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

Abstract

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.

1. 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 $J\ge 1$ demes and is large enough so that we can ignore stochastic fluctuations. With $n$ alleles, $A_1,\dots ,A_n$, at a given locus, we are interested in the evolution in discrete time (nonoverlapping generations) of the genetic composition

$$p^{\left(j\right)}={\left(p_i^{\left(j\right)}\right)}_{i=1}^n\in \Delta =\{p^{\left(j\right)}\in {\mathbb{R}}^n:p^{\left(j\right)}\ge 0,\sum _{i=1}^n{p}_i^{\left(j\right)}=1\}$$

of the population, where $p_i^{\left(j\right)}$ denotes the relative frequency of allele $i$ in deme $j$ at zygote stage ($j=1,\dots ,J$). We denote by $a_{ik}^{\left(j\right)}$the fitness (viability) of an $A_i{A}_k$ individual in deme $j$, where we require $a_{ik}^{\left(j\right)}=a_{ki}^{\left(j\right)}$. For each deme $j$ we collect the fitness values $a_{ik}^{\left(j\right)}$ in symmetric fitness matrices $A^{\left(j\right)}={\left(a_{kl}^{\left(j\right)}\right)}_{k,l=1}^n$. In line with the standard selection model, the gene frequencies in deme $j$ after selection, denoted by ${\left(p^{\left(j\right)}\right)}^{\ast }$, are given by

$${\left(p_i^{\left(j\right)}\right)}^{\ast }=p_i^{\left(j\right)}\frac{{\left(A^{\left(j\right)}{p}^{\left(j\right)}\right)}_i}{{\left(p^{\left(j\right)}\right)}^T{A}^{\left(j\right)}{p}^{\left(j\right)}}\text{,}$$ (1a)

where ${\left(A^{\left(j\right)}{p}^{\left(j\right)}\right)}_i={\sum }_{k=1}^n{a}_{ik}^{\left(j\right)}{p}_k^{\left(j\right)}$ is the mean fitness of individuals in deme $j$ carrying allele $A_i$, and ${\left(p^{\left(j\right)}\right)}^T{A}^{\left(j\right)}{p}^{\left(j\right)}={\sum }_{k,l=1}^n{a}_{kl}^{\left(j\right)}{p}_k^{\left(j\right)}{p}_l^{\left(j\right)}$ is the mean fitness of the subpopulation in deme $j$. After selection adults migrate independently of their genotypes. For the gene frequencies in the next generation, ${\left(p^{\left(j\right)}\right)}^{\prime }$, we obtain

$${\left(p_i^{\left(j\right)}\right)}^{\prime }=\sum _{k=1}^J{m}_{kj}{\left(p_i^{\left(k\right)}\right)}^{\ast }\text{,}$$ (1b)

where the $m_{kj}$ denote the backward migration rates, i.e., the probability that an individual in deme $j$ migrated from deme $k$.

The central assumption of the Levene model is that migration rates do not depend on the deme of origin. Hence we set $m_{kj}=c_j$ for all $k,j$. 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 $\left(j\right)$ of the allele frequencies and write $p_i$ instead of $p_i^{\left(j\right)}$. Therefore, we can represent the recursion for the Levene model as

$$p_i^{\prime }=p_i\sum _{j=1}^J{c}_j\frac{{\left(A^{\left(j\right)}p\right)}_i}{p^T{A}^{\left(j\right)}p}\text{,}i=1,\dots ,n\text{.}$$ (2)

If $n=2$, i.e., for two alleles, this recursion reduces to a single equation. We set $x=p_1$, hence $p_2=1-x$ and $x={(x,1-x)}^T$. Then (2) simplifies to

$$x^{\prime }=F\left(x\right)\text{,}$$ (3a)

where

$$F\left(x\right)=x\sum _{j=1}^J{c}_j\frac{{\left(A^{\left(j\right)}x\right)}_1}{x^T{A}^{\left(j\right)}x}=x\sum _{j=1}^J{c}_j\frac{a_{11}^{\left(j\right)}x+a_{12}^{\left(j\right)}(1-x)}{a_{11}^{\left(j\right)}{x}^2+2x(1-x)a_{12}^{\left(j\right)}+a_{22}^{\left(j\right)}{(1-x)}^2}\text{.}$$ (3b)

Determining the fixed points of (3) means solving $x=F\left(x\right)$ which, after multiplying by the denominators, takes the form

$$0=x(1-x)P\left(x\right)\text{,}$$

where $P$ is a polynomial of degree $\le 2J-1$. Hence the maximum possible number of internal equilibria of (3) is $2J-1$. Whether this upper bound can be attained for arbitrary $J$ has been an open problem so far. The situation is clear only for $J=1,2$. Then, every possible number ($\le 2J-1$) of fixed points with a feasible¹ stability configuration is known to occur in concrete examples. Furthermore, Karlin (1977) presents several examples for $J=3$ producing four internal fixed points, as well as one configuration for $J=7$ with five.

