Branching-diffusion model for the formation of distributional patterns in populations
The authors are interested in the behaviour of some commonly used dispersion indices for the distribution of individuals moving and reproducing in the plane according to a branching Brownian process. They assume an initial Poisson distribution of individuals in the plane, and that individuals reproduce according to independent Markov branching processes and move according to independent Brownian motions. Suppose there are \(Z_ t\) individuals alive at time t occupying positions \(X_ 1,...,X_{Z_ t}\) and let f be a function defined on \({\mathbb{R}}^ 2\). The authors' fundamental result shows that \(u(t,x,\epsilon)=E_ x[\exp(-\epsilon \sum^{Z_ t}_{i=1}f(X_ i))]\) satisfies a diffusion equation containing an additional non-linear term in u. This theorem is used to derive expressions for the mean and variance of the number of individuals in a square and the expected value of the square of the distance from a randomly chosen point to the nearest individual. These results are used to obtain qualitative insight into the behavior of \textit{E. Kuno's} \(C_ A\)-index [Studies on the population dynamics of rice leafhoppers in a paddy field. Bull. Kyushu Agricultural and Experimental Statistics 14, 137-246 (Japanese) (1968)] and \textit{C. E. Pielou's} \(\alpha\)-index [Mathematical ecology. J. Wiley and Sons (1977)]. Not surprisingly, these results indicate a clumpy distribution of individuals.
scientific article; zbMATH DE number 3849047