Polymorphism with selection and genotype-dependent migration

J. Math. Biology 11,245-267 (1981) Journal of

Mathematical 131ology

�9 by Springer-Verlag 1981

Polymorphism with Selection and Genotype-Dependent Migration*

Michael Moody**

Department of Biophysics and Theoretical Biology, The University of Chicago, 920 E. 58th Street, Chicago, IL 60637, USA

Abstract. For a single autosomal locus with multiple alleles both an island and a multiple-niche model with discrete nonoverlapping generations are formulated for the maintenance of genetic variability. Both models incorporate viability selection in an arbitrary way and allow for genotypic differences in the pertinent migration structure. Random drift is ignored, and mating is at random. A global analysis is given for the island model in the neutral case. For a subdivided population, conditions are derived for the existence of a protected polymorphism, and the model is examined in some special two-niche cases. Of particular consideration is the loss of neutral alleles due solely to population regulation and genotype-dependent migration, and the possible existence of equilibrium clines without selection.

Key words: Poiaulation genetics - Evolution - Migration - Geographical variation - Habitat choice - Polymorphism

I. Introduction

Since motility of most organisms is mediated by some morphological manifestation such as wings, fins, and limbs, one intuitively suspects that genetic differences in migration rates should at the least occasionally exist. Wingless varieties of certain insect species come readily to mind as examples, albeit extreme. Perhaps due to technical difficulties associated with determining migration rates in natural populations, let alone simultaneously measuring gene frequencies, few instances of such genotype-dependence have been recorded. Dispersal studies of the field voles Microtus ochrogaster and Microtuspennsylvanicus by Meyers and Krebs (1971) and Krebs et al. (1973) have yielded plausible evidence for genotype-dependent migration rates with respect to certain plasma protein markers.

The elucidation of those mechanisms which alter the genetic composition of populations is one of the central tasks of evolutionary theory. It is of particular importance in this regard to ascertain the possible extent to which genotype-

* M. M. was supported by USPHS Pre-doctoral training grant No. GM 7197 to the University of Chicago; this work represents part of the author's Doctoral dissertation. ** Present address: University of Wisconsin-Madison, Laboratory of Genetics, 509 Genetics Building, Madison, WI 53706, USA

0303 - 6812/81/0011/0245/$04.60

246 M. Moody

dependent migration and population regulation alone can cause loss of genetic variability by elimination of alleles. Should this phenomenon occur, it would be of immediate empirical consequence: systematic loss of alleles in a pammictic geographically structured population could not then be attributed to natural selection without direct evidence.

That geographical variation in selection intensity, in conjunction with genotype-independent migration, can lead to stable monotone gene frequency configurations is well known (Haldane, 1948 ; Fisher, 1950; Slatkin, 1973; Fleming and Su, 1974; Fleming, 1975; May et al., 1975; Nagylaki, 1975, 1976, 1978; Fife and Peletier, 1977). Of some interest then is the possible mimicking of such clines by genotype-dependent migration when selection is absent. If such clines may in principle exist, the interpretation of empirical data on geographical variation of gene frequency must be done with care.

Although much recent attention has been given to migration and selection in the theoretical literature, essentially all existing models assume that migration is independent of genotype. Edwards (1963) and Parsons (1963) considered for two subpopulations some particular examples of genotype dependence, but their models impose a rather severe symmetry condition on the gene frequencies, and generally suffer from formulational and analytical inexactitude. There appears to have been no subsequent work on this problem. Since genetic variation in migration rates is likely to occur, perhaps frequently, in nature, it is some interest to assess the evolutionary consequences of this phenomenon more completely.

Towards this end, we will construct several deterministic models with discrete nonoverlapping generations for a single autosomal locus with multiple alleles in a monoecious population, incorporating in a general way the effects of natural selection, population regulation and genotype-dependent migration (also called selective migration). The first is an "island" model; selection acts only through differences in viability, and each generation an arbitrary but fixed fraction of each genotype is exchanged with a large population ("continent") in genetic equilibrium. In the diallelic case, conditions are derived for the maintenance of rare alleles, and a global analysis is presented in the absence of selection with only homozygote immigrants of a single type.

The second model pertains to populations that are divided into an arbitrary number of subpopulations, with random mating within each niche. Natural selection is manifested only in viability differences, and environmental diversity is reflected in different selection schemes for the various niches. Migration between subpopulations occurs each generation according to any fixed pattern, which may differ for distinct genotypes, and population regulation operates to stabilize the relative niche sizes. Adult migration with soft- and hard-selection (Christiansen, 1975) and juvenile dispersion are treated. With soft-selection (Wallace, 1968), the production of adults in each niche is prescribed and the relative deme sizes are unchanged by selection. For hard-selection, the number of zygotes is fixed, and the relative subpopulation sizes after selection are proportional to the mean fitnesses of the various niches. As Dempster (1955) has observed, soft-selection should be a good approximation if the population is subject to local regulation, whereas hard-selection would be more appropriate should be population be regulated as a whole.

Polymorphism with Selection and Genotype-Dependent Migration 247

For two alleles, conditions analogous to the standard results (Prout, 1968; Bulmer, 1972; Christiansen, 1974, 1975) for the existence of a protected polymorphism are derived and particular attention is given to the maintenance of rare neutral alleles. The model is considered for the special cases of symmetric migration and generalizations of the models of Levene (1953) and Edwards (1963). Numerical examples are discussed regarding the existence of equilibrium monotone clines with genotype-dependent migration in the absence of natural selection.

II. An Island Model

The classical island model is perhaps the simplest conceptual and analytical description of migration in natural populations and, as such, motivates the following generalization to selective migration. Consider a single locus in a diploid, monoecious, panmictic population with alleles Ai, i = 1, 2 , . . . , ordered genotypes A~Aj, and discrete nonoverlapping generations. Natural selection is present only insofar as there are differences in viability, and the population is of sufficient size as to make our deterministic treatment reasonable. Denote by Nij, N*, Nij**, and Nij, respectively the numbers of ordered AiAj genotypes in zygotes before selection, adults after selection, adults after migration, and zygotes after reproduction. The total number of individuals, and the frequencies of allele Ai and of ordered A~Aj genotypes at stage (.) will be written as N~'),PI0, and PI], respectively (- = , , , , * * , ' ) .

With viability wlj accorded to AiAj (wij = wjO, we have immediately in adults

N* = wi~Nij = wijplpjN, N* = #N, (1)

where ~ = ~,ij wi~pipj. We posit that a fraction aij of AiAj adults is removed each generation from our island by emigration, and is replaced from an independent source in equilibrium with constant frequencies P;j by a fraction flij of the adult population. Hence, after migration

N** = (1 - cqj)N* + filjN*Pij, (2)

N** = Z(1 - aij)N* + N*fi, (3) it

where ~ = ~. ij Pijflij. Dividing (2) by (3), summing overj and employing (1) with the identity Pi = ~,j Pij yields

Pi Z (1 - ~ij)wijpj + wfii'fli p' = p** = J , (4a)

i 2 (1 - o~ij)wijpipj -[- wfl ij

where

!Sifii = ~ flijPi2. (4b) J

Equations (4a) and (4b) comprise a system of difference equations completely specifying the evolution of the gene frequencies. As is trivially verified, setting ei~ = c~, flij = fl for all i, j recovers the standard genotype-independent model

248 M. Moody

(Nagylaki, 1977). The general system (4), though displaying the formal structure of the model, seems to be analytically intractable. Our results are thus confined to the diallelic case.