2. Preparatory results

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

Definition 2.1

Consider the function $F:[0,1]\to [0,1]$. We call $F$, or the dynamical system $x^{\prime }=F\left(x\right)$, symmetric if $F\left(x\right)=1-F(1-x)$ for all $x\in [0,1]$.

A simple observation gives.

Lemma 2.2

Let $F_j$, $j=1,\dots ,J$, be symmetric and ${\sum }_j{\epsilon }_j=1$, ${\epsilon }_j\ge 0\forall j$ . Then ${\sum }_j{\epsilon }_j{F}_j$ is symmetric.

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

Lemma 2.3

For $\zeta \ge 1$ and any of the matrices

$$\mathrm{(a)}Z=\left(\begin{array}{cc}\hfill \zeta \hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\text{,}\mathrm{(b)}Z=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill \zeta \hfill \end{array}\right)\text{,}\mathrm{(c)}Z=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \zeta \hfill \\ \hfill \zeta \hfill & \hfill 1\hfill \end{array}\right)\text{,}\mathrm{(d)}Z=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \zeta \hfill \\ \hfill \zeta \hfill & \hfill \zeta \hfill \end{array}\right)$$

define the function $s_{\zeta }:[0,1]\to [0,1]$ by

$$s_{\zeta }\left(x\right)=x\frac{{\left(Zx\right)}_1}{x^{T}Zx}=x\frac{z_{11}x+z_{12}(1-x)}{z_{11}{x}^2+2x(1-x)z_{12}+z_{22}{(1-x)}^2}$$

(compare Eq. (3b)), where $z_{ij}=z_{ij}\left(\zeta \right)$ is the entry in row $i$ and column $j$ of the matrix $Z$ . Then, $\frac{d}{dx}{s}_{\zeta }\left(x\right)$ is bounded for $\zeta \in [1,\infty )$ and $x\in [a,1-a]$, $0<a<\frac{1}{2}$.

Proof

Differentiating yields

$$\frac{d}{dx}{s}_{\zeta }\left(x\right)=\frac{z_{11}{z}_{12}{x}^2+2z_{11}{z}_{22}x(1-x)+z_{12}{z}_{22}{(1-x)}^2}{{(z_{11}{x}^2+2x(1-x)z_{12}+z_{22}{(1-x)}^2)}^2}\text{.}$$

• (a)
  We set $g(x,\zeta )=\frac{d}{dx}{s}_{\zeta }\left(x\right)(=\frac{{(1-x)}^2+\zeta x(2-x)}{{(1-x^2+\zeta x^2)}^2})$ and show that

  $$g:[a,1-a]\times [1,\infty )\to \mathbb{R}$$

  is bounded for any $0<a<\frac{1}{2}$. Clearly, $g$ is continuous in both variables and nonnegative. Furthermore, for every $x\in [a,1-a]$ we have

  $$g(x,\zeta )\stackrel{\zeta \to \infty }{⟶}0$$

  because $g$ is a quotient of two polynomials where the numerator has a lower degree in $\zeta$ than the denominator. Some computations yield

  $$\frac{\partial }{\partial \zeta }g(x,\zeta )=0\iff \zeta =\zeta \left(x\right)=\frac{x^3-2x^2+3x-2}{x^2(x-2)}\text{,}$$

  which is bounded for $x\in [a,1-a]$ (Fig. 1). Set

  $$\overline{\zeta }=\underset{a\le x\le 1-a}{max}\zeta \left(x\right)\text{,}$$

  then, by the statements above, $g$ is decreasing if $\zeta >\overline{\zeta }$ (for any fixed $x\in [a,1-a]$). Since $[a,1-a]\times [1,\overline{\zeta }]$ is compact, there must be some $M>0$ such that $g(x,\zeta )\le M$ on $[a,1-a]\times [1,\overline{\zeta }]$ and therefore $g(x,\zeta )\le M$ on $[a,1-a]\times [1,\infty )$.

