Geodesic
Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model
A.E.F. Burgess1 ; P.G. Schofield2 ; S.F. Hubbard2 ; M.A.J. Chaplain3 ; T. Lorenzi3
1 Division of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland
2 College of Life Sciences, University of Dundee, Dundee DD1 4HN, Scotland
3 School of Mathematics and Statistics, University of St Andrews St Andrews KY16 9SS, Scotland
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 5, pp. 49-64.

Voir la notice de l'article provenant de la source EDP Sciences

We present a novel hybrid modelling framework that takes into account two aspects which have been largely neglected in previous models of spatial evolutionary games: random motion and chemotaxis. A stochastic individual-based model is used to describe the player dynamics, whereas the evolution of the chemoattractant is governed by a reaction-diffusion equation. The two models are coupled by deriving individual movement rules via the discretisa- tion of a taxis-diffusion equation which describes the evolution of the local number of players. In this framework, individuals occupying the same position can engage in a two-player game, and are awarded a payoff, in terms of reproductive fitness, according to their strategy. As an example, we let individuals play the Hawk-Dove game. Numerical simulations illustrate how random motion and chemotactic response can bring about self-generated dynamical patterns that create favourable conditions for the coexistence of hawks and doves in situations in which the two strategies cannot coexist otherwise. In this sense, our work offers a new perspective of research on spatial evolutionary games, and provides a general formalism to study the dynamics of spatially-structured populations in biological and social contexts where individual motion is likely to affect natural selection of behavioural traits.
DOI : 10.1051/mmnp/201611504

A.E.F. Burgess 1 ; P.G. Schofield 2 ; S.F. Hubbard 2 ; M.A.J. Chaplain 3 ; T. Lorenzi 3

1 Division of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland
2 College of Life Sciences, University of Dundee, Dundee DD1 4HN, Scotland
3 School of Mathematics and Statistics, University of St Andrews St Andrews KY16 9SS, Scotland 
@article{MMNP_2016_11_5_a3,
     author = {A.E.F. Burgess and P.G. Schofield and S.F. Hubbard and M.A.J. Chaplain and T. Lorenzi},
     title = {Dynamical {Patterns} of {Coexisting} {Strategies} in a {Hybrid} {Discrete-continuum} {Spatial} {Evolutionary} {Game} {Model}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {49--64},
     publisher = {mathdoc},
     volume = {11},
     number = {5},
     year = {2016},
     doi = {10.1051/mmnp/201611504},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611504/}
}
TY  - JOUR
AU  - A.E.F. Burgess
AU  - P.G. Schofield
AU  - S.F. Hubbard
AU  - M.A.J. Chaplain
AU  - T. Lorenzi
TI  - Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model
JO  - Mathematical modelling of natural phenomena
PY  - 2016
SP  - 49
EP  - 64
VL  - 11
IS  - 5
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611504/
DO  - 10.1051/mmnp/201611504
LA  - en
ID  - MMNP_2016_11_5_a3
ER  - 
%0 Journal Article
%A A.E.F. Burgess
%A P.G. Schofield
%A S.F. Hubbard
%A M.A.J. Chaplain
%A T. Lorenzi
%T Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model
%J Mathematical modelling of natural phenomena
%D 2016
%P 49-64
%V 11
%N 5
%I mathdoc
%U https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611504/
%R 10.1051/mmnp/201611504
%G en
%F MMNP_2016_11_5_a3
A.E.F. Burgess; P.G. Schofield; S.F. Hubbard; M.A.J. Chaplain; T. Lorenzi. Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 5, pp. 49-64. doi : 10.1051/mmnp/201611504. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611504/

[1] A.R.A Anderson, M.A.J Chaplain Continuous and discrete mathematical models of tumor-induced angiogenesis Bulletin of Mathematical Biology 857 899 1998

[2] D. Basanta, R.A. Gatenby, A.R.A. Anderson Exploiting evolution to treat drug resistance: combination therapy and the double bind Mol. Pharm. 914 921 2012

[3] D. Basanta, H. Hatzikirou, A. Deutsch Studying the emergence of invasiveness in tumours using game theory Euro. Phys. J. B 393 397 2008

[4] D. Basanta, J.G. Scott, M.N. Fishman, G. Ayala, S.W. Hayward, A.R.A. Anderson Investigating prostate cancer tumour-stroma interactions: clinical and biological insights from an evolutionary game Br. J. Cancer 174 181 2012

[5] N. Champagnat, R. Ferrière, S. Méléard Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models Theoretical population biology 297 321 2006

[6] Z. Chen, J. Gao, Y. Cai, X. Xu Evolution of cooperation among mobile agents Physica A: Statistical Mechanics and its Applications 1615 1622 2011

[7] R.H. Chisholm, T. Lorenzi, L. Desvillettes, B.D. Hughes Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences Zeitschrift für angewandte Mathematik und Physik 100 2016

[8] R.H. Chisholm, T. Lorenzi, A. Lorz, A.K. Larsen, L. Neves De Almeida, A. Escargueil, J. Clairambault Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation Cancer research 930 939 2015

[9] P.R. De Andrade, A.M.V. Monteiro, G. Ĉamara, S. Sandri Games on cellular spaces: How mobility affects equilibrium Journal of Artificial Societies and Social Simulation 5 2009

[10] C. Deroulers, M. Aubert, M. Badoual, B. Grammaticos Modeling tumor cell migration: from microscopic to macroscopic models Physical Review E 031917 2009