With two alleles we make the identifications p ----Pl,/3 -/31, and consider first the protection of allele A1. Should ]~ -- 0, (4) reduces to the usual constant-fitness selection model for a monoecious population with fitnesses (1 - a~)wij for AiAj. This is expected, for when fl = 0, the island is effectively uncoupled from the immigration source and (1 - c~,j)w~j is just the probability that an A~A~ zygote survives to reproduce on the island.

Henceforth we assume flii> O, i,j = 1,2. Thus, At and A2 are obviously protected if 0 </3 < 1, since they are introduced each generation via immigration. We therefore suppose that only A2A2 individuals appear among immigrants, i.e., /3 = 0. Equation (4) now reads

ullp 2 + ulzp(1 - p ) p' = -= f(p), (5a)

@11 -- 2V12 + V22)P 2 + 2@12 -- V22)P + 1)22

where (i, j = 1, 2)

ulj = (1 - ~ij)wlj, vii = (1 - c~ij + f122)wij. (5b)

That A2 cannot be lost is confirmed by noting that f(1) < 1. As p ~ 0, we obtain from (5)

p ' = (v22)-*u12p + O(p2), (6)

and thereby deduce v22 < ut2 as a sufficient condition for the protection of A1. Using (5b), this becomes

(1 - ~22 -~ f122)w22 < (1 - o~t2)wt2. (7)

Condition (7) agrees with our biological intuition: to protect A 1, the ratio of the number of AzA 2 individuals after selection and migration to the number of A2A2 zygotes must be less than the corresponding ratio for heterozygotes, when At is sufficiently rare (p ~ 0).

If there is no selection, or if A2 is dominant, we infer from (7) that f l22 < ~ 2 2 - - Cgt2 suffices to protect At, in contradistinction to the usual model where it is not protected (Nagylaki, 1977). Equation (7) also permits the inference that At is protected for arbitrary positive heterozygote fitness, provided selection for AzA2 is sufficiently weak, w22 ~ 0, or if the emigration rate of A2A2 is sufficiently large (ezz ~ 1) and its immigration rate (//22) is sufficiently small. That condition (7) is strengthened as the emigration rate ofheterozygotes increases and is weakened as the corresponding rate for the immigrant homozygote is increased, also agrees with our biological expectations.

Returning to Eq. (5), we note that f(p) is unaffected when all parameters are rescaled by the same constant factor; hence (5) is essentially a problem with four free parameters, Due to the consequent algebraic complexities associated with such a problem, the general model (5) has not been analyzed globally. We will present below a global analysis for the neutral case.

Suppose wij = w > 0 for all i,j, and put ?~j = 1 - ~ij. We suppose that not all AtA2 individuals emigrate, i.e., 7tz > 0. By the aforementioned scaling property,


we posit, wi thout loss of generali ty

~11 = 1 + s ,

where s, a / > - 1, and define

~12 = 1, ~22 = 1 + ~ (8a)

v =/~2/~12. (Sb)

The paramete r v is the rat io o f the immigra t ion rate of A2A2 to the p ropor t ion of heterozygotes who do not emigrate. Substi tut ing (8) into (5) and se t t i ng f (p ) = p yields the following quadrat ic equat ion for the equilibria

with solutions

9(P) -= (s + a )p 2 - (s + 2a)p + a + v = 0, (9)

/3+ = [2(s + a ) ] - a { s + 2a _+ Is 2 - 4v(s + a)]1/2}. (10)

Observe tha t (10) admits the correct limits for the case of additive emigra t ion rates (s + o- = 0;s , a ~ 0)

v /~+ = 1 - - , s > 0, ( l l a )

s

y /3_ = 1 - - , s < 0 , ( l l b )

s

as s + cr ---~ 0.

We fix s, o-, and study/3_+ as functions of v, subject to the restrictions that/~_+ be real and that 0 ~</~_+ ~< 1. When s = a = 0, Ap < 0 for all 0 < p < 1 (v > 0); therefore p(t) ~ 0 as t ~ ~ . We assume hencefor th that s and ~ are not bo th 0. I f s ~< 0, a > 0, g(P) has no roots in [0, 1] for any v > 0, and, by dint o f the relat ionships sgn Ap = - sgng(p ), g(1) = v > O,p( t )~ 0 as t ~ ~ . This is expected on biological grounds, as A 1 is at a d isadvantage bo th f rom the s tandpoint o f emigra t ion and immigrat ion.

F o r the remaining cases,/~+ and/~_ are displayed explicitly as functions of v in Fig. 1 ; the results are ob ta ined by a s t ra ight forward but tedious appl icat ion of the me thods used by Nagylak i (1977, pp. 1 2 6 - 129) for the usual model. The stability proper t ies o f the indicated equilibria were proved directly for s ~< 0, and indirectly for s > 0 by not ing tha t f ( p ) is in this case m o n o t o n e increasing.

In the figure, we m a k e use of the definitions

vl = s2/4(s + a), p+(v,) = 1 - [s/2(s + o')]; (12)

note tha t 1o-[ < Vx when s + a > 0. Certain quali tat ive features are immedia te f rom Fig. l a - d . When a > 0 ( A z A 2

emigrates at a lower rate than A~A2), Aa remains in the popula t ion only if a is sufficiently small (G < va), A,A1 also emigrates at a lower rate than A1A 2 (s > 0), and A1 is initially at high enough frequency [p(0) >/~_(v)] . I f heterozygotes emigrate more slowly than AzA 2 homozygo tes (a < 0) a globally stable po lymor - ph ism always exists for v ~< - a, independent ly of the sign of s. I f a < 0 and v exceeds - a, then A~ is main ta ined only if A~A~ is " f a v o r e d " sufficiently over

250 M. Moody

A" ~.p p.

s

m q:-O"

0 - 0 " I

[0) 0 " ~ 0 , S < - O r

0" S+O"

S, 0 - o r I

(b)o" < O, - o ' < s ~ - 2 o "

I I

I+ (3" s + o "

S / / -o- S v~ 0 0

(c) O'~ O, S > - 2 0 "

o . ~ ~ - * [ v 0 ................ $ + o "

$ s ='i

(d) o- > O, s > O

Fig. 1. The equilibrium and stability structure of the neutral island model with selective migration. S and U denote stability or instability, respectively, of the indicated equilibria

A1A2 (s > - 20-), v is not too large ( - a ~< v ~< vl) and A1 is initially at sufficiently high frequency, p(0) >/~_ (v).

For most parameter values, allele A1 is either present at high frequency or absent al together; it is rare only if A2A2 emigrates more than heterozygotes (a < 0), and v is slightly less than - a. This suggests, as for the genotype-independent model (Nagylaki, 1977), the improbabil i ty of maintaining rare alleles at many loci, thereby possibly limiting response to environmental changes.

