Branching-diffusion model for the formation of distributional patterns in populations

A continuous time branching-diffusion process model is presented to describe the development of spatial distributional patterns of a biological population. In the model each unit moves independently following diffusion processes on a plane, and multiplies or goes extinct at random times. Standard methods for measuring the degree of aggregation used in field ecology are applied to this model population. Kuno's C_Aindex using quadrat sampling is calculated, and the dependence of the index on time, quadrat size, initial density, and diffusion and branching rules, is discussed. Pielou's α index based on distance measurement is evaluated by the solution of a nonlinear partial differential equation. Both methods show that continuous-time branching-diffusion processes produce a contagious spatial pattern; as in a discrete-time model studied by Iwasa and Teramoto (1977).