In this chapter applications of stochastic calculus to population models are given. Diffusion models, Markov Jump process models, age-dependent non-Markov models and stochastic models for competition of species are presented. Diffusion Models are used in various areas of Biology as models for population growth and genetic evolution. Birth and Death processes are random walks in continuous time and are met in various applications. A novel approach to the age-dependent branching (Bellman-Harris) process is given by treating it as a simple measure-valued process. The stochastic model for interacting populations that generalizes the Lotka-Volterra prey-predator model is treated by using a semimartingale representation. It is possible to formulate these models as stochastic differential or integral equations. We demonstrate how results on stochastic differential equations and martingales presented in earlier chapters are applied for their analysis.

13.1 Feller's Branching Diffusion

A simple branching process is a model in which individuals reproduce independently of each other and of the history of the process. The continuous approximation to branching process is the branching diffusion. It is given by the stochastic differential equation for the population size X(t), 0 < X(t) < ?,

In this model the infinitesimal drift and variance are proportional to the population size. The corresponding forward (Kolmogorov or Fokker-Plank) equation for the probability density of X(t) is

Analysis of the process was done by solving the partial differential equation (13.2) (Feller (1951)). Here we demonstrate the stochastic calculus approach by obtaining the information directly from the stochastic equation...