Tha t the behavior of this model is qualitatively similar to the s tandard case is not surprising: f rom a genetical viewpoint, viability selection and selective emigration essentially serve the same purpose.

III. Subdivided Populations A. The General Model

a. Adult Migrat ion

We consider a single locus in a monoecious popula t ion with alleles Ai, i = 1, 2 , . . . and ordered genotypes AiAj. The popula t ion is assumed to be part i t ioned into K > 1 panmictic subpopulat ions (niches, demes), within which selection acts


only through viability differences. Generations are discrete and nonoverlapping; random drift is ignored. Denote by ~,+),~i,~,n(') _ij,~,p() and Nij,~,() respectively, the proportion of the total population in niche ~:, the frequency of allele Ai, the frequency of ordered A~Aj genotypes, and the number of ordered A~Aj genotypes, in niche ~c at the point of the life-cycle indicated by the superscript and defined by the following diagram:

selection migration regulation reproduction Zygotes --* Adults ~ Adults ~ Adults ~ Zygotes.

( ) (,) (**) (***) (')

We posit that population regulation does not change the genotypic frequencies in any niche, but may modify the relative subpopulation sizes. Hence, reproduction acts only to return the zygotic population to Hardy-Weinberg proportions each generation. As in the standard model, we may consider the case of zygotic regulation simply by interchanging the last two phases of the life cycle, with no consequent modification of our equations (Nagylaki, 1977).

To derive a system of difference equations for the p~,~, we first suppose that the fitness of genotype Av4 j in niche lc is w~j,~. It then follows for adults that

P * , K = P i , g P J , x W i j , x ( l ~ r ) - 1, (13)

where ~ = ~jp~,~pj,~wi~,~. Allowing for genotypic differences in migration, we posit that for each i, j there is a forward migration matrix (Bodmer and Cavalli- Sforza, 1968) 37/~j), such that an element rn ~j) of)l~ ~j) represents the probability that r2 an A~Aj individual originating in niche tc migrates to niche 2. With extremely large population numbers, as we suppose, me" r,,~Z'*"t~J) can be interpreted as proportions. The obvious normalization

~ (ij) ~ r n a = 1, (14) 2

holds for all i, j and K; by our ordering convention we have 3~r (i j) = 37/~J0. Since migration does not alter the total population size, we can consider the proportion, c'P* of AiAj genotypes in niche 2. Hence, after migration, 2 i j , 2 '

* * * * = Z (15) d r P i j , r .1, " 2

But Pi,~' =P***'i,~ , thus summing (15) over j and then i yields

C * * D t ~ : r ~ ( i j ) rc ~ i , r ~ / '~ t '2 z i j , 2 " " 2 r ' (16a)

(16b)

j 2

* * ~ - ~ * O * rTn(iJ) Cr ~ .7, t~2 Jt i j , 2 " ' ~ . r "

i j 2

Substituting (13) into the ratio of (16a) to (16b) leads to

2 ~ ~ C * W { ~ ; ~ - 1H~/(IJ) Fi,.~.//J,2 2 ij , AI, ).J 2~c

p, = jx . (17) i , r A.~ Z t P k , , ~ P l , z C * W k l , Z f f f ; 2 "~ - l r ~ ( k l ) ""2to

k l2

~ ( k l ) We assume that for all k, ! and lc, m~ > 0 for some 2, i.e., for each genotype every niche can be reached from some other niche.

252 M. M o o d y

If lff/~) = if/, independent of genotype, (17) is easily reduced to the standard model (Nagylaki, 1977). We remark that, in contrast to the genotype-independent case, no simpler formulation in terms of backward migration matrices (Nagylaki, 1977) appears to exist for this general problem.

Since the post-migration niche proportions are by (16b) generally dependent upon the gene frequencies, it is highly unlikely that c**(t) = e*(t) for all t. We therefore suppose that population regulation returns the niche distribution each generation to some constant e. With the niche distribution regulated to constancy, we need only stipulate the relationship between c* and c, in accordance with the nature of selection, to obtain from (21) a complete system of difference equations for the p~. We will consider only the cases of soft- or hard-selection, proceeding with the former, and derive protection conditions for the diallelic case.

1. Soft-Selection. For soft-selection, we specify that e* = c. In a diallelic population with alleles A1 and A2, put p~ = Pl,K, qK = 1 - p ~ , and define the fitness ratios

W12,r W12,r u~ = - - v~ --- (18)

W22, K ' W l l , x '

for w22,~, w11,~ r 0. Should A2A2 be lethal in every niche (w22,~ = 0), At, being present in every adult of each generation, cannot be lost. Similarly, if A 1A ~ is lethal in each niche, A2 cannot be lost. Henceforth we assume w~2,,cw11,,~ > 0 for all ~:. Expanding (17) in the norm IIPll (I[Pll = ~ P K , for example), we find

.~(12) ~p~uzc~rrtl K p' - x + O(llpll~), (19)

~'~,. /07/(22) ~'2 ),r

Z

as I IP[I --' O. Denote the diagonal matrix with elements (u~5~) by U. The matrix with elements

v~.(12) V ~ ~ ~v7,(2 2)-] - 1 "'~am(1) = ~z'"~K ]",~,'"n~ J (20)

will be written as M1. Note that MI is not generally a stochastic matrix. Equation (19) is now conveniently expressed to leading order as

P ' = Qlp, (21)

where Q1 = M1U. If 57/(i~ is independent of genotype, 57I (12) =/17/(11) =/17/~22), 3711 is just the usual backward migration matrix (Nagylaki, 1977).

We deduce at once from (21) a sufficient condition for the instability ofp = 0, namely, that Q1 have at least one eigenvalue exceeding one in modulus. If heterozygotes are lethal in each niche, Q1 = 0 and A~ is evidently unprotected; henceforth we assume wa2,~ > 0 for all x. As first observed by Nagylaki (1976, 1977), instability of the origin does not alone imply protection of A 1, since certain pathological behavior leading to the loss of A ~ may apriori occur. In particular, we must rule out the possibility of a transition to 0 in a single generation. That this is impossible is obvious from (21) and the positivity of the fitnesses. Likewise, p(t)


cannot approach 0 along any subsequence of time points, for then there would exist an accumulation point in the state space from which the system would converge to 0 in a single step. We appeal to the biological certainty of small perturbations to dismiss the possible convergence to 0 along the stable manifold corresponding to an eigenvalue of Q1 less than one in modulus. If Q1 is irreducible (Gantmacher, 1959), i.e., if there do not exist complimentary sets of integers land Jsuch that a!. ~) = 0 for i

e l . ]

in I andj in J, we invoke the argument of Nagylaki (1977, pp. 134 - 135) to preclude convergence to 0 from directions tending to such a stable manifold.

