Dynamics for a type of general reaction-diffusion model

In this paper, we discuss the following reaction-diffusion model which is a general form of many population models partial derivative u(t,x)/partial derivative t = Delta u(t,x) - delta u(t,x) + f(u(t - tau,x)). We study the oscillatory behavior of solutions about the positive equilibrium K of system (*) with Neumann boundary conditions. Sufficient and necessary conditions are obtained for global attractivity of the zero solution and acceptable conditions are established for the global attractivity of K. These results improve and complement existing results for system (*) without diffusion. Moreover, when these results are applied to the diffusive Nicholson's blowflies model and the diffusive model of Hematopoiesis, some new results are obtained for the latter. (C) 2006 Elsevier Ltd. All rights reserved.