Dynamics of a Memory-Based Diffusion Model With Maturation Delay and Spatial Heterogeneity

Abstract

In this paper, we consider a single memory-based diffusion population model with maturation delay, spatial heterogeneity, and Neumann boundary condition. When the integral of the intrinsic growth rate over the domain is nonnegative, we obtain sufficient conditions for the local stability of the positive steady state and the critical values of maturation delay for the associated Hopf bifurcation. When the integral of the intrinsic growth rate over the domain is negative, considering that the characteristic equation involves a non-self-adjoint operator and two delays, we utilize a geometric method to determine all bifurcation points in terms of memory and maturation delays. The impact of spatial heterogeneity on the distribution of solutions is also examined via numerical simulations. It is found that the core area of high population density is coincident with the source area of growth rate. This suggests the importance of spatial heterogeneity in shaping the distribution and dynamics of the species.