We suppose henceforth that ~/~12) is irreducible, which for positive U (u~ > 0 for all to) implies the irreducibility of QI ; consequently no pathological behavior can occur. Since Q~ is also non-negative, we infer from a theorem of Frobenius (Gantmacher, 1959) that Q~ has a real, positive, nondegenerate eigenvalue not exceeded in absolute value by any other eigenvalue, and to which corresponds a positive eigenvector. Denoting this eigenvalue by 2o(Q0, we then conclude that

2o(Q1) > 1 (22)

suffices for the protection of Aa. Reversing this inequality will imply that A1 is not protected. To protect A2, we define the matrix M2 by interchanging 1 and 2 in (20), the diagonal matrix V with elements (v~6~z), and stipulate that 2o(Q2) > 1, where Q2 = M2 V. Hence, a protected polymorphism will exist if min[2o(Q1), 2o(Q2)] >1 .

if either A~ is recessive or selection is absent, then (25) reduces to

2o(M1) > 1; (23)

A~ may be lost if this inequality is reversed. For dominant A~, •o(M2) > 1 suffices to protect A2, and for neutral genes a protected polymorphism exists if

min[-2o(M0, 2o(M2)] > 1. (24)

It is instructive to compare conditions (22) and (23) to the standard model. Without selective migration, A a will be protected if the leading eigenvalue of the matrix Q = M U exceeds one in modulus, where M is the (irreducible) backward migration matrix (Bulmer, 1972; Christiansen, 1974; Nagylaki, 1977). Should AI be recessive, Q = M and 2o(M) = 1 (Nagylaki, 1977); hence linear local analysis is insufficient to conclude protection and higher order terms must be considered (Karlin, 1977; Nagylaki, 1977). Furthermore, for neutral genes the standard model is completely linear, and it is easy to see that a protected polymorphism always exists (Nagylaki, 1979). However, with selective migration, M1 and M2 are not generally stochastic, so that conditions (23) and (24) are not generally vacuous.

It is perhaps surprising that 2o(M~) and 20(Mz) are not always at least as large as 1. Hence, in marked contrast to the standard model, p = 0 (and q = 0) can be locally asymptotically stable despite the absence of natural selection. This result may contradict one's biological intuition: since migration leaves the overall frequency of neutral genes unchanged, ~ c~p~ = V c**n** while regulation does not alter the gene frequency in any niche, one might naively expect that no gene initially present could be lost, as is true for the standard model. Thus, at least for some initial conditions, population regulation and selective migration can jointly cause elimination of genes from a population.

254 M. Moody

We discuss the possible loss of alleles in two part icular cases.

(i) Two Niches with Neut ra l Alleles Suppose the niches are equally populous , Cl = c2 = �89 and write the 37/tu) in the

fo rm

Y2 1 - - YZJ" Yl 1 '

M ~ 1 2 ) = ( 1 - w z 1-zW ) . (25)

Hence

(i -w t Mi = + ~ 1 + (26) w 1 '

-~i 1 - - ~ /

where ~i = Y~ - xi and 0 ~< x~, y/, w, z ~< 1, i = 1,2. Consider the possible loss of A 1. I f A2Az migrates symmetrical ly, ~i = 0, and we easily deduce tha t 2o(M1) = 1, so that stability of the origin is indeterminate by linear analysis. Hencefor th suppose ~1 ~ 0, and exclude the singular cases of complete AzAz homozygo te migra t ion to a given niche (~1 = +_ 1). Thus (26) yields

;b(M1) = [2(1 - ~ 2 ) ] - l { h + [72 - 4(1 - w - z)(1 - ~2)]~/2) (27a)

where

71 = (1 - w)(1 - ~ ) + (1 - z)(1 + ~,). (27b)

Allele A ~ will then be lost when initially sufficiently rare if 2o(M1) < 1, which f rom (27) necessitates h < 2(1 - ~ ) . This is true only if ~_ < ~l < ~+ where

0~• = 0~_+(W,Z) -~" �88 -- W ~ [-(Z -- W) 2 + 8(Z -~- W)]1/2}, (28)

and - 1 < ~_ < 0 < ~+ < 1. Fo r s u c h a n ~l, we deduce f rom(27) that 2o(M,) < 1 if, and only if, ~ [ ~ x - ( z - w ) ] < 0 . But it is readily verified tha t ~ _ < z - w < ~ + ; hence we infer that A1 is lost if ]~lL<Jz-wl and sgu ~ = sgn(z - w), which is equivalent to (see Fig. 2)

0 < [Yl - xil < Iz - wl, sgn(yi - x l ) = sgn(z - w). (29)

Allele A, is therefore lost when sufficiently rare if the difference in the migra t ion rates between the two niches for A2A2 homozygotes is of the same sign and smaller in absolute value than the corresponding difference for heterozygotes. By continuity, this will also be true for sufficiently weak selection in every niche. Hence A1 m a y be lost even when favored in each deme, u~ - 1 > 0 for all tc, provided these differences are sufficiently small and (29) is fulfilled. Similar analysis implies that A 2 is not protected if (29) is satisfied with 1 replaced by 2. Hence, if for i = 1,2

0 < maxlyl - xil < [z - wl, sgn(yl - x0 = sgu(z - w), (30) i


Fig. 2. Region of non-protection (shaded) for allele A1 in the neutral, two-niche, diallelic model with equal subpopulation sizes

Z - - W

�9 ,f I

/ /

' / /

o

- I

I �9 Y l - X ' i 1

then p = 0 and q = 0 are simultaneously asymptotically stable, in contrast to the standard model with recessive (Karlin, 1977) or neutral alleles.

Numerical analysis of this model with the general migration matrices (25) reveals quite complex behavior, even when complete exchanges between niches are excluded. Either allele can be lost globally for an appropriate choice of the migration matrices, though no clearly systematic pattern could be discerned. Unique, globally asymptotically stable polymorphisms were also observed, as well as unique unstable polymorphic equilibria, for which convergence to one fixation state or another always occurred, depending on the initial conditions. Both an unstable and a locally asymptotically stable polymorphic equilibrium can simultaneously exist; when this occurred, one of the fixation equilibria was always locally stable and the other unstable.

Due to the complexity of the recursion relations even in this case, no global analytical results for general migration patterns have been derived. We therefore present a particular example which bears upon the issue just discussed. Suppose only A2A2 individuals can go to or remain in deme 2:

j~r(11) = ~f(12)_~_ ( : ~) ' ]~/(22I = ( 1-Xy 1--yX ) , (31)

where 0 < x, y < 1. From (17) we infer that

t p~ =pl[xp~(1 - p ~ ) + 1 - x + y] 1, (32a)

P2 = 0, t ~> 1; (32b)

thus, after one generation there are no A~ genes in niche 2 and thereafter only AzA2 homozygotes are exchanged with niche 1. By the methods used in Nagylaki (1977, pp. 124- 129), we deduce that a unique, globally, asymptotically stable polymorphism exists in niche I if x > y, while x ~< y implies 91obal loss of A1.

