Branching-diffusion model for the formation of distributional patterns in populations
type: Item
instance of: scholarly article
title: Branching-diffusion model for the formation of distributional patterns in populations
zbMATH Open document ID: 0534.92028
DOI: 10.1007/BF00275934
author: N. E. Zubov
published in: Journal of Mathematical Biology
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 60J85
zbMATH DE Number: 3849047
zbMATH Keywords: branching diffusion processes
zbMATH Keywords: dynamics of spatial pattern
zbMATH Keywords: contagious distribution
zbMATH Keywords: quadrat sampling
zbMATH Keywords: distance sampling
zbMATH Keywords: Pielou alpha index
zbMATH Keywords: Kuno index
zbMATH Keywords: dispersion indices
zbMATH Keywords: initial Poisson distribution
zbMATH Keywords: Brownian motions
zbMATH Keywords: mean
zbMATH Keywords: variance
Wikidata QID: Q57065688
MaRDI profile type: Publication
cites work: The spectral analysis of two-dimensional point processes
cites work: On branching Markov processes
cites work: Fundamental equations of branching Markov processes
cites work: Q5675177
cites work: Branching Diffusion Processes
cites work: SPATIAL POINT PROCESSES, WITH APPLICATIONS TO ECOLOGY
full work available at URL: https://doi.org/10.1007/bf00275934
OpenAlex ID: W1967111343
description: scientific article; zbMATH DE number 3849047