Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-28T07:55:20.766Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluid limit for the Poisson encounter-mating model

Part of: Limit theorems Markov processes

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
Get access Rights & Permissions [Opens in a new window]

Abstract

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 spaces 60F15: Strong theorems
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1201 - 1229
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bomze, I. M. (1983). Lotka–Volterra equation and replicator dynamics: a two-dimensional classification. Bio. Cybernetics 48, 201–211. CrossRefGoogle Scholar
[2] Courtiol, A. et al. (2016). The evolution of mutual mate choice under direct benefits. Amer. Naturalist 188, 521–538. CrossRefGoogle ScholarPubMed
[3] Dietz, K. and Hadeler, K. P. (1988). Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1–25. CrossRefGoogle ScholarPubMed
[4] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York. CrossRefGoogle Scholar
[5] Etienne, L., Rousset, F., Godelle, B. and Courtiol, A. (2014). How choosy should I be? The relative searching time predicts evolution of choosiness under direct sexual selection. Proc. R. Soc. B 281, 20140190. CrossRefGoogle ScholarPubMed
[6] Ewens, W. J. (2004). Mathematical Population Genetics. I. Theoretical Introduction (Interdisciplinary Appl. Math. 27), 2nd edn. Springer, New York. Google Scholar
[7] Gillespie, J. H. (1998). Population Genetics: A Concise Guide. Johns Hopkins University Press, Baltimore, MD. Google Scholar
[8] Gilpin, M. E. (1979). Spiral chaos in a predator–prey model. Amer. Naturalist 113, 306–308. Google Scholar
[9] Gimelfarb, A. (1988). Processes of pair formation leading to assortative mating in biological populations: encounter-mating model. Amer. Naturalist 131, 865–884. CrossRefGoogle Scholar
[10] Gün, O. and Yilmaz, A. (2017). The stochastic encounter-mating model. Acta Appl. Math. 148, 71–102. Google Scholar
[11] Hofbauer, J. and Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press. Google Scholar
[12] Kirkpatrick, M. (1982). Sexual selection and the evolution of female choice. Evolution 36, 1–12. Google Scholar
[13] Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7, 49–58. CrossRefGoogle Scholar
[14] Kurtz, T. G. (1978). Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl. 6, 223–240. Google Scholar
[15] Lande, R. (1981). Models of speciation by sexual selection on polygenic traits. Proc. Nat. Acad. Sci. USA 78, 3721–3725. Google Scholar
[16] Lee, A. L., Engen, S. and Sæther, B.-E. (2008). Understanding mating systems: a mathematical model of the pair formation process. Theoret. Pop. Biol. 73, 112–124. Google Scholar
[17] Taylor, C. E. (1975). Differences in mating propensities: some models for examining the genetic consequences. Behavior Genetics 5, 381–393. Google Scholar
[18] Taylor, P. D. and Jonker, L. B. (1978). Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156. CrossRefGoogle Scholar