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Preface
January
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 |
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.
Mathematics Subject Classification:
Primary: 37C05, 91A05.
Citation:
Włodzimierz Bąk, Tadeusz Nadzieja, Mateusz Wróbel. Models of the population playing the rock-paper-scissors game. Discrete and Continuous Dynamical Systems - B,
2018, 23
(1)
: 1-11.
doi: 10.3934/dcdsb.2018001
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References:
| [1] |
K. Barański and 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 and 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 and 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 and 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 2.
Sample trajectories of $V_{\lambda}$ .
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