Reduction of structured population models to threshold-type delay equations and functional differential equations: A case study
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.
hyperbolic partial differential equation
scientific article; zbMATH DE number 219754
scientific article