• Previous Article
    Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary
  • DCDS-B Home
  • This Issue
  • Next Article
    Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity
February  2021, 26(2): 1061-1082. doi: 10.3934/dcdsb.2020153

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  February 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (2) : 1061-1082. 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.

[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. SchusterK. 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.

show all references

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. SchusterK. 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.

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

Włodzimierz Bąk, Tadeusz Nadzieja, Mateusz Wróbel. Models of the population playing the rock-paper-scissors game. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 1-11. doi: 10.3934/dcdsb.2018001

[2]

Peter Bednarik, Josef Hofbauer. Discretized best-response dynamics for the Rock-Paper-Scissors game. Journal of Dynamics and Games, 2017, 4 (1) : 75-86. doi: 10.3934/jdg.2017005

[3]

Gunter Neumann, Stefan Schuster. Modeling the rock - scissors - paper game between bacteriocin producing bacteria by Lotka-Volterra equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 207-228. doi: 10.3934/dcdsb.2007.8.207

[4]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic and Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051

[5]

Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic and Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187

[6]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981

[7]

Yadong Shu, Ying Dai, Zujun Ma. Evolutionary game theory analysis of supply chain with fairness concerns of retailers. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022098

[8]

Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200

[9]

King-Yeung Lam. Dirac-concentrations in an integro-pde model from evolutionary game theory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 737-754. doi: 10.3934/dcdsb.2018205

[10]

Ross Cressman, Vlastimil Křivan. Using chemical reaction network theory to show stability of distributional dynamics in game theory. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021030

[11]

Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic and Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109

[12]

Matthieu Alfaro, Mario Veruete. Density dependent replicator-mutator models in directed evolution. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2203-2221. doi: 10.3934/dcdsb.2019224

[13]

Hao Wang, Yang Kuang. Alternative models for cyclic lemming dynamics. Mathematical Biosciences & Engineering, 2007, 4 (1) : 85-99. doi: 10.3934/mbe.2007.4.85

[14]

Scott G. McCalla. Paladins as predators: Invasive waves in a spatial evolutionary adversarial game. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1437-1457. doi: 10.3934/dcdsb.2014.19.1437

[15]

William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics and Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485

[16]

John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291

[17]

Tao Li, Suresh P. Sethi. A review of dynamic Stackelberg game models. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 125-159. doi: 10.3934/dcdsb.2017007

[18]

Gaidi Li, Jiating Shao, Dachuan Xu, Wen-Qing Xu. The warehouse-retailer network design game. Journal of Industrial and Management Optimization, 2015, 11 (1) : 291-305. doi: 10.3934/jimo.2015.11.291

[19]

T. S. Evans, A. D. K. Plato. Network rewiring models. Networks and Heterogeneous Media, 2008, 3 (2) : 221-238. doi: 10.3934/nhm.2008.3.221

[20]

Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (172)
  • HTML views (288)
  • Cited by (0)

Other articles
by authors

[Back to Top]