"The two-size Wright-Fisher model"Dopmeyer, HannahConsider a population with two types of individuals, where type is interpreted as size: large individuals are of size $1$ and small individuals are of size $1-\kappa$, $\kappa \in (0,1)$. In each generation there is a constant amount of $K$ resource units ($=$ space) available. To form a new generation, individuals from the current generation are sampled one by one, and if there is at least some available space, they reproduce and their offspring are added to the new generation. The probability of sampling an individual whose offspring is small is given by $\mu^K(x)$, where $x$ is the proportion of small individuals in the current generation. This mechanism allows the available space $K$ to be exceeded by at most $1$ resource unit and leads to varying population sizes. We call this stochastic model in discrete time the two-size Wright-Fisher model. The function $\mu^K$ can be used to model mutation and/or various forms of frequency-dependent selection. Denoting by $(X_t)_{t \geq 0}$ the frequency process of small individuals, we show convergence on the evolutionary time scale $Kt$ to the solution of the SDE $$\mathrm{d} X_t = \left(-\kappa X_t(1-X_t)+\mu(X_t)\right)\mathrm{d} t + \sqrt{X_t(1-X_t)(1-\kappa X_t)}\, \mathrm{d} B_t,$$ where $\mu(x)=\lim_{K \to \infty} K(\mu^K(x)-x)$ and $B_t$ is a standard Brownian motion. The main difference from other models is that the parameter $\kappa$ appears in the drift, but also in the diffusion, making it an intriguing evolutionary force to study. To prove this statement, methods from renewal theory are applied, with special attention to the stopping summand and its size-biased distribution. Properties of the diffusion are analyzed and in particular parameter regimes moment duality is established. Joint work with Fernando Cordero. |
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