Gene frequency patterns in the levene subdivided population model

THEORETICAL

POPULATION

BIOLOGY

11,

356-385

(I 977)

Gene Frequency Levene

Subdivided SAMUEL

Department

of Theoretical Department

Mathematics,

of Mathematics,

Patterns Population

in the Model*

KARLIN

The Weizmann Institute of Science, Rehovot, Israel, and Stanford University, Stanford, California

Received

October

1976

In the context of a multideme population structure subject to selection, migration, and mating forces, it is desired to ascertain the stable evolutionary outcomes for various classes of selection regimes. These include selection patterns as (i) a mosaic of local directed selection effects, (ii) overdominance throughout with varying intensities of the local heterozygote advantage, (iii) varying degrees of underdominance throughout, (iv) a mixed underdominant-overdominant regime. An accounting of the nature of the equilibrium configurations in the Levene population subdivision model was done with respect to the above classes of selection regimes. In particular, it is established that multiple polymorphic equilibria do not arise for selection structures (i) and (ii), while for the mixed underdominant-overdominant selection form (iv) with appropriate parameter ranges there can exist two stable internal equilibria. A surprising finding is that the number and/or character of the equilibria is not changed by increased population division beyond that of two habitats, while there is a significant difference in the equilibrium possibilities for a one- as against a two-deme population. These results appear to be a special limiting feature of the Levene population subdivision formulation.

1. INTRODUCTION

The attempt to explicate levels and forms of spatial variation in gene frequency arrays necessarily depends on the nature of the selection regime, migration pattern, population structure, and other ecological and environmental factors. There are a number of formulations of the dynamic multideme population model that incorporate the foregoing forces and factors in various ways and each may be relevant in appropriate situations. Contributors to the theory of selection migration interaction include Christiansen (1974, 1975) Deakin (1966, 1968), Karlin (1976, 1977, 1978), Levene (1953), Maynard Smith (1966), Prout (1968) Strobeck (1974) Slatkin (1973) among others. Recent * Research MPS71-02905

supported A03.

in

part

by

NIH

Grant

USPHS

10452-13,

and

NSF

Grant

356 Copyright All rights

C 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0040-5809

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reviews pertaining to genetic polymorphisms under conditions of variable selection and migration are given by Hedrick (1976) and Felsenstein (1976), which also list numerous references to experimental, field, and theoretical studies. In this paper, we concentrate on a formulation originally proposed by Levene (1953) and usually called the Levene population subdivision model. We present a detailed description of its stable equilibrium configurations and discuss some of their implications for evolutionary theory. A number of qualitative questions of interest to be dealt with include: (i) When and to what extent is there a multiplicity of stable polymorphisms ? (ii) How are the stable equilibrium configurations affected by the number and relative sizes of the demes ? (iii) What are the domains of attraction, significance, and biological implications of different polymorphic patterns ? (iv) To what extent can we infer the nature of the underlying selection regime from the observed gene frequency array ? The usual multideme population genetic model incorporating selection and migration forces has the following formulation. A population composed of separate units (demes CYpl, ,Y, ,... , :P,) is distributed over a finite region connected by interdeme gene flow. We consider a trait with two possible alleles labeled .L2 and a. The following assumptions apply: (i) Successive generations are discrete and nonoverlapping; (ii) mating and natural selection operate independently in each deme prior to the migration of individuals to other demes; and (iii) each subpopulation is sufficiently large that genetic drift effects can be ignored. The transformation of gene frequency in deme Y’i is expressed by the relation x’ --fi(x), signifying that where x is the A-frequency in 9% at the start of a generation, after the operation of mating and selection, the resulting A-frequency prior to migration is x’. Obviously, each local selection function satisfies 0 :--f,(x) :rl I for 0 < x < 1 and, generally, fi( x ) is continuous and increasing. nlutation is not considered (or is insignificant during the time frame involved) so that f,(O) := 0 and f*(l) =:= 1. An important choice for fi(x) arising from the classical diploid one-locus two-allele random mating viability model has the form six2 + X(1 - X) six2 + 2x( 1 - x) + rJi(l -

fiCx) = where the viability

Of special of additive 1d+ 1 < 1. we would

X)” ’

parameters of the genotypes are AA

Au

au

Si

I

ci >

0 < si ,

ui < co.

interest is the case of a dominant trait (si = 1, for all ;) and the case allelic selection effects corresponding to si =: I + di , gi == I - di , In a haploid situation, where the relative fitness of A to a is si : 1, take fi(x) = sjx/(l - x + sjx). Other relevant determinations for

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KARLIN

fi(x) can be generated by superimposing expressions of frequency-dependent selection, degrees of assortative or disassortative mating propensities, incompatibility mechanisms, etc. The following extensions of classical concepts and representations for selection functions are important to our later deliberations. (a) Local directed selection favors allele A (synonymously allele A is advantageous) in deme 9”i means that, where such a deme is isolated from the other demes or other influences of external populations, from any starting positive frequency of allele A repeated mating and selection would bring the local population of cP’~to fix on the A-allele. A corresponding notion pertains to the case where allele a is advantageous. Notice that for a haploid trait with constant fitness values, only local directed selection is possible favoring allele A(a) according as si > 1 (si < 1). (p) A local selection function for a deme is said to express overdominance provided the gene frequency evolution in the deme when isolated has the property that, from any initial state having both alleles present, no allele can ever go extinct. (y) Underdominance is said to operate locally in deme ~7~ such that if -/Pi were isolated both fixation outcomes could be attained, and which, depends on initial conditions. In analytic terms with only two alternative alleles involved, local overdominance occurs in deme P’i provided the pertinent selection function fi(x) satisfies min(fi’(0),fi’(l)) > 1 (f’( x ) is the derivative of f evaluated at x). A situation of local underdominance exists where max(fi’(0),fi’(l)) < 1. Directed selection favoring allele A (allele a) in deme .Yi corresponds to the analytic property fi(X) > x (fi(x) < x) f or all 0 < x < 1. In the concrete case of the two-allele viability expression (I), local directed selection is operating provided either si < 1 < (T{ or si > 1 > oi holds, where si and g’i are not both 1, signifying that the heterozygote carries an intermediate fitness value between the two homozygotes. Overdominance is in force where max(si , 0,) < 1 and underdominance where min(s, , ui) > I. The global selection regime is described by the collection of local selection functions {fi(x),...,fhi(x)}. When all local selection functions are of the form (l), the selection regime is equivalently delimited by the fitness parameter array @I , 4, (% >4Y.7 (sN, oN)}. Four cases of global selection patterns of special interest are distinguished here. (i) A selection regime involving local directed selection in all demes: This depicts a situation when in each deme either allele A or allele a is advantageous. Equivalently, if migration ceases, then a mosaic of pure populations would emerge with allele A or a established in its natural habitat (i.e., at those demes where allele A or a is advantageous, respectively). Where the same allele is

