978)), an ESS can be defined as an individual strategy s* such thatF(s, (1 - E) s* + E:) > F(s, (1 -&) s* + EB) (5.5)

for every strategy 5 # s* and every E > 0 sufficiently small, whereF(i, (1 - E) s* + 6) represents the fitness of an s strategist in a populationwith mean individual strategy (1 - E) s* + 6. With two pure possiblestrategies (or phenotypes) F1 and g2 whose fitnesses are general functionsF,(w) and F2(w), respectively, where w is the frequency of @, in the currentpopulation, an individual adopting g1 with probability 6 and V2 withprobability 1 - 6 has fitness

F(G, w) = biw,(w) + (1 - G) F,(w). (5.6)


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224 SABIN LESSARDIn this case, condition (5.5) becomes

F(w, w) - P(t3, w) = (w* - G)(F,(w) -F,(w)) > 0, (5.7)where w = (1 - E) w* + F@with E > 0 arbitrarily small. This entails that theESSs are the zeros of F,(w) - F2(w) from plus to minus, not to mention thepure strategies w * =0 and w* = 1 if F,(O) F,(l),respectively.We have shown that the phenotypic equilibria corresponding to the ESSsin a general frequency-dependent selection context of mixed strategieswith two components are evolutionary attractive (cf. Definition 1) overnonoverlapping discrete generations in diploid populations with individualstrategy determined at a multiallelic autosomal locus subject to mutations.This extends Eshels result (1982) that a mixture of strategies s protected ina population with an associated inear game allowing two pure strategies andhaving a (strictly) mixed ESS. Note that the one-side-inequality (A6) in theAppendix defining locally adaptive systems was proved by Eshel but is notsufficient to conclude about convergence. Our results show that stablegenotypic equilibria are possible in presence of genetically accessible ESSseven in the linear case (cf. Corollary 1). Nevertheless, genetic mutations willeventually destabilize any genotypic equilibrium and a phenotypicequilibrium corresponding to an ESS will ultimately be reached although thedynamical tendency may be temporarily stopped due to genetic constraints.It is worth noting that not only the phenotypic composition of the populationwill converge, but also its genotypic composition. As a matter of fact, amixed ESS generally corresponds to a continuum of genetic polymorphisms:there is not a unique population composition of mixed strategies that canproduce the phenotypic equilibrium but infinitely many. Such an equilibriumis stable as a manifold in a very strong sense since it is stable against anyallelic perturbations including mutations. We may mention that it is alsostructurally stable meaning that small perturbations on the phenotypicparameters of the model would cause small displacements of the originalequilibrium but preserve its stability properties in generic cases (i.e., with(3.2) and (3.3) in force).The basic fact that a balance of fitnesses in dimorphic populations isestablished at equilibrium under the effects of frequency-dependent selectionwas mentioned early by Fisher (1930) in discussing the maintenance ofmimetic forms. Moreover the equilibration principle is well known for themarginal allelic fitnesses n standard (frequency-independent) diploid models,and more generally in linear dynamical games, and recognized topredominate within populations over the maximization principle of the meanfitness as pointed out by Cockerham and Burrows (1971). That static prin-ciple was restated for any number of phenotypic classes in response to


