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EVOLUTIONARY DYNAMICS IN DISCRETE TIME FOR THE PERTURBED POSITIVE DEFINITE REPLICATOR EQUATION

Published online by Cambridge University Press:  09 December 2020

AMIE ALBRECHT
Affiliation:
Scheduling and Control Group (SCG), Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, SA5095, Australia; e-mail: amie.albrecht@unisa.edu.au, phil.howlett@unisa.edu.au and geetika.verma@unisa.edu.au.
KONSTANTIN AVRACHENKOV
Affiliation:
INRIA, Sophia Antipolis, COSTNET CA15109, France; e-mail: k.avrachenkov@sophia.inria.fr.
PHIL HOWLETT
Affiliation:
Scheduling and Control Group (SCG), Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, SA5095, Australia; e-mail: amie.albrecht@unisa.edu.au, phil.howlett@unisa.edu.au and geetika.verma@unisa.edu.au.
GEETIKA VERMA
Affiliation:
Scheduling and Control Group (SCG), Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, SA5095, Australia; e-mail: amie.albrecht@unisa.edu.au, phil.howlett@unisa.edu.au and geetika.verma@unisa.edu.au.

Abstract

The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation.

Type
Research Article
Copyright
© Australian Mathematical Society 2020

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References

Avrachenkov, K. and Borkar, V. S., “Metastability in stochastic replicator dynamics”, Dyn. Games. Appl. 9 (2019) 366390; doi:10.1007/s13235-018-0265-7.CrossRefGoogle Scholar
Cabrales, A., “Stochastic replicator dynamics”, Internat. Econom. Rev. 41 (2000) 451481; doi:10.1111/1468-2354.00071.CrossRefGoogle Scholar
Cooper, R. W., Coordination games: Complementarities and macroeconomics (Cambridge University Press, Cambridge, 1999); doi:10.1017/CBO9780511609428..CrossRefGoogle Scholar
Cressman, R. and Tao, Y., “The replicator equation and other game dynamics”, Proc. Natl. Acad. Sci. USA 111 (2014) 1081010817; doi:10.1073/pnas.1400823111.CrossRefGoogle ScholarPubMed
Fisher, R. A., The genetical theory of natural selection (Clarendon Press, Oxford, 1930); doi:10.5962/bhl.title.27468. CrossRefGoogle Scholar
Foster, D. and Young, P., “Stochastic evolutionary game dynamics”, Theor. Popul. Biol. 38 (1990) 219232; doi:10.1016/0040-5809(90)90011-J [Corrigendum: Theor. Popul. Biol. 51 (1997) 77–78].CrossRefGoogle Scholar
Fudenberg, D. and Harris, C., “Evolutionary dynamics with aggregate shocks”, J. Econ. Theory 57 (1992) 420441; doi:10.1016/0022-0531(92)90044-I.CrossRefGoogle Scholar
Hofbauer, J. and Imhof, L. A., “Time averages, recurrence and transience in the stochastic replicator dynamics”, Ann. Appl. Probab. 19 (2009) 13471368; doi:10.1214/08-AAP577.CrossRefGoogle Scholar
Hofbauer, J. and Sigmund, K., “Evolutionary game dynamics”, Bull. Amer. Math. Soc. 40 (2003) 479519; doi:10.1090/S0273-0979-03-00988-1.CrossRefGoogle Scholar
Imhof, L. A., “The long-run behavior of the stochastic replicator dynamics”, Ann. Appl. Probab. 15 (2005) 10191045; doi:10.1214/105051604000000837.CrossRefGoogle Scholar
Khasminskii, R. and Potsepun, N., “On the replicator dynamics behavior under Stratonovich type random perturbations”, Stoch. Dyn. 6 (2006) 197211; doi:10.1142/S0219493706001712.CrossRefGoogle Scholar
Kingman, J. F. C., “A mathematical problem in population genetics”, Math. Proc. Cambridge 57 (1961) 574582; doi:10.1017/S0305004100035635.CrossRefGoogle Scholar
Kleshnina, M., “Evolutionary games under incompetence and foraging strategies of marine bacteria”, Ph.D. Thesis, University of Queensland, Australia, 2019; doi:10.14264/uql.2019.339.Google Scholar
McGehee, R., “Attractors for closed relations on compact Hausdorff spaces”, Indiana Univ. Math. J. 41 (1992) 11651209.CrossRefGoogle Scholar
Mertikopoulos, P. and Viossat, Y., “Imitation dynamics with payoff shocks”, Int. J. Game Theory 45 (2016) 291320; doi:10.1007/s00182-015-0505-7.CrossRefGoogle Scholar
Mulholland, H. P. and Smith, C. A. B., “An inequality arising in genetical theory”, Am. Math. Monthly 66 (1959) 673683; doi:10.1080/00029890.1959.11989387.CrossRefGoogle Scholar
Sandholm, W. H., Population games and evolutionary dynamics (MIT Press, Cambridge, MA, 2011).Google Scholar
Schuster, P. and Sigmund, K., “Replicator dynamics”, J. Theor. Biol. 100 (1983) 533538; doi:10.1016/0022-5193(83)90445-9.CrossRefGoogle Scholar
Taylor, P. D. and Jonker, L. B., “Evolutionary stable strategies and game dynamics”, Math. Biosci. 40 (1978) 145156; doi:10.1016/0025-5564(78)90077-9.CrossRefGoogle Scholar
Weibull, J., Evolutionary game theory (MIT Press, Cambridge, MA, 1995).Google Scholar
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EVOLUTIONARY DYNAMICS IN DISCRETE TIME FOR THE PERTURBED POSITIVE DEFINITE REPLICATOR EQUATION
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