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

• (b)$\zeta \left(x\right)=\frac{x-x^2+x^3}{{(x-1)}^2(x+1)}$ (Note that this case can be obtained from (a) by the transformation $x\mapsto 1-x$.)
• (c)$\zeta \left(x\right)=\frac{1-4x+8x^3-4x^4}{2x(1-3x+4x^2-2x^3)}$ and
• (d)$\zeta \left(x\right)=\frac{2x^2-x^3}{2-3x+2x^2-x^3}$, where all denominators are $>0$ on the area of interest.

Remark

In case (d) we do not have $g(x,\zeta )\stackrel{\zeta \to \infty }{⟶}0$, but

$$g(x,\zeta )=\frac{\zeta x(2-x)+{\zeta }^2{(1-x)}^2}{{(x^2+\zeta (1-x^2))}^2}\stackrel{\zeta \to \infty }{⟶}\frac{1}{{(1+x)}^2}\text{.}$$

Nevertheless, since $\frac{1}{{(1+x)}^2}<\infty$, the above arguments hold with some obvious adaptations. □

Fig. 1.

The function $\zeta \left(x\right)$ for the cases (a)–(d). The value $\overline{\zeta }$ is attained at either of the endpoints $x=a$ or $x=1-a$ of the interval of interest. We note that the maximum of $\frac{d}{dx}{s}_{\zeta }\left(x\right)$ is located on the solid boundary of the gray filled area.

3. Case I: both monomorphic equilibria are stable

In this section we treat the case of asymptotically stable monomorphic equilibria $x=0$ and $x=1$. The idea is the following: We start out by choosing $F$ in (3b) with $J\ge 1$ demes and $k$ 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 $J+2$ demes has $k+4$ 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 $s_{\zeta }$, which behaves sufficiently well on a compact interval containing all internal fixed points of $F$ 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 $\zeta ⟶\infty$, $s_{\zeta }$ becomes discontinuous at $x=0$ and $x=1$, which allows us to generate additional fixed points near the end points of $[0,1]$.

We consider (3) for given $J\ge 1$ and suppose that

• •$F$ is symmetric,
• •$x=0$ and $x=1$ are asymptotically stable, i.e., $\left|\frac{d}{dx}F{|}_{x=0}\right|<1$, $\left|\frac{d}{dx}F{|}_{x=1}\right|<1$, and
• •
  $F$ has $k$ hyperbolic, internal fixed points.² We label them

  $$0<x_1<\cdots <x_k<1\text{.}$$

Note that under these conditions two adjacent fixed points always have opposite stability properties; i.e., $x_1$ is repelling, $x_2$ is attracting, …, $x_k$ is repelling. In particular, $k$ must be odd.

For $\zeta \ge 1$ define the following two fitness matrices

$$Z^{(J+1)}=\left(\begin{array}{cc}\hfill \zeta \hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\text{,}{Z}^{(J+2)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill \zeta \hfill \end{array}\right)\text{.}$$

Then, for $0<\epsilon <1$, define the recursion

$$x^{\prime }=F_{\epsilon ,\zeta }\left(x\right)=(1-\epsilon )F\left(x\right)+\epsilon s_{\zeta }\left(x\right)\text{,}$$ (4)

where

$$s_{\zeta }\left(x\right)=\frac{x}{2}(\frac{{\left(Z^{(J+1)}x\right)}_1}{x^T{Z}^{(J+1)}x}+\frac{{\left(Z^{(J+2)}x\right)}_1}{x^T{Z}^{(J+2)}x})\text{.}$$

Clearly, $F_{\epsilon ,\zeta }$ is of the form (3b) with $J+2$ demes. The perturbation function $s_{\zeta }$ is symmetric and therefore, by Lemma 2.2, $F_{\epsilon ,\zeta }$ is symmetric. Furthermore, $s_{\zeta }$ is continuous in $\zeta$. Thus, by choosing $\epsilon$ sufficiently small we can ensure that for all $\zeta \ge 1$ the new dynamical system (4) is similar to (3) in the following sense:

