November  2005, 5(4): 1015-1026. doi: 10.3934/dcdsb.2005.5.1015

Comparison and convergence to equilibrium in a nonlocal delayed reaction-diffusion model on an infinite domain

1. 

Department of Mathematics and Statistics, University of Surrey, GU2 7XH, United Kingdom

2. 

Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom

Received  September 2004 Revised  April 2005 Published  August 2005

We study a nonlocal time-delayed reaction-diffusion population model on an infinite one-dimensional spatial domain. Depending on the model parameters, a non-trivial uniform equilibrium state may exist. We prove a comparison theorem for our equation for the case when the birth function is monotone, and then we use this to establish nonlinear stability of the non-trivial uniform equilibrium state when it exists. A certain class of non-monotone birth functions relevant to certain species is also considered, namely birth functions that are increasing at low densities but decreasing at high densities. In this case we prove that solutions still converge to the non-trivial equilibrium, provided the birth function is increasing at the equilibrium level.
Citation: Michele V. Bartuccelli, S.A. Gourley, Y. Kyrychko. Comparison and convergence to equilibrium in a nonlocal delayed reaction-diffusion model on an infinite domain. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1015-1026. doi: 10.3934/dcdsb.2005.5.1015
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