Permanence in polymatrix replicators

Telmo Peixe

doi:10.3934/jdg.2020032

Article Contents

2021, Volume 8, Issue 1: 21-34. Doi: 10.3934/jdg.2020032

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Permanence in polymatrix replicators

Telmo Peixe

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

Early access: November 2020

Published: January 2021

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

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Abstract

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.

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Mathematics Subject Classification: 34D05, 34D20, 37B25, 37C75, 37N25, 37N40, 91A22.

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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 $

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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 $

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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 $

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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 $

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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 $

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