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FREQUENCY-DEPENDENTTWO-PHENOTYPE MODELS 225frequency-dependent selection (Lloyd, 1977; Slatkin, 1978, 1979). Ourdynamical analysis has specified the exact conditions on the fitness functionsfor actual realization and maintenance of equilibration in the case of twophenotypic classes (see (5.7) and Proposition 2). In sex-differentiatedpopulations, we must resort to the concept of reproductive value (Fisher,1930) to average the within-sex fitness functions in order to equalize thecontributions by males and females to the next generation. Such models areintrinsically frequency-dependent and nonlinear unless the within sex fitnessparameters are constant. (For this special case in haplo-diplonts, seeGregorius (1982).) As previously mentioned, the equilibration principle issubsumed n the concept of ESS. Some extensions of this concept to two-rolemodels were proposed (Taylor, 1979; Schuster and Sigmund, 1981).Maynard Smith (1982, pp. 199-202) suggested hat the mean Iitnesses of thetwo groups should be taken into account. This is in agreement with ourProposition 4 in a more general framework of frequency-dependence or two-sex haploid populations. Weaker analytical results supporting similar prin-ciples for general two-sex models in diploid populations including parentcontrol versions for which a global dynamical approach might be prohibitivewill be published separately.It is interesting to compare our results for discrete generations withcontinuous approximations. The analyses of Zeeman (1981) and Hines(1982) for haploid populations and diploid populations with additive alleliceffects, respectively, revealed that the ESSs defined by (5.1) in a context oflinear games allowing mixed strategies with r components are evolutionaryattractive (in our terminology) in continuous time. Moreover, as pointed outby Eshel (1982) (and after Ewens, 1969), the additive case over loci inmultilocus systems for our model is formally equivalent to a one-locusproblem with regard to the allelic and phenotypic frequencies. Other geneticassumptions remain to be considered.

APPENDIX: PROOF OF PROPOSITION 1For ease of exposition, the following notation will be used:

F=llf;-jll~,j=l = a bI Id (a,b, c, d > 01, (Ala)R=F,=d+(c-d)w, (Alb)

Q=F,-FF,=(b-d)+(a-b-c+d)w. (Ale)In this notation, transformation (2.4) on the frequency vectors p = (p, ,..., p,)takes the form


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226 SABIN LESSARDp! =PiCwiQ + R)1 wQ+R i = I,..., n,

where wi=Cj=I Vijpj and ~=Cy,j=IVijpipj with V=j(~~j(l~,~=, as aphenotypic determination matrix. The next value of w is determined asfollows:nw = 1 vijp;pi

i,j=l

= CL=1 VijPipj(w/Q +R)(wjQ + R) i piwi = w*,,z i 1=l

643)

644)while Fishers fundamental theorem of natural selection with V = )(vijllf.,i=,as a viability matrix guarantees that

2 VijPiPjWiWj > w3.i,j= 1 (A5)

Both inequalities are strict unless wi = w where pi > 0. Therefore, if Q > 0,we havew, > Q2w3 + 2QRw + R2w/ (wQ+R) =w G46)

with equality only at a genotypic equilibrium of (A2). By symmetry betweenthe two phenotypic classes (with phenotypic determination matrix(1 - zliJll~j,l instead of llz~~~ll~,~=~),he reverse inequality holds if Q < 0.Note that these monotonicity properties do not require any linearitycondition on Q as a function of W. They define locally adaptive systems inEshels terminology (1982). They stipulate that the frequency of a phenotypeincreases when the phenotype is selectively favored.


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FREQUENCY-DEPENDENT TWO-PHENOTYPE MODELS 221Note also that the next value of Q defined by (Ale) can be expressed nthe formQ'=(b-d)+(a-b-c+d)w'

(b-d)(~~Q~+2~QR+R*)+(u-b-c+d)X (Q Cy,j=l VijPipjWiWj + 2QR CI=, PiWf + R*W) J 0. Owing to the monotonicity property (A6)and the linear form of Q as defined by (Ale), we can conclude immediatelythat Q > Q where Q > 0 with equality only at a genotypic equilibrium, andsymmetrically Q < Q where W < 0, i.e.,lQ'l>lQI (w

with equality only at an equilibrium point of (A2).Case (ii). (a-b-c + d) < 0. Without loss of generality, we mayassume Q > 0. In this case, (A6) ensures Q < Q, and it remains to showthat Q > -Q to conclude that

with strict inequality at the nonequilibrium points of (A2). The fact that 0 Q R2 + (a - c)(w*Q + 2wR)(wQ+R>

(All)

6412)


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228 SAWN LESSARDwhere Q is positive. Then it will be sufftcient to show that

