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Polymorphic equilibria with assortative mating and selection in subdivided populations
THRORBTICAL
POPULATION
BIOLOGY
Polymorphic
94-111
18,
Equilibria
and Selection
of
Mathematics, of Mathematics,
with
Assortative
in Subdivided R. B.
Department Department
(1980)
Mating
Populations*
CAMPBELL
Stanford Purdue
University, University,
Received
April
Stanford, California 94305, and West Lafayette, Indiana 47907
18,
1980
Two modes of assortative mating, partial selling and assorting by phenotypic classes, are studied in a subdivided population. Differential viability is allowed and the selection intensities and assorting tendencies are permitted to vary among the habitats. There exists a symmetric polymorphism; we delimit its level of heterozygosity and stability nature (dependent on the selection intensities and assorting propensities). This complements studies of the fixation states and thereby provides further insight into the global equilibrium structure in subdivided populations. Circumstances are given where the fixation states and symmetric polymorphism comprise the global equilibrium structure. Examples are also given stable polymorphic migration.
where
migration equilibrium
engenders cycles which
stable polymorphic are absent in single
equilibria demes
and without
Although random mating (which produces Hardy-Weinberg frequencies in gametes) appears to govern many genetic systems, nonrandom union of gametes (which produces nonrandom associations of alleles in zygotes at all or part of the genome), perhaps based on similar or dissimilar parental phenotype (assortative mating), is also well documented. One form of assortative mating is selfing which occurs in many plants; self-sterility mechanisms and host plant selection by pollenating vectors will cause different manifestations of assortative mating in plants. The ability of animals, including humans, to select mates actively allows a myriad of assorting schemes. Of course, various ecological parameters (e.g., density of the species, density and species (race) of pollenating vectors, wind) may affect the assorting mechanisms and tendencies. We investigate the effect of spatial variation in assorting tendencies on the genetic structure of populations. Nonrandom mating is one of the original threads of population genetics research (e.g., Wright, 1921). Several models of assortative mating are presented in extensive papers by Rarlin (1968, 1978); see also O’Donald (1960, 1977), * Supported
in part
by NIH
Grant
GM
10452-
94 0040-5809/80/040094-18$02.00/O Copyright All rights
8 1980 by Academic Press, Inc. of reproduction in any form reserved.
16 and
NSF
Grant
MCS
76-80624-AOl.
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95
Parsons (1962) Crow and Felsenstein (1968), Ghai (1974) and Matessi and Scudo (1975). These models have generally been analyzed in the context of a single deme, but the multideme case has not been ignored (e.g., Karlin and McGregor, 1972; Moody, 1979). In this work we concentrate on extending some single-deme analyses to the multideme context. The global equilibrium structure for multideme systems is in general difficult to ascertain. One facet which is tractable is “protection” or the stability nature of the fixation states. This has been analyzed for partial selfing by Moody (1979) and other mating systems can be treated analogously. We complement these analyses with the study of the stability and genotype frequencies of certain polymorphic equilibria. In order to study the stability of equilibria it is necessary to characterize them. To this end, we assume symmetric mating and selection patterns. In particular, we consider only one locus with two alleles under the assumption that both homozygotes have the same viability and same assorting propensity. Thus, we are not considering the general case, but the results provide insight into the effect of selection-assortative mating-migration interaction on polymorphic equilibria because the results are structurally stable permitting limited qualitative inferences. We also restrict our attention to the standard model of selection migration interaction which has been dubbed soft selection by Christiansen (1975). This model posits that deme sizes are regulated prior to migration which allows the construction of a constant backward migration matrix (Bodmer and CavalliSforza, 1968) for describing the averaging of allele frequencies due to migration. Although the class of symmetric equilibria studied here will exist under, e.g., hard selection (Dempster, 1955; Christiansen, 1975), the explicit characterization and stability nature of the equilibria will differ from those for the case we are considering.
THE
MODEL
We have chosen from among the many models of assortative mating with selection two which are particularly suited to the study of polymorphic equilibria in multideme systems. These describe the change in allele frequencies due to mating, reproduction, and viability selection within demes. The effect of migration on the allele frequencies is addressed presently. A crux of our models is the division of each genotypic class into individuals which mate assortatively and individuals which mate at random in accord with specified proportions. This can also be posed as the probability that an individual will mate assortatively versus mate at random. These tendencies are constants which depend on the genotype and deme, but are independent of allele frequencies or densities.
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The first model (1) manifests assorting based on phenotypic classes. Specifically, assorting homozygotes mate- only with homozygotes, but do not discriminate between the two homozygous types (AA and aa). In the case of a single deme the global equilibrium structure entailing up to five equilibria is fully delimited by Karlin and Farkash (1978) along with a brief discussion of the biological context of the model. The equilibrium structure is dependent on (Y, /3, and s. The second (2) corresponds to partial selfing as characterizes many plant species; Homozygotes which assort mate only with their own genotype, but we assume the same fraction of each homozygous type assorts. This model has been studied by Karlin (1968, 1978) and Moody (1979) among others. Explicitly, the allele frequency transformation equations are (1 +s&;[$-]
-+j
6 = f(w, 2) = 1+&[&]
-+L2] (1)
(1 -B)V;+aZ+(l
-,)[(I
-++;a]
1 - g(w 2) = 1 +.s[&[&]
+%a]
’
for assortative mating and
- w>+ ; ((1- w)”- 2q + ;] i - w)’- 22,]+ ;] ’ -4 +jcu
(1 + 4 [Cl - 4 [&(l i 6 = f(w, 2) = 1 + s [(l - 4 [;.(I
- w) +-2w1 2 1 -- 4 + #
m + (1- 4 [au 2 = g(e), 2) = 1 + s [(I - a) [&(l
(2) I
+ (1 -+ --
4 2- 2’)] + ;] ’
for partial selfing, where w = frequency of Aa heterozygotes z =.frequency of AA homozygotes 1 f s = heterozygote (Aa) viability
minus frequency of aa homozygotes
1 = homozygote (AA or aa) viability (Y = fraction of homozygotes (AA or aa) which mate assortatively females which discriminate in choosing a mate)
(e.g.,
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1 - 01 = fraction of homozygotes which randomly mate (e.g., females; infinite males are assumed so that there is no change in the composition of the mating pool during mating) /3 = fraction of heterozygotes heterozygotes)
which
mate assortatively
(i.e., only with
I - p = fraction of heterozygotes which randomly mate. Thus, f and g specify the change in allele frequencies due to assortative mating, reproduction, and selection and in the case of a single deme the tilde (“) designates the next generation. These single-deme transformations are generalized to the multideme context in the same manner as when random mating is acting (e.g., Christiansen, 1974; Karlin, 1976). All parameters (‘u, Z, s, 01,/3) and the transformations (f, g) are subscripted to indicate to which of the n demes (habitats) they refer and are formed into n-vectors in the natural manner. Migration is specified by the constant n x 11 backward migration matrix M which is stochastic. The backward migration matrix specifies with its zjth entry the fraction of the ith deme which immigrated from the jth deme (Bodmer and Cavalli-Sforza, 1968). It is calculated from the forward migration matrix (which specifies with its zjth entry the fraction of the ith deme which migrates to the jth deme) and the deme sizes preceding migration (cf., e.g., Christiansen, 1975; Karlin, 1976; Nagyalaki, 1977). The assumption that deme sizes are fixed preceding migration allows that the backward migration matrix is constant. We assume a life history of mating, reproduction, and viability selection within demes (habitats) followed by migration among the demes. The change in genotype frequencies in a generation is given by v’ = MC = Mf(v, z),
z’ = MCi = Mg(v, z)
(3)
or in components n 0:
=
2
j=l
tQjfj(Vj
9 Zj),
&
=
1 j=l
??lijgj(Vj
, X;),
where the prime (‘) denotes the subsequent generation (i.e., we choose as our census time that point in the life cycle following migration but preceding mating so that the prime reflects the sequential action of mating-reproduction, viability selection, and migration on the unprimed parameters in accordance with the life history given above) and the tilde (“) specifies the population state after mating-reproduction and viability selection but preceding migration. We only consider the cases where assorting by phenotypic classes is manifested in all the habitats (i.e., fi and gi are of the form (1) for all ;) or partial selfing is occurring in all habitats (i.e., fi and gi are of the form (2) for all i). Assorting propensities and relative viabilities are assumed constant with respect to time.
