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Fluid limit for the Poisson encounter-mating model

Part of:Markov processesLimit theorems

Published online by Cambridge University Press:  17 November 2017

Onur Gün and
Atilla Yilmaz
Onur Gün*
Affiliation:
Weierstrass Institute
Atilla Yilmaz*
Affiliation:
Koç University
*
* Postal address: Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany. Email address: onur.guen@wias-berlin.de
** Postal address: Department of Mathematics, Koç University, 34450 Sarıyer, Istanbul, Turkey. Email address: atillayilmaz@ku.edu.tr
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Abstract

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Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this paper we study SEM models with Poisson firing times. First, we prove that the model enjoys a fluid limit as the population size diverges, that is, the stochastic dynamics converges to a deterministic system governed by coupled ordinary differential equations (ODEs). Then we convert these ODEs to the well-known Lotka–Volterra and replicator equations from population dynamics. Next, under the so-called fine balance condition which characterizes panmixia, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of female and male. Without the fine balance assumption, but under certain symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to characterize assortative mating.

Keywords

Density dependent population processpair formationassortative matingpanmixiamating preferencemating patternfluid approximationLotka–Volterra equationreplicator equation

MSC classification

Primary:92D25: Population dynamics (general)
Secondary:60J28: Applications of continuous-time Markov processes on discrete state spaces60F15: Strong theorems

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017, pp. 1201 - 1229
Copyright
Copyright © Applied Probability Trust 2017 

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