Survival and extinction results for a patch model with sexual reproduction

TL;DR

This paper analyzes a spatially structured population model with sexual reproduction, identifying conditions for survival and extinction, and finds that intermediate dispersal ranges optimize population spread.

Contribution

It introduces a mean-field limit for a patch-based sexual reproduction model and derives explicit survival and extinction criteria, linking stochastic and deterministic dynamics.

Findings

Survival and extinction conditions are explicitly characterized.

Population survives with nearest neighbor dispersal but not with long-range dispersal.

Intermediate dispersal ranges are optimal for population spread.

Abstract

This article is concerned with a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents' patch or are sent to a nearby patch with some fixed probabilities. As the patch size tends to infinity, we identify a mean-field limit consisting of an infinite set of coupled differential equations. For the mean-field equations, we find explicit conditions for survival and extinction that we call expansion and retreat. Using duality techniques to compare the stochastic model to its mean-field limit, we find that expansion and retreat are also precisely the conditions needed to ensure survival and extinction of the stochastic model when the patch size is large. In addition, we study the dependence of survival on the dispersal range. We find that, with probability close to one and for a certain set of parameters, the metapopulation survives in the presence of nearest neighbor interactions while it dies out in the presence of long range interactions, suggesting that the best strategy for the population to spread in space is to use intermediate dispersal ranges.