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CAMPBELL
PROBLEM
As we stated in the introduction our objective is to analyze a polymorphic equilibrium (i.e., determine its level of heterozygosity and stability nature dependent on the parameters) as a complement to the available analyses of fixation states. In the event that a polymorphic equilibrium is unstable it cannot be expected to occur in natural populations, but is of interest because it helps define the boundaries of the domains of attraction of stable equilibria. If it is stable it may occur in natural populations and the level of heterozygosity is of interest because it is one of the standard statistics which are employed to characterize natural populations. Inspection of (1) and (2) sh ows that f = 0 if z = 0 and it follows by (3) that z’ = 0 if z = 0 for either phenotypic assorting (1) or partial selfing (2). .This suggests looking for an equilibrium with z = 0 (which implies polymorphism). We demonstrate existence of such a symmetric equilibrium, show that there is only one equilibrium characterized by z = 0 (quite generally), and then delimit its stability nature and level of heterozygosity dependent on the parameter values. Phenotypic Assorting Consider first the case where the local genotype frequency transformations are of the form (1). The existence and uniqueness of an equilibrium with z = 0 follows by its explicit construction. If z = 0, (1) simplifies to bi = fi(vi,
0) = t: z ziz 2,
,
i = 1,2 ,..., n
(which lies between 0 and 1). Hence v’ = IM+ is independent of v. Substituting (4) into (3) provides an explicit expression for the equilibrium oi
=ktnij(*),
i=
1,2,...,n,
which shows that it exists and is unique. Stability The population is described by the 2n-vector y = (wl , w2 ,..., v, , z1 ,..., zJ. For notational convenience we shall designate the subsequent generation with Y = (v; , w; ,..., v:, , z; ,..., z;). It is a standard result that stability of an equilibrium depends on the spectral radius of the 2n x 2n gradient matrix J = !I aY,/ay, (1 evaluated at that equilibrium (all perturbations in v and z are admissible at z = 0 and 0 < v < 1). This matrix can be evaluated using
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(3) producing a composite of four n x n matrices of the form (z 3, where Q = Mdiag[af,/&,],
b = M diag[af,j%,],
c = M diag[agi/&+],
d = M diag[8giiazi].
(5)
(diag[.] designatesthe matrix with the indicated entries on the diagonal and O’s off the diagonal.) Taking the partial derivatives of (1) and evaluating at zi = 0 yields
Thus, stability of the symmetric equilibrium is dependent on p(M diag[ag,/&,]) or explicitly p (M diag [ ’ ’ (~i~[$(ui’2)
])
where p(.) designatesthe spectral radius. Recall that the explicit construction of v from (4) and (3) enablesthe evaluation of (7). Thus, stability only depends on the spectral radius of one 12x n matrix which we can explicitly display rather than a 2n x 2n matrix which the dimension of the perturbation space would suggestor two II x 7tmatriceswhich a nice eigenstructurein the perturbation spacecould provide. Dependence on the Parameters
We note from (4) that 4 hence v is independent of a and P (if z = 0) but increasing in s. Thus increasedselection favoring the heterozygotes increases the heterozygote fraction acrossthe population range as occurs in a single demc. However, we can conclude from (7) that if s is fixed the stability of the symmetric equilibrium increaseswith 13and decreaseswith a in the sense that decreasing(7) may causestability if it becomeslessthan one; if it is further decreasedbeyond one, random fluctuations will be more rapidly eliminated by deterministic forces providing greater robustness against perturbations (i.e., the greater the tendency for heterozygotes to assort as compared to homozygotesthe better the prospectsfor stability of the symmetric equilibrium). If l3 > a the fact that v increaseswith s implies that increasing the selection intensity favoring the heterozygote increasesthe prospects for stability of the symmetric equilibrium along with increasing its inherent heterozygosity. EXAMPLE 1. We specialize to the casesi = 0 for all i entailing no viability selection. Then the criterion for stability is
p (Mdiag [l -
(” 4 “$)I)
< 1.
loo
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CAMPBELL
Remark 1. The term “no selection” for the casesi = 0 for all i is in a sensea misnomer. Unless a s P the allele frequencies are not in general con-. served. In fact, one has protection with global convergence to the symmetric equilibrium if f3 > a and both fixation states are stable concordant with an unstable symmetric equilibrium if IX > p. An heuristic argument states that the classwhich assortsless (e.g., heterozygotes if p < a) are taken as mates lessthan the other class.Therefore, whichever alleleis more prevalent (compared to l/2, the frequency in heterozygotes) in the homozygous class is favored. (However, if two demesare present polymorphism can be maintained analogous to the case of disruptive selection (Karlin and McGregor, 1972; Bazykin, 1972).) (If a > @equal frequencies are favored since heterozygotes have equal frequenciesof both alleles.) Partial
Selfng
When the local transformation equations are of the form (2) existence of a symmetric equilibrium is most easily demonstrated by considering (2) at a = 0. Since this maps the closed unit interval (in V) into itself continuously, (3) maps the closed unit hypercube into itself continuously guaranteeing the existence of a fixed point. We demonstrate the uniquenessof this fixed point in two cases. Casei. - 1 < si < 1 for all i. This parameter range encompassesmild selection, in particular when the heterozygote is never more than twice as viable as the homozygote. (Extreme selection favoring the homozygotes is allowed, i.e., the heterozygote can be lethal.) The constraint - 1 < s < 1 precludes an equilibrium involving heterozygotes in a single deme when homozygotes are complete selfers (a = 1). Recall that assortative mating precludes considering just whether heterozygotes or homozygotes are more viable (S > 0 or s < 0) for deciding stability of equilibria, but the assorting tendencies must also be considered. For this casewe consider
(1 + 4;
af av’ [
1 fs
++]
1-w
2
at z=O.
(9
(
This lies between0 and 1 for a! and w between0 and 1 and - 1 < s < 1. (The necessarycalculations are outlined in the Appendix.) The fact 1afi/aoi 1< 1 for all i suffices to demonstrate uniqueness.(We employ the monotonicity of f in order to give a more elegant proof of uniquenessin the Appendix.) Case ii. si > 0 for all i. This caserequires that the heterozygote be more viable than the homozygotes in all habitats. However, this doesnot necessarily imply that the symmetric equilibrium is stable becausethe “overdominant”
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selection can be countered by the assorting pattern. Extreme selection entailing near lethality of the homozygotes is included in this case. Note that the circumstances of this case overlap case i, but environmental heterogeneity manifesting near lethality of homozygotes in some environments and near lethality of heterozygotes in other environments is not included in either of the two cases. We prove uniqueness in this case by noting from (9) that f is monotone increasing in v and the second derivative
+ s)(G)(T)
-31
ay -=
(10)
iiV2
l+s
;-aly2))3
i
at
z="
(
is negative which provides that f is concave in v on the interval (0, 1) for s > 0, 0 < (Y < 1. In conjunction with the fact that f(0, 0) > 0 this implies that there is at most one equilibrium with z = 0. We outline a proof of this implication in the Appendix. A noteworthy part of the proof shows that ~(Mdiag[af/av~]) < 1 under the hypotheses of case ii. This is useful in the stability analysis of the symmetric equilibrium. Stability The stability of the symmetric equilibrium is dependent on the spectral radius of the matrix J in (5) as in the phenotypic assorting case. However, under partial selfing we only have (cf. (6)) afi G
--1 -
Qi &Ii
0
for all i,
and
p (Mdiag
and hence p(J) < 1 is equivalent to p (Mdiag
[$$-I)
< 1
[%I)
< 1
(12)
because these matrices lie on the diagonal. Thus stability depends on the spectral radii of two n x n matrices rather than one 2n x 2n matrix as one would a priori expect. We have remarked above that under the hypotheses of cases i and ii (and in fact whenever there is a unique symmetric equilibrium) the first inequality in (12) is necessarily satisfied. Hence the criterion for stability of the symmetric equilibrium is quite generally
-=c 1, II
(13)
which
depends on one n
x
n matrix as in the phenotypic assorting case.