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not favored throughout then for some levels of migration a globally stable polymorphism may be achieved. Definitely, under small migration flux, a unique polymorphic globally stable state is attained. However, with a more substantial gene flow the effects of selection blend in a complex fashion and the evolutionary outcomes are less predictable. We refer to the underlying selection pattern described above as a mosaic pattern of directional selection. Where the selection strength is correlated with some environmental parameter over a population range (e.g., temperature, water availability, background coloration, soil millieux parameters, etc.), a mosaic of directed selection patterns may be appropriate. A case in point involving generally one to three loci determinants is dorsal coloration in lizards adapting to background colors. (ii) A global selection pattern with local overdominance manifested throughout the range is probably of common occurrence. In this circumstance, the heterozygote is advantageous in each deme where the strength or degree of its advantage can vary spatially and/or temporal1y.l It can be surmised that for this geographical selection regime qualitatively a global unique stable polymorphism exists for any migration structure. The validity of this principle is corroborated for the Levene population model in this paper and in the context of several other migration patterns in Karlin (1978). (iii) Underdominant local selection forces throughout the population range reflects a situation where the heterozygote is deleterious compared to both homozygotes. The degree of heterozygote disadvantage may vary from locality to locality. In such a model of N habitats, entailing very slight migration flow connecting the separate demes, 2N different stable polymorphisms can coexist where in each deme one of the two allelic types predominates. Usually, with moderate or substantial gene flow and some degree of underdominance expressed in each locality, the possibilities of polymorphism are significantly reduced (see Karlin and McGregor, 1972). There are many writings that ascribe the distribution of plant allelemorphs for certain traits to microgeographical adaptations exhibiting patches of different homozygote genotypes consonant to a regime of underdominant selection effects. Other circumstances reflecting underdominance throughout (heterozygote inferiority) may be associated with the prevalence of hybrids between species. (iv) Another important selection regime has each local selection function expressing either overdominance or underdominance. It is of interest to discern the equilibrium gene frequency patterns in the presence of such a mixed underdominant-overdominant regime. i It is important to emphasize that the phenomenon of temporal (systematic or random) changes in selection pattern can be mostly incorporated into the structure of spatial heterogeneity in selection pattern and, accordingly, temporal selection variation is a special situation of spatial selection variation. An elaborate discussion of this matter will be presented elsewhere.

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KARLIN

Gillespie and Langley (1974) emphasized a selection regime of a mosaic of directional selection effects engendered by additive allelic contributions to fitness as models to explain biochemical allelic diversity. (They concentrate mostly on temporal rather than spatial selection variation.) Although such a model may be germaine to some situations, the biochemical properties of heterozygotes are usually not exactly intermediate either for multimeric enzymes and even for monomeric enzymes. There are many known cases where the properties of heterozygote enzymes are not nearly an average of the corresponding homozygotes. For example, cases of nonintermediacy of biochemical properties are known for heat stability and specific activity of esterase-5. A discussion of this matter is presented in Berger (1976). Furthermore, Berger advances a number of cases of biochemical genes pointing up heterosis as the likely local operative selection manifestation where the heterozygote compared with homozygotes maintains a higher level of catalytic activity and/or increased efficiency in conserving metabolic energy under suitable conditions of temperature, chemical, electrical, or environmental backgrounds. We cannot discuss the validity or nonvalidity of overdominance for these cases, but it is pertinent to our theoretical analysis that a number of biochemical traits cited in Berger suggest a mixed under-overdominant selection regime associated with different performance and function levels dependent on the environmental milleux. Th e preceding remarks indicate that the four classes of selection patterns singled out in this paper have independent interest and possible relevance for certain natural multideme population genetic structures. It is important to realize that the gene frequency equilibrium outcomes differ qualitatively with respect to (i)-(iv). It is only the detailed characterizations of the stable equilibrium possibilities that allow suitable discrimination of a germaine (or falsifiable) underlying global selection regime. In order to write the transformation relations connecting gene frequencies over successive generations and to take proper account of such factors as variable relative deme sizes and the influence of local differential viability forces intertwined with gene flow, the concept of the backward migration matrix is fundamental. The elements of the backward migration matrix M = !I m,, i/ after selection and migration specify mij = the fraction in the ith deme originating in a given generation.

from the jth deme

Let xi denote the frequency of type A in deme Yi at the start of a generation and xi’ the frequency for the next generation. The standard transformation equations connecting x (xi ,..., xN) to x’ = (x1’,..., xN’) over two successive generations is given by N

Xi’

=

1 i=l

FLijfj(X,),

1’ =

I,...,

N.

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For more on the general model see, e.g. Christiansen (1974) and Karlin (1976), and references therein. The property of persistence of an allele A (not going ultimately extinct) even when it is initially rare, is now commonly called protection of the A-allele or A-protection. If both alleles are protected, then a “protected polymorphism” is said to exist. A protected polymorphism may entail one or several stable oscillatory (or even unpredictable) internal equilibria states or, conceivably, changes of gene frequency values can occur over successive generations. What is definite is that a protected polymorphism precludes the fixation of either allele from any polymorphic initial state with both alleles involved. An important objective of the study of geographical population genetic models is to relate the observed gene frequency distributions with possible background selection migration structures. Except for some examples of twodeme populations and cases of clinal gradients virtually all studies of multideme models have focused on deriving conditions of protection for some special forms of migration flow. Levene (1953) was among the earliest to study selection heterogeneity in a subdivided population. He determined some conditions of protection for a two-allele trait. Prout (1968) ascertained criteria for protection in the same model for a dominant trait. Deakin (I 966, 1968) and Christiansen (I 974) discussed a specific extension superimposing a uniform homing propensity on the Levene population subdivision model. In Karlin (1976, 1978), natural conditions establishing the existence of a protected polymorphism for a multideme population model with general migration patterns are set forth. In these works interpretations and contrasts are made for relating degrees of environmental heterogeneity and levels of migration mixing. The determination of the equilibrium configurations in some two-deme models have been investigated by Moran (1959), EyIand (1971), and Karlin and McGregor (1972), among others. Recently, analytic insights into the equilibrium possibilities with cline models are accumulating. Slatkin (1973), Nagylaki (1975) Fleming (1975), Conley (1975) May et al. (1976) and others have studied some cases of selection cline structures with a continuum of continuously along a curve. Migration demes, i.e., a population distributed has been represented by diffusion motion, which is the continuous analog of a stepping-stone migration mode. Endler (1973) has conducted computer runs on some regular patterns of finite deme cline models. The characterization of the possible stable polymorphic states and discussion of related qualitative questions for finite deme cline models is reported in a series of papers by Karlin and Richter-Dyn (e.g., see 1976). Finally, we mention that an abundance of long-standing literature exists concerning open and closed migration systems without selection. Pertaining to this last topic consult, for example, the book by Jacquard (1974, Chap. 12), Malecot (1951, 1959) Carmelli and Cavalli-Sforza (1976) and the extensive bibliographies in Hedrick (1976) and Felsenstein (1976).

362

SAMUEL

KARLIN

It is useful now to set forth the precise formulation of the Levene model. Recall first that the Wright Island model depicts a situation where the population is divided into panmictic units, each receiving an equal proportion of the total population. Levene generalized the Wright model allowing that the population, after mating at random in a common area, distributes itself into N separate habitats, a fraction ci going into the ith habitat. Then selection occurs according to the state of the environment in each habitat. Notice that after migration the subpopulations involve the same mixture of the whole population for each generation. It has been suggested that the foregoing setup may be appropriate for a species whose numbers are regulated within each of the separate demes, but not on the whole population. In the Levene population subdivision model, it is more appropriate to refer to the subpopulation .Pi as a patch or habitat since breeding occurs at a common locality, while only differential selection effects (varying spatially and/or temporally) discriminate among genotypes. The Levene subdivision model is characterized by three main features: (a) Numerous microhabitats are available for the population; (/3) mating occurs at random across the local habitat structure; and (y) the output from each site is locally set. Some classes of organisms that possibly fit this life-style include the polychaetes (marine worms), which are principally sessile, but in mating engage in swarming maneuvers and then mostly settle back to available habitats. Other possible cases approximating the Levene subdivision model may include seabird populations that nest in large rookeries. A number of fish populations (e.g., the American eel and herring) breed together in spawning areas and then disburse back to habitats located up various streams, somewhat reminiscent of the foregoing population subdivision structure. Some discussion of the inherent limitations in the Levene model is found in the accompanying paper (Karlin and Kenett, 1977). Finally, it should be realized that the Levene structure constitutes the most homogeneous environment of a whole hierarchy of heterogeneous environmental selection migration patterns (on this concept an evaluation of the attendant see Karlin, 1976). Despite these limitations, stable gene frequency configurations in this case, apart from their own interest, serves as a control in the analysis and interpretation of other appropriate (more robust) migration selection structures. The Levene model fits the general framework of (2) under the following identification. The backward migration matrix reduces to mij = ci independent of i, and the transformation of gene frequency can be expressed in terms of a single variable, viz., six2 + x(1 - x) x’

