Alternate Models of Replicator Dynamics

Models of evolutionary dynamics are often approached via the replicator equation, which in its standard form is given by ẋi = xi ( fi (x)−φ) , i = 1, . . . ,n, where xi is the frequency, or relative abundance, of strategy i, fi is its fitness, and φ = ∑i=1 xi fi is the average fitness. A game-theoretic aspect is introduced to the model via the payoff matrix A, where Ai, j is the expected payoff of i vs. j, by taking fi(x) = (A · x)i. This model is based on the exponential model of population growth, ẋi = xi fi, with φ introduced in order both to hold the total population constant and to model competition between strategies. We analyze the dynamics of analogous models for the replicator equation of the form ẋi = g(xi)( fi−φ), for selected growth functions g. INTRODUCTION The field of evolutionary dynamics combines game theory with ordinary differential equations to model Darwinian evolution via competition between adaptive strategies. A common approach [1] uses the replicator equation, which modifies the exponential model of population growth, ẋi = xi fi, where fi is the fitness of strategy i, by introducing the average fitness over all strategies, φ . The change in the relative abundance, xi, is then ẋi = xi( fi−φ), (1) where φ is chosen so that {x ∈ Rn : ∑xi = 1,0≤ xi ≤ 1} is an invariant manifold. This means that ∑ ẋi = 0, so φ = ∑ xi fi ∑xi = ∑xi fi. (2) In essence, φ acts as a coupling term that introduces dependence on the abundance and fitness of other strategies. In this work, we generalize the replicator model by replacing the base model ẋi = xi fi by ẋi = g(xi) fi, where g is a natural growth function. The replicator equation for each strategy becomes ẋi = g(xi)( fi−φ), (3) where φ is now a modified average fitness, again chosen so that ∑xi = 1. The game-theoretic component of this model lies in the choice of fitness functions. Take the payoff matrix A, whose (i, j)-th entry is the expected reward for strategy i when it competes with strategy j. The fitness fi of strategy i is then (A · x)i, 1 Copyright c © 2013 by ASME where x ∈ Rn is the vector of frequencies xi. In this work, we use a payoff matrix representing a game analogous to rock-paperscissors (RPS): there are three strategies, each of which has an advantage versus one other and a disadvantage versus the third. Each strategy is neutral versus itself. Analysis of the resulting dynamical system is presented. We find that for the logistic model g(x) = x−ax2, (4) with appropriate choices of the parameter a, there are multiple fixed points of the system that do not exist in the usual model g(x) = x. We will show that when A is chosen so that the RPS game is zero-sum, there are 13 equilibria: one neutrally stable equilibrium with all three strategies surviving; three saddle points with all three strategies surviving; three saddles with only one surviving strategy; and three attracting and three repelling fixed points where two strategies survive. The system exhibits both periodic motion and convergence to attractors. We analyze the symmetries of this system, and its bifurcations as the entries of A vary. This alternate formulation may be useful in modeling natural or social systems that are not adequately described by the usual replicator dynamics. DERIVATION Let us review the usual replicator dynamics. We have fi = fi (x), where fi (x) = (A · x)i, where A is the payoff matrix. The average payoff is thus φ =∑i xi fi, and the change in frequency of strategy i is given by the product of the frequency xi and its payoff relative to the average. In this model, all population-dependence of the effectiveness (hence growth rate) of strategy i is accounted for by fi. However, we wish our fitness functions fi to represent the game-theoretic payoff of individual-level competition. We therefore include some of the population dependence in a growth function g(xi); this represents the growth rate of the raw population using strategy i, in the absence of competition. Thus the expected population-level payoff of strategy i is g(xi) fi, and the average population-level payoff is φ = ∑i g(xi) fi ∑i g(xi) . (5) We require that in this model, φ (and hence ẋ) is only defined for growth functions g such that the denominator does not vanish for any x in the region of interest. With that caveat, using this definition of φ , the replicator equation becomes ẋi = g(xi)( fi−φ) , i = 1, . . . ,n. (6) We can verify that ∑ i ẋi = ∑ i g(xi)( fi−φ) = ∑ i g(xi) fi−∑ i g(xi) ∑i g(xi) fi ∑i g(xi) = 0 (7) so the total population over all strategies is constant, and it is valid to say that each xi represents the frequency of strategy i. We will use the term relative abundance for xi whenever there is ambiguity between xi and the time-frequency of any periodic motion in the dynamics. ROCK-PAPER-SCISSORS We consider the game-theoretic case in which n = 3 and fi is given by fi (x) = (A · x)i, where A is the payoff matrix A =  0 −1 +1 +1 0 −1 −1 +1 0  , (8) representing a zero-sum rock-paper-scissors game. That is, writing (x1,x2,x3) as (x,y,z), f1 = z− y, f2 = x− z, f3 = y− x. (9) We note that this model has been shown to be relevant to biological applications [2], [3], and to social interactions [4]. Note that the dynamics of the 3-strategy game takes place on the triangle in R3 (in fact, the three-dimensional simplex) Σ = { (x,y,z) ∈ R3 : x+ y+ z = 1 and x,y,z≥ 0 } . (10) Therefore we can eliminate z using z = 1− x− y. This reduces the problem to two dimensions, so that (6) becomes ẋ = g(x)((1−3y)g(1− x− y)+(2−3x−3y)g(y)) g(x)+g(y)+g(1− x− y) (11) ẏ = −g(y)((1−3x)g(x)+(2−3x−3y)g(1− x− y)) g(x)+g(y)+g(1− x− y) (12) where we have used φ as defined in Eqn (5). This vector field is defined on the projection of Σ onto the x−y plane. We will refer to this region as T = {(x,y) : (x,y,1− x− y) ∈ Σ} . (13) 2 Copyright c © 2013 by ASME