Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size

Abstract

We are interested in the long-time behavior of a diploid population with sexual reproduction and randomly varying population size, characterized by its genotype composition at one bi-allelic locus. The population is modeled by a 3-dimensional birth-and-death process with competition, weak cooperation and Mendelian reproduction. This stochastic process is indexed by a scaling parameter $K$ that goes to infinity, following a large population assumption. When the individual birth and natural death rates are of order $K$, the sequence of stochastic processes indexed by $K$ converges toward a new slow-fast dynamics with variable population size. We indeed prove the convergence toward 0 of a fast variable giving the deviation of the population from quasi Hardy–Weinberg equilibrium, while the sequence of slow variables giving the respective numbers of occurrences of each allele converges toward a 2-dimensional diffusion process that reaches (0,0) almost surely in finite time. The population size and the proportion of a given allele converge toward a Wright-Fisher diffusion with stochastically varying population size and diploid selection. We insist on differences between haploid and diploid populations due to population size stochastic variability. Using a non trivial change of variables, we study the absorption of this diffusion and its long time behavior conditioned on non-extinction. In particular we prove that this diffusion starting from any non-trivial state and conditioned on not hitting (0,0) admits a unique quasi-stationary distribution. We give numerical approximations of this quasi-stationary behavior in three biologically relevant cases: neutrality, overdominance, and separate niches.

Appendices

Appendix 1: Extension to non-neutral leading-order dynamics

The scaling assumption presented in Eq. (3.1) and considered throughout the whole article can be extended as presented in this section. Let us consider $\gamma _1>0$, $\gamma _2>0$ and $\gamma _3>0$ such that $2\gamma _2=\gamma _1+\gamma _3$, and let us assume

$$\begin{aligned} \begin{aligned} b^{i,K}&= \gamma _i K+\beta _i\in [0,\infty [\\ d^{i,K}&= \gamma _i K+\delta _i\in [0,\infty [\\ c_{ij}^K&=\frac{\alpha _{ij}}{K}\in \mathbb {R}\\ Z^K_0&\underset{K\rightarrow \infty }{\rightarrow } Z_0 \quad \hbox {in law,} \end{aligned} \end{aligned}$$

where $Z_0$ is a $(\mathbb {R}_+)^3$-valued random variable. Under this more general scaling, the decomposition of the infinitesimal generator $L^K$ presented in Eq. (3.2) becomes:

$$\begin{aligned} L^Kf(z)= & {} K^2y\left[ \gamma _1f\left( z-\frac{e_1}{K}\right) -2\gamma _2f\left( z-\frac{e_2}{K}\right) +\gamma _3f\left( z-\frac{e_3}{K}\right) \right] \\&+\,\gamma _1 K^2n\,(x)^2\left[ f\left( z+\frac{e_1}{K}\right) +f\left( z-\frac{e_1}{K}\right) -2f\left( z\right) \right] \\&+\,\gamma _2 K^22nx(1-x)\left[ f\left( z+\frac{e_2}{K}\right) +f\left( z-\frac{e_2}{K}\right) -2f\left( z\right) \right] \\&+\,\gamma _3 K^2n\left( 1-x\right) ^2\left[ f\left( z+\frac{e_3}{K}\right) +f\left( z-\frac{e_3}{K}\right) -2f(z)\right] \\&+\,\beta _1Kn\,(x)^2\left[ f\left( z+\frac{e_1}{K}\right) -f(z)\right] \\&+\,\beta _2K2nx(1-x)\left[ f\left( z+\frac{e_2}{K}\right) -f(z)\right] \\&+\,\beta _3Kn(1-x)^2\left[ f\left( z+\frac{e_3}{K}\right) -f(z)\right] \\&+\,K\underset{i\in \{1,2,3\}}{\sum }\left( \delta _i+\underset{j\in \{1,2,3\}}{\sum }\alpha _{ji}z_j\right) z_i\left[ f\left( z-\frac{e_i}{K}\right) -f(z)\right] \\&+\,K\underset{i\in \{1,2,3\}}{\sum }\left( \gamma _i K+\delta _i+\underset{j\in \{1,2,3\}}{\sum }\alpha _{ji}z_j\right) ^-z_i\left[ f\left( z-\frac{e_i}{K}\right) -f(z)\right] . \end{aligned}$$

