Dynamics for a type of general reaction–diffusion model

Abstract

In this paper, we discuss the following reaction–diffusion model which is a general form of many population models (∗) ∂ u ( t , x ) ∂ t = △ u ( t , x ) − δ u ( t , x ) + f ( u ( t − τ , 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.