A stochastic model for speciation by mating preferences

Abstract

Mechanisms leading to speciation are a major focus in evolutionary biology. In this paper, we present and study a stochastic model of population where individuals, with type a or A, are equivalent from ecological, demographical and spatial points of view, and differ only by their mating preference: two individuals with the same genotype have a higher probability to mate and produce a viable offspring. The population is subdivided in several patches and individuals may migrate between them. We show that mating preferences by themselves, even if they are very small, are enough to entail reproductive isolation between patches, and we provide the time needed for this isolation to occur as a function of the carrying capacity. Our results rely on a fine study of the stochastic process and of its deterministic limit in large population, which is given by a system of coupled nonlinear differential equations. Besides, we propose several generalisations of our model, and prove that our findings are robust for those generalisations.

A Technical results and reduction of the system

This section is dedicated to some technical results needed in the proofs, as well as the reduction of the system to the minimal number of effective parameters. We first prove the convergence when K goes to infinity of the sequence of rescaled processes $\mathbf {Z}^K$ to the solution of the dynamical system (7) stated in Lemma 1.

Proof (Proof of Lemma 1) The proof relies on a classical result presented in Chapter 11 of the book by Ethier and Kurtz (1986). Let $\mathbf {z}$ be in $\mathbb {N}^{\mathcal {E}}/K$. According to (2)–(5), the rescaled birth, death and migration rates

$$\begin{aligned} \widetilde{\lambda }_{\alpha ,i}(\mathbf {z})=\frac{1}{K}\lambda _{\alpha ,i}(K\mathbf {z})=\lambda _{\alpha ,i}(\mathbf {z}), \quad \widetilde{d}_{\alpha , i}(\mathbf {z})=\frac{1}{K}d^K_{\alpha ,i}(K\mathbf {z})=\left[ d+cz_{A,i}+cz_{a,i}\right] {z_{\alpha ,i}}, \end{aligned}$$

(67)

and

$$\begin{aligned} \widetilde{\rho }_{\bar{i}\rightarrow i}(\mathbf {z})=\frac{1}{K}\rho _{\bar{i}\rightarrow i}(K\mathbf {z})=\rho _{\bar{i}\rightarrow i}(\mathbf {z}), \quad (\alpha , i) \in {\mathcal {E}} \end{aligned}$$

are Lipschitz and bounded on every compact subset of $\mathbb {N}^{\mathcal {E}}$, and do not depend on the carrying capacity K.

Let $(Y_{\alpha ,i}^{(\lambda )},Y_{\alpha ,i}^{(d)},Y_{\alpha ,i}^{(\rho )},(\alpha ,i)\in {\mathcal {E}})$ be twelve independent standard Poisson processes. From the representation of the stochastic process $(\mathbf {N}^{K}(t),t\ge 0)$ in (6) we see that the stochastic process $(\bar{\mathbf {Z}}^{K}(t), t \ge 0)$ defined by

$$\begin{aligned} \bar{\mathbf {Z}}^{K}(t)= & {} \mathbf {Z}^K(0)\\&+ \underset{(\alpha ,i)\in {\mathcal {E}}}{\sum }\frac{\mathbf {e}_{\alpha ,i}}{K} \Big [{Y}_{\alpha ,i}^{(\lambda )}\Big ( \int _0^tK \widetilde{\lambda }_{\alpha ,i}(\bar{\mathbf {Z}}^{K}({s})) ds\Big )- {Y}_{\alpha ,i}^{(d)}\Big ( \int _0^t K \widetilde{d}_{\alpha ,i}(\bar{\mathbf {Z}}^{K}({s})) ds\Big )\Big ]\\&+ \underset{(\alpha ,i)\in {\mathcal {E}}}{\sum } \frac{(\mathbf {e}_{\alpha ,i}-\mathbf {e}_{\alpha ,\bar{i}}) }{K} Y_{\alpha ,i}^{(\rho )} \Big ( \int _0^t K \widetilde{\rho }_{\alpha ,i}(\bar{\mathbf {Z}}^{K}({s})) ds\Big ), \end{aligned}$$

has the same law as $(\mathbf {Z}^{K}(t), t \ge 0)$. Moreover, a direct application of Theorem 2.1 p 456 in the book by Ethier and Kurtz (1986) gives that $(\bar{\mathbf {Z}}^{K}(t), t \le T)$ converges in probability to $(\mathbf {z}^{(\mathbf {z}^0)}(t), t \le T)$ for the uniform norm. As a consequence, $(\mathbf {Z}^K(t), t \le T)$ converges in law to $(\mathbf {z}^{(\mathbf {z}^0)}(t), t \le T)$ for the same norm. But the convergence in law to a constant is equivalent to the convergence in probability to the same constant. The result follows.

We now recall a well known fact on branching processes which can be found in the book by Athreya and Ney (1972), p. 109.

Lemma 4

• Let $Z=(Z_t)_{t \ge 0}$ be a birth and death process with individual birth and death rates b and d. For $i \in \mathbb {Z}^+$, $T_i=\inf \{ t\ge 0, Z_t=i \}$ and $\mathbb {P}_i$ is the law of Z when $Z_0=i$. If $d\ne b \in \mathbb {R}_+^*$, for every $i\in \mathbb {Z}_+$ and $t \ge 0$,

  $$\begin{aligned} \mathbb {P}_{i}(T_0\le t )= \Big ( \frac{d(1-e^{(d-b)t})}{b-de^{(d-b)t}} \Big )^{i}. \end{aligned}$$

  (68)

As we mentioned in Sect. 2, it is possible to reduce the number of parameters b, c, d, p, $\beta$ by using a change of time and a scaling. Let us introduce the new variables

$$\begin{aligned} \tilde{z}_{\alpha ,i}(t):= \frac{c}{b} z_{\alpha ,i} \Big (\frac{t}{b}\Big ), \end{aligned}$$

for all $\alpha \in \{A,a\}$, $i \in \{1,2\}$ and $t \ge 0$, and the parameters

$$\begin{aligned} \tilde{d}:= \frac{d}{b}, \quad \tilde{p}:= \frac{p}{b}. \end{aligned}$$

Then the new variables satisfy the following dynamical system

$$\begin{aligned} \frac{d}{dt}\tilde{z}_{\alpha ,i}(t)= \tilde{z}_{\alpha ,i}\left[ \frac{\beta \tilde{z}_{\alpha ,i}+\tilde{z}_{\bar{\alpha },i}}{\tilde{z}_{\alpha ,i}+\tilde{z}_{\bar{\alpha },i}} -\tilde{d}-(\tilde{z}_{\alpha ,i}+\tilde{z}_{\bar{\alpha },i})- \tilde{p}\frac{\tilde{z}_{\bar{\alpha },i}}{\tilde{z}_{\alpha ,i}+\tilde{z}_{\bar{\alpha },i}}\right] +\tilde{p}\frac{\tilde{z}_{\alpha ,\bar{i}}\tilde{z}_{\bar{\alpha },\bar{i}}}{\tilde{z}_{\alpha ,\bar{i}} +\tilde{z}_{\bar{\alpha },\bar{i}}}, \end{aligned}$$

for $\alpha \in \{A,a\}$, $\bar{\alpha }=\{A,a\}\setminus \alpha$, $i \in \{1,2\}$ and $\bar{i}=\{1,2\} \setminus i$.