Boundary dynamics of the replicator equations for neutral models of cyclic dominance

Eric Foxall

doi:10.3934/dcdsb.2020153

Article Contents

2021, Volume 26, Issue 2: 1061-1082. Doi: 10.3934/dcdsb.2020153

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Boundary dynamics of the replicator equations for neutral models of cyclic dominance

Eric Foxall

University of British Columbia, Okanagan Campus, 3333 University Way, Kelowna BC, Canada V1V1V7

Received: June 30, 2019

Revised: January 31, 2020

Early access: May 2020

Published: February 2021

The author is supported by an NSERC Discovery grant

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Abstract

We study the replicator equations, also known as mean-field equations, for a simple model of cyclic dominance with any number $ m $ of strategies, generalizing the rock-paper-scissors model which corresponds to the case $ m = 3 $. Previously the dynamics were solved for $ m\in\{3,4\} $ by consideration of $ m-2 $ conserved quantities. Here we show that for any $ m $, the boundary of the phase space is partitioned into heteroclinic networks for which we give a precise description. A set of $ {\lfloor} m/2{\rfloor} $ conserved quantities plays an important role in the analysis. We also discuss connections to the well-mixed stochastic version of the model.

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Mathematics Subject Classification: Primary: 92D40.

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Figure 1. Simulation output of $ u_0(t) $ from (1) for various $ m $, with initial data $ u(0) = (0.01,0.99/(m-1),\dots,0.99/(m-1)) $, chosen to be close to $ \partial S_m $

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Figure 2. Simulation output of (1) with initial data $ u(0) = (0,1/(m-1),\dots,1/(m-1)) $

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References

[1]	H. Amann, Ordinary Differential Equations: An Introduction to Nonlinear Analysis, Translated from the German by Gerhard Metzen. De Gruyter Studies in Mathematics, 13. Walter de Gruyter & Co., Berlin, 1990. doi: 10.1515/9783110853698.
[2]	M. Bramson and D. Griffeath, Flux and fixation in cyclic particle systems, The Annals of Probability, 17 (1989), 26-45. doi: 10.1214/aop/1176991492.
[3]	M. Berr, T. Reichenbach, M. Schottenloher and E. Frey, Zero-one survival behavior of cyclically competing species, Physical Review Letters, 102 (2009), 048102.
[4]	S. O. Case, C. H. Durney, M. Pleimling and R. K. P. Zia, Cyclic competition of four species: Mean-field theory and stochastic evolution, Europhysics Letters, 92 (2011), 58003.
[5]	O. Diekmann and S. A. van Gils, On the cyclic replicator equation and the dynamics of semelparous populations, SIAM J Appl Dyn Sys, 8 (2009), 1160-1189. doi: 10.1137/080722734.
[6]	H. I. Freedman and P. Waltman, Persistence in a model of three competitive populations, Mathematical Biosciences, 73 (1985), 89-101. doi: 10.1016/0025-5564(85)90078-1.
[7]	E. Foxall and H. Lyu, Clustering in the three and four color cyclic particle systems in one dimension, Journal of Statistical Physics, 171 (2018), 470-483. doi: 10.1007/s10955-018-2004-2.
[8]	M. E. Gilpin, Limit cycles in competition communities, The American Naturalist, 109 (1975), 51-60.
[9]	T. G. Kurtz, Strong approximation theorems for density-dependent Markov chains, Stochastic Processes and their Applications, 6 (1978), 223-240. doi: 10.1016/0304-4149(78)90020-0.
[10]	R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM Journal on Applied Mathematics, 29 (1975), 243-253. doi: 10.1137/0129022.
[11]	M. Mobilia, Oscillatory dynamics in rock-paper-scissors games with mutations, Journal of Theoretical Biology, 264 (2010), 1-10. doi: 10.1016/j.jtbi.2010.01.008.
[12]	A. Szolnoki, M. Mobilia, L. L. Jiang, B. Szczesny, A. M. Rucklidge and M. Perc, Cyclic dominance in evolutionary games: A review, Journal of the Royal Society Interface, 11 (2014), 20140735.
[13]	A. Szolnoki and M. Perc, Correlation of Positive and Negative Reciprocity Fails to Confer an Evolutionary Advantage: Phase Transitions to Elementary Strategies, Physical Review X, 3 (2013), 041021.
[14]	A. Szolnoki, M. Perc and G. Szabó, Defense Mechanisms of Empathetic Players in the Spatial Ultimatum Game., Physical Review Letters, 109 (2012), 078701.
[15]	A. Szolnoki and M. Perc, Evolutionary dynamics of cooperation in neutral populations, New Journal of Physics, 20 (2018), 013031, 9pp. doi: 10.1088/1367-2630/aa9fd2.
[16]	P. Schuster, K. Sigmund and R. Wolff, On $\omega$-limits for competition between three species, SIAM Journal on Applied Mathematics, 37 (1979), 49-54. doi: 10.1137/0137004.