# American Institute of Mathematical Sciences

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

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

Received  July 2019 Revised  February 2020 Published  May 2020

Fund Project: The author is supported by an NSERC Discovery grant

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.

Citation: Eric Foxall. Boundary dynamics of the replicator equations for neutral models of cyclic dominance. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020153
##### 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.  Google Scholar [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.  Google Scholar [3] M. Berr, T. Reichenbach, M. Schottenloher and E. Frey, Zero-one survival behavior of cyclically competing species, Physical Review Letters, 102 (2009), 048102. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. E. Gilpin, Limit cycles in competition communities, The American Naturalist, 109 (1975), 51-60.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [14] A. Szolnoki, M. Perc and G. Szabó, Defense Mechanisms of Empathetic Players in the Spatial Ultimatum Game., Physical Review Letters, 109 (2012), 078701. Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] M. Berr, T. Reichenbach, M. Schottenloher and E. Frey, Zero-one survival behavior of cyclically competing species, Physical Review Letters, 102 (2009), 048102. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] M. E. Gilpin, Limit cycles in competition communities, The American Naturalist, 109 (1975), 51-60.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [14] A. Szolnoki, M. Perc and G. Szabó, Defense Mechanisms of Empathetic Players in the Spatial Ultimatum Game., Physical Review Letters, 109 (2012), 078701. Google Scholar [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.  Google Scholar [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.  Google Scholar
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 $\partial S_m$
Simulation output of (1) with initial data $u(0) = (0,1/(m-1),\dots,1/(m-1))$
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