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Dependence on the Parameters The stability nature of the symmetric equilibrium is readily ascertained in the following cases ensuing from the single-deme analysis by the Frobenius theory of positive matrices which provides that if a matrix is componentwise greater than a positive matrix, then its spectral radius is also greater, Recall that the spectral radius of a stochastic matrix (i.e., M) is one. Case i. s = 0 (no viability selection). The symmetric equilibrium is stable if pi > ai for all i and unstable if ,& < ai for all i. If oli = pi for all i the situation is quite neutral. Case ii. s > 0, p > a (overdominant viability selection, heterozygotes have a greater propensity to self than homozygotes in all habitats). Under these circumstances the symmetric equilibrium is stable and in fact globally attracting. (The latter fact can be demonstrated by showing / I 1 < ( z 1 with equality only if z = 0.) Case iii. s < 0, f3 < a (underdominant viability selection, homozygotes have a greater propensity to self than heterozygotes in all habitats). The symmetric equilibrium is unstable and the fixation states are stable, but there may be many more polymorphic equilibria depending on the migration structure. However, if z > 0 or z < 0 the multideme population will converge to z = 1 or z = -1, respectively (i.e., fixation of the A allele and a allele, respectively), precluding the existence of any nonfixation equilibria with z > 0 or z < 0. (This can be demonstrated by showing 1f / > ( z 1 in (2) for this case.) Aside from the above cases where we can delineate the nature of the equilibrium, we make some general comments on the influence of the parameters on the symmetric equilibrium. p: The equilibrium heterozygote frequencies v (level of heterozygosity) are independent of the selfing propensities among heterozygotes p. This entails that P(M diag[@J%J) is independent of f3, but P(M diag[%gi/3zi]) decreases with p and hence the prospects for stability of the symmetric equilibrium increase with f3. a: The equilibrium heterozygote frequencies v (level of heterozygosity) decrease as a increases. This provides that P(M diag[afi/L&]) increases with a, but if there are never multiple symmetric equilibria this quantity is always less than one and its dependence on a is of limited interest. Unfortunately, p(Mdiag[agJ&]) is not monotone in a. s: The heterozygote frequencies v increase with s but the stability criteria are not monotone in s. EXAMPLE
structure
2. In special cases it is possible to delineate the global equilibrium rather than just the stability nature of the central equilibrium. As
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an illustrative example we consider a system with assorting by phenotypic classes composed of just two habitats with sr = sa = 0 (i.e., no selection, but cf. Remark 1). We know that there is exactly one equilibrium with z = 0; we investigate the existence of other equilibria. For this purpose we may rewrite
(3) at equilibrium
as
- 1 (1- kT;*- 1) zy M= (’ irn 1Y,), o=( (1- 4h 9% pi = gi’zi
) i = 1,2.
(14)
For a solution other than z = 0 the determinant of the matrix in (14) must be zero. Solving for ~a when this determinant is 0 we have
which prescribes a hyperbola with negative slope for ~~ as a function of v1 . We are interested in determining when this hyperbola passesthrough the admissiblerange of values for ~~ and F*. We discussthe prospectsfor solutions. Fuller elaboration is given in the Appendix. Casei. 0~~< ,&, i = 1,2. This circumstance allows only the existence of the two fixation states (v = 0, z = fl) and the symmetric equilibrium in the presenceof migration. This is not surprising since the fixation states are unstable in the absenceof migration. Case ii. ai > pi, i = 1, 2. With these parameter values the ftxation states are stable in each habitat when isolated and hence equilibria with z # 0, f 1 must exist for sufficiently small migration. However, these equilibria do not persist for all migration flow. If f(l - m) > 1 - m - k or f(l - k) >, 1 - nt - k (these are not the tightest bounds, but are necessarily satisfied if m + k 2 $), then the symmetric equilibrium (which is unstable) and the two fixation states(which are stable) are the only equilibria. Case iii. % > PI, 'ye
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there is no selection (s = 0) and the migration structure is Levene (rank one backward migration matrix). Because all the demes receive the same immigrant fraction from each deme (which with assorting cannot be due to a common mating area because the assorting tendency varies with the habitat), this circumstance reduces to a single-deme analysis. The entries in the backward migration matrix weight the importance of the various demes, very possibly as a direct measure of their size. Although total panmixia is often an appropriate description of the Levene model (selection in habitats followed by a common mating pool), its use merits qualification when assortative mating is in force because assorting by definition precludes total mixing. We elaborate for two demes and phenotypic assortative mating. The backward migration matrix A4 can be represented as M = (i z), m + tt = 1. This provides that the genotype frequencies will be the same in all demes following migration and the transformation equations governing these frequencies are
(16) 1 - ka, - ma2 22 2 .
1
This is manifestly the transformation Aal + ma2 and 4% + mS2 .
for a single deme with
parameters
Remark 2. Cyclical migration with M = (y :) (which can represent temporal variation in assorting propensities for two season cycles) has the periodic equilibrium with v = 0 and z alternating between (I, - 1) and (-1, 1). This represents two isolated demes and the cyclic equilibrium is stable if f3 < a and unstable if f3 > a in accordance with Remark 1. This is also true for partial selfing (2) because Remark 1 applies to partial selfing as well as phenotypic assorting. Perturbation arguments (Karlin and McGregor, 1972) guarantee that if this equilibrium cycle is stable, it will remain with slight homing tendencies. Thus population subdivision allows equilibrium cycles not observed in single demes. DISCUSSION
Nonrandom mating affects the genotype frequencies at some or all of the loci in many natural populations. It has been studied extensively in single demes and also in subdivided populations (see references in the introduction).
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We focus on the existence and stability nature of certain polymorphic equilibria in subdivided populations. In analogy to studies of multi-locus interactions (Lewontin and Kojima, 1960; Christiansen and Feldman, 1975; Karlin and Campbell, 1978) we employ symmetry assumptions in order to characterize polymorphic equilibria and therefrom delimit their stability nature. The conclusions remain qualitatively valid under slight deviations from the symmetr! assumptions and hence provide meaningful insight into the general problem. We consider two forms of assortative mating: partial selfing and partial assorting based on phenotypic classes. A viability regime based on the level types have the same viability) is of heterozygosity (i.e., both homozygous superimposed on both assorting schemes. We highlight some of the results. Level
of Heterozygosit3
We study polymorphic equilibria which have equal frequency (one-half) of both alleles in each deme and hence these equilibria can be characterized by the level of heterozygosity (frequency of heterozygotes) in each deme. It is easy to show that such an equilibrium exists from the symmetry assumptions. The fact that there is only one such equilibrium in a multideme system is neither intuitively obvious nor trivial to prove. (Even in a single deme there can be multiple symmetric equilibria under the action of partial selfing if homozygotes are obligate selfers, cf. Remark Al in the Appendix.) But there is quite generally only one such symmetric equilibrium (e.g., for assorting by phenotypic classes or partial selfing with mild or overdominant selection) and we display the dependence of its level of heterozygosity on the parameters. The following discussion essentially repeats the single-deme case. The frequency of heterozygotes at the symmetric equilibrium increases with s under both models. This is expected since 1 -t- s specifies the viabilities of the heterozygotes in the various habitats entailing that increasing s increases their relative viabilities. Under phenotypic assorting (1) the levels of heterozygosity are independent of the assorting tendencies a and p. This follows from the assumption z == 0 and the inability to distinguish between the homozygous types which provide equal allele frequencies (l/2) in both assorting classes and the whole population. Under partial selfing (2) the level of heterozygosity is also independent of b the heterozygote assorting tendency (p) ecause both the heterozygotes and the whole population have equal allele frequencies (l/2) when z = 0. However, the level of heterozygosity (v) decreases when the selfing tendency among homozygotes (a) increases because selfing homozpgotes do not produce an! heterozygotes.
The stability analysis of the symmetric equilibrium reduces to considering only perturbations in the z (deviation from equal allele frequencies) parameter
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space whenever there is only one symmetric equilibrium. Even if there are multiple symmetric equilibria, stability can be determined by considering the z and v (level of heterozygosity) perturbation spaces separately. Hence one only need evaluate the spectral radii of n x 7t rather than 2n x 2n matrices to determine stability. The dependence of stability upon the parameters follows as in the singledeme case. For fixed viability (s) the prospects for stability of the symmetric equilibrium (i.e., the reciprocal of the spectral radius of J in (5)) increase with heterozygote assorting (p) and decrease with homozygote assorting (a) when phenotypic assorting (1) is in force. If the assorting tendencies are fixed and p > a increasing the relative viability of heterozygotes (s) increases the prospects for stability of the symmetric equilibrium. When partial selfing of the form (2) is in force throughout the population range the prospects for stability of the symmetric equilibrium increase with the selfing tendency among heterozygotes (p), but the dependence of the stability on a and s is more recondite. Other Equilibria Circumstances were provided in Example 2 under which there are no equilibria other than the fixation states and symmetric equilibrium. However, even in a single deme there can be polymorphisms other than the central equilibrium (Karlin and Farkash, 1978). In analogy to disruptive selection (Karlin and McGregor, 1972), if the fixation states are stable in each habitat without migration there can be 212- 2 stable polymorphisms for sufficiently slight migration. However, as noted in the discussion of the dependence on parameters for partial selfing the regions where other equilibria may occur are sometimes restrictive. Example 3 merits mention because it provides a stable equilibrium cycle. Hard Selection As mentioned in the introduction, we have restricted ourselves to soft selection. The effect of fluctuations in deme sizes preceding migration due to genotypic composition has been studied (Dempster, 1955). This produces a nonconstant backward migration matrix which affects the stability of fixation equilibria under random mating (Christiansen, 1975; Karlin, 1976) as well as in the presence of assortative mating (Moody, 1979). When polymorphic equilibria are well defined under both modes of selection, their stability also differs. The equilibria studied here exist under hard selection, and their characterization and stability will be presented in a subsequent paper. Varying Migration Our focus has been on varying selection intensities and assorting propensities. The effect of changing the migration pattern (e.g., more migration sensu Karlin
ASSORTING IN SUBDIVIDED POPULATIONS
107
(1976)) on polymorphic equilibria also merits study (e.g., Karlin and Campbell, 1978). This perspective is briefly illustrated in Example 2. Multiple
Loci
This aspect of the problem is not addressedhere. An interesting facet of this aspect where one locus controls the assorting tendency was considered by Karlin and McGregor (1974). The extreme caseof quantitative polygenic traits is currently an active area of investigation (e.g., Karlin (1979) and references therein).