• •
  Since $s_{\zeta }\left(0\right)=0$, the point $x=0$ remains a fixed point (and by symmetry of $F_{\epsilon ,\zeta }$ the same holds for $x=1$). Moreover, these monomorphic states remain stable because

  $${\frac{d{s}_{\zeta }}{dx}|}_{x=0}={\frac{d{s}_{\zeta }}{dx}|}_{x=1}=\frac{1}{2}(1+\frac{1}{\zeta })\text{,}$$

  which clearly is bounded for $\zeta \ge 1$.
• •Since $s_{\zeta }$ is continuous and bounded ($0\le s_{\zeta }\le 1\forall \zeta \ge 1$) on $[0,1]$, $F_{\epsilon ,\zeta }$ has $k$ fixed points ${\tilde{x}}_i$, $i=1,\dots ,k$, where ${\tilde{x}}_i$ is close to the fixed point $x_i$ of $F$ for each $i$. In particular, all ${\tilde{x}}_i$ are within the interval $[a,1-a]$ for some sufficiently small $a>0$.
• •Since $\frac{d{s}_{\zeta }}{dx}$ is uniformly bounded in $\zeta \in [0,\infty )$ for $x\in [a,1-a]$ (see Lemma 2.3(a) and (b)), the ${\tilde{x}}_i$ are unique, i.e., no additional equilibria emerge in $[a,1-a]$. In particular, $\frac{d{F}_{\epsilon ,\zeta }}{dx}{|}_{x=\widetilde{x_i}}$ is close to $\frac{dF}{dx}{|}_{x=x_i}$ and therefore, the local stability properties of the equilibria ${\tilde{x}}_i$, $i=1,\dots ,k$, are the same as the corresponding $x_i$’s; i.e., ${\tilde{x}}_i$ is repelling for $i$ odd and attracting for $i$ even.

By a short calculation we find $s_{\zeta }\left(x\right)\to \frac{1}{2}$ if $\zeta \to \infty$ for all $x\in (0,1)$ (compare Fig. 2), which motivates the next step: Set ${\epsilon }^{\ast }=min(\epsilon ,x_1)$ and fix a $\delta \in (0,\frac{{\epsilon }^{\ast }}{2})$. Since $s_{\zeta }\left(x\right)\to \frac{1}{2}$ ($\zeta \to \infty$) we can find a ${\zeta }^{\ast }>1$ such that

$$\delta <{\epsilon }^{\ast }{s}_{{\zeta }^{\ast }}\left(\delta \right)\text{.}$$

Because $s_{\zeta }\left(x\right)\nearrow \frac{1}{2}$ for $x\in (0,\frac{1}{2})$ and ${\tilde{x}}_1$ is close to $x_1$, we conclude

$$\delta <{\epsilon }^{\ast }{s}_{{\zeta }^{\ast }}\left(\delta \right)<\frac{{\epsilon }^{\ast }}{2}<{\tilde{x}}_1\text{.}$$

Therefore we have $F_{{\epsilon }^{\ast },{\zeta }^{\ast }}\left(\delta \right)=(1-{\epsilon }^{\ast })F\left(\delta \right)+{\epsilon }^{\ast }{s}_{{\zeta }^{\ast }}\left(\delta \right)>\delta$ and $\delta \in (0,{\tilde{x}}_1)$. From the stability configuration above (i.e., 0 is repelling, ${\tilde{x}}_1$ attracting) we know that $F_{{\epsilon }^{\ast },{\zeta }^{\ast }}\left(x\right)<x$ on an interval $(0,{\epsilon }_1)$ as well as on $({\tilde{x}}_1-{\epsilon }_2,{\tilde{x}}_1)$ for suitable ${\epsilon }_1,{\epsilon }_2>0$. Thus, by the intermediate value theorem it follows that two additional fixed points exist in $(0,{\tilde{x}}_1)$. Since $F_{{\epsilon }^{\ast },{\zeta }^{\ast }}$ is symmetric, we automatically have two additional equilibria in $({\tilde{x}}_k,1)$ as well. Therefore we have proved

Fig. 2.

The function $s_{\zeta }$ for some values of $\zeta$. Note that $\frac{d{s}_{\zeta }}{dx}{|}_{x=0}=\frac{d{s}_{\zeta }}{dx}{|}_{x=1}<1$ for all $\zeta >1$, which becomes visible only after zooming in.

Lemma 3.1

Let $F$ be given by (3b) in $J$ demes with $k$ internal fixed points and suppose that $F$ is symmetric and $x=0$ and $x=1$ are asymptotically stable. Then we can always find $F_{\epsilon ,\zeta }$ with $J+2$ demes and $k+4$ internal fixed points, which is symmetric, and $x=0$ as well as $x=1$ are asymptotically stable.

