January  2021, 8(1): 21-34. doi: 10.3934/jdg.2020032

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

Received  May 2020 Published  November 2020

Fund Project: The author was supported by FCT-Fundação para a Ciência e a Tecnologia, under the project CEMAPRE - UID/MULTI/00491/2013 through national funds

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 [15] introduced the concept of permanence as a stability notion for systems that models the self-organization of biological macromolecules. After, in 1987 W. Jansen [9], and J. Hofbauer and K. Sigmund [6] give sufficient conditions for permanence in the replicator equations. In this paper we extend these results for polymatrix replicators.

Citation: Telmo Peixe. Permanence in polymatrix replicators. Journal of Dynamics & Games, 2021, 8 (1) : 21-34. doi: 10.3934/jdg.2020032
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.  Google Scholar

[2]

H. N. AlishahP. 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.  Google Scholar

[3]

H. N. AlishahP. Duarte and T. Peixe, Asymptotic Poincaré maps along the edges of polytopes, Nonlinearity, 33 (2020), 469-510.  doi: 10.1088/1361-6544/ab49e6.  Google Scholar

[4]

P. DuarteR. 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.  Google Scholar

[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.  Google Scholar

[6]

J. Hofbauer and K. Sigmund, Permanence for Replicator Equations, Springer Berlin Heidelberg, 1987. doi: 10.1007/978-3-662-00748-8_7.  Google Scholar

[7] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[8]

J. Hofbauer, A general cooperation theorem for hypercycles, Monatsh. Math., 91 (1981), 233-240.  doi: 10.1007/BF01301790.  Google Scholar

[9]

W. Jansen, A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25 (1987), 411-422.  doi: 10.1007/BF00277165.  Google Scholar

[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.  Google Scholar

[11]

J. M. Smith, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.  Google Scholar

[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.  Google Scholar

[13]

P. Schuster and K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538.  doi: 10.1016/0022-5193(83)90445-9.  Google Scholar

[14]

P. SchusterK. SigmundJ. Hofbauer and R. Wolff, Self-regulation of behaviour in animal societies, Biol. Cybernet., 40 (1981), 9-15.  doi: 10.1007/BF00326676.  Google Scholar

[15]

P. SchusterK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944.   Google Scholar

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.  Google Scholar

[2]

H. N. AlishahP. 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.  Google Scholar

[3]

H. N. AlishahP. Duarte and T. Peixe, Asymptotic Poincaré maps along the edges of polytopes, Nonlinearity, 33 (2020), 469-510.  doi: 10.1088/1361-6544/ab49e6.  Google Scholar

[4]

P. DuarteR. 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.  Google Scholar

[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.  Google Scholar

[6]

J. Hofbauer and K. Sigmund, Permanence for Replicator Equations, Springer Berlin Heidelberg, 1987. doi: 10.1007/978-3-662-00748-8_7.  Google Scholar

[7] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[8]

J. Hofbauer, A general cooperation theorem for hypercycles, Monatsh. Math., 91 (1981), 233-240.  doi: 10.1007/BF01301790.  Google Scholar

[9]

W. Jansen, A permanence theorem for replicator and Lotka-Volterra systems, J. Math. Biol., 25 (1987), 411-422.  doi: 10.1007/BF00277165.  Google Scholar

[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.  Google Scholar

[11]

J. M. Smith, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.  Google Scholar

[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.  Google Scholar

[13]

P. Schuster and K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538.  doi: 10.1016/0022-5193(83)90445-9.  Google Scholar

[14]

P. SchusterK. SigmundJ. Hofbauer and R. Wolff, Self-regulation of behaviour in animal societies, Biol. Cybernet., 40 (1981), 9-15.  doi: 10.1007/BF00326676.  Google Scholar

[15]

