Deterministic approximations for stochastic processes in population biology

Differential equations are frequently used as deterministic models for stochastic processes. They approximate the expectations of the modeled stochastic processes, and are presumably good approximations when the state space (population) is sufficiently large. Although experimental simulations often support this approach, the justification of the use of deterministic equations for describing stochastic processes is not obvious. Some examples show that the deterministic approximation can deviate considerably from the exact result even for large populations. In this paper, we study two model examples: (a) A one sex pair formation process occurring in a closed population, with a nonlinear transition rate that results from assuming the mass action law for the pairing rule. This process has a finite state space. (b) A one sex pair formation process obtained from the first scenario by adding the effects of immigration and death. This process has an infinite state space. For both scenarios we compute analytically the equilibrium behavior of the stochastic and the deterministic models. We use these computations to compare the equilibrium expectations (in the infinite case) and the asymptotic expansions of the equilibrium expectations (in the finite state space model) of the stochastic processes with their deterministic counterparts. We use this comparison to study the appropriate representation of the mass action type reactions, and to study the quality of the deterministic model as an approximation for the stochastic scenario.