Persistent distribution functions for the dispersion of structured populations
type: Item
instance of: scholarly article
title: Persistent distribution functions for the dispersion of structured populations
zbMATH Open document ID: 0646.92017
DOI: 10.1016/0025-5564(87)90073-3
author: M. O. Vlad
published in: Mathematical Biosciences
publication date: 1987-01-01T00:00:00Z
review text: This paper is concerned with the large-population dynamics of individuals that are characterized by a varying state vector \b{X}, whose birth intensity depends only on parental age and their state at birth, whose mortality function is a sum of an age-dependent component and a component depending on absolute time and the evolution of the population density, and which diffuse in a crowd-avoiding manner. More specifically, the state \b{X} of individuals is supposed to vary randomly throughout the individual's lifetime. This variation is age dependent: The state \b{X} of an individual aged a changes to \b{X}' by \(a+\Delta a\) with probability W(\b{X}\(\to \underline X',a)\Delta a\). Apart from this the author's model is a standard one. Thus if \(\rho\) (a,\b{X},ṟ,t) is the population density of individuals aged a, having state \b{X} and situated at position ṟ, all at time t, then \[ (\partial_ t+\partial_ u)\rho +div_{\underline r}\Phi +\mu \rho =\int W(\underline X'\to \underline X,a)\rho (a,\underline X,\underline r,t)d\underline X'-\int W(\underline X\to \underline X',a)\rho (a,\underline X,\underline r,t)d\underline X' \] \[ and\quad \rho (a=0,\underline X,\underline r,t)=\int \int^{\infty}_{0}\lambda (a',\underline X',\underline X,...)\rho (a',\underline X',\underline r,t)da'd\underline X', \] where \(\lambda\) (\(\cdot)\) and \(\mu\) (\(\cdot)\) are the birth and death intensities, respectively, and \(\Phi\) is the diffusion flux. Under the above conditions on \(\lambda\), \(\mu\) and \(\Phi\), the author seeks formal product solutions of this system along lines explored by \textit{M. E. Gurtin} and \textit{R. C. MacCamy}, ibid. 62, 157-167 (1982; Zbl 0505.92019). He then calculates several demographic averages from these solutions. As a particular example he considers the case where there is a single state variable, body size, and he computes expressions for body size and energy production rate averages. Finally, the author considers a generalization where the i…
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 60J85
zbMATH DE Number: 4055439
zbMATH Keywords: dispersion of structured populations
zbMATH Keywords: age-dependent models
zbMATH Keywords: persistent age-stage distribution functions
zbMATH Keywords: nonlinear population model
zbMATH Keywords: age- dependent master equations
zbMATH Keywords: large-population dynamics
zbMATH Keywords: formal product solutions
zbMATH Keywords: demographic averages
zbMATH Keywords: body size
zbMATH Keywords: energy production
MaRDI profile type: Publication
full work available at URL: 0025-5564(87)90073-3
OpenAlex ID: W2051749056
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description: scientific article; zbMATH DE number 4055439