Hopf Bifurcations in Two-player Delayed Replicator Dynamics

We investigate the dynamics of two-strategy replicator equations in which competition between strategies is delayed by a given time interval T . Taking T as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems, and present an analysis of the resulting limit cycles using Lindstedt’s method. INTRODUCTION The field of evolutionary dynamics uses both game theory and differential equations to model population shifts among competing adaptive strategies. One standard approach [1, 2] uses the replicator equation, which modifies the exponential model of population growth, ξ̇i = ξigi (i = 1, . . . ,n) (1) where ξi is the population of strategy i and gi(ξ1, . . . ,ξn) is the fitness of that strategy. The replicator equation [3] results from equation (1) by changing variables from the populations ξi to the relative abundances, defined as xi ≡ ξi/p where p is the total population: p(t) = ∑ i ξi(t). (2) ∗Address all correspondence to this author. We see that ṗ = ∑ i ξ̇i = ∑ i ξigi (3) = p∑ i ξi p gi = p∑ i xigi (4) = pφ (5) where φ ≡ ∑i xigi is the average fitness of the whole population. By the product rule, ẋi = ξ̇i p − ξi ṗ p2 (6) = ξi p gi− ξi p ṗ p (7) = xi (gi−φ) . (8) Therefore ∑ i ẋi = ∑ i xigi−φ ∑ i xi (9) = ∑ i xigi−∑ j x jg j ∑ i xi. (10) So, using the fact that ∑ i xi = ∑i ξi p = p p ≡ 1 (11) 1 Copyright c © 2014 by ASME equation (10) reduces to the identity ∑ i ẋi = 0. (12) The fitness of a strategy is assumed to depend only on the relative abundance of each strategy in the overall population, since the model only seeks to capture the effect of competition between strategies, not any environmental or other factors. Therefore, we assume that gi has the form gi(ξ1, . . . ,ξn) = fi ( ξ1 p , . . . , ξn p ) = fi(x1, . . . ,xn). (13) Under this assumption, equation (8) is the replicator equation, ẋi = xi( fi−φ), (14) where φ is now expressed entirely in terms of the xi, as φ = ∑ i xi fi. (15) Mathematically, φ is a coupling term that introduces dependence on the abundance and fitness of other strategies. The game-theoretic component of the replicator model lies in the choice of fitness functions. Take the payoff matrix A = (ai j), where ai j is the expected reward for strategy i when it competes with strategy j. Then the fitness fi is the total expected payoff of strategy i vs. all strategies, weighted by their frequency: