Separable models in age-dependent population dynamics
type: Item
instance of: scholarly article
title: Separable models in age-dependent population dynamics
zbMATH Open document ID: 0574.92025
DOI: 10.1007/BF00276489
published in: Journal of Mathematical Biology
publication date: 1985-01-01T00:00:00Z
review text: This paper treats nonlinear models of age structured populations. The models have a certain separable form and include equations such as \[ u_ a+u_ t+\mu (a)u=F(t,P(t)),\quad u(a,0)=u_ 0(a),\quad u(0,t)=\int^{\infty}_{0}\beta (a)u(a,t)da, \] where \(P(t)=\int^{\infty}_{0}u(a,t)da\). The problem is recast in terms of a partial differential equation for the age profile \(w(a,t)=u(a,t)/P(t)\) and an ordinary nonlinear differential equation for P(t). The equation for w(a,t) is analyzed using classical linear renewal theory. Results concerning asymptotic behavior of solutions are obtained for a variety of models. In particular the effects of maturation periods are studied for interacting species and predator-prey models. Hopf bifurcation theory is used to establish that increases in maturation period can have stabilizing or destabilizing consequences, and in certain cases successive exchanges in stability. This paper develops a very useful approach to a general class of nonlinear models of age dependent population dynamics.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 3918170
zbMATH Keywords: existence
zbMATH Keywords: global attractivity
zbMATH Keywords: separable equations
zbMATH Keywords: nonlinear models of age structured populations
zbMATH Keywords: linear renewal theory
zbMATH Keywords: maturation periods
zbMATH Keywords: interacting species
zbMATH Keywords: predator-prey models
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: stability
author: Stavros N. Busenberg
author: M. Iannelli
reviewed by: Glenn Webb
MaRDI profile type: Publication
Wikidata QID: Q52679323
cites work: A global stability criterion for simple control loops
cites work: Periodic solutions of a nonlinear second order differential equation with delay
cites work: Global asymptotic stability of certain models for protein synthesis and repression
cites work: Interaction of spatial diffusion and delays in models of genetic control by repression
cites work: Periodic solutions of difference-differential equations
cites work: Q3892115
cites work: Q4189109
cites work: Periodic solutions for certain protein synthesis models
cites work: Models of genetic control by repression with time delays and spatial effects
cites work: Qualitative analysis of a coupled reaction-diffusion model in biology with time delays
cites work: Semigroups of linear operators and applications to partial differential equations
description: scientific article; zbMATH DE number 3918170