A stochastic discrete generation birth, continuous death population growth model and its approximate solution
type: Item
instance of: scholarly article
title: A stochastic discrete generation birth, continuous death population growth model and its approximate solution
zbMATH Open document ID: 0548.92010
DOI: 10.1007/BF00275862
author: Helle Aagaard-Hansen
author: Geoffrey F. Yeo
published in: Journal of Mathematical Biology
publication date: 1984-01-01T00:00:00Z
review text: The authors consider a stochastic generalisation of a population model first introduced by \textit{E. T. Poulsen}, ibid. 8, 325-343 (1979; Zbl 0442.92020). Birth events occur at times \(t_ n\) where \(\tau_ n=t_ n- t_{n-1}\) (\(n\geq 1\), \(t_ 0=0)\) are independent and identically distributed random variables. Let X(t) denote the population size at time t, \(X_ n=X(t_ n-)\) and \(Z_ n=X(t_ n+).\) Given \(Z_ n\), the population changes deterministically during \((t_ n,t_{n+1})\) according to the differential equation \(X'(t)=-r(X(t))\) where \(r(0-)=0\) and \(0<r(x)<\infty\) if \(0<x\leq N\leq\infty \). Given \(X_ n=x_ n\) the population size just after the birth event at \(t_ n\) is a random variable \(S_ n(x_ n)\) and \(P(S_ n(x_ n)\leq y)=\int^{y}_{0}b(u,x_ n)du\). The authors pay particular attention to the cases where the \(\tau_ n\) are equal and constant and \(r(\cdot)\) corresponds to logistic or Gompertzian growth. The sequence \(\{Z_ n\}\) satisfies a first order non-linear stochastic difference equation and, apart for a couple of special cases, it is impossible to find tractable expressions for the distributions of the \(Z_ n\), or even the limiting distribution if it exists. They show how to obtain good approximations by using Gram-Charlier expansions and illustrate their methodology with several numerical examples. Comparisons are also made with some specific Ito stochastic differential equation models for population growth in random environments.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 3875028
zbMATH Keywords: single species growth model
zbMATH Keywords: integral equation
zbMATH Keywords: discrete birth
zbMATH Keywords: continuous death
zbMATH Keywords: limiting population size
zbMATH Keywords: extinction time
zbMATH Keywords: logistic growth
zbMATH Keywords: Gompertzian growth
zbMATH Keywords: first order non-linear stochastic difference equation
zbMATH Keywords: Gram-Charlier expansions
zbMATH Keywords: numerical examples
zbMATH Keywords: Ito stochastic differential equation models
zbMATH Keywords: population growth in random environments
Wikidata QID: Q113909242
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3875028