A stochastic discrete generation birth, continuous death population growth model and its approximate solution
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.
first order non-linear stochastic difference equation
Ito stochastic differential equation models
population growth in random environments
scientific article; zbMATH DE number 3875028