APPENDIX:
UNIQUENESS OF SYMMETRIC
EQUILIBRIA
In the caseof partial selfing we demonstrate uniquenessof the symmetric equilibrium by considering the stability of symmetric equilibria restricted to the manifold z = 0. We demonstrate that too many of the equilibria would need to be stable on that manifold and hence there can be only one. We first note that (3) restricted to the manifold z = 0 (i.e., considered as a function of v only) is monotone (v’ > v2 implies v’(v’) > v’(+)) in the unit hypercube and v’(0) > 0, v’(1) < 1. This provides that points in a neighborhood of 0 will converge monotonely to a stable equilibrium which is componentwise lessthan or equal to the stable equilibrium to which points in a neighborhood of 1 converge monotonely. If these two equilibria are equal, that equilibrium is the only equilibrium on the manifold z = 0. If they are not equal, topological considerations require that there be an unstable equilibrium between them componentwise. This is ruled out in case i since af/av, < 1 for all i and w, , thereby providing that all equilibria are stable. In case ii the monotonicity and concavity of f mandate that any equilibrium componentwisegreater than a stable equilibrium must be stable (the criterion for stability is P(M diag[ajJaz)i]) < 1). M ore general circumstancesproviding a unique equilibrium under circumstances of concavity can be found in Krasnoselskii(1964). The convergenceto an equilibrium from a neighborhood of 0 showsstability of the symmetric equilibrium (on the manifold z = 0) when it is unique. Verification
of Partial
Case (i)
Se&g
Demonstrating
(1 +sj;
af x&j’
,tl
[
1 +s
(
f-+)]
(Al)
108 for --I
R.
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CAMPBELL
< s < 1 is routine. Note that
a;if__[
a: --
2
as aa
(s+1)01 1 1-v ---a2 (2 2 13 1-v 3 ‘0 1 1 +s ---a-----1 (2 2 >I
if 1s ) < 1. Hence, it &ices to consider the case s = 1. It is maximized at s = 1 by choosing a = 1 and v = 0 for which and the inequality (Al) holds if either (Y < 1 or v > 0.
642)
is
clear that a?/&
choice
af/& - 1
(b) m+k>l
m+k=l
FIG. Al. asymptotes
Graph of (15) are included.
for
(a) det(M)
> 0; (b)
det(M)
< 0; (c) det(il4)
= 0. The
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Remark Al. If CY= 1, n = z = 0 is always an equilibrium. If s > I there is another symmetric equilibrium characterized by v = 1 - l/s (which equals 0 when s = 1) so that if CL= 1 we cannot expect there to be a unique symmetric equilibrium in general. Veri’cation
of Example2
The discussion of Example 2 can be validated by considering the graphs of the hyperbolas in (15) which depend on AZ, and the values which vi = gJzi may assumeover the admissiblevalues of a, p, v, and z (i.e., / OL /, ( /3 /, 1v /, 1z 1 < 1). Figure Al gives the graph of (15) for det(M)>, <, and =O. Casei. % c Pi f i=l,2: In this case +-<,pli 0 entails z1 = za = 51). The three graphs demonstrate that there are no solutions with 0 < vi < 1. Hence this case, which provided instability of the fixation states with no migration, allows only the single polymorphic (i.e., z f +I) equilibrium with z = 0 if any migration is introduced. Caseii. cyi> pi , i = 1, 2: These assumptions provided stability of the fixation statesin isolated demesand hence we know that for sufficiently small migration there are equilibria with z # 0 or i-1. These are consistent with the first graph since, as m, k -+ 0, the asymptotes approach ~a = 1, vi = 1 and indeed the upper curve va(pi) passesarbitrarily closeto (1, 1) asm, k --L 0-k. However, because 1 ,< vi < 8, in this case we can read off the graphs a sufficient condition (in terms of migration) that z = 0 be the only polymorphic equilibrium. If 3(1 - m) > 1 - m - k or $(l - k) 2 1 - m - k, then neither branch of the curve passesthrough the square 1 < vi, v2 .< 8 and hence z = 0 is the only polymorphic equilibrium. Caseiii. CQ< /$ , aa > /3a:In this casethe graphs cannot be used to rule out the existence of equilibria since v1 < 1 and ~a > 1. The theory of small parameters assuresthe existence of multiple polymorphic equilibria for sufficiently small migration since the fixation states are stable in the first deme isolated and the symmetric equilibrium is stable in the second. If migration is introduced and z # (1, 1) or (- 1, - 1) there cannot be an equilibrium with v = 0. This follows from considering
21;= (1 - m) [k - [ 2(1 T z’l) + +]
z12]
(43)
110
R.
B.
CAMPBELL
which nessitates xi2 = 1 (i = 1,2), if v 1: 0 = v‘. z; = (1 -m)z,
[l + 3 (0~1- PI)] + mzz [ 1 + 5 (as - A,]
(cf. (3)) requires that .z, = z2 if v = 0 (at equilibrium
(-44)
zi = xi).
ACKNOWLEDGMENTS Acknowledgment is especially due Samuel Karlin for suggesting the problem for useful discussions. Useful suggestions toward elucidating the manuscript were supplied by an anonymous referee.
and also
A. D. 1972. Disadvantages of heterozygotes in a system of two adjacent populaGenetika 8 (1 l), 155-161. BODMER, W. F., AND CAVALLI~FORZA, L. L. 1968. A migration matrix model for the study of random genetic drift, Genetics 59, 562-592. CHRISTIANSEN, F. B. 1974. Sufficient conditions for a protected polymorphism in a subdivided population, Amer. Nutur. 108, 157-166. CHRISTIANSEN, F. B. 1975. Hard and soft selection in a subdivided population, Am. Natur. 109, 11-16. CHRISTIANSEN, F. B., AND FELDMAN, M. W. 1975. Subdivided populations: A review of the one- and two-locus deterministic theory, Theor. Pop. Biol. 6, 76-91. CROW. J. F., AND FELSENSTEIN, J. 1968. The effect of assortative mating on the genetic composition of a population, Eugenics Quart. 15, 85-97. DEMPSTER, E. R. 1955. Maintenance of genetic heterogeneity, Cold Spring Harbor Symp. Quant. Biol. 20, 25-32. GHAI, G. L. 1974. Analysis of some non-random mating models, Theor. Pop. Biol. 6, 7691. KARLIN, S. 1968. Equilibrium behavior of population genetic models with non-random mating, J. Appl. Prob. 5, 231-313. KARLIN, S. 1976. Population subdivision and selection migration interaction, in “Population Genetics and Ecology” (S. Karlin and E. Nevo, Eds.), pp. 617-657, Academic Press, New York. KARLIN, S. 1978. Comparisons of positive assortative mating and sexual selection models, Theor. Pop. Biol. 14, 281-312. KARLIN, S. 1979. Models for multifactorial inheritance. I. Multivariate formulations and basic convergence result, Theor. Pop. Biol. 15, 308-355. KARLIN, S., AND CAMPBELL, R. B. 1978. Analysis of central equilibrium configurations for certain multi-locus systems in subdivided populations, Genet. Res. (Cambridge) 32, 151-169. KARLIN, S., ANJJ FARKASH, S. 1978. Analysis of partial assortative mating and sexual selection models for a polygamous species involving a trait based on levels of heterosygosity, Theor. Pop. Biol. 14, 430-445. BAZYKIN,
tions,
ASSORTING
IN
SUBDIVIDED
POPULATIONS
111
J. L. 1972. Application of method of small parameters to models, Theor. Pop. Biol. 3, 186-208. KARLIN, S., AND MCGREGOR, J. L. 1974. Towards a theory of the evolution of modifier genes, Theou. Pop. Biol. 5, 59-103. KRASNOSELSKII, hl. A. 1964. “Positive Solutions of Operator Equations,” Noordhoff, Groningen. Z.EWONTIN, R. C., AND KoJrnrA, K. 1960. The evolutionary dynamics of complex poij-morphisms, Evolution 14, 458-472. MATESSI, C., AND SCUDO, IT. M. 1975. The population genetics of assortative mating based in imprinting, Theor. Pop. Biol. 7, 306-337. MOODY, hl. 1979. Polymorphism with migration and selection, J. Math. Biol. 8, 73-109. -~AGYLAlCI, T. 1977. “Selection in One- and Two-Locus Systems,” Springer-Verlag. Berlin. O’DONALD, P. 1960. Xssortative mating in a population in which two alleles are segregating, Heredity 15, 389-396. O’DONALD, P. 1977. Theoretical aspects of sexual selection, Theor. Pop. Biol. 12, 298-334. PARSONS, P. 4. 1962. The initial increase of a new gene under positive assortative mating, Heredity 17, 267-276. WRIGHT, S. 1921. Systems of mating. III. Assortative mating based on somatic resemblance, Genetics 6, 144-161.
