Stochastic Population Theory: Diffusion Processes

In the previous contribution we discussed birth and death processes as models of populations subject to random growth. There, the population size at each instant was represented as a discrete random variable labeled by the considered instant. Since the probabilistic description was characterized by a straightforward integration, we purposely avoided spending time on definitions and mathematical preliminaries. However, it is sometimes convenient to model population growth by continuous differential equations and by their stochastic counterparts. This implies that the population size at each instant can be any nonnegative real number or else a continuous space-continuous time stochastic process. It is to be stressed that this is evidently an approximation which, however, may prove useful in making inferences about global properties of the population dynamics such as stability, extinction, etc.