=

5

i=l

CLW)

=

igl

(where x refers to the A-allele

ci s.x2

z

+

frequency

Zx(l

-

x)

+

q(l

of the mating

-

x)”

’

(3)

pool in the current

FREQUENCY PATTERNS IN SUBDIVIDED POPULATION

363

generation). We have specified the local selection functions to correspond to the classical diploid one-locus two-allele viability function (1) with fitness parameters varying from habitat to habitat. The character and numbers of equilibrium states for various types of geographical selection regimes are set forth in the next section. Some qualitative interpretations and implications of the results are discussed subsequently. It is of interest to contrast and compare the dynamics and equilibrium structure of the population evolution for a selection regime with either allele A or a advantageous in each habitat with a mixed underdominant-overdominant selection pattern. This inquiry can be expressed as an evaluation of the effects of disruptive selection between versus within habitats. A number of mathematical proofs are relegated to the appendixes.

2. RESULTS AND SOME IMPLICATIONS I. Equilibrium Advantage

States for the Levene Model with Directed Selection or Heterozygote Throughout

The evolutionary (transformation) equation (3) in the Levene population subdivision model can be traced in terms of the single variable, the A-allele frequency in the whole population. The equilibrium states correspond to the solutions of the equation

restricted to 0 < x < 1 (cf. (3)). There are two trivial equilibria, viz., the fixation states x = 0 and x = 1. Levene (1953) provided the local analysis of the fixation states, showing that a sufficient condition for A-protection is f ‘(0) =J$, cilui > 1 and for a-protection f’(l)= CL1ci/si> 1. Becauseeachfi( x )’is monotone increasing over 0 < x < 1, so is f (x) and no oscillatory behavior of the iterates x, = f (xnP1), n = 1, 2,..., occurs. Thus, convergence to some equilibrium takes place from any starting state. OUY principle objective is to delineate the equilibrium possibilities, their dependence on the selection regime, and deme sixes.

The solutions of (4) apart from R = 0 and 4 = 1, i.e., the possible polymorphic equilibrium frequencies, are roots of a polynomial of degree at most 2N - 1, namely, the numerator polynomial of the equation [(Si + ui - 2)x + 1 - UC]

g(x) = i=l5 ci Six2 Manifestly,

+ 2X(1 -

X) + Oi(l

-

X)”

= O.

an upper bound to the number of possible polymorphisms is

364

SAMUEL KARLIN

2N - I (consult Appendix A). A pertinent inquiry is: Does there exist a set of selection parameters and arrangement of habitat sizes such that (4) admits exactly 2N - I polymorphisms in addition to the two fixation states and therefore, N + 1 or N stable equilibria with bona fide domains of attraction? Equivalently can the realizations in this multiple habitat model exhibit evolutionary outcomes highly sensitive to initial conditions. It is a striking fact that in the case at hand, generally, at most three internal equilibrium points exist independent of the speci)ication of the jitness components. Now we describe results on the numbers of polymorphic equilibria corresponding to various configurations of selection parameters. To this end, consider a selection regime characterized by the fitness array

{(Sl, ‘T1);(s2, FJ;...; hi , 9);.-; (SN,%>>, where si , 1, (TVare the viability RESULT I.

selection values operating

in habitat Qi .

Where sp’i < 1

persist for all

i = 1, 2 ,..., N,

at most one internal equilibrium state exists. Moreover, when a polymorphism is present, this state is globally stable, i.e., it is approached from any initial internal state. If some jixation is locally stable, then it is globally stable. A proof of this result is given in Appendix A. Condition (7) describes a sufficient, but not necessary, condition guaranteeing at most one polymorphic equilibrium. It is worthwhile highlighting some important cases of Result I. (a)

Underlying

heterozygote advantage in each habitat. Then

si < 1

and

for each

ui < 1

z = 1,2,..., N

and plainly (7) holds. In this situation both fixation a unique globally stable polymorphism is established. (b)

Additive

allelic effects.

states are unstable

and

The fitness values have the form

AA

Aa

si==l+di,

1,

aa (ii = 1 -

di

where ( di ( < 1 for all i = 1, 2,..., N. Clearly siui = 1 applies. The pertinence of the assumption of additive morphological and enzyme electrophoretic characters and Langley (1974) although its prevalence is contested (e.g., see Berger, 1976).

(9)

- di” < 1 and Result I allelic effects for some is argued in Gillespie by a number of authors

365

FREQUENCY PATTERNS IN SUBDIVIDED POPULATION

(c)

Multiplicative

The fitness coefficients now take the form

allelic effect. AA sj = yj

Aa 1

aa uj = l/Yi

(10)

Equivalently, a multiplicative factor yi accrues to each substitution of allele A for allele a. Renormalization of the heterozygote fitness to value 1 produces the equivalent fitness array (IO). With (IO) we have siui == I and Result I is in force. (d) It is of interest to point out that the Levene population model for a haploid trait and selection regime

Fitnesses with viability

function

A

a

si

oi

subdivision

in b, ,

in habitat Yi ,

fib9 =

SjX

sjx -f- Ui(l -

X) ’

transforms equivalently as a diploid viability scheme with multiplicative allelic effects having yi = si/ai . Thus, Result I tells us that for the evolution of a haploid trait in a subdivided population of the Levene form, at most one polymorphic equilibrium (when present, it is globally stable) can be generated (cf. Glidden and Strobeck, 197.5’). RESULT IA.

siui -

Opposite to (7), suppose that totally siui > 1 such that 1 > max[(s, -

1)2, (oi -

l)2]

for all

i = I, 2 ,..., N,

(11)

and assume that at least one internal polymorphism exists. Then both Jixation states are stable and a unique internal unstable equilibrium is present. A partially dominant trait. Consider the situation where the heterozygote has a constant degree of dominance throughout the population range. More specifically, assume the fitness values have the form AA 1 +d<,

Aa l+hd,,

l?&

in habitat .Y.1 ,

(12)

where / di 1 < 1. Observe that h is independent of the habitat site. When j h 1 < I, the heterozygote has an intermediate fitness value between the two homozygotes in every locality. Allele A is dominant (recessive) when h 2 1 (h == - 1). It is worth pointing out that the representation (12) does not include all the cases of dominance since h is constrained to satisfy h > ~ 1/di when di is positive and h < -l/d, when dj is negative. Additive allelic effects corresponds to h == 0. When h > I, di > 0, overdominance is in force at habitat ,Yi and for h < - 1, di > 0 we have local underdominance.

