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

Part of: Control systems Stochastic systems and control

Published online by Cambridge University Press:  09 December 2020

AMIE ALBRECHT Open the ORCID record for AMIE ALBRECHT [Opens in a new window] ,
KONSTANTIN AVRACHENKOV Open the ORCID record for KONSTANTIN AVRACHENKOV [Opens in a new window] ,
PHIL HOWLETT Open the ORCID record for PHIL HOWLETT [Opens in a new window]  and
GEETIKA VERMA Open the ORCID record for GEETIKA VERMA [Opens in a new window]
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.
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Abstract

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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.

Keywords

replicator equationdiscrete timerandom perturbations

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 93C10: Nonlinear systems 93C55: Discrete-time systems 93C73: Perturbations 93E15: Stochastic stability

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 2 , April 2020 , pp. 148 - 184
Copyright
© Australian Mathematical Society 2020

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