P. SchusterK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey, 1944.   Google Scholar
Figure 1.  Two different perspectives of the polytope $ \Gamma_{(2,2,2)} $ from Example 2 where the polymatrix replicator given by the payoff matrix $ A $ is defined. Namelly, the plot of its equilibria and three interior orbits (with initial conditions near the boundary of the polytope) that converge to the unique interior equilibrium, $ q $
Table 1.  The vertices of $ \Gamma_{(2,2,2,2)} $ and the value of $ f(v_i) $, where $ f(x) = (x-q)^TAx\, $ and $ q = \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2} \right) \in {\rm int}\left(\Gamma_{(2,2,2,2)}\right) . $
Vertices of $ \Gamma_{(2,2,2,2)} $ $ f(v_i) $
$ v_1=\left(1,0,1,0,1,0,1,0\right) $ $ -394 $
$ v_2=\left(1,0,1,0,1,0,0,1\right) $ $ -4 $
$ v_3=\left(1,0,1,0,0,1,1,0\right) $ $ -392 $
$ v_4=\left(1,0,1,0,0,1,0,1\right) $ $ -6 $
$ v_5=\left(1,0,0,1,1,0,1,0\right) $ $ -602 $
$ v_6=\left(1,0,0,1,1,0,0,1\right) $ $ -592 $
$ v_7=\left(1,0,0,1,0,1,1,0\right) $ $ -204 $
$ v_8=\left(1,0,0,1,0,1,0,1\right) $ $ -198 $
$ v_9=\left(0,1,1,0,1,0,1,0\right) $ $ -198 $
$ v_{10}=\left(0,1,1,0,1,0,0,1\right) $ $ -204 $
$ v_{11}=\left(0,1,1,0,0,1,1,0\right) $ $ -592 $
$ v_{12}=\left(0,1,1,0,0,1,0,1\right) $ $ -602 $
$ v_{13}=\left(0,1,0,1,1,0,1,0\right) $ $ -6 $
$ v_{14}=\left(0,1,0,1,1,0,0,1\right) $ $ -392 $
$ v_{15}=\left(0,1,0,1,0,1,1,0\right) $ $ -4 $
$ v_{16}=\left(0,1,0,1,0,1,0,1\right) $ $ -394 $
Vertices of $ \Gamma_{(2,2,2,2)} $ $ f(v_i) $
$ v_1=\left(1,0,1,0,1,0,1,0\right) $ $ -394 $
$ v_2=\left(1,0,1,0,1,0,0,1\right) $ $ -4 $
$ v_3=\left(1,0,1,0,0,1,1,0\right) $ $ -392 $
$ v_4=\left(1,0,1,0,0,1,0,1\right) $ $ -6 $
$ v_5=\left(1,0,0,1,1,0,1,0\right) $ $ -602 $
$ v_6=\left(1,0,0,1,1,0,0,1\right) $ $ -592 $
$ v_7=\left(1,0,0,1,0,1,1,0\right) $ $ -204 $
$ v_8=\left(1,0,0,1,0,1,0,1\right) $ $ -198 $
$ v_9=\left(0,1,1,0,1,0,1,0\right) $ $ -198 $
$ v_{10}=\left(0,1,1,0,1,0,0,1\right) $ $ -204 $
$ v_{11}=\left(0,1,1,0,0,1,1,0\right) $ $ -592 $
$ v_{12}=\left(0,1,1,0,0,1,0,1\right) $ $ -602 $
$ v_{13}=\left(0,1,0,1,1,0,1,0\right) $ $ -6 $
$ v_{14}=\left(0,1,0,1,1,0,0,1\right) $ $ -392 $
$ v_{15}=\left(0,1,0,1,0,1,1,0\right) $ $ -4 $
$ v_{16}=\left(0,1,0,1,0,1,0,1\right) $ $ -394 $
Table 2.  The equilibria on $ 3d $-faces of $ \Gamma_{(2,2,2,2)} $ and the value of $ f(q_i) $, where $ f(x) = (x-q)^TAx\, $ and $ q = \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2} \right) \in {\rm int}\left(\Gamma_{(2,2,2,2)}\right) . $
Equilibria on $ 3d $-faces of $ \Gamma_{(2,2,2,2)} $ $ f(q_i) $
$ q_1=\left(0.05266, 0.9473, 0.93275, 0.0672483, 0.991199,\frac{9049}{1028189}, 0, 1\right) $ $ -201.7 $
$ q_2=\left( 0.9473, 0.05266, 0.0672483, 0.93275,\frac{9049}{1028189}, 0.991199, 1, 0 \right) $ $ -201.7 $
Equilibria on $ 3d $-faces of $ \Gamma_{(2,2,2,2)} $ $ f(q_i) $
$ q_1=\left(0.05266, 0.9473, 0.93275, 0.0672483, 0.991199,\frac{9049}{1028189}, 0, 1\right) $ $ -201.7 $
$ q_2=\left( 0.9473, 0.05266, 0.0672483, 0.93275,\frac{9049}{1028189}, 0.991199, 1, 0 \right) $ $ -201.7 $
Table 3.  The equilibria on $ 2d $-faces of $ \Gamma_{(2,2,2,2)} $ and the value of $ f(q_i) $, where $ f(x) = (x-q)^TAx\, $ and $ q = \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2} \right) \in {\rm int}\left(\Gamma_{(2,2,2,2)}\right) . $
Equilibria on $ 2d $-faces of $ \Gamma_{(2,2,2,2)} $ $ f(q_i) $
$ q_3=\left(0,1,0,1,\frac{9803}{29100},\frac{19297}{29100},\frac{893}{2910},\frac{2017}{2910}\right) $ $ -19.2 $