366

SAMUEL KARLIN

Normalizing the fitness values to bring the heterozygote converts (12) to the parameters

1+ 4

fiZrihd,,

c?i=---

fitness value to 1,

1 - di 1 + hdi

and &ei = (1 - di2)/(1 + hd$, which can exceed 1. Therefore, Result I does not always apply and in fact the possible outcomes are more varied, as described below: RESULT II. (i) With the selection regime of (12) where allele A has the same degree of dominance to allele a over all habitats of the population range, there exist at most two internal equilibria. When two internal equilibria exist, one of the $xation states and one of the polymorphisms are simultaneously stable. (For the last possibility, see Appendix C.)

(ii) With complete dominance, at most a single internal and it is globally stable when present.

equilibrium

We next discuss the model where in each habitat the selection satisfy either favoring allele A, si > 1 > ui ) or favoring allele a. si < 1 < ui )

exists

coefficients (13a) (13b)

This is a fitness regime involving a mosaic of local directed selection forces throughout. Thus, in each locality one or other of the alleles would fix under the action of selection and reproduction if insulated from the other habitats of the environment. PRINCIPLE I. With a mosaic of directed selection, there can exist at most one stable polymorphic equilibrium.2

Concordant to this principle there can occur directed selection regimes throughout, entailing both fixation events locally stable and a stable polymorphic equilibrium. An explicit analytic construction with outcomes producing such equilibrium configurations is given in Appendix A (the discussion of Eqs. (A. 12)-(A. 14)). This situation requires concomitantly the existence of two unstable equilibria which separate the domains of attraction to the fixation states and the stable polymorphism. In order for such equilibrium configurations to arise, the selection differentials between A- and a-types are necessarily very pronounced. In violation of the principle above, with very extreme 2 An explanatory note on our terminology distinguishing results as against principles. A result embodies a rigorous conclusion precluding any counterexamples. A principle is valid covering a large spectrum of natural situations, but there can be extreme cases violating its statement. Thus, a principle (as in physics theory) is approximately valid and enjoys wide scope of applications.

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(unrealistic) fitness values, there can arise rare cases of two stable polymorphisms exhibiting frequency patterns generally located close to fixation outcomes; see the numerical cases at the close of this section. In contrast, selection engendered by additive (or multiplicative) allelic effects or in the case of a dominant trait, the phenomenon of three coexisting stable equilibria involving both fixations plus a polymorphism cannot occur. A numerical example of three polymorphisms for five subpopulations subject to local directed selections involving stable fixation states and one stable internal equilibrium between two unstable ones is as follows: Cl = c3 = c3 = c4 = cg = 0.20 s1 = 0.75

s2 = 1.4

s3 = 0.63

sq = 9.9

s5 = 0.85

u1 = 6.4

(72 = 0.45

us = 2.0

a4 = 0.8

us = 3.8

The internal equilibria values are xi m 0.11 (unstable), xa m 0.275 (stable), xa M 0.98 (unstable); see also the close of this section concerning further numerical output on this model. II.

Selection Regime Expressing Local Heterozygote

Advantage

Throughout

We already pointed out in connection with Result I that if heterozygote advantage is in force in every habitat (with fitness coefficients allowed to differ between habitats), then a unique globally stable polymorphism occurs. This conclusion appears to be robust independent of the migration structure. III.

Underdominance or Overdominance in Each Deme

In this situation the feasible equilibrium configurations are more variable. Nevertheless, a number of principles emerge. We first present a complete description of the stable equilibrium configurations for two specific selection patterns in this category. Homozygotes carry equal fitnesses, so that Model (i)

si = U{

for each

z = 1,2,..., N.3

(14)

Then in a habitat where si < 1, overdominance selection operates, and where si > 1, underdominance is in force. In the case at hand, the frequency value x = 8 is obviously a common fixed point of each local selection function. Therefore, the polymorphic equilibrium is always present.

3 Our results are structurally stable, meaning where (14) approximately holds.

that

the same

(15) qualitative

results

apply

368 This equilibrium

SAMUEL KARLIN

is stable provided f’(3)

= f

Ci(2SJ(l + Si)) < 1

(16)

i=l

and unstable if the reverse inequality holds. (The equality event Cci(2sJ( 1 -t si)) -- 1 is nongeneric (i.e., degenerate, unlikely to occur) and therefore irrelevant in real cases. Recall that the fixation states for the case at head are stable (unstable) if

Condition (17) applies symmetrically following result has interest.

to fixation

of allele A or allele a. The

RESULT III. For the selection regime corresponding to (14), where the environment confers in each habitat the same fitness expression on both homozygotes which may vary from habitat to habitat, then at most three polymorphic equilibria exist whose delineation is as follows.

(a) If xr=, (ci/si) < 1, then only the fixation equilibrium x * = 3 is the unique interior equilibrium (b) If Cc, (2c,sJ(l globally stable equilibrium.

states are locally stable. The and is, of course, unstable.

-(- si)) < 1, then x* = i

is the unique

(internal)

(c) Where xy==, (cJsi) > 1 and 2 ~~~, (c,s,/(l + si)) > 1 hold, there exist two stable polymorphic equilibria Z?and 2 -= 1 - 2 (0 < .? < h < A < 1) such that 9 is approached from any initial state x0 , 0 < x,, < 4 and x” is reached from x0 with 4 < x0 < 1. The points A?and 2 are calculated as the values yielding is the unique solution in 0 < 5 < $ of the equation

x( 1 - x) ~:=d where 1

It is helpful to depict the possibilities of (a)-(c) in graphical form in the case of two habitats (Fig. 1). Consider, for simplicity, cr = c2 = 0.5. In region I, case (b) holds and the unique globally stable equilibria x* = & prevails. In region III, both fixation states are locally stable in conformance with (a). In region II, there exist two stable polymorphic states SZand 2 =:- 1 - 9 per the dictates of case (c). The following elementary inference holds (see Appendix B):

(18)

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Sl FIGURE

1

Thereby, we deduce that where the fixation possibilities are stable, x* of (15) is unstable. A selection regime amenable to complete analysis, extending that of (14), has the fitness values in habitat Yi of the form AA

Aa

Si

1

aa (19)

l--+$-i4

where 01is a fixed parameter obeying the restriction 0 < 01< 1. Observe that with a = 1 we recover the symmetric case (14). The choice LX= 0 has allele A as a recessive trait. In the model of (19) the fitnesses of the two homozygotes are linearly dependent, displaying a constant relationship throughout the operates at range. With 0 < 01< 1, heterozygote advantage (disadvantage) habitat ,Bi for si < 1 (sI > 1) having the selection coefficient of genotype aa bearing a stronger impact than that of the au-genotype. The viability functions fi(x) associated with (19) sh are the fixed point ~/(cx + 1) independent of i. Thus, the three equilibria x* = 0, x** = 1, and 3i:= ~/(a + 1) are always present for this model. In some circumstances of the parameters, there may exist up to two additional polymorphic equilibria. A summary of the complete dynamic and equilibrium behavior is given in Table I. All four cases can occur. It seems unintuitive that independent of the multiplicity of habitats and flexibility in specification of fitness coefficients, at most three internal equilibria can arise (provided (19) holds), these appearing already in the situation of two habitats. The following principle emerges. * The

results

of Table

I are structurally

stable;

see (14).

