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

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  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.

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

[2]

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

[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.

[4]

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

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

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

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[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.

[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.

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

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

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

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[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.

[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.

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

Peter Bednarik, Josef Hofbauer. Discretized best-response dynamics for the rock-paper-scissors game. Journal of Dynamics & Games, 2017, 4 (1) : 75-86. doi: 10.3934/jdg.2017005

[2]

Gunter Neumann, Stefan Schuster. Modeling the rock - scissors - paper game between bacteriocin producing bacteria by Lotka-Volterra equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 207-228. doi: 10.3934/dcdsb.2007.8.207

[3]

Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305

[4]

Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943

[5]

Gabriela Planas, Eduardo Hernández. Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1245-1258. doi: 10.3934/dcds.2008.21.1245

[6]

Juan C. Jara, Felipe Rivero. Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4127-4137. doi: 10.3934/dcds.2014.34.4127

[7]

Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277

[8]

Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019

[9]

Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197

[10]

Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279

[11]

Minvydas Ragulskis, Zenonas Navickas. Hash function construction based on time average moiré. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 1007-1020. doi: 10.3934/dcdsb.2007.8.1007

[12]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1

[13]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[14]

María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure & Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157

[15]

Jorge Ferreira, Mauro De Lima Santos. Asymptotic behaviour for wave equations with memory in a noncylindrical domains . Communications on Pure & Applied Analysis, 2003, 2 (4) : 511-520. doi: 10.3934/cpaa.2003.2.511

[16]

Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206

[17]

Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119

[18]

Teresa Faria. Asymptotic behaviour for a class of delayed cooperative models with patch structure. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1567-1579. doi: 10.3934/dcdsb.2013.18.1567

[19]

Giovambattista Amendola, Sandra Carillo, John Murrough Golden, Adele Manes. Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815

[20]

Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (8)
  • HTML views (29)
  • Cited by (0)

[Back to Top]