Reduction of a metapopulation genetic model to an effective one island model

TL;DR

This paper develops a simplified one-island genetic model from a complex metapopulation framework using time-scale separation, accurately predicting fixation probabilities and times.

Contribution

It introduces a novel reduction technique that simplifies a multi-island metapopulation model to a single-variable model, enhancing analytical tractability.

Findings

Effective one-island model matches numerical simulations

Accurate predictions of fixation probabilities

Reliable estimates of mean fixation times

Abstract

We explore a model of metapopulation genetics which is based on a more ecologically motivated approach than is frequently used in population genetics. The size of the population is regulated by competition between individuals, rather than by artificially imposing a fixed population size. The increased complexity of the model is managed by employing techniques often used in the physical sciences, namely exploiting time-scale separation to eliminate fast variables and then constructing an effective model from the slow modes. Remarkably, an initial model with 2$\mathcal{D}$ variables, where $\mathcal{D}$ is the number of islands in the metapopulation, can be reduced to a model with a single variable. We analyze this effective model and show that the predictions for the probability of fixation of the alleles and the mean time to fixation agree well with those found from numerical simulations of the original model.