SAMUEL

KARLIN

0 II

.9 0

0”

+ u I 3

(See Appendix

and

B for their proofs.)

a The four cases exhaust all the possibilities

zzz 0, x**

1

all unstable

=

x2 + a(1 - $+

1+a

by virtue of the following

O
--E-1

‘a+

located in the intervals

-

izz
(I/($

The points 2 and x” are

g

Obtained as the unique solutions of the equation

0-l 2, x” stable

O,l,A

x*

oi 2 = a+1

1)) = O

stable

the polymorphism

(20)

< x0 < 1 the polymorphism For -% a+1 2 is established

For 0 < x,, < -% a+1 S is established

Unstable

Globally

312

SAMUEL KARLIN

PRINCIPLE II. For any selection regime in the Levene population subdivision model with a general set of local viability parameters there exist at most two stable polymorphic equilibria (and with multi-alleles at most one closed curve of stable equilibria).

Result I and the cases of the selection regimes having only either underdominant or overdominant local selection expression (e.g., cases (14) and (19)) testify to the validity of Principle II. We have conducted many numerical computations covering an assortment of selection prescriptions involving 3 to 7 habitats and the conclusion of Principle II holds up. In the presence of Principle II, the previous considerations show that in the Levene population subdivision structure already the circumstance of two habitats produces the scope of qualitative stable configurations that may occur with increased selection heterogeneity and population subdivision. This is definitely not the case in the context of a general migration pattern (Karlin, 1978), which attests to the special nature of the Levene population subdivision model. Some Numerical

Calculations

We establish in Appendix

on the Levene Subdivision Model A that the equilibria

correspond

to the zeros of

X(S( t ui - 2) + 1 - (Ti g(x) == i: ci i=l X2($ + ui - 2) + 2x(1 - Ui) + u’i = O (formula (A.4)). We considered a population subdivision of seven habitats and made 70,000 independent choices of selection regimes and relative deme sizes. In order to ascertain the number of internal equilibrium points for a particular model determination, the value of g(x) was calculated successively at each grid point Gh 2oooof a partition of the unit interval into 2000 equal parts (0.0005 distance apart). The count in changes of sign along the sequence (g([JjFoo estimates quite accurately the number of internal equilibrium points. We should emphasize that for moderate selection coefficients, the dictates of Principles I and II will apply, and certainly Results I-III are in force. Cases of more than three polymorphisms arise only in the presence of extreme selection effects operating at some of the habitats. A number of rare cases of four polymorphisms (only two stable) were observed and one instance of five polymorphisms (only two stable) occurred. Actually, of lo6 numerical studies, no exceptions to the phenomenon of only two stable polymorphisms were detected. The explicit constructions and outcomes proceeded as follows. Each si and ci was chosen between 603 and 60e3 by the prescription x,t _

603-S”

ui _ 603-6vi,

where xi and yi are independent

random

i = 1) 2 ,..., 7,

values between

0 and 1. Of the

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70,000 model determinations, 13 cases with 4 polymorphisms rejlecting 2 stable plus a stable Jixation state were found for an average of l/6000 of the model constructions. By reducing the range to si -;

3()(3-64,

Oi z

303-6”i

with mi , yi independent uniformly distributed on [0, 11, we uncovered eight cases entailing four polymorphisms for an average of l/8000. The single example in this collection with five polymorphisms came from the following parameter specifications: cr -0.07