KARLIN,
S.,
multiniche
653/18/1-S
AND
MCGREGOR,
population
genetic
POPULATION
BIOLOGY
Polymorphic
94-111
18,
Equilibria
and Selection
of
Mathematics, of Mathematics,
with
Assortative
in Subdivided R. B.
Department Department
(1980)
Mating
Populations*
CAMPBELL
Stanford Purdue
University, University,
Received
April
Stanford, California 94305, and West Lafayette, Indiana 47907
18,
1980
Two modes of assortative mating, partial selling and assorting by phenotypic classes, are studied in a subdivided population. Differential viability is allowed and the selection intensities and assorting tendencies are permitted to vary among the habitats. There exists a symmetric polymorphism; we delimit its level of heterozygosity and stability nature (dependent on the selection intensities and assorting propensities). This complements studies of the fixation states and thereby provides further insight into the global equilibrium structure in subdivided populations. Circumstances are given where the fixation states and symmetric polymorphism comprise the global equilibrium structure. Examples are also given stable polymorphic migration.
where
migration equilibrium
engenders cycles which
stable polymorphic are absent in single
equilibria demes
and without
Although random mating (which produces Hardy-Weinberg frequencies in gametes) appears to govern many genetic systems, nonrandom union of gametes (which produces nonrandom associations of alleles in zygotes at all or part of the genome), perhaps based on similar or dissimilar parental phenotype (assortative mating), is also well documented. One form of assortative mating is selfing which occurs in many plants; self-sterility mechanisms and host plant selection by pollenating vectors will cause different manifestations of assortative mating in plants. The ability of animals, including humans, to select mates actively allows a myriad of assorting schemes. Of course, various ecological parameters (e.g., density of the species, density and species (race) of pollenating vectors, wind) may affect the assorting mechanisms and tendencies. We investigate the effect of spatial variation in assorting tendencies on the genetic structure of populations. Nonrandom mating is one of the original threads of population genetics research (e.g., Wright, 1921). Several models of assortative mating are presented in extensive papers by Rarlin (1968, 1978); see also O’Donald (1960, 1977), * Supported
in part
by NIH
Grant
GM
10452-
94 0040-5809/80/040094-18$02.00/O Copyright All rights
8 1980 by Academic Press, Inc. of reproduction in any form reserved.
16 and
NSF
Grant
MCS
76-80624-AOl.
ASSORTING
IN
SUBDIVIDED
POPULATIONS
95
Parsons (1962) Crow and Felsenstein (1968), Ghai (1974) and Matessi and Scudo (1975). These models have generally been analyzed in the context of a single deme, but the multideme case has not been ignored (e.g., Karlin and McGregor, 1972; Moody, 1979). In this work we concentrate on extending some single-deme analyses to the multideme context. The global equilibrium structure for multideme systems is in general difficult to ascertain. One facet which is tractable is “protection” or the stability nature of the fixation states. This has been analyzed for partial selfing by Moody (1979) and other mating systems can be treated analogously. We complement these analyses with the study of the stability and genotype frequencies of certain polymorphic equilibria. In order to study the stability of equilibria it is necessary to characterize them. To this end, we assume symmetric mating and selection patterns. In particular, we consider only one locus with two alleles under the assumption that both homozygotes have the same viability and same assorting propensity. Thus, we are not considering the general case, but the results provide insight into the effect of selection-assortative mating-migration interaction on polymorphic equilibria because the results are structurally stable permitting limited qualitative inferences. We also restrict our attention to the standard model of selection migration interaction which has been dubbed soft selection by Christiansen (1975). This model posits that deme sizes are regulated prior to migration which allows the construction of a constant backward migration matrix (Bodmer and CavalliSforza, 1968) for describing the averaging of allele frequencies due to migration. Although the class of symmetric equilibria studied here will exist under, e.g., hard selection (Dempster, 1955; Christiansen, 1975), the explicit characterization and stability nature of the equilibria will differ from those for the case we are considering.
THE
MODEL
We have chosen from among the many models of assortative mating with selection two which are particularly suited to the study of polymorphic equilibria in multideme systems. These describe the change in allele frequencies due to mating, reproduction, and viability selection within demes. The effect of migration on the allele frequencies is addressed presently. A crux of our models is the division of each genotypic class into individuals which mate assortatively and individuals which mate at random in accord with specified proportions. This can also be posed as the probability that an individual will mate assortatively versus mate at random. These tendencies are constants which depend on the genotype and deme, but are independent of allele frequencies or densities.
96
R.
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CAMPBELL
The first model (1) manifests assorting based on phenotypic classes. Specifically, assorting homozygotes mate- only with homozygotes, but do not discriminate between the two homozygous types (AA and aa). In the case of a single deme the global equilibrium structure entailing up to five equilibria is fully delimited by Karlin and Farkash (1978) along with a brief discussion of the biological context of the model. The equilibrium structure is dependent on (Y, /3, and s. The second (2) corresponds to partial selfing as characterizes many plant species; Homozygotes which assort mate only with their own genotype, but we assume the same fraction of each homozygous type assorts. This model has been studied by Karlin (1968, 1978) and Moody (1979) among others. Explicitly, the allele frequency transformation equations are (1 +s&;[$-]
-+j
6 = f(w, 2) = 1+&[&]
-+L2] (1)
(1 -B)V;+aZ+(l
-,)[(I
-++;a]
1 - g(w 2) = 1 +.s[&[&]
+%a]
’
for assortative mating and
- w>+ ; ((1- w)”- 2q + ;] i - w)’- 22,]+ ;] ’ -4 +jcu
(1 + 4 [Cl - 4 [&(l i 6 = f(w, 2) = 1 + s [(l - 4 [;.(I
- w) +-2w1 2 1 -- 4 + #
m + (1- 4 [au 2 = g(e), 2) = 1 + s [(I - a) [&(l
(2) I
+ (1 -+ --
4 2- 2’)] + ;] ’
for partial selfing, where w = frequency of Aa heterozygotes z =.frequency of AA homozygotes 1 f s = heterozygote (Aa) viability
minus frequency of aa homozygotes
1 = homozygote (AA or aa) viability (Y = fraction of homozygotes (AA or aa) which mate assortatively females which discriminate in choosing a mate)
(e.g.,
ASSORTING
IN
SUBDIVIDED
97
POPULATIONS
1 - 01 = fraction of homozygotes which randomly mate (e.g., females; infinite males are assumed so that there is no change in the composition of the mating pool during mating) /3 = fraction of heterozygotes heterozygotes)
which
mate assortatively
(i.e., only with
I - p = fraction of heterozygotes which randomly mate. Thus, f and g specify the change in allele frequencies due to assortative mating, reproduction, and selection and in the case of a single deme the tilde (“) designates the next generation. These single-deme transformations are generalized to the multideme context in the same manner as when random mating is acting (e.g., Christiansen, 1974; Karlin, 1976). All parameters (‘u, Z, s, 01,/3) and the transformations (f, g) are subscripted to indicate to which of the n demes (habitats) they refer and are formed into n-vectors in the natural manner. Migration is specified by the constant n x 11 backward migration matrix M which is stochastic. The backward migration matrix specifies with its zjth entry the fraction of the ith deme which immigrated from the jth deme (Bodmer and Cavalli-Sforza, 1968). It is calculated from the forward migration matrix (which specifies with its zjth entry the fraction of the ith deme which migrates to the jth deme) and the deme sizes preceding migration (cf., e.g., Christiansen, 1975; Karlin, 1976; Nagyalaki, 1977). The assumption that deme sizes are fixed preceding migration allows that the backward migration matrix is constant. We assume a life history of mating, reproduction, and viability selection within demes (habitats) followed by migration among the demes. The change in genotype frequencies in a generation is given by v’ = MC = Mf(v, z),
z’ = MCi = Mg(v, z)
(3)
or in components n 0:
=
2
j=l
tQjfj(Vj
9 Zj),
&
=
1 j=l
??lijgj(Vj
, X;),
where the prime (‘) denotes the subsequent generation (i.e., we choose as our census time that point in the life cycle following migration but preceding mating so that the prime reflects the sequential action of mating-reproduction, viability selection, and migration on the unprimed parameters in accordance with the life history given above) and the tilde (“) specifies the population state after mating-reproduction and viability selection but preceding migration. We only consider the cases where assorting by phenotypic classes is manifested in all the habitats (i.e., fi and gi are of the form (1) for all ;) or partial selfing is occurring in all habitats (i.e., fi and gi are of the form (2) for all i). Assorting propensities and relative viabilities are assumed constant with respect to time.