From this we directly get

Proposition 3.2

For (3), the theoretical upper bound of $2J-1$ internal fixed points can be achieved.

Proof

By induction: For $J=1,2$ the configurations

$$A=\left(\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right)\text{,}\{A^{\left(1\right)}=\left(\begin{array}{cc}\hfill 6\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right),A^{\left(2\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 6\hfill \end{array}\right),c_1=c_2=\frac{1}{2}\}$$

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 $J>2$. □

4. 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.

Proposition 4.1

The theoretical upper bound of $2J-1$ internal fixed points in (3) can be realized with both alleles protected, i.e., with repelling monomorphisms $x=0$ and $x=1$.

Proof

For $J=1$, symmetric dynamics with protected alleles and one internal fixed point are known to exist (e.g., consider $A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right)$).

The following procedure is the same as in the argumentation leading to Lemma 3.1: Let $F$ be defined by (3) for some $J\ge 1$ such that $x=0$ and $x=1$ are attracting. Furthermore, assume that $F$ is symmetric and (3a) bears the maximum possible number of $2J-1$ internal fixed points. Such $F$ exists by Proposition 3.2.

Consider Eq. (4) with

$$s_{\zeta }\left(x\right)=x\frac{{\left(Zx\right)}_1}{x^{T}Zx}\text{,}$$

where $Z=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \zeta \hfill \\ \hfill \zeta \hfill & \hfill 1\hfill \end{array}\right)$, $\zeta \ge 1$. The idea here is to introduce one additional niche that favors heterozygotes and therefore inhibits the loss of either allele.

We find that $s_{\zeta }$ is symmetric, continuous, and $s_{\zeta }\left(x\right)\to \frac{1}{2}$ on $(0,1)$ for $\zeta \to \infty$. Furthermore, $s_{\zeta }\le 1$ and the derivative $\frac{d{s}_{\zeta }}{dx}$ is bounded for $\zeta \ge 1$ on every compact interval $[a,1-a]$, $0<a<\frac{1}{2}$ (Lemma 2.3(c)). Therefore, by choosing $\epsilon$ small and $\zeta$ large–for the things to come $\zeta >\frac{1}{\epsilon }$ will suffice–we obtain analogously to the previous construction that all internal fixed points of $F$ and their stability properties are conserved. In particular, ${\tilde{x}}_1$ remains a repellor. On the other hand, $\frac{d{s}_{\zeta }}{dx}{|}_{x=0}=\zeta$ and thus, by the choice of $\zeta$ above, we have

$${\frac{d{F}_{\epsilon ,\zeta }}{dx}|}_{x=0}=(1-\epsilon ){\frac{dF}{dx}|}_{x=0}+\epsilon \zeta >0+\epsilon \frac{1}{\epsilon }=1\text{.}$$

Hence, ${\tilde{x}}_0=0$ is also a repellor and it follows that there is an additional fixed point in $(0,{\tilde{x}}_1)$. Since $F_{\epsilon ,\zeta }$ is symmetric we automatically get another equilibrium in $({\tilde{x}}_{2J-1},1)$, and therefore have constructed a dynamics with protected alleles, $J+1$ demes and $2J+1$ internal fixed points. □

5. 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 $J\ge 1$ we can find dynamics with $2k-1$, $k\in \{1,\dots ,J\}$, 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 $F$ be defined by (3), with $2k-1$ hyperbolic internal equilibria for some $k\in \{1,\dots ,J\}$. Furthermore, presume that $x=0$ and $x=1$ are attractors.

For $Z=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \zeta \hfill \\ \hfill \zeta \hfill & \hfill \zeta \hfill \end{array}\right)$, $\zeta \ge 1$, $\epsilon$ sufficiently small and $s_{\zeta }\left(x\right)=x\frac{{\left(Zx\right)}_1}{x^{T}Zx}$ consider Eq. (4). Lemma 2.3(d) and some basic algebra show that

• •$s_{\zeta }$ is monotone, continuous, $s_{\zeta }\le 1$, and $\frac{d{s}_{\zeta }}{dx}$ is uniformly bounded in $\zeta$ on every compact interval $[a,1-a]$, $0<a<\frac{1}{2}$,
• •$\frac{d{s}_{\zeta }}{dx}{|}_{x=0}=1$ and $\frac{d{s}_{\zeta }}{dx}{|}_{x=1}=\zeta$.