c,==O.lO

c,=O.O6

c,=O.27

c,==O.25

cs =0.22

c7-0.03

~~-130.75

s,=O.14

s,=100.14

s,=2.18

s,=205.75

s,=33.38

s,=15.65

or==63187

u2=28.42

0,=0.08

~~~9.39

a,=243.15

a,=15.06

a,=37.20

The pal!-morphisms .cj a 0.01,

occur at the points

.$ m 0.04,

23 hy 0.795,

id m 0.905,

S5 a 0.980.

The stable equilibria correspond to i,, = 0, S$ , 3i’, , and S$ = 1. We corroborated Principle I with 100,000 cases of seven populations with ci =- 6, i = 1, 2 ,..., 5. The number of situations with three polymorphisms, one stable, appeared an average of l/4000 runs. Another 200,000 choices of selection regimes exhibiting more extreme heterogeneous directed selection pressures throughout were made. A total of 14 cases involving 2 stable polymorphic points with a third unstable polymorphism appeared. In these cases, the fixation states were unstable. In all examples, the polymorphic points tended to be located quite close to 0 or 1. The validity of these stable configurations was checked independently by iteration of the transformation relation (3). U’e also explored numerous cases of underdominant selection in operation at all of the seven habitats. It is known that for general migration structures, and with underdominant selection in each local habitat, a greater proliferation of polymorphisms are realized. After 100,000 model determinations, we found 273 cases of 5 polymorphisms for an average of l/240 per run. Thus as anticipated, an underdominant selection regime yields the most polymorphic picture consoant to the analytic findings in Karlin (1977b) and Karlin and McGregor (1972). (The author is much indebted to Ron Kenett for assistance in conducting the computations). Another cautionary comment is the following. Where the local selection functions are not based simply on the standard viability expressions of the form (1) the qualitative conclusions discussed above do not carry over. It is already possible to construct, with two habitats, local selection functions

374

SAMUEL KARLIN

jr(x), fa(x) favoring alleles A and a in demes 1 and 2, respectively (actually f,(x) and fi(.z) can even be determined so that fi(x) is concave with jr(x) > x for all 0 < x < 1, f,(x) is convex with f2(x) < x for all 0 < x < l), so that with uniform mixing (exchange rate m = + between the demes) a multiplicity of K polymorphisms exist, K arbitrarily large. This is in sharp contrast with the range of possibilities occurring for standard viability functions of the form (1).

3. SUMMARY AND DISCUSSION Most theoretical studies of selection-migration interaction concentrate on ascertaining conditions for the existence of a “protected polymorphism” (exceptions include cline models). The affirmation of a protected polymorphism gives relatively little information concerning the nature of the gene frequency equilibrium configurations and how they correlate with the underlying selection regime. An important objective of a theoretical model is to discern the possible forms of the underlying selection regimes concomitant with an observed gene frequency pattern. We singled out for special attention in the Levene populations subdivision structure, four classes of selection regimes, as follows: (a) a mosaic of directed selection expression, i.e., in each habitat one of the alleles is advantageous and would fix were this habitat insulated; (b) some degree of overdominance operates in each habitat; (c) a pattern of heterozygote disadvantage at each habitat with varying intensity from locality to locality; and (d) mixed overdominantunderdominant selection effects, reflecting a situation where in some localities there is heterozygote advantage while in other habitats heterozygote disadvantage is in force. These cases are of manifest biological relevance although other more complicated selection structures can be envisioned. The key qualitative implications of the results reported in Section 2 are as follows: (1) It is a familiar proposition that for a two-allele population limited to a single habitat subject to viability selection, at most one polymorphic equilibrium can be maintained. With two habitats, we found that at most two distinct stable polymorphisms can occur. We expected, ab initio, that with increased population subdivision allowing heterogeneous viability expression over the population range but maintaining a single mating area, a further proliferation of possible stable gene frequency patterns would occur. It is perhaps surprising that for N > 3 habitats, only at most two stable polymorphisms apparently exist, each endowed with its own domain of attraction. Thus, for a selection regime involving local differential viability expression of the form (1) and (6), the number of equilibria possible is not qualitatively changed by

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375

increased population division beyond that of two habitats. There is clearly a pronounced quantum jump of equilibrium possibilities when passing from one to two habitats, but it appears that the nature of the equilibrium configurations with any number (N 3 2) of habitats is virtually fully represented by appropriate selection regimes prescribed in a two-deme population. The bound 2 on the number of possible stable polymorphisms is intimately tied to the Levene subdivision model. The contrast is pronounced for the situations of local underdominant selection expression everywhere, interacting with a general migration flow among N habitats where up to 2N stable polymorphic states can be manifested, particularly with slight but positive interdeme gene flow. (2) For a selection regime involving only a mosaic of local directed selection expression so that either allele A or allele a is favored in each habitat, there can occur at most one stable polymorphism. In other words, independent of the distribution and selection intensity at the habitats conferring advantage to allele A as against those favoring allele a, the overall balance can generally only maintain a single stable polymorphic state. Thus, the selection expression attendant to a mosaic of local directed differential viability forces is insufficient in plasticity to engender more than a single global polymorphic balance. However, where directed selection is sufficiently intense at a number of localities, an equilibrium configuration can emerge where both fixation states in conjunction with a single polymorphic state coexist locally stable. With such a contingency there actually exist three internal equilibria states (two are unstable) and then the realized outcome depends on initial conditions, founder effects, and sampling fluctuations individually, or in combination. For very intense selection coefficients, the case of two or more stable equilibria can arise in the presence of only locally directed selection effects. The numerical results at the close of Section 2 are pertinent. The qualitative fact entailing at most one locally stable polymorphism resulting from the influence of local heterogeneous directed selection throughout the population range appears to be substantially valid with any migration pattern. (3) We raised a number of questions in Section 1. Of some interest is the situation of a mixed underdominant-overdominant selection regime. Here three categories of equilibrium outcomes generally arise. Explicitly, there coexist no more than two stable equilibrium points embracing the possibilities of: (a) one or both fixation states; (b) one fixation and/or polymorphic realization; and (c) two stable polymorphic equilibria coexist. On the other hand, as mentioned already, for an environmental selection regime where directed selection hoIds at each deme of the population range, up to three stable equilibria coexist, but at most one is polymorphic. The realization of both fixation events and a stable polymorphism requires extreme intra and interhabitat selection heterogeneity. In practical terms, if frequency observations implicate two stable

376

SAMUEL

KARLIN