$ q_4=\left(0,1,1,0,\frac{9649}{14550},\frac{4901}{14550},\frac{994}{1455},\frac{461}{1455}\right) $ $ -76.7 $
$ q_5=\left(0,1,\frac{171}{400},\frac{229}{400},\frac{29}{40},\frac{11}{40},0,1\right) $ $ -197.4 $
$ q_6=\left(1,0,0,1,\frac{4901}{14550},\frac{9649}{14550},\frac{461}{1455},\frac{994}{1455}\right) $ $ -76.7 $
$ q_7=\left(1,0,1,0,\frac{19297}{29100},\frac{9803}{29100},\frac{2017}{2910},\frac{893}{2910}\right) $ $ -19.2 $
$ q_8=\left(1,0,\frac{229}{400},\frac{171}{400},\frac{11}{40},\frac{29}{40},1,0\right) $ $ -197.4 $
Equilibria on $ 2d $-faces of $ \Gamma_{(2,2,2,2)} $ $ f(q_i) $
$ q_3=\left(0,1,0,1,\frac{9803}{29100},\frac{19297}{29100},\frac{893}{2910},\frac{2017}{2910}\right) $ $ -19.2 $
$ q_4=\left(0,1,1,0,\frac{9649}{14550},\frac{4901}{14550},\frac{994}{1455},\frac{461}{1455}\right) $ $ -76.7 $
$ q_5=\left(0,1,\frac{171}{400},\frac{229}{400},\frac{29}{40},\frac{11}{40},0,1\right) $ $ -197.4 $
$ q_6=\left(1,0,0,1,\frac{4901}{14550},\frac{9649}{14550},\frac{461}{1455},\frac{994}{1455}\right) $ $ -76.7 $
$ q_7=\left(1,0,1,0,\frac{19297}{29100},\frac{9803}{29100},\frac{2017}{2910},\frac{893}{2910}\right) $ $ -19.2 $
$ q_8=\left(1,0,\frac{229}{400},\frac{171}{400},\frac{11}{40},\frac{29}{40},1,0\right) $ $ -197.4 $
Table 4.  The equilibria on $ 1d $-faces of $ \Gamma_{(2,2,2,2)} $ and the value of $ f(q_i) $, where $ f(x) = (x-q)^TAx\, $ and $ q = \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2} \right) \in {\rm int}\left(\Gamma_{(2,2,2,2)}\right) . $
Equilibria on $ 1d $-faces of $ \Gamma_{(2,2,2,2)} $ $ f(q_i) $
$ q_9=\left(0,1,0,1,1,0,\frac{97}{100},\frac{3}{100} \right) $ $ -5.94 $
$ q_{10}=\left(1,0,1,0,0,1,\frac{3}{100},\frac{97}{100} \right) $ $ -5.94 $
$ q_{11}=\left(0,1,1,0,0,1,\frac{1}{50},\frac{49}{50} \right) $ $ -593.96 $
$ q_{12}=\left(1,0,0,1,1,0,\frac{49}{50},\frac{1}{50} \right) $ $ -593.96 $
$ q_{13}=\left(0,1,1,0,\frac{47}{95},\frac{48}{95},1,0 \right) $ $ -207.1 $
$ q_{14}=\left(1,0,0,1,\frac{48}{95},\frac{47}{95},0,1 \right) $ $ -207.1 $
$ q_{15}=\left(0,1,\frac{2}{5},\frac{3}{5},1,0,0,1 \right) $ $ -307.2 $
$ q_{16}=\left(1,0,\frac{3}{5},\frac{2}{5},0,1,1,0 \right) $ $ -307.2 $
$ q_{17}=\left(0,1,0,1,\frac{1}{2},\frac{1}{2},0,1 \right) $ $ -203 $
$ q_{18}=\left(1,0,1,0,\frac{1}{2},\frac{1}{2},1,0 \right) $ $ -203 $
$ q_{19}=\left(0,1,\frac{1}{2},\frac{1}{2},0,1,0,1 \right) $ $ -488 $
$ q_{20}=\left(1,0,\frac{1}{2},\frac{1}{2},1,0,1,0 \right) $ $ -488 $
Equilibria on $ 1d $-faces of $ \Gamma_{(2,2,2,2)} $ $ f(q_i) $
$ q_9=\left(0,1,0,1,1,0,\frac{97}{100},\frac{3}{100} \right) $ $ -5.94 $
$ q_{10}=\left(1,0,1,0,0,1,\frac{3}{100},\frac{97}{100} \right) $ $ -5.94 $
$ q_{11}=\left(0,1,1,0,0,1,\frac{1}{50},\frac{49}{50} \right) $ $ -593.96 $
$ q_{12}=\left(1,0,0,1,1,0,\frac{49}{50},\frac{1}{50} \right) $ $ -593.96 $
$ q_{13}=\left(0,1,1,0,\frac{47}{95},\frac{48}{95},1,0 \right) $ $ -207.1 $
$ q_{14}=\left(1,0,0,1,\frac{48}{95},\frac{47}{95},0,1 \right) $ $ -207.1 $
$ q_{15}=\left(0,1,\frac{2}{5},\frac{3}{5},1,0,0,1 \right) $ $ -307.2 $
$ q_{16}=\left(1,0,\frac{3}{5},\frac{2}{5},0,1,1,0 \right) $ $ -307.2 $
$ q_{17}=\left(0,1,0,1,\frac{1}{2},\frac{1}{2},0,1 \right) $ $ -203 $
$ q_{18}=\left(1,0,1,0,\frac{1}{2},\frac{1}{2},1,0 \right) $ $ -203 $
$ q_{19}=\left(0,1,\frac{1}{2},\frac{1}{2},0,1,0,1 \right) $ $ -488 $
$ q_{20}=\left(1,0,\frac{1}{2},\frac{1}{2},1,0,1,0 \right) $ $ -488 $
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