A two-size Wright--Fisher model: asymptotic analysis via uniform renewal theory

TL;DR

The paper analyzes a two-size Wright--Fisher model in which reproduction consumes a fixed resource $R$, producing a diffusion limit for the frequency of small individuals on an accelerated time scale. By linking one-step dynamics to a Bernoulli-type renewal process and employing uniform renewal theory, the authors prove that $X_{\lfloor Rt\rfloor}^{R}$ converges to a Wright--Fisher-type diffusion with drift $d(x)=-(1-\vartheta)x(1-x)+\rho(x)$ and diffusion coefficient $\sqrt{x(1-x)(1-(1-\vartheta)x)}$, where $\rho$ summarizes selection and mutation effects. A key contribution is the development of uniform renewal theorems and a uniform convergence analysis of the stopping-summand, which enable rigorous generator convergence and the diffusion limit. The work also explores a model variant without the stopping-bias term and discusses long-term behavior, including extinction probabilities and mean absorption times, under various selection and mutation regimes. Together, these results provide a rigorous bridge between resource-based reproductive strategies and diffusion limits in population genetics, correcting prior drift claims and clarifying the role of stopping-time bias.

Abstract

We consider a population with two types of individuals, distinguished by the resources required for reproduction: type-$0$ (small) individuals need a fractional resource unit of size $\vartheta \in (0,1)$, while type-$1$ (large) individuals require $1$ unit. The total available resource per generation is $R$. To form a new generation, individuals are sampled one by one, and if enough resources remain, they reproduce, adding their offspring to the next generation. The probability of sampling an individual whose offspring is small is $ρ_{R}(x)$, where $x$ is the proportion of small individuals in the current generation. We call this discrete-time stochastic model a two-size Wright--Fisher model, where the function $ρ_{R}$ can represent mutation and/or frequency-dependent selection. We show that on the evolutionary time scale, i.e. accelerating time by a factor $R$, the frequency process of type-$0$ individuals converges to the solution of a Wright--Fisher-type SDE. The drift term of that SDE accounts for the bias introduced by the function $ρ_{R}$ and the consumption strategy, the latter also inducing an additional multiplicative factor in the diffusion term. To prove this, the dynamics within each generation are viewed as a renewal process, with the population size corresponding to the first passage time $τ(R)$ above level $R$. The proof relies on methods from renewal theory, in particular a uniform version of Blackwell's renewal theorem for binary, non-arithmetic random variables, established via $\varepsilon$-coupling.

Figures (7)

• Figure 1: Simulation of the population size $M_{n}^R$ (left) and the proportion of type-$0$ individuals $X_{n}^R$ (right) for six populations of a two-size Wright--Fisher model with parameters ${R}=5$, ${\vartheta}=0.3$, $\rho_{R}(x)=x$ and $x_{0}=0.5$. The horizontal gray lines show the respective codomain.
• Figure 2: Illustration of a sample construction of generation $n+1$ in a two-size Wright--Fisher model. The two sizes of the rectangles correspond to the two types: small rectangles represent small individuals. Each individual in generation $n$ is assigned a color and a pattern for identification. The individuals from generation $n$ are sampled, and their offspring are placed one after another from left to right until the capacity $R$ is reached. In this example three individuals from generation $n$ place exactly one offspring in generation $n+1$, one individual places two offspring and two individuals do not place any offspring. Note that the offspring of the small yellow individual mutates from small to large.
• Figure 3: Simulations of the evolution of the proportion of small individuals in the finite model (left) with $\rho_{{R}}(x) = x$, and sample trajectories of the limiting SDE (right) with $\rho \equiv 0$. In both cases, the parameters are ${\vartheta} = 0.6$ and $x_{0} = 0.5$. The finite model was simulated for 3000 generations with $R = 3000$. SDE trajectories were generated using the Euler method with step size $h = 1/3000$.
• Figure 4: Approximation (blue) of $R\,{\mathbb E}_{x}[X_{1}^{R} - x_{0}]$ in the two-size Wright--Fisher model without selection (left) and with genic selection favoring small individuals (right). For each $x_{0} = i/100$, $i \in \{0,\dots,100\}$, we approximated $R\, {\mathbb E}_{x}[X_{1}^{R} - x_{0}]$ by simulating $R(X_{1}^{R} - x_{0})$$10^6$ times and computing the mean. The small individuals' size parameter was set to ${\vartheta} = 0.3$, and the resource capacity was $R = 1000$. The theoretical drift term $d(x_{0}) = (-(1 - \vartheta) + s)x_{0}(1 - x_{0})$ from the SDE \ref{['eq.SDE']} is plotted in green.
• Figure 5: Approximations (blue) of $R, {\mathbb E}_{x}[{\overline{X}}_{1}^{R} - x_{0}]$ (left) and $R, {\mathbb E}_{x}[({\overline{X}}_{1}^{R} - x_{0})^2]$ (right) for the variant of the two-size Wright--Fisher model with $\rho_{R}(x) = x$. For each $x_{0} = i/100$, $i \in \{0, \dots, 100\}$, we estimated the expectations by simulating $R({\overline{X}}_{1}^{R} - x_{0})$ and $R({\overline{X}}_{1}^{R} - x_{0})^2$ over $n_{\text{sim}}$ runs and computing the sample means. The size of small individuals was set to ${\vartheta} = 0.3$, and the resource capacity to $R = 1000$. The drift function $d(x)$ (left) and the diffusion coefficient $\sigma^2(x)$ (right) from the SDE in Theorem \ref{['thm:model variant']} are shown in green.

Theorems & Definitions (33)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.1
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.2
• Remark 2.8