polymorphic outcomes and the Levene population subdivision structure is pertinent, then we can infer that the underlying selection regime cannot be a mosaic of directed selection effects in each habitat. For a trait with a constant degree of dominance throughout the habitat range, there can exist at most a single stable polymorphism and both fixation states can never be simultaneously locally stable. However, two stable equilibria may occur together, involving one fixation possibility and a stable poly-morphic state (for more on this, consult the analysis of (12)). With additive allelic eflects, viz., si = 1 + di , o, = 1 - d, , 1dj :c< 1 or multiplicative allelic ejfects, viz., si : yi , vi = l/yi , condition (7) manifestly holds and in these cases either a globally stable polymorphism is established or one of the fixation events is realized. More generally, if the selection intensities for the homozygotes in each locality are not intense or excessive compared to the heterozygote, such that siui < 1 for all i = 1, 2,..., A-, then the dynamic behavior is like that of a large single population admitting only one of the fixation possibilities or a globally stable polymorphism. (4) The validity of the results discussed in paragraphs 2 and 3 decisively rests on the property that each local selection function has the specific form (1) general reflecting differential viability pressures. For the cases entailing forms of local selection functions, perhaps, corresponding to frequencydependent selection or more complex selection interaction effects, then even with local directed selection in two demes, the possibility of a myriad of stable equilibrium configurations is real. (5) The general principle that any migration structure cannot dissipate the effects of a selection regime expressing underlying overdominance throughout its range is corroborated in the Levene population subdivision model. Then, a “central” globally stable polymorphism is established irrespective of the subpopulation )relative sizes or the intensity or variation in the viability- values expressing local heterozygote advantage. (6) Cannings (1971) has discussed some examples of a multi-allele version of the Levene model. For a case of two habitats, he constructs a viability array leading to a stable surface of equilibria. This result can be construed as a multidimensional version of the phenomenon involving two interior stable equilibria where underdominance is effective in one habitat and overdominance in the second habitat. Our experience in passing from the two-allele model to the corresponding multi (three or more)-allele model indicates that the existence of multi-stable polymorphic states is reflected in surfaces of stable equilibrium points (compare to Cannings, 1971; Carmelli and Karlin, 1975; Karlin and Farkas, 1978). (7) The Levene population subdivision model is the most homogeneous selection regime of a broad hierarchy of environmental selection patterns and

FREQUENCY PATTERNS IN SUBDIVIDED POPULATION

317

concomitantly the scope of the stable equilibrium configurations is limited compared to the general migration selection interaction model. The meaning, precision, and implications of this concept are elaborated in Karlin (1976 and

1978). (8) In the Levene population subdivision structure, the influence of relative deme sizes as reflected in various specifications of (ci} seems to be minor. More fundamental is the heterogeneity in the selection regime as distributed over the different population habitats.

APPENDIX

A:

THE

NUMBERS

OF POLYMORPHIC

Let the selection regime be characterized values {(sl , 4;

EQUILIBRIA

by the array of homozygote

fitness

(s2 , a,);...; (SN, 0~11: genotypes

AA

Aa

aa

titnesses

xi

1

ci

operating

in habitat .Ypi

(A.1)

If x represents the A-allele frequency in the current generation available at the common mating area, then the transformation of gene frequency resulting from mating and local selection is given by x’ = f

c&x)

=f(x),

64.2)

i=l

where

Si5” + ((1 - S) fi(E) = sit2 + 2[(1 - 5) + Ui(l The equilibria

frequency

6)” ’

i = 1, 2 ,..., 1V.

values solve the equation

Roots of (A.3) not between 0 and I have no biological relevance. Factoring out x and subsequently combining both sides, using the fact of 1 = Cy=, ci , reduces (A.3) to x(1 - x)g(x) where

= 0,

G4.4)

378

SAMUEL

KARLIN

The zeros x* = 0 and 1 correspond to the fixation states. The zeros of g(x) are those of a polynomial P(x) of degree 2N - 1 so that a priori, up to 2N - 1 polymorphic equilibria can exist in the presence of the environmental selection regime (A.l). However, the multiplicity of possible polymorphisms in general is not essentially altered by increased population subdivision beyond two demes: Result I is germane to this assertion. Proof of Result I.

We can execute a partial fraction decomposition and write 1 - ‘Ji $ + Ui - 2 l-oad 2 si + Ui - 21 + (Q sf”b,--l2,2 x+

g(x) = 5 ci i=l

x+ [ (

+L[ .lyl. + 3’,]~ z

2=1

z

where for 2 > si + oi , Ki

=

1 - ui - (1 - Sic#‘2 2 - (Si + Ui) ’

y2. = 1 - C7i+ (1 - S&)l’2 z 2 - (Si + Ui) .

The hypotheses uisi < 1 imply the relations yi’liy2i < 0 and either

yli > 1 or

y2yzi> 1.

64.6)

The result of (A.6) holds also if si + oi > 2. Thus, g(x) admits the representation (A-7) where di are all positive and e, real. Observe thatg’(x) = --xEr (C&/(X- e,)“) < 0 for all x # ei . Therefore g(x) possess one zero between consecutive ei’s. Since ei are either <0 or > 1, it follows that g(x) involves at most one zero in the interval 0 to 1. Therefore, under the conditions of (7), at most one polymorphism can exist. Both fixation states cannot be simultaneously stable if (l/ur) + (I/Q) > 2 for all i = 1, 2,..., N (with strict inequality for at least one z). Indeed, suppose to the contrary that f ‘(0) = %$$j

z

< 1

and

f’(1) = $$i $ < 1, z

and therefore

: ci (+ i=l

+ +,

< 2,

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which is manifestly absurd. Now since $((l/sJ + (l/uJ) >, l/(~~a~)l/~ with inequality if si # aa , we see that siui < 1 implies (l/Si) + (l/ui) > 2. Thus, if SiUi < 1, i E 1, 2,.**, N, and si # ui for some i, then both fixations are not locally stable together. As the polymorphism is unique when present and since both fixation possibilities cannot be locally stable simultaneously, the only consistent state of affairs compels both fixation states to be unstable, and concomitantly the unique polymorphism if extant is globally stable. Remark A.1. Where siui < 1 for i = 1,2,..., K, a similar analysis of the partial fraction representation (A.7) establishs the bound 2N - (2K - 1) for the number of feasible polymorphisms. With general fitness parameters the practical attainable bound appears to be 3 independent of N and the specifications of the fitness parameters si and (TV. Remark A.2.

A direct calculation yields

siui - 1 (si+ui-2)2i x+s,:,“2 g’(x) = i ci l-u{ 2 i=l x+ Si + Ui - 21 + (Si yuy2 I K

i2 92

(A.81

I 2

With si”i - 1 > max((s, - 1)2, (ui - 1)2),

i = 1, 2,..., N,

(A.%

we find thatg’(x) > 0 on [0, 11, implying at most one zero of g(x) in [0, 11. In order for a solution to exist, we must have g(0) < 0 and g(1) > 0, which exactly implies that the fixation states are stable. Therefore, in the circumstance of (A.9), any internal equilibrium is unstable and merely demarcates the domains of convergence to the fixation states. The situation of (A.9) can be construed as a multideme case of disruptive selection. The number of internal equilibria can be three with N = 2 habitats even with allele A favored in habitat 1 and the alternative allele a favored in habitat 2. The following construction produces an example with A-fixation, a-fixation, and some polymorphic equilibrium simultaneously locally stable. In order to achieve this example, the intensity of selection favoring allele A in g1 and that for allele a in 8, is necessarily quite strong. In this situation there must exist two other internal unstable equilibria points which serve to demarcate the domains of attraction to the stable equilibria. The polymorphic equilibria are obtained as the solutions located inside the unit interval for the equation

I& i=l

SfX + (1 - x) SIX2+ 241 - x) + (Ti(1 - x)” .

(A.lO)

380

SAMUEL KARLIN

Transposing 1 = Cr=, ci to the left, removing the factor 1 - x, and finally, passing to the variable w = x/(1 - x), the polymorphic equilibria correspond to the solutions in w > 0 of

With N = 2, ci = c, = -& we can determine sa < 1 < u2 and

si , cri to satisfy S, > 1 > CQ,

(A.12) The last inequalities of (A.ll) is explicitly

imply

W(SI -

that both fixation

1) + 1 -

01

s1w2+ 2w + ‘51 Where

u2

+ co, 1 >

+

states are stable. The expression

w(s2--1)+l--z=o

> $, sr > 10, the numerator 1 M 0, whose roots are

u1

(A.13)

*

s2w2+ 2w + u2

polynomial

behaves like

w2s1+ (3 - sJ w + 2u, -

(A.14) and 0 < A < (si - 3)2. Both roots are positive if si > 10 and therefore (A.13) admits three roots provided u2 is large enough. Because both fixations are stable, as guaranteed by (A.12), the central root of (A.13) induces a locally stable polymorphism. The above specifications lead to five equilibria such that both fixations and an interior point are stable.

APPENDIX

B:

UNDERDOMINANCE

THE

EQUILIBRIUM

OR OVERDOMINANCE

To ease the exposition, form

POSSIBILITIES OPERATING

WITH

EITHER

IN EACH HABITAT

the analysis is done for the selection regime of the

genotypes

AA

Aa

fitnesses

si

1

aa 1-

u + olsi operating

in habitat Bi .

(B-1)

(a is a constant parameter, 0 < 01< 1.) The fitness values reflect heterozygote advantage in those habitats where si < 1 and heterozygote disadvantage where si > 1. The environmental fitness gradient depicts a melange of overdominance and underdominance selection effects. The results described below for the model (B.l) probably reflect the qualitative equilibrium behavior engendered

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by a general mixed underdominant-overdominant selection regime. In order to determine the equilibria apart from x* = 0, 1, and a/(1 + o(), we reduce the equation x =f(x) to the form N X-e= ( x-e

11

x(1

-

oI)(Si