256 M. Moody

(ii) Protection of a Recessive Allele in a Special Case We turn now to the case of recessive A1 when all niches are equally sized,

c l c2 "" cK 1 / K , and migration is d o u b l y ~ ~ ( u ) = = �9 = = s t o c h a s t i c , i.e., ~ , , . ~ = 1 for all k, l, ~:. Note that symmetric migration is doubly stochastic. With these assumptions, Q1 = )17/~2), and hence )~o(Q1) = )~o(~/~z)) = 1, since JlT~ ~2) is stochastic; thus (23) cannot obtain, and we must consider the quadratic terms. Expanding (17) to second order and recalling that wt2,~ = w22,~ for all ~c yields

p, = ~/(12)p + f(p), (33a)

where

fr(P) ~-~ F~71(11)" - 1 ff/(12)].2 ~-~ ,~7,(12)I-~(2 2) ~(12) L"'~ ~ ~ j~.~ + 2 (33b) = - - m ~ ] P A P ,

and v~ is given in (18). We further suppose 37/~12) to be aperiodic, which implies that all eigenvalues other than 20[~/(~2)] are less than 1 in modulus. Observing that V r = K-x(1, 1 , . . . , 1) is the normalized left eigenvector of ~(~2) corresponding to 1, the argument of Nagylaki (1977, pp. 138 - 139) applies straightforwardly to yield for the protection of A~,

K- 1 ~ v~- ~ > 1. (34) rr

Equation (34) is equivalent to 6 < 1, where v is the harmonic mean of the v~. This agrees with the corresponding condition for genotype-independent migration (Karlin, 1977; Nagylaki, 1977). In the absence of selection, the left-hand side of (34) is 1 ; consequently protection of a neutral allele with doubly stochastic migration cannot be determined to this order.

* c ~ ( # ) - 1 where 2 . H a r d - S e l e c t i o n . For hard-selection, we suppose c~ = = ~ c ~ , and derive from (17) for two alleles the recursion relations

p ' = J~ . ( 3 5 )

k12

Defining the matrix Rt with elements

,o) ~a~l 2,~,,,a~ (36) ~ C~lW22,~lHI~I r

(35) admits the linearization

p = R~p, (37)

as Ilpll --, 0. Hence A~ will be protected if 2o(R~) > 1, the technical considerations being as for soft-selection. Replacing 1 by 2 in equation (36) and denoting the resulting matrix by R2 yields the criterion 2o(R2) > 1 for the protection of A2.

We will compare hard- and soft-selection under the assumption that e, the wu,~, and the migration matrices are the same in both instances. Let the diagonal matrices D1 and D2 have respective elements

Polyrnorphism with Selection and Genotype-Dependent Migration 257

and write R1 in the form R1 = D1M1D2. Since DaMaD2 and M~D2DI have the same eigenvalues, we obtain the hard-selection protection criterion from the soft- selection condition (22) by the formal replacement of u~ by

#K ~ Wl2,x - ( 2 2 ) - ( 2 2 ) ,

completely analogous to the standard case (Nagylaki, 1977). As for genotype- independent migration, no general statement can be made about the relative stringency of the hard- and soft-selection protection conditions.

Observe from (36) that, in contrast to soft-selection, for hard-selection the linearized equations, and hence the protection conditions, generally depend explicitly on the fitnesses rather than just on the fitness ratios. Only if either the heterozygote or AzA2 homozygote fitnesses are the same in each niche will this not be so.

b. Juvenile Dispersion

If juveniles rather than adults disperse, the model requires reformulation. Let P*,~ denote the frequency of AiA; among post-dispersal zygotes of niche ~:. Since pre- dispersal zygotes are in Hardy-Weinberg equilibrium, we argue as in (15) and (16b) to infer

p.. = -(c#) . (39) ,1,~ pi,zpj,zc2mz~ pk,zpz,xcath ) I .-2 J I-kl~. J

Hence, after selection, regulation and reproduction,

* w P'i,~ = (ff~,,) - 1 ~ Pij,~ iJ,~, (40) J

p* where #, = Z i j ij,KWij,r. One sees at a glance that the model described by (39) and (40) is quite different

from that of (17). Note that with e regulated to constancy, the recursion relations, and hence the protection conditions, are independent of the nature of selection.

Consider the protection of allele A 1 in the diallelic case. As I lpll --, 0 (p~ - pl,~), we obtain from (39) and (40) the linearized system

p'= Tip, (41)

where T1 = UM~, and U and Ma are defined as for adult migration. Hence A1 will be protected for irreducible Tx if

2o(T1) > 1, (42)

the technical considerations being as before. To protect A2, we replace 7"1 by T2 = VM2 in (42).

Since the eigenvalues of UM1 (VM2) are the same as those of Ma U (M= V), we infer that 2o(Ti) = 2o(Q~), i = 1,2. Hence the protection conditions are the same for

258 M. Moody

juvenile dispersion as for adult migration with soft-selection. Note especially that the discussion pertaining to the loss of rare neutral alleles applies unaltered here. If all the niches are of equal size, A1 is recessive, and migration of each genotype is doubly stochastic, then T1 = AT~ 12), implying 20(T1) = 1. Expanding (39) and (40) to second order yields (33) with vz replaced by v~. If all other eigenvalues are less than 1 in modulus, we apply the argument of Nagylaki (1977, pp. 138- 139) and obtain the sufficient condition

< 1, (43)

for the protection of A 1. This is the same as (34) for adul t migration with soft- selection.

That the protection conditions for juvenile dispersion are the same as for soft- selection with adult migration agrees with the standard model (Nagylaki, 1977).

B. Special Cases

For the following special cases only adult migration will be considered.

a. The Levene Model

We posit that migration of each genotype is independent of the niche of origin. That ~(ij) = m~j) for non-negative constants m~ j) such that is, for each i, j, and K, ,,0~

~zm(] j) = 1. This is a generalization to selective migration of the model introduced by Levene (1953). For genotype-independent migration, the post-migration gene frequencies are the same in each deme, and the problem is univariate (Nagylaki, 1977). It is also true of the standard Levene model that migration conserves the subpopulation distribution, and therefore regulation is unnecessary (Nagylaki, 1977). However, with genotype-dependent migration rates, such simplifications are generally not available; the protection conditions, though, are obtained from (22) and (37) by invoking the special form of the )9/~u).

1. Soft-Selection. Defining m~ = ,~(12)/via(22) we infer from the definition of M1 that = _(i) Since Q1 has rank 1, its only nonzero eigenvalue m(1) c~m~. Hence v,,a = c~u~m~.

is tr Q1. Thus

czuzm~ > 1 (44) 2

suffices to protect A 1. A 1 is unprotected if this inequality is reversed. To protect A2, ~(12)/~(11) in (44). we replace uz by vz, and ma by #4 . . . . z ~,,~z

If migration is independent of genotype, mz = 1 for all 2 and (44) reduces to the usual result, ~7 = ~ z c~u~ > 1 (Levene, 1953). Observe that (44) can hold when

< 1 and can fail when t7 > 1. Hence protection with genotype dependence need not imply protection for the standard model with the same fitnesses and niche distribution and vice-versa.

