Discrete Evolutionary Birth-Death Processes and Their Large Population Limits

Abstract

We study a class of discrete-state Markovian models of evolutionary population dynamics for k types of organisms, called discrete evolutionary birth-death processes (EBDP). The organisms in a model environment interact with each other by playing a certain game. Birth rates for type i organisms at time t are determined by their expected payoff in the game against an opponent chosen randomly from the environment at time t. Death rates at time t for all the organism types are equal, and proportional to the total population at time t. A discrete EBDP is therefore a continuous-time Markov chain on the nonnegative k-dimensional integer lattice, with state transitions to neighboring vertices only. A certain system of k nonlinear ordinary differential equations (ODE) can be derived from the discrete EBDP, and used as a deterministic approximation. We prove that a properly scaled sequence of EBDP's converges (in probability, uniformly on bounded sets) to the solution of the system of ODE's. We also prove that a different scaling of the sequence converges (in distribution) to a certain system of stochastic differential equations (SDE). An example based on reactive strategies for iterated prisoner's dilemma is used to illustrate population dynamics for discrete EBDP's, as well as the dynamics for the ODE and SDE limits.