Evolution of populations with strategy-dependent time delays

TL;DR

This paper introduces a novel replicator dynamics model for evolving populations with strategy-dependent time delays, revealing that stationary states depend continuously on delays and can lead to new equilibrium behaviors.

Contribution

It develops a microscopic model leading to a new type of replicator dynamics where stationary states vary with time delays, unlike previous models.

Findings

Stationary states depend continuously on strategy-dependent time delays.

Time delays can cause the disappearance or emergence of interior stationary states.

In Prisoner's Dilemma, specific delays lead to stable cooperation states.

Abstract

We study effects of strategy-dependent time delays on equilibria of evolving populations. It is well known that time delays may cause oscillations in dynamical systems. Here we report a novel behavior. We show that microscopic models of evolutionary games with strategy-dependent time delays lead 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, stationary states of such dynamics depend continuously on time delays. We show that in games with an interior stationary state (a globally asymptotically stable equilibrium in the standard replicator dynamics), at certain time delays, it may disappear or there may appear another interior stationary state. In the Prisoner's Dilemma game, for time delays of cooperation smaller than time delays of defection, there appears an ustable interior equilibrium and therefore for some initial conditions, the population converges to the homogeneous state with just cooperators.