WIAS Preprint No. 2039, (2014)

Fluid and diffusion limits for the Poisson encounter-mating model



Authors

  • Gün, Onur
  • Yilmaz, Atilla

2010 Mathematics Subject Classification

  • 92D25 60F37 60J28

Keywords

  • population dynamics, fluid limit, diffusion limit, Lotka-Volterra equations, replicator equations, pair formation, encounter-mating, assortative mating, random mating, heterogamy, panmixia, homogamy, monogamy, mating preferences, mating pattern, contingency table, Poisson process

Abstract

Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this article we study SEM with Poisson firing times. We prove that an infinite population corresponds to a fluid limit, i.e., the stochastic dynamics converges to a deterministic system governed by coupled ODEs. Moreover, we establish a functional central limit theorem and give a diffusion approximation for the model. Next, we convert the fluid limit ODEs to the well-known Lotka-Volterra and replicator equations from population dynamics. Under the so-called fine balance condition, which characterizes panmixia for finite populations, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of females and males. Without the fine balance assumption, but under some symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to fully characterize assortative mating.

Download Documents