98
R.
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THE
CAMPBELL
PROBLEM
As we stated in the introduction our objective is to analyze a polymorphic equilibrium (i.e., determine its level of heterozygosity and stability nature dependent on the parameters) as a complement to the available analyses of fixation states. In the event that a polymorphic equilibrium is unstable it cannot be expected to occur in natural populations, but is of interest because it helps define the boundaries of the domains of attraction of stable equilibria. If it is stable it may occur in natural populations and the level of heterozygosity is of interest because it is one of the standard statistics which are employed to characterize natural populations. Inspection of (1) and (2) sh ows that f = 0 if z = 0 and it follows by (3) that z’ = 0 if z = 0 for either phenotypic assorting (1) or partial selfing (2). .This suggests looking for an equilibrium with z = 0 (which implies polymorphism). We demonstrate existence of such a symmetric equilibrium, show that there is only one equilibrium characterized by z = 0 (quite generally), and then delimit its stability nature and level of heterozygosity dependent on the parameter values. Phenotypic Assorting Consider first the case where the local genotype frequency transformations are of the form (1). The existence and uniqueness of an equilibrium with z = 0 follows by its explicit construction. If z = 0, (1) simplifies to bi = fi(vi,
0) = t: z ziz 2,
,
i = 1,2 ,..., n
(which lies between 0 and 1). Hence v’ = IM+ is independent of v. Substituting (4) into (3) provides an explicit expression for the equilibrium oi
=ktnij(*),
i=
1,2,...,n,
which shows that it exists and is unique. Stability The population is described by the 2n-vector y = (wl , w2 ,..., v, , z1 ,..., zJ. For notational convenience we shall designate the subsequent generation with Y = (v; , w; ,..., v:, , z; ,..., z;). It is a standard result that stability of an equilibrium depends on the spectral radius of the 2n x 2n gradient matrix J = !I aY,/ay, (1 evaluated at that equilibrium (all perturbations in v and z are admissible at z = 0 and 0 < v < 1). This matrix can be evaluated using
ASSORTING
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POPULATIONS
(3) producing a composite of four n x n matrices of the form (z 3, where Q = Mdiag[af,/&,],
b = M diag[af,j%,],
c = M diag[agi/&+],
d = M diag[8giiazi].
(5)
(diag[.] designatesthe matrix with the indicated entries on the diagonal and O’s off the diagonal.) Taking the partial derivatives of (1) and evaluating at zi = 0 yields
Thus, stability of the symmetric equilibrium is dependent on p(M diag[ag,/&,]) or explicitly p (M diag [ ’ ’ (~i~[$(ui’2)
])
where p(.) designatesthe spectral radius. Recall that the explicit construction of v from (4) and (3) enablesthe evaluation of (7). Thus, stability only depends on the spectral radius of one 12x n matrix which we can explicitly display rather than a 2n x 2n matrix which the dimension of the perturbation space would suggestor two II x 7tmatriceswhich a nice eigenstructurein the perturbation spacecould provide. Dependence on the Parameters
We note from (4) that 4 hence v is independent of a and P (if z = 0) but increasing in s. Thus increasedselection favoring the heterozygotes increases the heterozygote fraction acrossthe population range as occurs in a single demc. However, we can conclude from (7) that if s is fixed the stability of the symmetric equilibrium increaseswith 13and decreaseswith a in the sense that decreasing(7) may causestability if it becomeslessthan one; if it is further decreasedbeyond one, random fluctuations will be more rapidly eliminated by deterministic forces providing greater robustness against perturbations (i.e., the greater the tendency for heterozygotes to assort as compared to homozygotesthe better the prospectsfor stability of the symmetric equilibrium). If l3 > a the fact that v increaseswith s implies that increasing the selection intensity favoring the heterozygote increasesthe prospects for stability of the symmetric equilibrium along with increasing its inherent heterozygosity. EXAMPLE 1. We specialize to the casesi = 0 for all i entailing no viability selection. Then the criterion for stability is
p (Mdiag [l -
(” 4 “$)I)
< 1.
loo
R.
B.
CAMPBELL
Remark 1. The term “no selection” for the casesi = 0 for all i is in a sensea misnomer. Unless a s P the allele frequencies are not in general con-. served. In fact, one has protection with global convergence to the symmetric equilibrium if f3 > a and both fixation states are stable concordant with an unstable symmetric equilibrium if IX > p. An heuristic argument states that the classwhich assortsless (e.g., heterozygotes if p < a) are taken as mates lessthan the other class.Therefore, whichever alleleis more prevalent (compared to l/2, the frequency in heterozygotes) in the homozygous class is favored. (However, if two demesare present polymorphism can be maintained analogous to the case of disruptive selection (Karlin and McGregor, 1972; Bazykin, 1972).) (If a > @equal frequencies are favored since heterozygotes have equal frequenciesof both alleles.) Partial
Selfng
When the local transformation equations are of the form (2) existence of a symmetric equilibrium is most easily demonstrated by considering (2) at a = 0. Since this maps the closed unit interval (in V) into itself continuously, (3) maps the closed unit hypercube into itself continuously guaranteeing the existence of a fixed point. We demonstrate the uniquenessof this fixed point in two cases. Casei. - 1 < si < 1 for all i. This parameter range encompassesmild selection, in particular when the heterozygote is never more than twice as viable as the homozygote. (Extreme selection favoring the homozygotes is allowed, i.e., the heterozygote can be lethal.) The constraint - 1 < s < 1 precludes an equilibrium involving heterozygotes in a single deme when homozygotes are complete selfers (a = 1). Recall that assortative mating precludes considering just whether heterozygotes or homozygotes are more viable (S > 0 or s < 0) for deciding stability of equilibria, but the assorting tendencies must also be considered. For this casewe consider
(1 + 4;
af av’ [
1 fs
++]
1-w
2
at z=O.
(9
(
This lies between0 and 1 for a! and w between0 and 1 and - 1 < s < 1. (The necessarycalculations are outlined in the Appendix.) The fact 1afi/aoi 1< 1 for all i suffices to demonstrate uniqueness.(We employ the monotonicity of f in order to give a more elegant proof of uniquenessin the Appendix.) Case ii. si > 0 for all i. This caserequires that the heterozygote be more viable than the homozygotes in all habitats. However, this doesnot necessarily imply that the symmetric equilibrium is stable becausethe “overdominant”
ASSORTING
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POPULATIONS
selection can be countered by the assorting pattern. Extreme selection entailing near lethality of the homozygotes is included in this case. Note that the circumstances of this case overlap case i, but environmental heterogeneity manifesting near lethality of homozygotes in some environments and near lethality of heterozygotes in other environments is not included in either of the two cases. We prove uniqueness in this case by noting from (9) that f is monotone increasing in v and the second derivative
+ s)(G)(T)
-31
ay -=
(10)
iiV2
l+s
;-aly2))3
i
at
z="
(
is negative which provides that f is concave in v on the interval (0, 1) for s > 0, 0 < (Y < 1. In conjunction with the fact that f(0, 0) > 0 this implies that there is at most one equilibrium with z = 0. We outline a proof of this implication in the Appendix. A noteworthy part of the proof shows that ~(Mdiag[af/av~]) < 1 under the hypotheses of case ii. This is useful in the stability analysis of the symmetric equilibrium. Stability The stability of the symmetric equilibrium is dependent on the spectral radius of the matrix J in (5) as in the phenotypic assorting case. However, under partial selfing we only have (cf. (6)) afi G
--1 -
Qi &Ii
0
for all i,
and
p (Mdiag
and hence p(J) < 1 is equivalent to p (Mdiag
[$$-I)
< 1
[%I)
< 1
(12)
because these matrices lie on the diagonal. Thus stability depends on the spectral radii of two n x n matrices rather than one 2n x 2n matrix as one would a priori expect. We have remarked above that under the hypotheses of cases i and ii (and in fact whenever there is a unique symmetric equilibrium) the first inequality in (12) is necessarily satisfied. Hence the criterion for stability of the symmetric equilibrium is quite generally
-=c 1, II
(13)
which
depends on one n
x
n matrix as in the phenotypic assorting case.