We infer from (44) that allele A 1 will always be protected for fixed e and u if for some 2 migration of AzA2 homozygotes into deme 2 is sufficiently weak compared to heterozygotes, i.e., mz > (czuz)- 1 for some 2. For fixed niche distribution and


migration rates, A1 is also protected for sufficiently strong selection favoring heterozygotes in some niche, u~ > (cxma)-~ for some 2.

It is instructive to consider the loss of A t for two niches without selection. Let X = m]12) ,y = m(122) and z = m(111) s o that

ma = x/y, m2 = (1 - x)/(1 - y), (45)

where 0 ~< x ~< 1 and 0 < y < 1. By (44), ~ c z m a < 1 suffices for stability of 0; hence A1 is lost when sufficiently rare if (c = ca)

x 1 - x c - + (1 - c) < 1, (46a)

y 1 - y

which is equivalent to

c < y < x or c > y > x. (46b)

With c fixed, the region for which A 1 is unprotected is depicted in Fig. 3. A 1 is not protected if c < y < x, i.e., the proportion of AzA2 homozygotes entering a niche exceeds the size of that niche and is exceeded by the fraction of heterozygotes migrating to that same subpopulation. Since both x and y exceed c, and AIAa individuals are essentially absent for Aa rare, population regulation decreases the relative size of this deme (Nagylaki, 1977, p. 142), thereby diminishing more the frequency of AaA2 than AzA2 genotypes. I fc > y > x, i.e., the deme size exeeds the proportion of AzA2 immigrants, which exceeds that of immigrant heterozygotes, A 1 is also not protected. Population regulation now increases the relative size of the deme (Nagylaki, 1977, p. 142), and hence increases more the frequency of AeAz than A1A2 individuals. Thus, both conditions in (46b) are biologically sensible.

Observe that (46b) and (29) agree when c = �89 with the appropriate identifications. Allele A2 is not protected if (46) holds with y replaced by z.

If migration is doubly stochastic, which for the Levene model implies m (i j) = K- 1 for all i,j, to, all demes are of equal size and A1 is recessive, we infer from (34) and the remark following (44) relating the protection of A2 that a protected

Fig. 3. Region of non-protection for the neutral, diallelic two-niche Levene model with selective migration (shaded)

Y I

x

0

260 M. Moody

polymorphism exists if

g < 1 < ~, (47)

precisely the result for the standard Levene model (Prout, 1968). Since g ~< t5 with equality if, and only if, the vx are the same in each niche, fitness ratios satisfying (47) can certainly be found�9 Also note that (44) may be satisfied even without heterozygote superiority in the mean of the u~, i.e., even if ~ < 1. As opposed to the usual Levene model, such superiority is not sufficient for protection�9

2. H a r d - S e l e c t i o n . Either proceeding directly from (37) or from the correspondence between soft- and hard-selection mentioned following (38), we obtain

E c2m2w12, 2 > W22 (48) 2

for the protection of A~, where 1~22 ~--- Es C2W22,. ~. The interchanges Wz2 ~ w~ and mx --* #z in (48) yield a sufficient condition for protecting A z .

Comparing (44) and (48), we see that the protection conditions for hard- and soft-selection are identical if the A z A 2 homozygote fitnesses are the same in each niche. These conditions are not always equivalent: for example, if the heterozygote fitnesses are all the same, and the w2~ ~ are uncorrelated with the m~ when weighted by relative subpopulation size, protection with hard-selection implies protection with soft-selection but not necessarily the converse�9 That the hard-selection protection criterion is not always more stringent than for soft-selection can be seen by putting w12,~ = m~-lw22,~ and arguing as in Nagylaki (1977, p. 146).

b. Symmetric Migration

Throughout this and the subsequent subsections, we assume that all demes are equally populous and that selection is absent�9

If each genotype migrates symmetrically, we �9 ~ (is) = ,~(~s) reqmre m ~ ...~ for all i , j , ~, 2.

It is then immediate from (17) that if each allele is equally frequent in every deme the population is in equilibrium and the relative deme sizes are constant�9 Due to the analytical complexity of the general symmetric problem, we will hereafter restrict our attention to two diaUelic niches.

Let rh(~ 1) = 1 - x, ffl]~ z) = 1 - y, rh(t2~ 2) = 1 - z and define the other elements of the migration matrices by normalization and symmetry�9

We suppose that at least one of the homozygote types migrates, x + z > 0, and that neither homozygote migrates completely, x, z < 1. From (17) we then deduce the recursion relations for two niches�9

( y -- x ) ( p ~ -- p~) -- Y ( P l -- Pz) + Pl P'~ (2y -- x -- z ) ( p ~ -- p~) + 2(z -- Y)(Pl -- P2) + 1 ' (49a)

( y - x)(p - - y(p - p 0 +

P~ (2y - x - z)(p 2 - p z) + 2(z - Y ) ( P z - Pl) + 1 ' (49b)

observe that (49a) is obtained from (49b) with the interchange Px ~-~Pz, and vice- versa. One easily sees from (49) that p~ = Pz = P implies p] --- p~ = p.


Denote by L1 that portion of the line PI = P2 contained in [0, 1] x [0, 1], and put ? = 2y - x - z. The matrix of the linearized system corresponding to (49) about any equilibrium point (p,p) on L1 has eigenvalues 2o = 1 and 21 = 1 - 2 y + 4 7 p ( 1 - p ) with corresponding eigenvectors Vo = ( 1 1) and V1 = (1 - 1). It is easy to show 1211 < 1 for 0 < p < 1 and x + z > 0. We therefore suspect L1 to be locally stable in the sense that [pa(0),p2(0)] sufficiently close to L1 will imply [px( t ) ,pz ( t ) ] --* (p ,p ) for some 0 ~<p ~< 1 as t ~ oe.

To show this, let P1 = Px - P z , P2 = Pl q-P2, and infer from (49) that

e ' l = f(P1, e 2 ) e l , (50a)

where

1 - 2y + 7P2(2 - P2) f (P~ , P2) - 1 - { [ 2 ( z - y) + 7P2]P1} 2 ' (50b)

Expanding (50) through first order in P1 yields

P'I = [1 - 2y + P2(2 - Pz)]P1 + O ( P 2) (5l)

as P1 ~ 0. It is easy to see that the coefficient of the term linear in P1 in (51) is strictly less than ! in absolute value for all 0 < P2, x + y < 2. Hence P~(t) ~ 0 as t --* oe for PI(0) sufficiently small, and L1 is therefore locally asymptotically stable as a locus of equilibria. Since 1 - 2y + 7P2(2 - P2) may be strictly negative for all P2 (e.g., 2y > x + y > 1), convergence can be oscillatory.