By the first point we may assume (making $\epsilon$ 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 $x=0$ being an attractor remains unchanged as well. On the other hand, enlarging $\zeta$ we get arbitrarily large values for $\frac{d{s}_{\zeta }}{dx}{|}_{x=1}$ and hence also for $\frac{d{F}_{\epsilon ,\zeta }}{dx}{|}_{x=1}$, eventually making $x=1$ 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 $s_{\zeta }$.

To sum up, we have a dynamics with $J+1$ demes and $2k$ ($k\in \{1,\dots ,J\}$) internal fixed points. Putting this together with what we know from the previous sections we get.

Theorem

If $n=2$, the Levene model (3) allows for any number $k\in \{1,\dots ,2J-1\}$ of hyperbolic internal fixed points with any feasible stability configuration.

6. Numerical examples

In this section, we back up some of our results by numerical examples. Consider the configuration

$$A^{\left(1\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right)\text{,}{A}^{\left(2\right)}=\left(\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\text{,}{A}^{\left(3\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill \end{array}\right)\text{,}$$

$$c_1=0.26\text{,}{c}_2=0.37\text{,}{c}_3=0.37\text{,}$$

then Eq. (3) has fixed points at approximately

$$x_0=0\text{,}{x}_1\approx 0.049\text{,}{x}_2\approx 0.308\text{,}{x}_3\approx 0.5\text{,}{x}_4\approx 0.692\text{,}{x}_5\approx 0.951\text{,}{x}_6=1\text{,}$$

where $x_1$, $x_3$, and $x_5$ are asymptotically stable. The reverse stability configuration can be found by setting

$$A^{\left(1\right)}=\left(\begin{array}{cc}\hfill 4\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 4\hfill \end{array}\right)\text{,}{A}^{\left(2\right)}=\left(\begin{array}{cc}\hfill 25\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\text{,}{A}^{\left(3\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 25\hfill \end{array}\right)\text{,}$$

$$c_1=0.52\text{,}{c}_2=0.24\text{,}{c}_3=0.24\text{.}$$

Inserting this into (3) produces equilibria at

$$x_0=0\text{,}{x}_1\approx 0.205\text{,}{x}_2\approx 0.328\text{,}{x}_3\approx 0.5\text{,}{x}_4\approx 0.672\text{,}{x}_5\approx 0.795\text{,}{x}_6=1\text{,}$$

where now $x_0$, $x_2$, $x_4$, and $x_6$ are asymptotically stable.

If $J=4$, we set

$$A^{\left(1\right)}=\left(\begin{array}{cc}\hfill 4\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 4\hfill \end{array}\right)\text{,}{A}^{\left(2\right)}=\left(\begin{array}{cc}\hfill 25\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\text{,}{A}^{\left(3\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 25\hfill \end{array}\right)\text{,}{A}^{\left(4\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 100\hfill \\ \hfill 100\hfill & \hfill 1\hfill \end{array}\right)\text{,}$$

$$c_1=0.51\text{,}{c}_2=0.24\text{,}{c}_3=0.24\text{,}{c}_4=0.01$$

and obtain the configuration: Here, filled circles “$•$” represent stable fixed points, whereas unfilled rings “$\circ$” stand for repellors.

As for the reversed stability properties:

$$A^{\left(1\right)}=\left(\begin{array}{cc}\hfill 90\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\text{,}{A}^{\left(2\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 90\hfill \end{array}\right)\text{,}{A}^{\left(3\right)}=\left(\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\text{,}{A}^{\left(4\right)}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill \end{array}\right)\text{,}$$

$$c_1=c_2=0.0564\text{,}{c}_3=c_4=0.4436$$

Table 1 displays the approximate frequencies of cases with $i$ internal fixed points for (3) with randomly chosen fitness matrices and niche proportions for $J=1,\dots ,4$. More precisely, for each row 10⁷ fitness configurations were created by choosing fitness values and niche proportions uniformly $[0,1]$-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 $J=3$ 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 $i$ internal fixed points for $n=2$. Note that a dash “—” signifies that this number of equilibria is impossible, whereas “0” means that no such number was found in 10⁷ examples.

	$i=0$	$i=1$	$i=2$	$i=3$	$i=4$	$i=5$	$i\ge 6$
$J=1$	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