Evolution of populations with strategy-dependent time delays

Abstract

We address the issue of stability of coexistence of two strategies with respect to time delays in evolving populations. It is well known that time delays may cause oscillations. Here we report a novel behavior. We show that a microscopic model of evolutionary games with a unique mixed evolutionarily stable strategy (a globally asymptotically stable interior stationary state in the standard replicator dynamics) and with strategy-dependent time delays leads to a new type of replicator dynamics. It describes the time evolution of fractions of the population playing given strategies and the size of the population. Unlike in all previous models, an interior stationary state of such dynamics depends continuously on time delays and at some point it might disappear, no cycles are present. In particular, this means that an arbitrarily small time delay changes an interior stationary state. Moreover, at certain time delays, there may appear another interior stationary state. Author summarySocial and biological processes are usually described by ordinary or partial differential 1 equations, or by Markov processes if we take into account stochastic perturbations. 2 However, interactions between individuals, players or molecules, naturally take time. 3Results of biological interactions between individuals may appear in the future, and in 4 social models, individuals or players may act, that is choose appropriate strategies, on 5 the basis of the information concerning events in the past. It is natural therefore to 6 introduce time delays into evolutionary game models. It was usually observed, and 7 expected, that small time delays do not change the behavior of the system and large 8 time delays may cause oscillations. Here we report a novel behavior. We show that 9 microscopic models of evolutionary games with strategy-dependent time delays, in 10 which payoffs appear some time after interactions of individuals, lead to a new type of 11 replicator dynamics. Unlike in all previous models, interior stationary states of such 12 dynamics depend continuously on time delays. This shows that effects of time delays 13 are much more complex than it was previously thought. 14 Introduction 15Many social and biological processes can be modeled as systems of interacting 16 individuals within the framework of evolutionary game theory [9, 11, 15-17, 22, 23, 26, 29]. 17 The evolution of very large (infinite) populations can be then given by differential 18 the basis of the information concerning events in the past. It is natural therefore to 25 introduce time delays into evolutionary game models. 26It is well known that time delays may cause oscillations in dynamical systems [2][3][4][5]. 27 One usually expects that interior equilibria of evolving populations, describing 28 coexisting strategies or behaviors, are asymptotically stable for small time delays and 29 above a critical time delay, where the Hopf bifurcation appears, they become unstable, 30 evolutionary dynamics exhibits oscillations and cycles. Here we report a novel behavior -31 ...