I f 7 = 0 ("no dominance"), then [ f (P1,Pz)[ < 1 for all 0 <[PI[ < 1 and 0 < P2, x -1- y < 2. Hence Pl ( t ) ~ 0 as t ~ oo globally. After moderate effort, a global p roof of convergence to L1 for 7 # 0 remains to be given. Numerical evidence suggests that such convergence always occurs under the above conditions. In support of this conjecture we will prove convergence for the weak migration, continuous-time approximation to (49). To this end, we rescale time and the migration rates in the following manner:

z = fit, x = 62, y = 6~, z = 6~, (52a)

where 0 ~< 2,y, ~ ~< 6-1, 6 > 0 is a small parameter, and the gene frequencies in the scaled variables are

ffi(z) = pi(t), i = 1,2. (52b)

Substituting (52) into (49), expanding the right-hand side to 0(62) , dividing by 6 and passing to the limit 6 ~ 0 yields for the variables Pl = t 5 1 - / ~ 2 and P2 =/51 +/52 the coupled ordinary differential equations

~fil = [~P2(2 - P2) - 229]P1, (53a)

P2 = [2(29 - e) - ~P2]P~, (53b)

where ~ = 237 - g - ~, and the superior dot indicates differentiation with respect to "c. As for the discrete case, we asume migration of at least one of the homozygotes, g + g > 0, and then deduce f rom (53) that the only points of equilibrium are preci.sely those satisfying P , = 0, i.e., fil =/52. It is easily inferred from (53) that sgnP1 = - s g n P x for all 0 < P2 < 2; hence Pl(z) ~ 0 as z--* oe and the gene frequencies are ultimately equal in the two subpopulations.

262 M. Moody

c. Another Two-Niche Model

Let the migration matrices be given by

z 1 - z ' y 1 - y '

2f /~22'=( 1 - z x 1 - x Z ) . (54)

Thus, the model is invariant under the simultaneous interchange of demes and alleles. This is a generalization of the model of Edwards (1963), to which (54) reduces on setting x = y and z = w + y, for some 0 ~< w ~< 1 - y. Parsons (1963) described models similar to Edwards (1963), and included various forms of selection.

Employing (54) in (17), we obtain the recursion relations for the gene frequencies

( y - x)pZl + (z - y ) p ~ + Y(P2 - P l ) + P l

P'I 7(P~ - p2) + 2(z - Y)Pl + 2(y - x)p2 + x - z + 1' (55a)

(X l Y)P~ + ~ -- z)P~ + Y ( P l - P2) + P2 P~ y(p~ _ p 2 ) + 2(y - z ) p l + 2 ( x - y ) p 2 + z - x + 1 ' (55b)

where again V = 2y - x - z. Observe that (55) is invariant under the simultaneous interchanges x ~ z and

1 ~ 2. Also note that setting x = z reduces (55) to a special case of(49); henceforth, we assume x • z, and that neither homozygote migrates completely, x, z < 1.

Routine calculations using the particular form of the ~/~is) for this model reveal that a protected polymorphism always exists with our parameter restrictions.

I f the natural symmetry condition p~ + P2 = 1 is satisfied, (55) informs us that p'~ + p~ = 1. Thus, populations which are initially on L2 = {(Pl,P2)]Pa + P2 = 1, P~,P2 >1 0} remain on L2 and we have a univariate problem. In this case we put P = Pl and infer from (55)

p ' = p + z q 2 - x p 2 =- f ( p ) , (56)

where the frequency of A1 in the other niche is q. Given the special symmetry of the problem, it is not surprising that (56) is

independent of the heterozygote migration rate for populations confined to L2. Upon setting Ap = zq 2 - x p 2 = O, we deduce that there exists a unique

equilibrium/3 with 0 ~</3 ~< 1 given by

Z - - ( X Z ) 1/2

/3 = (57) Z - - X

I f x = 0, then/3 = 1 and Ap = z q z > 0 for all 0 ~< p < 1, thereby implying fixation of A1 in niche 1 and A2 in niche 2. When z = 0,/3 = 0 and Ap = - zp z < 0 for all 0 < p ~< 1 ; therefore A2 is fixed in deme 1 and A~ in deme 2. Henceforth we assume x, z > 0, and thus 0 </3 < 1. Observe that/3 ~ �89 as either x --* z for fixed z or vice


versa. This is expected s incep i = P2 - - �89 is the intersection point o f Li and Lz, and migra t ion is symmetr ic in the limit.

We will p rove that/~ is a global asymptot ica l ly stable equil ibrium of (56). Let r = (x - z)[1 - 2(xz) 1/z] - 1 and for 4xz # 1 define the variable ~ = r(p - /~ ) . F r o m (56) and (57) we deduce for ~ the recursion relat ion

if' = [1 - 2(xz)1/2]~(1 - 0 = 9(0, (58)

where fix ~< ~ ~< if2 and

{ x ~ (xz) i/z, r < 0 , (59) 1-1 - 2 ( x z ) i / 2 ] ~ i = (XZ) l/z, r > 0;

~2 is defined by interchanging x and z when explicit in (59). Since 0 < f ( p ) < 1 for all 0 ~< p ~< 1 when 0 < x, z < 1, we deduce f rom the

linearity of the t r ans fo rmat ion tha t ~l < 9 ( 0 < if2 whenever ~1 ~< ~ ~ ~z. Cor responding to b is the unique equil ibrium ~" = 0 contained in (~l, ~2), and the global stability o f p can therefore be inferred f rom that o f ~" = 0. To prove ~ = 0 is globally stable, we first write 9 ( 0 = h(0~ where h (0 = [1 - 2(xz)a/2](1 - ~) and then observe tha t [h(01 < 1 for all ffl ~ ~ ~< ~2 when 0 < x, z < 1. Globa l convergence then follows as a consequence of a result o f Nagylak i (1977, p. 92). Convergence is ul t imately m o n o t o n e for 4xz < 1 and oscil latory if 4xz > 1. Edwards ' (1963) condi t ion for the instabili ty of/~ can never be satisfied since it is inconsistent with the requi rement 0 ~< x ,y , z ~< 1.

Should 4xz = 1, we deduce f rom (57) for ~ = p - /~ , 0 ~< [~1 < 1, the recursion relat ion 4' = (z - x)~ 2. Thus r --+ 0 as t -~ 0% thereby implying p(t) --,

- 1 1 /~ = 2z(1 + 2z) , ~ ~< z ~< 1. I f z > x, convergence is mono tone , whereas for x > z it is oscillatory.

Due to the cer ta inty of small per tu rba t ions in biological populat ions , no gene- f requency t ra jectory is likely to satisfy the symmet ry condi t ionpa + Pz = 1 for all t. Thus, the convergence result derived above is o f limited utility; we will "s t rength- en" our result by prov ing tha t the equil ibrium O r = (#, 1 - / ~ ) is locally asymptot i - cally stable for the general system (55). To this end, let ~ = p - ~ where ~r = (el, e2) and/~ is defined in (57). As Ile[[ ~ 0, (55) admits the l inearization

e' = H~ (60a)

abou t the poin t ~, where

( h 2 ( x z ) i / 2 + h - 1 ) (60b) H = 2(xz)1/2 + h - 1 h

and h = 1 - y + 27/3(1 - /~ ) . The eigenvalues or H are

~1 = 2(XZ) 1/2 -I- 2h - 1, (61a)

22 = 1 - 2(xz) 1/2. (61b)

O b v i o u s l y [JL21 < 1 when 0 < x, z < 1. Analyzing separately the cases 7 < 0 and 7 t> 0 yields, after ra ther tedious algebraic manipula t ions , the inequali ty [211 < I for 0 < x, z < 1. Hence 0 is locally asymptot ica l ly stable as an equil ibrium o f the general system.

