Stochastic multi-species population dynamics

A stochastic model is presented for a system consisting of several species. The processes involved are birth, death, and mutation. In general, these processes lead to evolution equations for l-point functions which are coupled. Conditions are obtained that make these autonomous, such that the evolution equation for any l-point function contains only l-or-less-point functions. The time evolution of the one- and two-point functions is explicitly calculated. More restrictive conditions are obtained so that the evolution equation for the generating function is a first-order differential equation. This allows for an explicit study of the large time behavior of the system in general. This is done, and the criteria for the stability (instability) of the system are discussed. While this has been extensively studied for single species systems, here, the machinery of complex vector fields and stability of dynamical systems is used to obtain the qualitative large time behavior of systems involving several species. These are applied to a concrete family of models the processes of which change the population by at most a single unit.