Nonlinear character dependent models with constant time delay in population dynamics
type: Item
instance of: scholarly article
title: Nonlinear character dependent models with constant time delay in population dynamics
zbMATH Open document ID: 0639.92014
DOI: 10.1016/0022-247X(87)90211-3
author: Carlos Castillo-Chavez
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: This paper is devoted to a nonlinear size structured model of population dynamics. An additional feature of the model includes a time delay in the birth process. The model has the following form: \[ p_ t(m,t)+g(m,t)p(m,t)+[g_ m(m,t)+\theta (m,P)]p(m,t)=0,\quad m>m_ 1, \] \[ P(t)=\int^{\infty}_{m_ 1}p(m',t)dm',\quad g(m_ 1,t)p(m_ 1,t)=P_ E\int^{\infty}_{m_ 1}\lambda (m',P)p(m',t-\tau_ E)dm',\quad t>\tau_ E, \] \[ p(m,0)=p_ 0(m),\quad m>m_ 1,\quad p_ m(m,t)=0,\quad t<0. \] Here n is the size variable, p(m,t) is the density of the population at time t with respect to size m, g(m,t) is the size and time dependent growth rate of individuals, P(t) is the total population at time t, \(\theta\) (m,P(t)) and \(\lambda\) (m,m',p(t)) are the size and population dependent mortality and fertility rates, respectively, \(m_ 1\) is the birth size, \(\tau_ E\) is the egg-stage duration, \(P_ E\) is the probability of survivorship, and \(p_ 0(m)\) is the initial size distribution. The properties studied are the local and global existence of solutions, the uniqueness of solutions, and the existence and stability of nontrivial equilibrium solutions. The method of characteristics is used to convert the problem to a pair of coupled nonlinear functional differential equations. The results generalize the age structured case of \textit{M. E. Gurtin} and \textit{R. C. MacCamy} [Arch. Rat. Mech. Analysis 54, 281-300 (1974; Zbl 0286.92005)] to a size structured case with time delay.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34K05
zbMATH DE Number: 4043670
zbMATH Keywords: time delayed birth process
zbMATH Keywords: nonlinear size structured model of population dynamics
zbMATH Keywords: local and global existence of solutions
zbMATH Keywords: uniqueness of solutions
zbMATH Keywords: existence and stability of nontrivial equilibrium solutions
zbMATH Keywords: method of characteristics
zbMATH Keywords: coupled nonlinear functional differential equations
Wikidata QID: Q56774074
reviewed by: Glenn Webb
MaRDI profile type: Publication
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full work available at URL: 0022-247x(87)90211-3
OpenAlex ID: W2003662007
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description: scientific article; zbMATH DE number 4043670