# American Institute of Mathematical Sciences

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, doi: 10.3934/jdg.2020032
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##### References:
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, $q$
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 $\Gamma_{(2,2,2,2)}$ $f(v_i)$ $v_1=\left(1,0,1,0,1,0,1,0\right)$ $-394$ $v_2=\left(1,0,1,0,1,0,0,1\right)$ $-4$ $v_3=\left(1,0,1,0,0,1,1,0\right)$ $-392$ $v_4=\left(1,0,1,0,0,1,0,1\right)$ $-6$ $v_5=\left(1,0,0,1,1,0,1,0\right)$ $-602$ $v_6=\left(1,0,0,1,1,0,0,1\right)$ $-592$ $v_7=\left(1,0,0,1,0,1,1,0\right)$ $-204$ $v_8=\left(1,0,0,1,0,1,0,1\right)$ $-198$ $v_9=\left(0,1,1,0,1,0,1,0\right)$ $-198$ $v_{10}=\left(0,1,1,0,1,0,0,1\right)$ $-204$ $v_{11}=\left(0,1,1,0,0,1,1,0\right)$ $-592$ $v_{12}=\left(0,1,1,0,0,1,0,1\right)$ $-602$ $v_{13}=\left(0,1,0,1,1,0,1,0\right)$ $-6$ $v_{14}=\left(0,1,0,1,1,0,0,1\right)$ $-392$ $v_{15}=\left(0,1,0,1,0,1,1,0\right)$ $-4$ $v_{16}=\left(0,1,0,1,0,1,0,1\right)$ $-394$
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 $\Gamma_{(2,2,2,2)}$ $f(q_i)$ $q_1=\left(0.05266, 0.9473, 0.93275, 0.0672483, 0.991199,\frac{9049}{1028189}, 0, 1\right)$ $-201.7$ $q_2=\left( 0.9473, 0.05266, 0.0672483, 0.93275,\frac{9049}{1028189}, 0.991199, 1, 0 \right)$ $-201.7$
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 $\Gamma_{(2,2,2,2)}$ $f(q_i)$ $q_3=\left(0,1,0,1,\frac{9803}{29100},\frac{19297}{29100},\frac{893}{2910},\frac{2017}{2910}\right)$ $-19.2$ $q_4=\left(0,1,1,0,\frac{9649}{14550},\frac{4901}{14550},\frac{994}{1455},\frac{461}{1455}\right)$ $-76.7$ $q_5=\left(0,1,\frac{171}{400},\frac{229}{400},\frac{29}{40},\frac{11}{40},0,1\right)$ $-197.4$ $q_6=\left(1,0,0,1,\frac{4901}{14550},\frac{9649}{14550},\frac{461}{1455},\frac{994}{1455}\right)$ $-76.7$ $q_7=\left(1,0,1,0,\frac{19297}{29100},\frac{9803}{29100},\frac{2017}{2910},\frac{893}{2910}\right)$ $-19.2$ $q_8=\left(1,0,\frac{229}{400},\frac{171}{400},\frac{11}{40},\frac{29}{40},1,0\right)$ $-197.4$
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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