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

TL;DR

This paper analyzes a two-resource type Wright--Fisher model, demonstrating that the frequency of small individuals converges to a Wright--Fisher SDE under certain scaling, using renewal theory techniques.

Contribution

It introduces a novel asymptotic analysis of a two-size resource model using uniform renewal theory, connecting discrete dynamics to continuous Wright--Fisher processes.

Findings

Frequency process converges to Wright--Fisher SDE

Drift accounts for resource-based bias and strategy

Diffusion term includes multiplicative factors from consumption strategy

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 $\rho_{R}(x)$, 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 $\rho_{R}$ 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 $\rho_{R}$ 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 $\tau(R)$ 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.