[11] L.A. Dugatkin, D.S. Wilson Rover: a strategy for exploiting cooperators in a patchy environment American Naturalist 687 701 1991

[12] R.T. Durrett, S.A. Levin The importance of being discrete (and spatial) Theoretical Population Biology 363 394 1994

[13] M. Enquist, O. Leimar The evolution of cooperation in mobile organisms Animal Behaviour 747 757 1993

[14] T. Erneux, G. Nicolis Propagating waves in discrete bistable reaction-diffusion systems Physica D: Nonlinear Phenomena 237 244 1993

[15] R. Ferriere, R.E. Michod Invading wave of cooperation in a spatial iterated prisoner's dilemma Proceedings of the Royal Society of London B: Biological Sciences 77 83 1995

[16] R. Ferriere, R.E. Michod The evolution of cooperation in spatially heterogeneous populations American Naturalist 692 717 1996

[17] I.M. Hamilton, M. Taborsky Contingent movement and cooperation evolve under generalized reciprocity Proceedings of the Royal Society of London B: Biological Sciences 2259 2267 2005

[18] C. Hauert Effects of space in 2_ 2 games International Journal of Bifurcation and Chaos 1531 1548 2002

[19] D. Helbing, W. Yu Migration as a mechanism to promote cooperation Advances in Complex Systems 641 652 2008

[20] G. Jian-Yue, W. Zhi-Xi, W. Ying-Hai Evolutionary snowdrift game with disordered environments in mobile societies Chinese Physics 3566 2007

[21] T. Killingback, M. Doebeli Spatial evolutionary game theory: Hawks and doves revisited Proceedings of the Royal Society of London B: Biological Sciences 1135 1144 1996

[22] J.C. Koella The spatial spread of altruism versus the evolutionary response of egoists Proceedings of the Royal Society of London B: Biological Sciences 1979 1985 2000

[23] J.-F. Le Galliard, R. Ferriere, U. Dieckmann Adaptive evolution of social traits: origin, trajectories, and correlations of altruism and mobility The American Naturalist 206 224 2005

[24] Y.-T. Lin, H.-X. Yang, Z.-X. Wu, B.-H. Wang Promotion of cooperation by aspirationinduced migration Physica A: Statistical Mechanics and its Applications 77 82 2011

[25] J. Maynard Smith, G.R. Price The logic of animal conflict Nature 15 1973

[26] S. Meloni, A. Buscarino, L. Fortuna, M. Frasca, J. Gómez-Gardeñes, V. Latora, Y. Moreno Effects of mobility in a population of prisoner's dilemma players Physical Review E 067101 2009

[27] M.A. Nowak, S. Bonhoeffer, R.M. May More spatial games International Journal of Bifurcation and Chaos 33 56 1994

[28] M.A. Nowak, S. Bonhoeffer, R.M. May Spatial games and the maintenance of cooperation Proceedings of the National Academy of Sciences 4877 4881 1994

[29] M.A. Nowak, R.M. May Evolutionary games and spatial chaos Nature 826 829 1992

[30] M.A. Nowak, R.M. May The spatial dilemmas of evolution International Journal of Bifurcation and Chaos 35 78 1993

[31] K.J. Painter, T. Hillen Navigating the flow: individual and continuum models for homing in flowing environments Journal of The Royal Society Interface 20150647 2015

[32] Catherine J Penington, Barry D Hughes, Kerry A Landman Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena Physical Review E 041120 2011

[33] E. Pienaar, V. Dartois, J.J. Linderman, D.E. Kirschner In silico evaluation and exploration of antibiotic tuberculosis treatment regimens BMC Sys. Biol. 79 2015

[34] P.G. Schofield, M.A.J. Chaplain, S.F. Hubbard Mathematical modelling of host- parasitoid systems: effects of chemically mediated parasitoid foraging strategies on within-and between-generation spatio-temporal dynamics Journal of Theoretical Biology 31 47 2002

[35] P.G. Schofield, M.A.J Chaplain, S.F. Hubbard Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations Journal of Mathematical Biology 559 583 2005

[36] E.A. Sicardi, H. Fort, M.H. Vainstein, J.J. Arenzon Random mobility and spatial structure often enhance cooperation Journal of Theoretical Biology 240 246 2009

[37] J.M. Smith. Evolution and the Theory of Games. Cambridge university press, 1982.

[38] A. Stevens The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems SIAM Journal on Applied Mathematics 183 212 2000

[39] M. Tomassini, L. Luthi, M. Giacobini Hawks and doves on small-world networks Physical Review E 016132 2006

[40] M.H. Vainstein, A.T.C. Silva, J.J. Arenzon Does mobility decrease cooperation? Journal of Theoretical Biology 722 728 2007

[41] M. Van Baalen, D.A. Rand The unit of selection in viscous populations and the evolution of altruism Journal of Theoretical Biology 631 648 1998

[42] B.N. Vasiev Classification of patterns in excitable systems with lateral inhibition Physics Letters A 194 203 2004

[43] O.O. Vasieva, B.N. Vasiev, V.A. Karpov, A.N. Zaikin A model of dictyostelium discoideum aggregation Journal of theoretical biology 361 367 1994

[44] H.-X. Yang, Z.-X. Wu, B.-H. Wang Role of aspiration-induced migration in cooperation Physical Review E 065101 2010

[45] J. Zhang, W.-Y. Wang, W.-B. Du, X.-B. Cao Evolution of cooperation among mobile agents with heterogenous view radii Physica A: Statistical Mechanics and its Applications 2251 2257 2011

Cité par Sources :