264 M . M o o d y

Numerical analysis of (55) suggests that ~ is globally stable and, furthermore, is unique. In support of this claim, at least for the case ? = 0, global convergence to the equilibrium ~ in the continuous time approximation can be established.

Though rather severely constrained by the specific nature of migration, this model evidences certain features with perhaps important biological consequences. Specifically, since the locally stable equilibrium ~ derived from (68) can coincide with any point of L2, for appropriate x and z, selective migration can create and maintain an arbitrary amount of genetic divergence between the two niches. This behavior disposes us to consider whether significant amounts of spatial genetic diversity can exist in habitats with many demes.

Equation (17) was analyzed numerically in the diallelic case with equal niche sizes, no selection, and nearest-neighbor migration. Stable monotone clines were observed with asymmetric migration for any number of niches, and considerable spatial variation in gene frequency could be produced even with very low dispersal rates and moderate asymmetry. Though the numerical study was not sufficiently extensive as to completely characterize all the possible equilibrium profiles, it was abundantly clear that selective migration can establish and maintain geographical variation in gene frequency for neutral alleles. Since such regional differentiation is integral to the process of speciation, this is of some biological consequence.

An immediate empirical implication is that natural selection cannot properly be ascribed as the causative factor for an observed monotone cline without direct evidence.

p(K) i.o

o.G IK 31 to)

p(,r ) I.o.

o.o

(b) 31

p(~) I.O =

51 (c)

p(,~)

(d)

F i g . 4. E q u i l i b r i u m c l ines w i t h a s y m m e t r i c se l ec t ive m i g r a t i o n . P a r a m e t e r v a l u e s as p e r (62) : 31 d e m e s ,

(a) x l = w l = 0.1, Yl = z l = 0 .05, x2 = Y2 = w2 = z2 = 0 .025 , x3 = wa = 0 .05, Y3 = za = 0.1, (b)

x l = wl = 0.1, Yl = zx = 0 .05 , x2 = w~ = 0 .07375, Y z = z 2 = 0.07625, x3 = w3 = 0 .025 ,

Y3 = z3 = 0 .125, (e) x l = wx = 0 .15, Yl = z l = 0 .05, x2 = w2 = 0 .1125 , Y2 = z2 = 0 .0875 ,

x 3 = w 3 = 0 : 0 2 5 , Y a = Z 3 = 0 . 1 2 5 , (ti) x l = w l = 0 . 1 5 , Y l = Z l = 0 . 0 5 , x 2 = 0 . 1 1 2 5 , y 2 = 0 . 0 8 7 5 ,

w2 = 0 .05, z2 = 0.1, x3 = 0 .075, Y3 = 0 .125 , w 3 = 0 .025 , za = 0.1


Figure 4 contains numerical examples of equilibrium clines with selective migration. For the figure, migration is between nearest neighbors according to the following scheme (D. = deme .):

X i x i $r Wi ',Vi Wi

D , ~ D a " " ~ - ' " D K - , ~ - D , ~ - D ~ + I " ' " ~ ' " D a ~ - D 2 ~ + I (62) Yi Yi Yi Zi Z i Zi

where xi, yl, etc., i = 1,2, 3, are the probabilities of a one-step displacement in the indicated direction for genotype i (1 = A1Aa, 2 = A1A2, 3 = AzA2). The probabilities of remaining in a given deme are determined by the obvious normalizations, and 0 ~< x~ + y~, w~ + zi, wi + yl ~< 1, i = 1, 2, 3. The migration pattern in (62) was chosen so as to allow internal geographical inhomogeneity of the dispersal rates.

IV. Concluding Remarks

We have considered various models with discrete, nonoverlapping generations for the effect of genotype-dependent migration on the evolution of natural populations. An island model incorporating natural selection and genotype dependent emigration and immigrating was constructed. With only homozygote immigrants of a fixed type, a condition was derived for the maintenance of rare alleles, and a global analysis given in the neutral case. It was concluded that selective migration alone would be unlikely to maintain alleles at low frequency.

For a subdivided population with an arbitrary number of demes, a general model incorporating population regulation, selection and genotype-dependent migration was presented. Adult migration with soft- and hard-selectio~a and juvenile dispersion were treated; conditions for the existence of a protected polymorphism were derived for two alleles. The protection conditions for juvenile dispersion were shown to be equivalent to those for adult migration with soft selection.

By means of two-niche examples it was demonstrated that elimination of neutral alleles can occur due solely to selective migration and population regulation. Conditions implying local stability of the fixation equilibria with two alleles were given for the neutral two-niche model, and in a special case global loss of an allele was proved analytically. If selection is sufficiently weak, population regulation and genotype-dependent migration can cause loss of alMes which are favored throughout the habitat; alleles which are uniformly but sufficiently weakly deleterious may be maintained by the joint action of these forces.

A generalization of the Levene model in which each genotype migrates independently of the niche of origin was formulated. With two alleles, an explicit protection criterion was derived and discussed with regard to the loss of neutral alleles.

The general multi-deme model was examined for two equally sized niches with no selection in the special cases of symmetric migration and a generalization of Edwards' (1963) scheme. With symmetric migration, if each alMe is equally frequent in every deme, the population is in equilibrium. In the diallelic two-niche case, populations initially close to gene-frequency equality between the two demes were shown to ultimately converge to such an equilibrium. Global convergence to a

266 M. Moody

geograph ica l ly u n i f o r m gene f r equency d i s t r i b u t i o n was p roved for two niches in the c o n t i n u o u s - t i m e a p p r o x i m a t i o n .

A diallel ic two-n iche m o d e l i n v a r i a n t u n d e r the s i m u l t a n e o u s i n t e r change o f demes a n d alleles was also presented . This example general izes the m o d e l o f E d w a r d s (1963). I t was s h o w n tha t a local ly a sympto t i ca l ly s table p o l y m o r p h i c e q u i l i b r i u m exists. A t e q u i l i b r i u m the s u m o f the f requencies of an allele in the two demes is one . I t was d e m o n s t r a t e d tha t a n a rb i t r a ry a m o u n t o f genet ic d ivergence be tween the two n iches c an occur even to the ex ten t o f comple t e f iss ion by f ixa t ion o f one allele in a n iche a n d the o ther allele in the c o m p l i m e n t a r y niche.

M o n o t o n e cl ines were s h o w n n u m e r i c a l l y to exist wi th neu t r a l genes a n d a symmet r i c g e n o t y p e - d e p e n d e n t m i g r a t i o n . Such cl ines c an occur wi th b io logica l ly r e a s o n a b l e m i g r a t i o n rates. Th i s impl ies t ha t m o n o t o n e cl ines do n o t per se d e m o n s t r a t e selection.

Acknowledgement. The author wishes to thank Professor Thomas Nagylaki for his careful reading of the manuscript and for many helpful suggestions throughout the progress of the work.

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Received May 28/Revised October 8, 1980

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Polymorphism with selection and genotype-dependent migration