The number of equilibria in the diallelic Levene model with multiple demes.

The Levene model is the simplest mathematical model to describe the evolution of gene frequencies in spatially subdivided populations. It provides insight into how locally varying selection promotes a population's genetic diversity. Despite its simplicity, interesting problems have remained unsolved even in the diallelic case. In this paper we answer an open problem by establishing that for two alleles at one locus and J demes, up to 2J-1 polymorphic equilibria may coexist. We first present a proof for the case of stable monomorphisms and then show that the result also holds for protected alleles. These findings allow us to prove that any odd number (up to 2J-1) of equilibria is possible, before we extend the proof to even numbers. We conclude with some numerical results and show that for J>2, the proportion of parameter space affording this maximum is extremely small.

Introduction

The Levene model, proposed by Levene (1953) in 1953, was introduced to describe the influence of spatially varying selection on a population’s genetic structure. It covers migration between a finite number of demes and the evolution of gene frequencies from one generation to the next. Therefore, it is formulated as a discrete-space, discrete-time migration–selection model.

It is an interesting fact that in Levene’s model the geometric mean of the average fitness in each deme is nondecreasing from one generation to the next and constant only at equilibrium (Cannings, 1971; Edwards, 1977; Li, 1955; Nagylaki, 1992); mathematically speaking, there is a strict Lyapunov function for the Levene model, which highly facilitates analytical investigations. In particular, this excludes complex dynamical behavior and implies that trajectories approach (sets of) equilibria. Previous research has focused on giving conditions for protectedness of alleles (e.g., Levene (1953), Nagylaki (1992) and Prout (1968)) as well as conditions for the existence and stability of polymorphic equilibria (e.g., Edwards (1977), Karlin (1977), Karlin and Campbell (1980), Nagylaki (1992), Nagylaki (2009), Nagylaki and Lou (2001) and Nagylaki and Lou (2008)). Results about the number of possible equilibria in the Levene model were derived in Karlin (1977) and Nagylaki and Lou (2006), where, amongst others, conditions for the existence of a unique polymorphic equilibrium were established.

The Levene model can be described as follows (see also, e.g., Nagylaki (1992)): Consider a population of diploid individuals, which is subdivided into demes and is large enough so that we can ignore stochastic fluctuations. With alleles, , at a given locus, we are interested in the evolution in discrete time (nonoverlapping generations) of the genetic composition of the population, where denotes the relative frequency of allele in deme at zygote stage (). We denote by the fitness (viability) of an individual in deme , where we require . For each deme we collect the fitness values in symmetric fitness matrices . In line with the standard selection model, the gene frequencies in deme after selection, denoted by , are given by where is the mean fitness of individuals in deme carrying allele , and is the mean fitness of the subpopulation in deme . After selection adults migrate independently of their genotypes. For the gene frequencies in the next generation, , we obtain where the denote the backward migration rates, i.e., the probability that an individual in deme migrated from deme .

The central assumption of the Levene model is that migration rates do not depend on the deme of origin. Hence we set for all . Substituting this into (1) we see that, after one generation, gene frequencies in zygotes are the same in all demes. This means that despite locally varying selection there is no spatial structure in the population. Thus, we drop the superscripts of the allele frequencies and write instead of . Therefore, we can represent the recursion for the Levene model as

If , i.e., for two alleles, this recursion reduces to a single equation. We set , hence and . Then (2) simplifies to where

Determining the fixed points of (3) means solving which, after multiplying by the denominators, takes the form where is a polynomial of degree . Hence the maximum possible number of internal equilibria of (3) is . Whether this upper bound can be attained for arbitrary has been an open problem so far. The situation is clear only for . Then, every possible number () of fixed points with a feasible1 stability configuration is known to occur in concrete examples. Furthermore, Karlin (1977) presents several examples for producing four internal fixed points, as well as one configuration for with five.

Preparatory results

To simplify subsequent arguments, let us settle on the following notion.

Consider the function . We call , or the dynamical system , symmetric if for all .

A simple observation gives.

Let , , be symmetric and , . Then is symmetric.

Moreover, we state a technical result which will be needed repeatedly in the arguments below:

For and any of the matricesdefine the function by(compare Eq. (3b)), where is the entry in row and column of the matrix . Then, is bounded for and , .