R2 + (a - c)(w2Q + 2wR) > -1(wQ+ RI2 , 6413)or equivalently,

t?(w) = R2 + (a - c)(wQ + 2wR) + (wQ + R) > 0. (A14)But since

Q = (a - c) - (a - b - c + d)( 1 - w) > 0, (A15)we have

g(w) = R2 t [Q t (a-b-c + d)(l - w)](wQ + 2wR) t (wQ + R)= 2(wQ + R) t (a - b - c + d)(l - w)(wQ + 2wR)> 2(wQ t R) t 2(a -b-c + d) w(1 - w)(wQ t R)=2(wQtR)[wQtRt(a-b-ctd)w(l-w)J=2(wQtR)((a-c)w+R]= 2(wQ t R)[aw + d(1 - w)] (416)

and therefore g(w) must be positive where Q > 0.Addendum to Appendix

Suppose that F,(w) and F,(w) are nonlinear positive fitness functions. LetQ(w) = F,(w) - F2(w) and 0 < w* < 1 be such thatQ(w*) = 0 and g (w*) < 0. 6417)

Let a, b, c, d be four positive constants defined from the Taylor expansionsof F,(w) and F,(w) about w* as follows:F,(w) = b + (a - b)w + O(l w - w* I),F,(w)=d+(c-d)wfO((w-w*(*).

(A18a)(A18b)

(A function O(E) is such that O(E)/& is bounded for E arbitrarily small.)Namely,


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FREQUENCY-DEPENDENT TWO-PHENOTYPE MODELS 229

a=F,(w*)+(l-W*)$(W*),

b = F,(w*) - w* gj (w*),

c=F,(w*) t (1- w*)$$w*),

d = F,(w*) - w* gf (w*),

(A19)

and the fitness functions F,(w) and F,(w) can be chosen sufficiently large toensure Q, b, c, d > 0. Furthermore

Q(w) = (b - d) t (a - b - c + d)w t O(l w - w* I) (A201with

(a-b-c+d)=$$(w*)


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230 SABIN LESSARDProof of Pointwise Convergence

It has been established that a phenotypic equilibrium w = w* (such thatQ(w*) = 0) with dQ(w*)/dw < 0 is locally stable as a manifold in thesimplex of frequency vectors A. We propose now to prove pointwiseconvergence, i.e., convergence of the iterates of transformation (A2) to apoint of the equilibrium manifold from any initial allelic frequency vector inthe domain of attraction of that manifold. The proof will be based on a moreaccurate estimate of the decay of 1Q ) near w *.

LEMMA A. Consider recurrence transformation (A2) and let w*(0 < w* < 1) be a zero of Q(w) with dQ(w *)/dw < 0. Then the successivevalues of Q near w* satisfy

where C is a positive constant and a2 = a(p) = Cf=, pi(wi - w).ProoJ Via a first-order approximation of the fitness functions near w* ifnecessary (cf. Addendum to Appendix), it will be sufficient to consider thelinear case for which Q = (b - d) + (a - b - c + d)w with(a - b - c + d) < 0. Moreover, recall that if a function f(x) is twice differen-tiable on some finite interval and the second derivative is bounded below bysome constant a > 0, then for any set of values x, ,..., x, in that interval withcorresponding frequencies p, ,..., p, (Of iit, Pixij ++,$, Pi(xi-iu)2af ii, Pixij3 (A26)i=l

where ,U= JJ=, pixi. Indeed the Taylor expansion off(x) about ,u, namely,

for some r between x and ,u, leads to

which, multiplied by pi and summed over i, yields (A26).Successive applications of (A26) give the following inequalities in thenotation of (A2):


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232 SABIN LESSARDo(p)) using the estimate Iw - w*l = O(a4!) near every genotypicequilibrium p* (i.e., a2(p*) = 0) with w(p*) = W* (cf. Lyubich et al., 1980).

Proof: From (A2), we havep! _ p, = Pi(Wi - w>QI I wQ+R i = l,..., n. (A33)

Using the norm /I . )I defined by I/x (I = Cr=, /xi I for x = (x, ,..., x,,), we haveHP-pII= ,(yjR $ Pilwi-wwJ< IQ10 (A34)I I wQ+R

where cr2= o(p) = C;=, pi(wi - w). I n some neighborhood of w(p) = w*,there exists a constant E > 0 such that o > E by continuity of u = a(p) whilelQl~2~~lQl-lQI>/~f or some constant C > 0 by (A25), so thatIQ10 IQIU'wQ+R %(wQ+R)\ < cg;;f;,


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234 SABIN LESSARDLLOYD, D. G. 1977. Genetic and phenotypic models of natural selection, J. Theor. Biol. 69.

