Growth with regulation in fluctuating environments. II. Intrinsic lower bounds to population size
type: Item
instance of: scholarly article
title: Growth with regulation in fluctuating environments. II. Intrinsic lower bounds to population size
zbMATH Open document ID: 0553.92017
DOI: 10.1007/BF00337078
published in: Biological Cybernetics
publication date: 1984-01-01T00:00:00Z
review text: [For part I see the preceding review, Zbl 0553.92016.] A third order deterministic model is proposed for population growth characterized by three equilibrium states: the zero population size, the carrying capacity and a ''threshold level'' after which ultimate extinction is secured. The authors analyze the above model when random fluctuations are introduced in the intrinsic fertility rate which then becomes a stationary normal process \(a+\Lambda (t)\) with \(\Lambda\) (t) being a white noise. A stochastic differential equation of Stratonovich type is thus obtained, which defines a temporally homogeneous diffusion process X(t). By means of a monotone transformation a Wiener process Y(t) is obtained such that its transition p.d.f. allows an easy computation of the transition p.d.f. of X(t); the asymptotic behaviour of that is analyzed. An alternative equation involving a logarithmic nonlinearity is also considered, which still allows three equilibrium states, and, as in the previous case, can be transformed into a stochastic differential equation for a Wiener process. Furthermore, a model which does not always allow a steady state distribution is analyzed; in the case in which, under suitable conditions, it does exist, this distribution is used to obtain the asymptotic moments of the population. Finally, a model with a logarithmic singularity is studied which always admits a steady state distribution, but on the other hand none of the moments always exist.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 92D40
zbMATH DE Number: 3882256
zbMATH Keywords: third order deterministic model
zbMATH Keywords: population growth
zbMATH Keywords: equilibrium states
zbMATH Keywords: zero population size
zbMATH Keywords: carrying capacity
zbMATH Keywords: threshold level
zbMATH Keywords: random fluctuations
zbMATH Keywords: intrinsic fertility rate
zbMATH Keywords: stationary normal process
zbMATH Keywords: white noise
zbMATH Keywords: stochastic differential equation of Stratonovich type
zbMATH Keywords: monotone transformation
zbMATH Keywords: Wiener process
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: logarithmic nonlinearity
zbMATH Keywords: asymptotic moments
zbMATH Keywords: logarithmic singularity
Wikidata QID: Q47245159
author: Amelia G. Nobile
author: L. M. Ricciardi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3882256