Differentiating yields

We set and show that is bounded for any . Clearly, is continuous in both variables and nonnegative. Furthermore, for every we have because is a quotient of two polynomials where the numerator has a lower degree in than the denominator. Some computations yield which is bounded for (Fig. 1). Set then, by the statements above, is decreasing if (for any fixed ). Since is compact, there must be some such that on and therefore on .

Fig. 1

The function for the cases (a)–(d). The value is attained at either of the endpoints or of the interval of interest. We note that the maximum of is located on the solid boundary of the gray filled area.

For the cases (b)–(d) analogous arguments apply with

(Note that this case can be obtained from (a) by the transformation .)

and

, where all denominators are on the area of interest.

In case (d) we do not have , but Nevertheless, since , the above arguments hold with some obvious adaptations. □

Case I: both monomorphic equilibria are stable

In this section we treat the case of asymptotically stable monomorphic equilibria and . The idea is the following: We start out by choosing in (3b) with demes and internal fixed points. Then we introduce two additional demes which contain only a small proportion of the population, but greatly favor homozygotes. After showing that this perturbation has a negligible impact on the equilibria we will prove that the resulting dynamics in demes has fixed points. From this we directly obtain our first result which demonstrates that the upper bound on the number of fixed points can be reached.

As to the mathematical procedure, we perturb Eq. (3) with a function , which behaves sufficiently well on a compact interval containing all internal fixed points of such that they are maintained and keep their stability properties (compare Karlin and McGregor (1972), Theorem 4.4). On the other hand, in the limit , becomes discontinuous at and , which allows us to generate additional fixed points near the end points of .

We consider (3) for given and suppose that Note that under these conditions two adjacent fixed points always have opposite stability properties; i.e.,  is repelling, is attracting, …, is repelling. In particular, must be odd.

is symmetric,

and are asymptotically stable, i.e., , , and

has hyperbolic, internal fixed points.2 We label them

For define the following two fitness matrices Then, for , define the recursion where Clearly, is of the form (3b) with demes. The perturbation function is symmetric and therefore, by Lemma 2.2, is symmetric. Furthermore, is continuous in . Thus, by choosing sufficiently small we can ensure that for all the new dynamical system (4) is similar to (3) in the following sense:

Since , the point remains a fixed point (and by symmetry of the same holds for ). Moreover, these monomorphic states remain stable because which clearly is bounded for .

Since is continuous and bounded () on , has fixed points , , where is close to the fixed point of for each . In particular, all are within the interval for some sufficiently small .

Since is uniformly bounded in for (see Lemma 2.3(a) and (b)), the are unique, i.e., no additional equilibria emerge in . In particular, is close to and therefore, the local stability properties of the equilibria , , are the same as the corresponding ’s; i.e., is repelling for odd and attracting for even.

By a short calculation we find if for all (compare Fig. 2), which motivates the next step: Set and fix a . Since () we can find a such that Because for and is close to , we conclude Therefore we have and . From the stability configuration above (i.e., 0 is repelling, attracting) we know that on an interval as well as on for suitable . Thus, by the intermediate value theorem it follows that two additional fixed points exist in . Since is symmetric, we automatically have two additional equilibria in as well. Therefore we have proved

Fig. 2

The function for some values of . Note that for all , which becomes visible only after zooming in.

Let be given by (3b) in demes with internal fixed points and suppose that is symmetric and and are asymptotically stable. Then we can always find with demes and internal fixed points, which is symmetric, and as well as are asymptotically stable.

From this we directly get

For (3), the theoretical upper bound of internal fixed points can be achieved.

By induction: For the configurations give the required result: Both dynamics are symmetric and produce the maximum number of 1 respectively 3 internal fixed points. Furthermore, both monomorphic equilibria are asymptotically stable for each configuration. By repeatedly applying Lemma 3.1 we can construct examples with the maximum possible number of fixed points for any . □

Case II: both monomorphic equilibria are repellors

So far, we have only shown that examples with stable monomorphisms and a maximum possible number of fixed points exist. The following proposition extends Proposition 3.2 to the case where both monomorphic equilibria are repellors, i.e., where there is a protected polymorphism.