In this case, setting $\gamma =\inf _i \gamma _i$, we still obtain as in the proof of Proposition 2,

$$\begin{aligned} \begin{aligned} \frac{d\mathbb {E}\left( (Y^K_t)^2\right) }{dt}&\le -2\gamma K\mathbb {E}\left( (Y^K_t)^2\right) +C \end{aligned} \end{aligned}$$

and therefore the convergence toward quasi Hardy–Weinberg equilibrium when $K\rightarrow \infty$. As in Theorem 1, the sequence of stochastic processes $(N^{A,K},N^{a,K})$ then converges toward a 2-dimensional diffusion process whose infinitesimal generator $L$ is defined for all function $f\in \mathcal {C}^2_b((\mathbb {R}_+)^2, \mathbb {R})$ for all $(n^A,n^a)\in (\mathbb {R}_+)^2$ by

$$\begin{aligned} Lf(n^A,n^a)= & {} \frac{n^A}{n^A+n^a}\Bigg [\Big (\beta _1-\delta _1-\frac{\alpha _{11}(n^A)^2+\alpha _{21}2n^An^a+\alpha _{31}(n^a)^2}{2(n^A+n^a)}\Big )n^A\\&+\,\Big (\beta _2-\delta _2-\frac{\alpha _{12}(n^A)^2+\alpha _{22}2n^An^a+\alpha _{32}(n^a)^2}{2(n^A+n^a)}\Big )n^a\Bigg ]\frac{\partial f}{\partial n^A}(n^A,n^a)\\&+\,\frac{n^a}{n^A+n^a}\Bigg [\Big (\beta _3-\delta _3-\frac{\alpha _{13}(n^A)^2+\alpha _{23}2n^An^a+\alpha _{33}(n^a)^2}{2(n^A+n^a)}\Big )n^a\\&+\,\Big (\beta _2-\delta _2-\frac{\alpha _{12}(n^A)^2+\alpha _{22}2n^An^a+\alpha _{32}(n^a)^2}{2(n^A+n^a)}\Big )n^A\Bigg ]\frac{\partial f}{\partial n^a}(n^A,n^a)\\&+\,\frac{n^A}{2}\Bigg [\gamma _1\Big (1+\frac{3n^A}{n^A+n^a}\Big )+\gamma _3\Big (\frac{n^a}{n^A+n^a}\Big )\Bigg ]\frac{\partial ^2 f}{\partial (n^A)^2}(n^A,n^a)\\&+\,(\gamma _1+\gamma _3)\frac{n^An^a}{n^A+n^a}\frac{\partial ^2 f}{\partial n^A\partial n^a}(n^A,n^a)\\&+\,\frac{n^a}{2}\Bigg [\gamma _3\Big (1+\frac{3n^a}{n^A+n^a}\Big )+\gamma _1\Big (\frac{n^A}{n^A+n^a}\Big )\Bigg ]\frac{\partial ^2 f}{\partial (n^a)^2}(n^A,n^a). \end{aligned}$$

Appendix 2: Calculations in the general case

1.1 Form of the function $\varvec{Q}$

If $\alpha$ is symmetric, we use Eqs. (4.6), (4.7) and (4.8) and look for a function $Q$ such that $\frac{\partial Q(s)}{\partial s_1}=q_1(S)$ and $\frac{\partial Q(s)}{\partial s_2}=q_2(S)$. After calculating the partial derivatives of functions of the form:

$$\begin{aligned} (s_1,s_2)\mapsto \left\{ \begin{array}{l} ((s_1)^2+(s_2)^2)^k\cos ^l\left( \sqrt{2}\arctan \left( \frac{s_2}{s_1}\right) \right) \quad \hbox {if}\, s_1\ge 0 \\ ((s_1)^2+(s_2)^2)^k\cos ^l\left( \sqrt{2}\arctan \left( \frac{s_2}{s_1}+\pi \right) \right) \quad \hbox {if}\, s_1\le 0 \end{array}\right. \end{aligned}$$

