Trait Evolution in two–sex Populations

P. Zwoleński

doi:10.1051/mmnp/20150611

P. Zwoleński ¹

¹ Institute of Mathematics, Polish Academy of Sciences Bankowa 14, 40–007 Katowice, Poland

Mathematical modelling of natural phenomena, Tome 10 (2015) no. 6, pp. 163-181

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Résumé

We present an individual–based model of phenotypic trait evolution in two–sex populations, which includes semi–random mating of individuals of the opposite sex, natural death and intra–specific competition. By passing the number of individuals to infinity, we derive the macroscopic system of nonlinear differential equations describing the evolution of trait distributions in male and female subpopulations. We study solutions, give criteria for persistence or extinction, and state a theorem on asymptotic stability, which we apply to particular examples of trait inheritance.

DOI : 10.1051/mmnp/20150611

Affiliations des auteurs :

P. Zwoleński ¹

¹ Institute of Mathematics, Polish Academy of Sciences Bankowa 14, 40–007 Katowice, Poland

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P. Zwoleński. Trait Evolution in two–sex Populations. Mathematical modelling of natural phenomena, Tome 10 (2015) no. 6, pp. 163-181. doi: 10.1051/mmnp/20150611

Bibliographie
Cité par

[1] O. Arino, R. Rudnicki C. R. Biologies 2004 961 969

[2] S. Asmussen Ann. Prob. 1980 727 744

[3] J. Banasiak, M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkháuser, 2014.

[4] F. Bolley, Separability and completeness for the Wasserstein distance, Séminaire de probabilités XLI, Lecture Notes in Mathematics 1934 (2008), 371–377.

[5] R. Bonduriansky, S. F. Chenoweth Trends in Ecology and Evolution 2009 280 288

[6] G. Busoni, A. Palczewski Applicationes Mathematicae 2000 21 34

[7] N. Champagnat, R. Ferrière, S. Méléard Stoch. Models 2008 2 44

[8] R. Collet, S. Méléard, J. A. J. Metz J. Math. Biol. 2013 569 607

[9] M. G. Crandall J. Math. Soc. Japan 1970 443 455

[10] K. Dietz, K. P. Hadeler J. Math. Biol. 1988 1 25

[11] R. Ferrière, V. C. Tran ESAIM: Proceedings 2009 289 310

[12] R. A. Fisher, The Genetical Theory of Natural Selection, Clarendon Press, Oxford, 1930.

[13] N. Fournier, S. Méléard Ann. Appl. Probab. 2004 1880 1919

[14] A. G. Fredrickson Mathematical Biosciences 1971 117 143

[15] G. Garnett Sex Transm. Inf. 2001 7 12

[16] S. Gavrilets, C. R. B. Boake The American Naturalist 1998 706 716

[17] L. Goodman Biometrics 1953 212 225

[18] K. P. Hadeler, R. Waldstätter, A. Wörz–Busekros J. Math. Biol. 1988 635 649

[19] D. G. Kendall J. Roy. Statist. Soc. Ser. B. 1949 230 264

[20] N. Keyfotz, The mathematics of sex and marriage, Proc. Sixth. Berkeley Symp. Math. Statist. Probability, Univ. California Press, 1972, 353–367.

[21] E. Kuno Researches on Population Ecology 1978 50 60

[22] M. Lachowicz Prob. Engin. Mech. 2011 54 60

[23] M. Lachowicz Nonlinear Anal. Real World Appl. 2011 2396 2407

[24] M. Lachowicz, M. Pulvirenti Arch. Ration. Mech. Anal. 1990 81 93

[25] M. Lachowicz, D. Wrzosek Math. Models Methods Appl. Sci. 2001 1393 1409

[26] A. Lasota J. Math. Anal. Appl. 2002 291 309

[27] H. Liu Chinese Journal of Ecology 2003 63 65

[28] J. H. Pollard, Mathematical Models for Growth of Human Populations, Cambridge Univ. Press, 1973.

[29] M. C. Mackey, R. Rudnicki J. Math. Biol 1994 89 109

[30] S. T. Rachev, Probability metrics and the stability of stochastic models, John Willey and Sons, Chichester 1991.

[31] K. H. Rosen Mathematical Modelling 1983 27 39

[32] R. Rudnicki, R. Wieczorek Bull. Polish Acad. Sci. 2006 175 191

[33] R. Rudnicki, R. Wieczorek Math. Mod. Nat. Phenomena 2006 83 100

[34] R. Rudnicki, P. Zwoleński J. Math. Biol. 2015 1295 1321

[35] A. Ulikowska Kinetic and Related Models 2012 873 900

[36] C. Villani, Optimal transport, old and new, Grundlehren der Mathematischen Wissenschaften 338, Springer–Verlag, 2008.

[37] K. Yang, F. Milner J. Biol. Dynam. 2009 252 270

[38] J. Yellin, P. A. Samuelson Proc. Nat. Acad. Sci. USA 1974 2813 2817

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