The theoretical upper bound of internal fixed points in (3) can be realized with both alleles protected, i.e., with repelling monomorphisms and .

For , symmetric dynamics with protected alleles and one internal fixed point are known to exist (e.g., consider ).

The following procedure is the same as in the argumentation leading to Lemma 3.1: Let be defined by (3) for some such that and are attracting. Furthermore, assume that is symmetric and (3a) bears the maximum possible number of internal fixed points. Such exists by Proposition 3.2.

Consider Eq. (4) with where , . The idea here is to introduce one additional niche that favors heterozygotes and therefore inhibits the loss of either allele.

We find that is symmetric, continuous, and on for . Furthermore, and the derivative is bounded for on every compact interval , (Lemma 2.3(c)). Therefore, by choosing small and large–for the things to come will suffice–we obtain analogously to the previous construction that all internal fixed points of and their stability properties are conserved. In particular, remains a repellor. On the other hand, and thus, by the choice of above, we have Hence, is also a repellor and it follows that there is an additional fixed point in . Since is symmetric we automatically get another equilibrium in , and therefore have constructed a dynamics with protected alleles, demes and internal fixed points. □

Case III: there is exactly one stable monomorphism

By Lemma 3.1 and following the proof of Proposition 4.1, we immediately see that we can construct examples for any possible configuration with an odd number of internal fixed points. In other words, for any we can find dynamics with , , internal fixed points and preset stability properties.3 To obtain an even number of internal fixed points, we must get rid of the symmetry in our examples. Still, the method is the same as conducted in Section 3.

As before, let be defined by (3), with hyperbolic internal equilibria for some . Furthermore, presume that and are attractors.

For , , sufficiently small and consider Eq. (4). Lemma 2.3(d) and some basic algebra show that By the first point we may assume (making smaller if necessary) that all internal fixed points and their stability properties are maintained. In particular, the rightmost internal equilibrium remains a repellor. By the second point, the property of being an attractor remains unchanged as well. On the other hand, enlarging we get arbitrarily large values for and hence also for , eventually making a repellor. As we cannot have two repellors side by side, one additional (asymptotically stable) internal fixed point must have emerged. Note that this is the only additional equilibrium by the monotonicity of .

is monotone, continuous, , and is uniformly bounded in on every compact interval , ,

and .

To sum up, we have a dynamics with demes and () internal fixed points. Putting this together with what we know from the previous sections we get.

If , the Levene model (3) allows for any number of hyperbolic internal fixed points with any feasible stability configuration.

Numerical examples

In this section, we back up some of our results by numerical examples. Consider the configuration then Eq. (3) has fixed points at approximately where , , and are asymptotically stable. The reverse stability configuration can be found by setting Inserting this into (3) produces equilibria at where now , , , and are asymptotically stable.

If , we set and obtain the configuration: Here, filled circles “” represent stable fixed points, whereas unfilled rings “” stand for repellors.

As for the reversed stability properties:

Table 1 displays the approximate frequencies of cases with internal fixed points for (3) with randomly chosen fitness matrices and niche proportions for . More precisely, for each row 107 fitness configurations were created by choosing fitness values and niche proportions uniformly -distributed (which is no restriction, because taking multiples of fitness matrices does not change the dynamics of the system). An algorithm following Sturm’s Theorem about zeros of polynomials produced the number of fixed points for each configuration. Note that already for the parameter region producing the maximum number of 5 equilibria is so small that only two examples with five internal fixed points occurred in 107 configurations. This gives a possible explanation why 30 years ago numerical examples for the maximum possible number of internal equilibria were not found by random searching.

Table 1

Approximate relative frequencies of parameter combinations yielding internal fixed points for . Note that a dash “—” signifies that this number of equilibria is impossible, whereas “0” means that no such number was found in 107 examples.

	i=0	i=1	i=2	i=3	i=4	i=5	i≥6
J=1	0.3333	0.6667	–	–	–	–	–
J=2	0.3183	0.6371	0.0405	0.0041	–	–	–
J=3	0.2949	0.6355	0.0608	0.0088	0.9×10−5	2.0×10−7	–
J=4	0.2699	0.6463	0.0703	0.0136	2.2×10−5	8.0×10−7	0