-

1)

+

1 -

01 +

s&x

i=l ci six2 + 2X(1 - x) + (1 - 01+ a$)( 1 - X)2

and cancel x - (a/(1 + a)). Afterward combining the two sides and inserting 1 = CL, ci leads to

Introducing O=+L

the variable y = x2 + ~(1 - x)~, the polymorphic solutions solve 1 i=l y - ei = m+

ei=z,

i = 1, 2,..., N

P.2)

and ei are all either 31 or GO. The analysis of (B.2) paralleling (A.7) leads to the property that P(y) admits at most one solution 5, 0 < 9 < 1 and in terms of x there arise at most two x values solving x2 + ol(l - x)” = 9. The curve x2 + ol(l - x)” = U(X) has the shape:

i ET La

1

It is clear from the graph that y can give rise to either 0, 1, or 2 admissible x values. When a! = 1, U(X) is symmetric about +. By convexity l/(1 - (Y+ 01s~)< 1 - (Y+ (ol/sJ, leading to the relation g+ (cf. (20)).

implies

Q+Wi
382

SAMUEL

Recall that the conditions hand are

of stability

KARLIN

of the fixation

states for the model at

Stability

(instability)

for fixation

of allele A occurs if iii -f-” < 1 2

Stability

(instability)

for fixation

of allele a occurs if 5

Evaluation

off’(a/(a Stability

“=*

(> 1)

(B.4)

+ 1)) produces the fact that (instability)

of the polymorphic

occurs if 5 ci 2a’kTi i=l

state

1 (y. < 1

(>l).

2

Implication (B.3), in conjunction with (B.4) and (B.5), shows that where fixation of allele A is stable, the event of a-fixation is a fortiori stable. On the other hand, the inequality

is feasible, indicating that u-fixation is not. The following implication holds:

may be a stable outcome

while A-fixation

(cf. (21)), showing that where both fixation possibilities are stable, the equilibrium state x* = a/(1 + CX)IS unstable. In this circumstance, a/(~ + I) is the unique internal equilibrium and is unstable. The proof of (B.7) goes as follows: Let 6 = 1 - a + (ys or s = ([ - 1 + ol)/ti, 7 --z l/t. Define ~(7) = (2 $ (a - I) ~)/(l + 7~). Straightforward verification reveals that p)(7) is strictly convex and decreasing for 7 > 0 and ~(1) = 1. The left inequality of p.7) assertsN XL, civi < 1, ((1/7J = & := 1 - OL+ Q-Q). Therefore vJ(Z&l Ci7i) G Ld W(7i). It f o 11ows that where y .= CG, ~~7~ < 1 since q(y) > 1 for y < 1 we have 1 < Xf, 47J = Et:, cj((201si + I - a)/(q + 1)) and (B.7) is proved. All the ingredients embodied by the foregoing facts are now available to implement the validation of the statements of Table I.

FREQUENCY PATTERNS IN SUBDIVIDED POPULATION

APPENDIX

C: THE EQUILIBRIA PARTIALLY DOMINANT

POSSIBILITIES TRAIT

383

FOR A

The fitness scheme is that of (12). The equilibria values are the solutions of x -f(x). Combining in the usual manner reduces this equation to

Let U(X) = 2x - 1 + 2hx(l - x), which for / h 1 < 1 varies monotonely from - 1 to 1 as x traverses 0 to 1. Observe that 1di / < I entails j l/d, ! 2 1 and therefore P(U) = cf, (c,/((l/dJ + u)) = 0 admits at most one zero in u satisfying j u 1 < 1. Therefore, in terms of x the sum quantity in (C.l) can produce at most one solution. The factor 1 + h(1 - 2x) of (C.1) gives us a legitimate equilibrium (x* = (1 + h)/(2h)) if and only if j h 1 > I whose value is independent of the di . For h > 0 this equilibrium is stable (unstable) provided

i

I + di((kr$

1)/(2/z))

< 0

(9).

These inequalities are reversed if h < 0 and x* = (1 + /2)/(2/z) exists. In the partially dominant case, where h exceeds I, it is possible to have two stable equilibria, a fixation state plus a polymorphism.

ACKNOWLEDGMENTS I am pleased to convey my indebtedness to J. Roughgarden several constructive comments on the manuscript.

and M. Feldman

for

REFERENCES BERGER, E. 1976. Heterosis and the maintenance of enzyme polymorphism, Amer. Naturalist 1 IO, 823-839. CANNINGS, C. 1971. Natural selection at a multiallelic autosomal locus with multiple niches, J. Genet. 60, 255-259. CARMELLI, D., AND KARLIN, S. 1975. Some population genetic models combining artificial and natural selection pressures. I. One locus theory, Theor. Pop. Biol. 7, 94-122. CARMELLI, D., AND CAVALLI-SFORZA, L. 1976. Some models of population structure and evolution, Theor. Pop. Biol. 9, 329-359. CHRISTIANSEN, F. B. 1974. Sufficient conditions for protected polymorphism in a subdivided population, Amer. Naturalist 108, 157-166. CHRISTIANSEN, F. B., AND FELDMAN, M. W. 1975. Subdivided populations: A review of the one two locus deterministic theory, Theor. Pop. Biol. 7, 13-38.

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CONEY, C. C. 1975. An application of Wazenski’s method to a non-linear boundary value problem which arises in population genetics, J. Math. Bid, 2, 241-249. DEAKIN, M. A. B. 1966. Sufficient conditions for genetic polymorphism, Amer. Naturalist 100, 690-692. DEARIN, M. A. B. 1968. Genetic poiymorphism in a subdivided population, Amt. 1. Bid. Sci. 21, 165-168. ENDIXR, J. 1973. Gene flow and population subdivision, Science 197, 243-250. EYLAND, E. A. 1971. Moran’s island migration model, Genetics 69, 399-403. FELSENSTEIN, J. 1976. The theoretical population genetics of variable selection and migration, Ann. Rev. Genet. 10, 253-280. FLEMING, 11.. H. 1974. “A Nonlinear Parabolic Equation Arising from a SelectionMigration Model in Genetics,” Institut de Recherche en Informatique et Automatique Seminars Review. FLEI\IING, \y. H. 1975. A selection-migration model in population genetics, I. Math. Biol. 2,

219-233.

FLEXI~XG, W., AND Su, C. H. 1974. One-dimensional migration models in population genetics theory, Theor. Pop. Biol. 5, 431-449. GILLESPIE, J. H., AND LANGLEY, C. H. 1974. A general model to account for enzyme variation in natural populations, Genetics 76, 837-848. GILLESI~IE, J. H., AND LANGLEY, C. H. 1976. Multi locus behavior in random environments, I, Random Levene models, Genetics 82, 123-137. GLIDDEN, C., AND STROBECK, C. 1975. Necessary and sufficient conditions for multipleniche polymorphism in haploids, Amer. Naturalist 109, 233-235. HEDRIC.R, P. W. 1976. Genetic polymorphism in heterogeneous environments, Annzr. Rev. Ed. Syst. 7, l-33. HOPPENSTEADT, F. C. 1975. Analysis of a stable polymorphism arising in a selectionmigration model in population genetics, J. Math. Biol. 2, 235-240. JACQVARD, -1. 1974. “The Genetic Structure of Populations,” Springer-Verlag, Berlin. KARI.IN, S. 1976. Population subdivision and selection migration interaction, in “Population Genetics and Ecology” (S. Karlin and E. Nevo, Eds.), Academic Press, Ne\v York. KARI~IN, S. 1977. Protection for recessive and dominant traits in a subdivided population with general migration structure, Amer. N&u&t, to appear. KAHLIN, S. 1978. Some Classifications of selection migration structure I. Conditions for protected polymorphisms, to appear. KARLIN, S., .~ND FARKAS, S. 1978. Some multi-locus assortative mating models, Theor. Pop. Lid., to appear. KARLIS, S., AXD KENNET, Ii. S. 1977. Variable spatial selection with two stages of migrations and comparisons between different timings, Theor. Pop. Bid. 11, 386-409. KARI.IN, S., AND MCGREGOR, J. L. 1972. Application of methods of small parameters to multi-niche population genetic models, Theor. Pop. Bid. 3, 186-208. KARLIN, S., .+ND RIGHTER-DYN, N. 1976. Some theoretical analyses of migration selection interacting in a cline, in Pop. Genetics and Ecology, (ed. S. Karlin and E. Nevo), Academic Press. LEVENE, H. 1953. Genetic equilibrium when more than one ecological niche is available, Amer. ~;ltwulist 87, 331-333. MALI~OT, G. 19.50. Quelques schbmas probabilistes sur la variabilitb des populations naturelles, z4nn. Univ. Lyon, Sci. Sec. A 13, 37-60. MALBCOT, G. 1951. Un traitement stochastique des problitmes 1inCaires (mutation, linkage, migration) en G&Ctique de Population, Ann. Univ. Lyon Sci. Sec. A 14,, 79-l

18.

FREQUENCY

PATTERNS

IN

SUBDIVIDED

POPULATION

385

MALBCOT, G. 1959. Les modeles stochastiques en g&-&ique de population, Publ. Inst. Stat. Univ. Paris 8, 173-210. MALBCOT, G. 1967. Identical loci and relationship, Proc. Fifth Berkeley Symp. Math. Stat. Prob. 4, 317-332. MALBCOT, G. 1969. “The Mathematics of Heredity,” Freeman, San Francisco. MAY, R. M., ENDLER, J. A., AND MCMURTRIE, R. E. 1976. Gene frequency clines in the presence of selection opposed by gene flow, Amer. Naturalist 109, 659-676. MAYNARD SMITH, J. 1966. Sympatric speciation, Amer. Naturalist 100, 637-650. MORAN, P. A. P. 1959. The theory of some genetical effects of population subdivision, Aust. J. Biol. Sci. 12, 109-116. MORAN, P. A. P. 1962. “The Statistical Processes of Evolutionary Theory,” Clarendon Press, Oxford. NAGYLAKI, T. 1974. The decay of genetic variability in geographically structured populations, Proc. Nat. Acad. Sci. USA 71, 2932-2936. NAGUYLAKI, T. 1977. The relation between distant individuals in geographically structured populations, Math. Bio. Sci., in press. PROUT, T. 1968. Sufficient conditions for multiple niche polymorphism, Amer. Naturalist 102, 493-496. SLATKIN, M. 1973. Gene flow and selection in a cline, Genetics 75, 733-756. STROBECK, C. 1974. Sufficient conditions for polymorphism with multiple niches and a genera1 pattern of mating, Amer. Naturalist 108, 152-I 56.