102
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CAMPBELL
Dependence on the Parameters The stability nature of the symmetric equilibrium is readily ascertained in the following cases ensuing from the single-deme analysis by the Frobenius theory of positive matrices which provides that if a matrix is componentwise greater than a positive matrix, then its spectral radius is also greater, Recall that the spectral radius of a stochastic matrix (i.e., M) is one. Case i. s = 0 (no viability selection). The symmetric equilibrium is stable if pi > ai for all i and unstable if ,& < ai for all i. If oli = pi for all i the situation is quite neutral. Case ii. s > 0, p > a (overdominant viability selection, heterozygotes have a greater propensity to self than homozygotes in all habitats). Under these circumstances the symmetric equilibrium is stable and in fact globally attracting. (The latter fact can be demonstrated by showing / I 1 < ( z 1 with equality only if z = 0.) Case iii. s < 0, f3 < a (underdominant viability selection, homozygotes have a greater propensity to self than heterozygotes in all habitats). The symmetric equilibrium is unstable and the fixation states are stable, but there may be many more polymorphic equilibria depending on the migration structure. However, if z > 0 or z < 0 the multideme population will converge to z = 1 or z = -1, respectively (i.e., fixation of the A allele and a allele, respectively), precluding the existence of any nonfixation equilibria with z > 0 or z < 0. (This can be demonstrated by showing 1f / > ( z 1 in (2) for this case.) Aside from the above cases where we can delineate the nature of the equilibrium, we make some general comments on the influence of the parameters on the symmetric equilibrium. p: The equilibrium heterozygote frequencies v (level of heterozygosity) are independent of the selfing propensities among heterozygotes p. This entails that P(M diag[@J%J) is independent of f3, but P(M diag[%gi/3zi]) decreases with p and hence the prospects for stability of the symmetric equilibrium increase with f3. a: The equilibrium heterozygote frequencies v (level of heterozygosity) decrease as a increases. This provides that P(M diag[afi/L&]) increases with a, but if there are never multiple symmetric equilibria this quantity is always less than one and its dependence on a is of limited interest. Unfortunately, p(Mdiag[agJ&]) is not monotone in a. s: The heterozygote frequencies v increase with s but the stability criteria are not monotone in s. EXAMPLE
structure
2. In special cases it is possible to delineate the global equilibrium rather than just the stability nature of the central equilibrium. As
ASSORTING
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POPULATIONS
an illustrative example we consider a system with assorting by phenotypic classes composed of just two habitats with sr = sa = 0 (i.e., no selection, but cf. Remark 1). We know that there is exactly one equilibrium with z = 0; we investigate the existence of other equilibria. For this purpose we may rewrite
(3) at equilibrium
as
- 1 (1- kT;*- 1) zy M= (’ irn 1Y,), o=( (1- 4h 9% pi = gi’zi
) i = 1,2.
(14)
For a solution other than z = 0 the determinant of the matrix in (14) must be zero. Solving for ~a when this determinant is 0 we have
which prescribes a hyperbola with negative slope for ~~ as a function of v1 . We are interested in determining when this hyperbola passesthrough the admissiblerange of values for ~~ and F*. We discussthe prospectsfor solutions. Fuller elaboration is given in the Appendix. Casei. 0~~< ,&, i = 1,2. This circumstance allows only the existence of the two fixation states (v = 0, z = fl) and the symmetric equilibrium in the presenceof migration. This is not surprising since the fixation states are unstable in the absenceof migration. Case ii. ai > pi, i = 1, 2. With these parameter values the ftxation states are stable in each habitat when isolated and hence equilibria with z # 0, f 1 must exist for sufficiently small migration. However, these equilibria do not persist for all migration flow. If f(l - m) > 1 - m - k or f(l - k) >, 1 - nt - k (these are not the tightest bounds, but are necessarily satisfied if m + k 2 $), then the symmetric equilibrium (which is unstable) and the two fixation states(which are stable) are the only equilibria. Case iii. % > PI, 'ye
104
R.
B.
CAMPBELL
there is no selection (s = 0) and the migration structure is Levene (rank one backward migration matrix). Because all the demes receive the same immigrant fraction from each deme (which with assorting cannot be due to a common mating area because the assorting tendency varies with the habitat), this circumstance reduces to a single-deme analysis. The entries in the backward migration matrix weight the importance of the various demes, very possibly as a direct measure of their size. Although total panmixia is often an appropriate description of the Levene model (selection in habitats followed by a common mating pool), its use merits qualification when assortative mating is in force because assorting by definition precludes total mixing. We elaborate for two demes and phenotypic assortative mating. The backward migration matrix A4 can be represented as M = (i z), m + tt = 1. This provides that the genotype frequencies will be the same in all demes following migration and the transformation equations governing these frequencies are
(16) 1 - ka, - ma2 22 2 .
1
This is manifestly the transformation Aal + ma2 and 4% + mS2 .
for a single deme with
parameters
Remark 2. Cyclical migration with M = (y :) (which can represent temporal variation in assorting propensities for two season cycles) has the periodic equilibrium with v = 0 and z alternating between (I, - 1) and (-1, 1). This represents two isolated demes and the cyclic equilibrium is stable if f3 < a and unstable if f3 > a in accordance with Remark 1. This is also true for partial selfing (2) because Remark 1 applies to partial selfing as well as phenotypic assorting. Perturbation arguments (Karlin and McGregor, 1972) guarantee that if this equilibrium cycle is stable, it will remain with slight homing tendencies. Thus population subdivision allows equilibrium cycles not observed in single demes. DISCUSSION
Nonrandom mating affects the genotype frequencies at some or all of the loci in many natural populations. It has been studied extensively in single demes and also in subdivided populations (see references in the introduction).
ASSORTING
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105
We focus on the existence and stability nature of certain polymorphic equilibria in subdivided populations. In analogy to studies of multi-locus interactions (Lewontin and Kojima, 1960; Christiansen and Feldman, 1975; Karlin and Campbell, 1978) we employ symmetry assumptions in order to characterize polymorphic equilibria and therefrom delimit their stability nature. The conclusions remain qualitatively valid under slight deviations from the symmetr! assumptions and hence provide meaningful insight into the general problem. We consider two forms of assortative mating: partial selfing and partial assorting based on phenotypic classes. A viability regime based on the level types have the same viability) is of heterozygosity (i.e., both homozygous superimposed on both assorting schemes. We highlight some of the results. Level
of Heterozygosit3
We study polymorphic equilibria which have equal frequency (one-half) of both alleles in each deme and hence these equilibria can be characterized by the level of heterozygosity (frequency of heterozygotes) in each deme. It is easy to show that such an equilibrium exists from the symmetry assumptions. The fact that there is only one such equilibrium in a multideme system is neither intuitively obvious nor trivial to prove. (Even in a single deme there can be multiple symmetric equilibria under the action of partial selfing if homozygotes are obligate selfers, cf. Remark Al in the Appendix.) But there is quite generally only one such symmetric equilibrium (e.g., for assorting by phenotypic classes or partial selfing with mild or overdominant selection) and we display the dependence of its level of heterozygosity on the parameters. The following discussion essentially repeats the single-deme case. The frequency of heterozygotes at the symmetric equilibrium increases with s under both models. This is expected since 1 -t- s specifies the viabilities of the heterozygotes in the various habitats entailing that increasing s increases their relative viabilities. Under phenotypic assorting (1) the levels of heterozygosity are independent of the assorting tendencies a and p. This follows from the assumption z == 0 and the inability to distinguish between the homozygous types which provide equal allele frequencies (l/2) in both assorting classes and the whole population. Under partial selfing (2) the level of heterozygosity is also independent of b the heterozygote assorting tendency (p) ecause both the heterozygotes and the whole population have equal allele frequencies (l/2) when z = 0. However, the level of heterozygosity (v) decreases when the selfing tendency among homozygotes (a) increases because selfing homozpgotes do not produce an! heterozygotes.
The stability analysis of the symmetric equilibrium reduces to considering only perturbations in the z (deviation from equal allele frequencies) parameter
106
R.
B.
CAMPBELL
space whenever there is only one symmetric equilibrium. Even if there are multiple symmetric equilibria, stability can be determined by considering the z and v (level of heterozygosity) perturbation spaces separately. Hence one only need evaluate the spectral radii of n x 7t rather than 2n x 2n matrices to determine stability. The dependence of stability upon the parameters follows as in the singledeme case. For fixed viability (s) the prospects for stability of the symmetric equilibrium (i.e., the reciprocal of the spectral radius of J in (5)) increase with heterozygote assorting (p) and decrease with homozygote assorting (a) when phenotypic assorting (1) is in force. If the assorting tendencies are fixed and p > a increasing the relative viability of heterozygotes (s) increases the prospects for stability of the symmetric equilibrium. When partial selfing of the form (2) is in force throughout the population range the prospects for stability of the symmetric equilibrium increase with the selfing tendency among heterozygotes (p), but the dependence of the stability on a and s is more recondite. Other Equilibria Circumstances were provided in Example 2 under which there are no equilibria other than the fixation states and symmetric equilibrium. However, even in a single deme there can be polymorphisms other than the central equilibrium (Karlin and Farkash, 1978). In analogy to disruptive selection (Karlin and McGregor, 1972), if the fixation states are stable in each habitat without migration there can be 212- 2 stable polymorphisms for sufficiently slight migration. However, as noted in the discussion of the dependence on parameters for partial selfing the regions where other equilibria may occur are sometimes restrictive. Example 3 merits mention because it provides a stable equilibrium cycle. Hard Selection As mentioned in the introduction, we have restricted ourselves to soft selection. The effect of fluctuations in deme sizes preceding migration due to genotypic composition has been studied (Dempster, 1955). This produces a nonconstant backward migration matrix which affects the stability of fixation equilibria under random mating (Christiansen, 1975; Karlin, 1976) as well as in the presence of assortative mating (Moody, 1979). When polymorphic equilibria are well defined under both modes of selection, their stability also differs. The equilibria studied here exist under hard selection, and their characterization and stability will be presented in a subsequent paper. Varying Migration Our focus has been on varying selection intensities and assorting propensities. The effect of changing the migration pattern (e.g., more migration sensu Karlin
ASSORTING IN SUBDIVIDED POPULATIONS
107
(1976)) on polymorphic equilibria also merits study (e.g., Karlin and Campbell, 1978). This perspective is briefly illustrated in Example 2. Multiple
Loci
This aspect of the problem is not addressedhere. An interesting facet of this aspect where one locus controls the assorting tendency was considered by Karlin and McGregor (1974). The extreme caseof quantitative polygenic traits is currently an active area of investigation (e.g., Karlin (1979) and references therein).
