Models of the population playing the rock-paper-scissors game

Włodzimierz Bąk; Tadeusz Nadzieja; Mateusz Wróbel

doi:10.3934/dcdsb.2018001

Article Contents

2018, Volume 23, Issue 1: 1-11. Doi: 10.3934/dcdsb.2018001

This issue Previous Article Preface Next Article Self-similar solutions of fragmentation equations revisited

Models of the population playing the rock-paper-scissors game

Instytut Matematyki i Informatyki, Uniwersytet Opolski, ul. Oleska 48, Poland

Received: October 2016

Revised: February 2017

Published: November 2017

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

We consider discrete dynamical systems coming from the models of evolution of populations playing rock-paper-scissors game. Asymptotic behaviour of trajectories of these systems is described, occurrence of the Neimark-Sacker bifurcation and nonexistence of time averages are proved.

Keywords:

Mathematics Subject Classification: Primary: 37C05, 91A05.

Citation:

Full Text(HTML)

Figure 1. Levels of Lyapunov function.

Download: Full-size image PowerPoint slide

Figure 2. Sample trajectories of $V_{\lambda}$.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	K. Barański and M. Misiurewicz , Omega-limit set for the Stein-Ulam spiral map, Topology Proceedings, 36 (2010) , 145-172.
	A. Gaunersdorfer , Time averages for heteroclinic attractors, SIAM J. Math. Anal., 52 (1992) , 1476-1489. doi: 10.1137/0152085.
	J. Guckenheimer, Bifurcations of dynamical systems, C. I. M. E Summer School Bressanone 1978, Progr. Math., Birkh'auser, Boston, Mass., 8 (1980), 115-231.
	G. N. Hardy , Mendelian proportions in a mixed population, Science, 28 (1908) , 49-50.
	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.
	Yu I. Lyubich , Basic concepts and theorem of the evolutionary genetics of free populations, Russian Math. Surveys, 26 (1971) , 51-116.
	M. T. Menzel, P. R. Stein and S. M. Ulam, Quadratic Transformations. Part 1, in Los Alamos Scientific Laboratory report LA-2305,1959.
	F. Takens , Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Bras. Mat., 25 (1994) , 107-120. doi: 10.1007/BF01232938.
	F. Takens , Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008) , T33-T36. doi: 10.1088/0951-7715/21/3/T02.
	S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, New York-London, Interscience Publishers, 1960.
	S. S. Vallander , The limiting behavior of the sequence of iterates of certain quadratic transformations, Dokl. Akad. Nauk SSSR, 202 (1972) , 515-517.
	W. Weinberg, Über den Nachweis der Vererbung beim Menschen, Jahreshefte Verein f. Vaterl. Naturk., in Würtembergh, 64 (1908), 368-383.
	D. Whitley , Discrete dynamical systems in dimensions one and two, Bull. London Math. Soc., 15 (1983) , 177-217. doi: 10.1112/blms/15.3.177.
	M. I. Zaharevič , On the behavior of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, Russian Math. Surveys, 33 (1978) , 265-266.