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January  2018, 23(1): 1-11. doi: 10.3934/dcdsb.2018001

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

Włodzimierz Bąk , Tadeusz Nadzieja and  Mateusz Wróbel

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

Received  October 2016 Revised  February 2017 Published  January 2018

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: Asymptotic behaviour of trajectories, time average, rock-paper-scissors game.
Mathematics Subject Classification: Primary: 37C05, 91A05.
Citation: Włodzimierz Bąk, Tadeusz Nadzieja, Mateusz Wróbel. Models of the population playing the rock-paper-scissors game. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 1-11. doi: 10.3934/dcdsb.2018001
References:
[1]

K. Barański and M. Misiurewicz, Omega-limit set for the Stein-Ulam spiral map, Topology Proceedings, 36 (2010), 145-172.   Google Scholar

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A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM J. Math. Anal., 52 (1992), 1476-1489.  doi: 10.1137/0152085.  Google Scholar

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G. N. Hardy, Mendelian proportions in a mixed population, Science, 28 (1908), 49-50.   Google Scholar

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Yu I. Lyubich, Basic concepts and theorem of the evolutionary genetics of free populations, Russian Math. Surveys, 26 (1971), 51-116.   Google Scholar

[7]

M. T. Menzel, P. R. Stein and S. M. Ulam, Quadratic Transformations. Part 1, in Los Alamos Scientific Laboratory report LA-2305,1959. Google Scholar

[8]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Bras. Mat., 25 (1994), 107-120.  doi: 10.1007/BF01232938.  Google Scholar

[9]

F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008), T33-T36.  doi: 10.1088/0951-7715/21/3/T02.  Google Scholar

[10]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, New York-London, Interscience Publishers, 1960. Google Scholar

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S. S. Vallander, The limiting behavior of the sequence of iterates of certain quadratic transformations, Dokl. Akad. Nauk SSSR, 202 (1972), 515-517.   Google Scholar

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W. Weinberg, Über den Nachweis der Vererbung beim Menschen, Jahreshefte Verein f. Vaterl. Naturk., in Würtembergh, 64 (1908), 368-383. Google Scholar

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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.  Google Scholar

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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.   Google Scholar

show all references

References:
[1]

K. Barański and M. Misiurewicz, Omega-limit set for the Stein-Ulam spiral map, Topology Proceedings, 36 (2010), 145-172.   Google Scholar

[2]

A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM J. Math. Anal., 52 (1992), 1476-1489.  doi: 10.1137/0152085.  Google Scholar

[3]

J. Guckenheimer, Bifurcations of dynamical systems, C. I. M. E Summer School Bressanone 1978, Progr. Math., Birkh'auser, Boston, Mass., 8 (1980), 115-231.  Google Scholar

[4]

G. N. Hardy, Mendelian proportions in a mixed population, Science, 28 (1908), 49-50.   Google Scholar

[5] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[6]

Yu I. Lyubich, Basic concepts and theorem of the evolutionary genetics of free populations, Russian Math. Surveys, 26 (1971), 51-116.   Google Scholar

[7]

M. T. Menzel, P. R. Stein and S. M. Ulam, Quadratic Transformations. Part 1, in Los Alamos Scientific Laboratory report LA-2305,1959. Google Scholar

[8]

F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Bras. Mat., 25 (1994), 107-120.  doi: 10.1007/BF01232938.  Google Scholar

[9]

F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008), T33-T36.  doi: 10.1088/0951-7715/21/3/T02.  Google Scholar

[10]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, New York-London, Interscience Publishers, 1960. Google Scholar

[11]

S. S. Vallander, The limiting behavior of the sequence of iterates of certain quadratic transformations, Dokl. Akad. Nauk SSSR, 202 (1972), 515-517.   Google Scholar

[12]

W. Weinberg, Über den Nachweis der Vererbung beim Menschen, Jahreshefte Verein f. Vaterl. Naturk., in Würtembergh, 64 (1908), 368-383. Google Scholar

[13]

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.  Google Scholar

[14]

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.   Google Scholar

Figure 1.  Levels of Lyapunov function.
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Figure 2.  Sample trajectories of $V_{\lambda}$.
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