APPENDIX:
UNIQUENESS OF SYMMETRIC
EQUILIBRIA
In the caseof partial selfing we demonstrate uniquenessof the symmetric equilibrium by considering the stability of symmetric equilibria restricted to the manifold z = 0. We demonstrate that too many of the equilibria would need to be stable on that manifold and hence there can be only one. We first note that (3) restricted to the manifold z = 0 (i.e., considered as a function of v only) is monotone (v’ > v2 implies v’(v’) > v’(+)) in the unit hypercube and v’(0) > 0, v’(1) < 1. This provides that points in a neighborhood of 0 will converge monotonely to a stable equilibrium which is componentwise lessthan or equal to the stable equilibrium to which points in a neighborhood of 1 converge monotonely. If these two equilibria are equal, that equilibrium is the only equilibrium on the manifold z = 0. If they are not equal, topological considerations require that there be an unstable equilibrium between them componentwise. This is ruled out in case i since af/av, < 1 for all i and w, , thereby providing that all equilibria are stable. In case ii the monotonicity and concavity of f mandate that any equilibrium componentwisegreater than a stable equilibrium must be stable (the criterion for stability is P(M diag[ajJaz)i]) < 1). M ore general circumstancesproviding a unique equilibrium under circumstances of concavity can be found in Krasnoselskii(1964). The convergenceto an equilibrium from a neighborhood of 0 showsstability of the symmetric equilibrium (on the manifold z = 0) when it is unique. Verification
of Partial
Case (i)
Se&g
Demonstrating
(1 +sj;
af x&j’
,tl
[
1 +s
(
f-+)]
(Al)
108 for --I
R.
B.
CAMPBELL
< s < 1 is routine. Note that
a;if__[
a: --
2
as aa
(s+1)01 1 1-v ---a2 (2 2 13 1-v 3 ‘0 1 1 +s ---a-----1 (2 2 >I
if 1s ) < 1. Hence, it &ices to consider the case s = 1. It is maximized at s = 1 by choosing a = 1 and v = 0 for which and the inequality (Al) holds if either (Y < 1 or v > 0.
642)
is
clear that a?/&
choice
af/& - 1
(b) m+k>l
m+k=l
FIG. Al. asymptotes
Graph of (15) are included.
for
(a) det(M)
> 0; (b)
det(M)
< 0; (c) det(il4)
= 0. The
ASSORTING
IN
SUBDIVIDED
109
POPULATIONS
Remark Al. If CY= 1, n = z = 0 is always an equilibrium. If s > I there is another symmetric equilibrium characterized by v = 1 - l/s (which equals 0 when s = 1) so that if CL= 1 we cannot expect there to be a unique symmetric equilibrium in general. Veri’cation
of Example2
The discussion of Example 2 can be validated by considering the graphs of the hyperbolas in (15) which depend on AZ, and the values which vi = gJzi may assumeover the admissiblevalues of a, p, v, and z (i.e., / OL /, ( /3 /, 1v /, 1z 1 < 1). Figure Al gives the graph of (15) for det(M)>, <, and =O. Casei. % c Pi f i=l,2: In this case +-<,pli
21;= (1 - m) [k - [ 2(1 T z’l) + +]
z12]
(43)
110
R.
B.
CAMPBELL
which nessitates xi2 = 1 (i = 1,2), if v 1: 0 = v‘. z; = (1 -m)z,
[l + 3 (0~1- PI)] + mzz [ 1 + 5 (as - A,]
(cf. (3)) requires that .z, = z2 if v = 0 (at equilibrium
(-44)
zi = xi).
ACKNOWLEDGMENTS Acknowledgment is especially due Samuel Karlin for suggesting the problem for useful discussions. Useful suggestions toward elucidating the manuscript were supplied by an anonymous referee.
and also
A. D. 1972. Disadvantages of heterozygotes in a system of two adjacent populaGenetika 8 (1 l), 155-161. BODMER, W. F., AND CAVALLI~FORZA, L. L. 1968. A migration matrix model for the study of random genetic drift, Genetics 59, 562-592. CHRISTIANSEN, F. B. 1974. Sufficient conditions for a protected polymorphism in a subdivided population, Amer. Nutur. 108, 157-166. CHRISTIANSEN, F. B. 1975. Hard and soft selection in a subdivided population, Am. Natur. 109, 11-16. CHRISTIANSEN, F. B., AND FELDMAN, M. W. 1975. Subdivided populations: A review of the one- and two-locus deterministic theory, Theor. Pop. Biol. 6, 76-91. CROW. J. F., AND FELSENSTEIN, J. 1968. The effect of assortative mating on the genetic composition of a population, Eugenics Quart. 15, 85-97. DEMPSTER, E. R. 1955. Maintenance of genetic heterogeneity, Cold Spring Harbor Symp. Quant. Biol. 20, 25-32. GHAI, G. L. 1974. Analysis of some non-random mating models, Theor. Pop. Biol. 6, 7691. KARLIN, S. 1968. Equilibrium behavior of population genetic models with non-random mating, J. Appl. Prob. 5, 231-313. KARLIN, S. 1976. Population subdivision and selection migration interaction, in “Population Genetics and Ecology” (S. Karlin and E. Nevo, Eds.), pp. 617-657, Academic Press, New York. KARLIN, S. 1978. Comparisons of positive assortative mating and sexual selection models, Theor. Pop. Biol. 14, 281-312. KARLIN, S. 1979. Models for multifactorial inheritance. I. Multivariate formulations and basic convergence result, Theor. Pop. Biol. 15, 308-355. KARLIN, S., AND CAMPBELL, R. B. 1978. Analysis of central equilibrium configurations for certain multi-locus systems in subdivided populations, Genet. Res. (Cambridge) 32, 151-169. KARLIN, S., ANJJ FARKASH, S. 1978. Analysis of partial assortative mating and sexual selection models for a polygamous species involving a trait based on levels of heterosygosity, Theor. Pop. Biol. 14, 430-445. BAZYKIN,
tions,
ASSORTING
IN
SUBDIVIDED
POPULATIONS
111
J. L. 1972. Application of method of small parameters to models, Theor. Pop. Biol. 3, 186-208. KARLIN, S., AND MCGREGOR, J. L. 1974. Towards a theory of the evolution of modifier genes, Theou. Pop. Biol. 5, 59-103. KRASNOSELSKII, hl. A. 1964. “Positive Solutions of Operator Equations,” Noordhoff, Groningen. Z.EWONTIN, R. C., AND KoJrnrA, K. 1960. The evolutionary dynamics of complex poij-morphisms, Evolution 14, 458-472. MATESSI, C., AND SCUDO, IT. M. 1975. The population genetics of assortative mating based in imprinting, Theor. Pop. Biol. 7, 306-337. MOODY, hl. 1979. Polymorphism with migration and selection, J. Math. Biol. 8, 73-109. -~AGYLAlCI, T. 1977. “Selection in One- and Two-Locus Systems,” Springer-Verlag. Berlin. O’DONALD, P. 1960. Xssortative mating in a population in which two alleles are segregating, Heredity 15, 389-396. O’DONALD, P. 1977. Theoretical aspects of sexual selection, Theor. Pop. Biol. 12, 298-334. PARSONS, P. 4. 1962. The initial increase of a new gene under positive assortative mating, Heredity 17, 267-276. WRIGHT, S. 1921. Systems of mating. III. Assortative mating based on somatic resemblance, Genetics 6, 144-161.
KARLIN,
S.,
multiniche
653/18/1-S
AND
MCGREGOR,
population
genetic