for $k\in \{1,2\}$ and $l\in \{1,2,3,4\}$, we find that

$$\begin{aligned} Q(s)=\left\{ \begin{array}{l} \frac{\ln ((s_1)^2+(s_2)^2)}{2}+\frac{1}{2}\ln \left( \sin \left( \sqrt{2}\arctan \left( \frac{s_2}{s_1}\right) \right) \right) \\ \quad -\frac{(s_1)^2+(s_2)^2}{4}\left[ \frac{\beta _1-\delta _1+2(\beta _2-\delta _2)+\beta _3-\delta _3}{4}-\frac{(s_1)^2+(s_2)^2}{4}\gamma \frac{\alpha _{11}+4\alpha _{12}+2\alpha _{13} +4\alpha _{23}+4\alpha _{22}+\alpha _{33}}{16}\right] \\ \quad -\,((s_1)^2+(s_2)^2)\left[ h(s)\frac{\beta _1-\delta _1-(\beta _3-\delta _3)}{8}+h(s)^2\frac{\beta _1-\delta _1-2(\beta _2-\delta _2)+\beta _3-\delta _3}{16}\right] \\ \quad +\,\frac{((s_1)^2+(s_2)^2)^2}{16}\gamma h(s)\left[ \frac{\alpha _{11}+2\alpha _{12}-2\alpha _{23}-\alpha _{33}}{4}+h(s)\frac{3\alpha _{11}-2\alpha _{13}-4\alpha _{22}+3\alpha _{33}}{8}\right. \\ \quad \left. +h(s)^2\frac{\alpha _{11}-2\alpha _{12}+2\alpha _{23}-\alpha _{33}}{4}+h(s)^3\frac{\alpha _{11}-4\alpha _{12} \quad +\,2\alpha _{13}-4\alpha _{23}+4\alpha _{22}+\alpha _{33}}{16}\right] \quad \hbox {if}\, s_1\ge 0 \\ \frac{\ln ((s_1)^2+(s_2)^2)}{2}+\frac{1}{2}\ln \left( \sin \left( \sqrt{2}\left( \arctan \left( \frac{s_2}{s_1}\right) +\pi \right) \right) \right) \\ \quad -\frac{(s_1)^2+(s_2)^2}{4}\left[ \frac{\beta _1-\delta _1+2(\beta _2-\delta _2)+\beta _3-\delta _3}{4}-\frac{(s_1)^2+(s_2)^2}{4}\gamma \frac{\alpha _{11}+4\alpha _{12}+2\alpha _{13} +4\alpha _{23}+4\alpha _{22}+\alpha _{33}}{16}\right] \\ \quad -\,((s_1)^2+(s_2)^2)\left[ h(s)\frac{\beta _1-\delta _1-(\beta _3-\delta _3)}{8}+h(s)^2\frac{\beta _1-\delta _1-2(\beta _2-\delta _2)+\beta _3-\delta _3}{16}\right] \\ \quad +\,\frac{((s_1)^2+(s_2)^2)^2}{16}\gamma h(s)\left[ \frac{\alpha _{11}+2\alpha _{12}-2\alpha _{23}-\alpha _{33}}{4}+h(s)\frac{3\alpha _{11}-2\alpha _{13}-4\alpha _{22}+3\alpha _{33}}{8}\right. \\ \quad \left. +\,h(s)^2\frac{\alpha _{11}-2\alpha _{12}+2\alpha _{23}-\alpha _{33}}{4}+h(s)^3\frac{\alpha _{11}-4\alpha _{12} +2\alpha _{13}-4\alpha _{23}+4\alpha _{22}+\alpha _{33}}{16}\right] \quad \hbox {if}\, s_1\le 0 \end{array}\right. \nonumber \\ \end{aligned}$$

(5.1)

where

$$\begin{aligned} h(s)=\left\{ \begin{array}{l}\cos \left( \sqrt{2}\arctan \left( \frac{s_2}{s_1}\right) \right) \quad \hbox {when}\, s_1\ge 0 \\ \cos \left( \sqrt{2}\left( \arctan \left( \frac{s_2}{s_1}\right) +\pi \right) \right) \quad \hbox {when}\, s_1\le 0.\end{array}\right. \end{aligned}$$

