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Generalized intransitive dice: Mimicking an arbitrary tournament
Permanence in polymatrix replicators
ISEG-Lisbon School of Economics & Management, Universidade de Lisboa, REM-Research in Economics and Mathematics, CEMAPRE-Centro de Matemática Aplicada à Previsão e Decisão Económica |
Generally a biological system is said to be permanent if under small perturbations none of the species goes to extinction. In 1979 P. Schuster, K. Sigmund, and R. Wolff [
References:
[1] |
H. N. Alishah and P. Duarte,
Hamiltonian evolutionary games, Journal of Dynamics and Games, 2 (2015), 33-49.
doi: 10.3934/jdg.2015.2.33. |
[2] |
H. N. Alishah, P. Duarte and T. Peixe,
Conservative and dissipative polymatrix replicators, Journal of Dynamics and Games, 2 (2015), 157-185.
doi: 10.3934/jdg.2015.2.157. |
[3] |
H. N. Alishah, P. Duarte and T. Peixe,
Asymptotic Poincaré maps along the edges of polytopes, Nonlinearity, 33 (2020), 469-510.
doi: 10.1088/1361-6544/ab49e6. |
[4] |
P. Duarte, R. L. Fernandes and W. M. Oliva,
Dynamics of the attractor in the Lotka-Volterra equations, J. Differential Equations, 149 (1998), 143-189.
doi: 10.1006/jdeq.1998.3443. |
[5] |
J. Hofbauer,
On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal., 5 (1981), 1003-1007.
doi: 10.1016/0362-546X(81)90059-6. |
[6] |
J. Hofbauer and K. Sigmund, Permanence for Replicator Equations, Springer Berlin Heidelberg, 1987.
doi: 10.1007/978-3-662-00748-8_7. |
[7] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9781139173179.![]() ![]() ![]() |
[8] |
J. Hofbauer,
A general cooperation theorem for hypercycles, Monatsh. Math., 91 (1981), 233-240.
doi: 10.1007/BF01301790. |
[9] |
W. Jansen,
A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25 (1987), 411-422.
doi: 10.1007/BF00277165. |
[10] |
A. J. Lotka, Elements of mathematical biology. (formerly published under the title Elements of Physical Biology), Dover Publications, Inc., New York, 1958.
doi: 10.1002/jps.3030471044. |
[11] |
J. M. Smith,
The logic of animal conflict, Nature, 246 (1973), 15-18.
doi: 10.1038/246015a0. |
[12] |
P. Schuster and K. Sigmund,
Coyness, philandering and stable strategies, Animal Behaviour, 29 (1981), 186-192.
doi: 10.1016/S0003-3472(81)80165-0. |
[13] |
P. Schuster and K. Sigmund,
Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538.
doi: 10.1016/0022-5193(83)90445-9. |
[14] |
P. Schuster, K. Sigmund, J. Hofbauer and R. Wolff,
Self-regulation of behaviour in animal societies, Biol. Cybernet., 40 (1981), 9-15.
doi: 10.1007/BF00326676. |
[15] |
P. Schuster, K. Sigmund and R. Wolff,
Dynamical systems under constant organization. ⅲ. cooperative and competitive behavior of hypercycles, Journal of Differential Equations, 32 (1979), 357-368.
doi: 10.1016/0022-0396(79)90039-1. |
[16] |
P. D. Taylor and L. B. Jonker,
Evolutionarily stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.
doi: 10.1016/0025-5564(78)90077-9. |
[17] |
V. Volterra, Leç cons sur la Théorie Mathématique de la Lutte pour la Vie (Reprint of the 1931 original), Éditions Jacques Gabay, Sceaux, 1990. |
[18] |
J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944.
![]() ![]() |
show all references
References:
[1] |
H. N. Alishah and P. Duarte,
Hamiltonian evolutionary games, Journal of Dynamics and Games, 2 (2015), 33-49.
doi: 10.3934/jdg.2015.2.33. |
[2] |
H. N. Alishah, P. Duarte and T. Peixe,
Conservative and dissipative polymatrix replicators, Journal of Dynamics and Games, 2 (2015), 157-185.
doi: 10.3934/jdg.2015.2.157. |
[3] |
H. N. Alishah, P. Duarte and T. Peixe,
Asymptotic Poincaré maps along the edges of polytopes, Nonlinearity, 33 (2020), 469-510.
doi: 10.1088/1361-6544/ab49e6. |
[4] |
P. Duarte, R. L. Fernandes and W. M. Oliva,
Dynamics of the attractor in the Lotka-Volterra equations, J. Differential Equations, 149 (1998), 143-189.
doi: 10.1006/jdeq.1998.3443. |
[5] |
J. Hofbauer,
On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal., 5 (1981), 1003-1007.
doi: 10.1016/0362-546X(81)90059-6. |
[6] |
J. Hofbauer and K. Sigmund, Permanence for Replicator Equations, Springer Berlin Heidelberg, 1987.
doi: 10.1007/978-3-662-00748-8_7. |
[7] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9781139173179.![]() ![]() ![]() |
[8] |
J. Hofbauer,
A general cooperation theorem for hypercycles, Monatsh. Math., 91 (1981), 233-240.
doi: 10.1007/BF01301790. |
[9] |
W. Jansen,
A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25 (1987), 411-422.
doi: 10.1007/BF00277165. |
[10] |
A. J. Lotka, Elements of mathematical biology. (formerly published under the title Elements of Physical Biology), Dover Publications, Inc., New York, 1958.
doi: 10.1002/jps.3030471044. |
[11] |
J. M. Smith,
The logic of animal conflict, Nature, 246 (1973), 15-18.
doi: 10.1038/246015a0. |
[12] |
P. Schuster and K. Sigmund,
Coyness, philandering and stable strategies, Animal Behaviour, 29 (1981), 186-192.
doi: 10.1016/S0003-3472(81)80165-0. |
[13] |
P. Schuster and K. Sigmund,
Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538.
doi: 10.1016/0022-5193(83)90445-9. |
[14] |
P. Schuster, K. Sigmund, J. Hofbauer and R. Wolff,
Self-regulation of behaviour in animal societies, Biol. Cybernet., 40 (1981), 9-15.
doi: 10.1007/BF00326676. |
[15] |
P. Schuster, K. Sigmund and R. Wolff,
Dynamical systems under constant organization. ⅲ. cooperative and competitive behavior of hypercycles, Journal of Differential Equations, 32 (1979), 357-368.
doi: 10.1016/0022-0396(79)90039-1. |
[16] |
P. D. Taylor and L. B. Jonker,
Evolutionarily stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.
doi: 10.1016/0025-5564(78)90077-9. |
[17] |
V. Volterra, Leç cons sur la Théorie Mathématique de la Lutte pour la Vie (Reprint of the 1931 original), Éditions Jacques Gabay, Sceaux, 1990. |
[18] |
J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944.
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