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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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