1.2 Form of the function $\varvec{q}$

Therefore if $s_1\ge 0$:

$$\begin{aligned} q_1(s)= & {} \frac{s_1}{(s_1)^2+(s_2)^2}-\frac{s_2}{(s_1)^2+(s_2)^2}\frac{1}{\sqrt{2}\tan \left( \sqrt{2}\arctan \left( \frac{s_2}{s_1}\right) \right) }\nonumber \\&-\,s_1\left[ \frac{\beta _1-\delta _1+2(\beta _2-\delta _2)+\beta _3-\delta _3}{8}\right. \nonumber \\&\left. -\,\frac{(s_1)^2+(s_2)^2}{4}\gamma \frac{\alpha _{11}+4\alpha _{12}+2\alpha _{13}+4\alpha _{23}+4\alpha _{22}+\alpha _{33}}{16}\right] \nonumber \\&-\,2s_1\left[ h(s)\frac{\beta _1-\delta _1-(\beta _3-\delta _3)}{8}+h(s)^2\frac{\beta _1-\delta _1-2(\beta _2-\delta _2)+\beta _3-\delta _3}{16}\right] \nonumber \\&+\,s_1\frac{((s_1)^2+(s_2)^2)}{4}\gamma h(s)\left[ \frac{\alpha _{11}+2\alpha _{12}-2\alpha _{23}-\alpha _{33}}{4}\right. \nonumber \\&+\,h(s)\frac{3\alpha _{11}-2\alpha _{13}-4\alpha _{22}+3\alpha _{33}}{8}\nonumber \\&+\,\left. h(s)^2\frac{\alpha _{11}-2\alpha _{12}+2\alpha _{23}-\alpha _{33}}{4}+h(s)^3\frac{\alpha _{11}\!-\!4\alpha _{12}+\!2\alpha _{13}-\!4\alpha _{23}+4\alpha _{22}\alpha _{33}}{16}\right] \nonumber \\&-\,\sqrt{2}s_2\sin \left( \sqrt{2}\arctan \left( \frac{s_2}{s_1}\right) \right) \left[ \frac{\beta _1-\delta _1-(\beta _3-\delta _3)}{8}\right. \nonumber \\&\left. +\,h(s)\frac{\beta _1-\delta _1-2(\beta _2-\delta _2)+\beta _3-\delta _3}{8}\right] \nonumber \\&+\,\frac{(s_1)^2+(s_2)^2}{16}\gamma \sqrt{2}s_2\sin \left( \sqrt{2}\arctan \left( \frac{s_2}{s_1}\right) \right) \left[ \frac{\alpha _{11}+2\alpha _{12}-2\alpha _{23}-\alpha _{33}}{4}\right. \nonumber \\&+\,h(s)\frac{3\alpha _{11}-2\alpha _{13}-4\alpha _{22}+3\alpha _{33}}{4}\nonumber \\&\left. +\,h(s)^2\frac{3(\alpha _{11}\!-2\alpha _{12}+\alpha _{23}\!-\alpha _{33})}{4}\right. \nonumber \\&\left. +\,h(s)^3\frac{\alpha _{11}-4\alpha _{12}+2\alpha _{13}-4\alpha _{23}+4\alpha _{22}+\alpha _{33}}{4}\right] . \end{aligned}$$

(5.2)

We have similar formulas for $q_2$ and when $s_1\le 0$.

1.3 Proof of Proposition 8

Recall that

$$\begin{aligned} F(s)=\vert \nabla Q(s)\vert ^2-{\varDelta } Q(s)=(q_1(s))^2+(q_2(s))^2-\frac{\partial q_1}{\partial s_1}(s)-\frac{\partial q_2}{\partial s_2}(s). \end{aligned}$$

Besides, note that under (H1), $\frac{\alpha _{11}+4\alpha _{12}+2\alpha _{13}+4\alpha _{23}+4\alpha _{22}+\alpha _{33}}{16}>0$. Therefore using Eqs. (4.6) and (4.7) we easily obtain that there exists a positive constant $C_1$ such that $(q_1(s))^2+(q_2(s))^2\ge C_1((s_1)^2+(s_2)^2)^3$. Finally, from Eq. (5.2), we obtain after some calculations that there exists a positive constant $C_2$ such that $\frac{\partial q_1}{\partial s_1}(s)+\frac{\partial q_2}{\partial s_2}(s)\le C_2((s_1)^2+(s_2)^2)^2$. Therefore Proposition 8 is true if $s_1\ge 0$. If $s_1\le 0$, the result is true as well by symmetry.