American Institute of Mathematical Sciences

Advanced
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

[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

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.
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[2]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012
[3]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365
[4]

Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020  doi: 10.3934/fods.2020017
[5]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381
[6]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[7]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[8]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218
[9]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[10]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[11]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[13]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[14]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[15]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336
[16]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[17]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[18]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[19]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[20]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (95)
  • HTML views (163)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences