Symmetric replicator dynamics with depletable resources.

The replicator equation is a standard model of evolutionary population game dynamics. In this article, we consider a modification of replicator dynamics, in which playing a particular strategy depletes an associated resource, and the payoff for that strategy is a function of the availability of the resource. Resources are assumed to replenish themselves, given time. Overuse of a resource causes it to crash. If the depletion rate is low enough, most trajectories converge to a stable equilibrium at which all initially present strategies are equally popular. As the depletion rate increases, these fixed points vanish in bifurcations. The phase space is periodic in each of the resource variables, and it is possible for trajectories to whirl around different numbers of times in these variables before converging to the stable equilibrium, resulting in a wide variety of topological types of orbits. Numerical solutions in a low-dimensional case show that in a cross section of the phase space, the topological types are separated by intricately folded separatrices. Once the depletion rate is high enough that the stable equilibrium in the interior of the phase space vanishes, the dynamics immediately become chaotic, without going through a period-doubling cascade; a numerical method reveals horseshoes in a Poincaré map. It appears that the multitude of topological types of orbits present before this final bifurcation generate this chaotic behavior. A periodic orbit of saddle type can be found using the symmetries of the dynamics, and its stable and unstable manifolds may generate a homoclinic tangle.