Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study
type: Item
instance of: scholarly article
title: Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study
zbMATH Open document ID: 0797.92024
DOI: 10.1016/0025-5564(93)90006-V
published in: Mathematical Biosciences
publication date: 1993-06-29T00:00:00Z
review text: A model system derived by \textit{J. A. J. Metz} and \textit{O. Diekmann} [in Lect. Notes Biomath. 68 (1986; Zbl 0614.92014)] consisting of a hyperbolic partial differential equation coupled with an ordinary differential equation was reduced, by integrating along characteristics, to a single threshold delay differential equation (TDE) and finally to a standard functional differential equation (FDE). The stability of equilibria and existence of Hopf bifurcations of the FDE are investigated and their implications for the original model are described. A brief discussion on a model of competition between juveniles and adults is also presented. The contributions to the theory and applications of FDEs of this paper, together with the author's other related work listed in the references, are significant. First of all, structured population models can be reduced to TDEs and to standard FDEs so that the extensive theory of FDEs can be applied to obtain long-time behavior of structured population models. Secondly, the obtained FDEs from structured population models are, in many cases, state-dependent delay equations and delay equations with distributed delays. This provides further profound biological motivations for the study of state-dependent delay equations and equations with distributed delays. The reviewer found it very benefical to read this paper before the study of the author's other related work aforementioned.
reviewed by: Jianhong Wu
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K25
zbMATH DE Number: 219754
zbMATH Keywords: hyperbolic partial differential equation
zbMATH Keywords: ordinary differential equation
zbMATH Keywords: threshold delay differential equation
zbMATH Keywords: functional differential equation
zbMATH Keywords: stability of equilibria
zbMATH Keywords: existence of Hopf bifurcations
zbMATH Keywords: model of competition
zbMATH Keywords: structured population models
zbMATH Keywords: long-time behavior
zbMATH Keywords: state- dependent delay equations
zbMATH Keywords: distributed delays
author: Hal L. Smith
MaRDI profile type: Publication
Wikidata QID: Q52404512
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full work available at URL: 0025-5564(93)90006-v
OpenAlex ID: W1993825961
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description: scientific article; zbMATH DE number 219754