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.

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Acknowledgements

The authors would like to warmly thank Sylvie Méléard for her continual guidance during their respective thesis works. They would also like to thank Pierre Collet for his help on the theory of dynamical systems, Sylvain Billiard for many fruitful discussions on the biological relevance of their model, Violaine Llaurens for her help during the revision of the manuscript, and the anonymous reviewers for their constructive comments that greatly contributed to improve the final version of the paper. C. C. and C. S. are grateful to the organizers of “The Helsinki Summer School on Mathematical Ecology and Evolution 2012: theory of speciation” which motivated this work. This work was partially funded by the Chair “Modélisation Mathématique et Biodiversité” of VEOLIA-Ecole Polytechnique-MNHN-F.X, and was also supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.

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Correspondence to Hélène Leman.

A Technical results and reduction of the system

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 \(\bar{i}=\{1,2\} \setminus i\).

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Coron, C., Costa, M., Leman, H. et al. A stochastic model for speciation by mating preferences. J. Math. Biol. 76, 1421–1463 (2018). https://doi.org/10.1007/s00285-017-1175-9

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Keywords

  • Birth and death process with competition
  • Mating preference
  • Reproductive isolation
  • Dynamical systems

Mathematics Subject Classification

  • 60J27
  • 37N25
  • 92D40