European Journal of Applied Mathematics



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A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay


J. F. M. AL-OMARI a1 and S. A. GOURLEY a2
a1 Faculty of Engineering Technology, PO Box 15008, Al-Balqa Applied University, Amman 11134, Jordan
a2 Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH, UK e-mail: s.gourley@surrey.ac.uk

Article author query
al-omari j   [Google Scholar] 
gourley sa   [Google Scholar] 
 

Abstract

We propose a delay differential equation model for a single species with stage-structure in which the maturation delay is modelled as a distribution, to allow for the possibility that individuals may take different amounts of time to mature. General birth and death rate functions are used. We find that the dynamics of the model depends largely on the qualitative form of the birth function, which depends on the total number of adults. If it is monotonic increasing and a non-zero equilibrium exists, then the equilibrium is globally stable for all maturation delay distributions with compact support. For the case of a finite spatial domain with impermeable boundaries, a reaction-diffusion extension of the model is rigorously derived using an approach based on the von Foerster diffusion equation. The resulting reaction-diffusion system is nonlocal. The dynamics of the reaction-diffusion system again depends largely on the qualitative form of the birth function. If the latter is non-monotone with a single hump, then the dynamics depends largely on whether the equilibrium is to the left or right of the hump, with oscillatory dynamics a possibility if it is sufficiently far to the right.

(Published Online March 23 2005)
(Received January 14 2004)
(Revised August 26 2004)