543-560.LYUBICH, Yu. I., MAISTROVSKII, G. D., AND OLKHOVSKII, Yu. G. 1980. Selection-inducedconvergence to equilibrium in a single-locus autosomal population, Problems Inform.

Transmission 16, 66-15.MACARTHUR, R. 1970. Species packing and competitive equilibrium for many species, Theor.

Pop. Biol. 1, l-11.MATESSI, C., AND JAYAKAR, S. D. 1976a. Models of density-frequency dependent selection

for the exploitation of resources. I. Intraspecific competition, in Population Genetics andEcology (S. Karlin and E. Nevo, Eds.), pp. 707-721, Academic Press, New York.

MATESSI. C., AND JAYAKAR, S. D. 1976b. Conditions for the evolution of altruism underDarwinian selection, Theor. Pop. Biol. 9, 36&387.

MAYNARD SMITH, J. 1974. The theory of games and the evolution of animal conflicts. J.Theor. Biol. 14, 209-221.MAYNARD SMITH, J. 1981. Will sexual population evolve to an ESS? Amer. Natur. 117.1015-1018.

MAYNARD SMITH, J. 1982. Evolution and the Theory of Games, Cambridge Univ. Press.Cambridge.

MAYNARD SMITH, J., AND PRICE. G. R. 1973. The logic of animal conflicts, Nature (London)246, 15-18.

NAGYLAKI, T. 1977. Selection in One- and Two-Locus Systems (Lecture Notes inBiomathematics, Vol. 15), Springer-Verlag, Berlin.

O'DONALD, P. 1980. A general analysis of genetic models with frequency-dependent mating,Heredity 44, 309-320.ROUGHGARDEN, J. 1972. Evolution of niche width, Amer. Natur. 106, 683-718.

SCHUSTER, P., AND SIGMUND, K. 1981. Coyness, philandering and stable strategies, Anim.Behau. 29, 186-192.

SCHUTZ, W. M.. BRIM, C. A.. AND USANIS, S. A. 1968. Inter-genotypic competition in plantpopulations. I. Feedback systems with stable equilibria in populations of autogamoushomozygous line, Crop. Sci. 8, 61-66.

SCHUTZ, W. M.. AND USANIS, S. A. 1969. Inter-genotypic competition in plant populations.-II. Maintenance of allelic polymorphisms with frequency-dependent selection and mixedselting and random mating, Genetics 61, 875-891.

SLATKIN, M. 1978. On the equilibrium of fitnesses by natural selection, Amer. Natur. 112,845-859.SLATKIN, M. 1979. The evolutionary response to frequency- and density-dependentinteractions, Amer. Natur. 114, 384-398.

TAYLOR, P. D. 1979. Evolutionary stable strategies with two types of player. J. Appl. Prob.16, 76-83.TAYLOR, P. D., AND JONKER, L. B. 1978. Evolutionarily stable strategies and game dynamics.Math. Biosci. 40, 145-156.

UYENOYAMA, M. K., FELDMAN, M. W., AND MUELLER, L. D. 1981. Population genetic theoryof kin selection. I. Multiple alleles at one locus, Proc. Nat. Acad. Sci. USA 78, 5036-5040.WRIGHT, S. 1955. Classification of the factors of evolution, Cold Spring Harbor Symp.Quant. Biol. 20, 16-24.

ZEEMAN, E. C. 1980. Population dynamics from game theory, in Global Theory ofDynamical Systems (Lecture Notes in Mathematics, Vol. 819), pp. 471-497, Springer-Verlag, Berlin.ZEEMAN, E. C. 1981. Dynamics of the evolution of animal conflicts, J. Theor. Biol. 89,249-270.

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