Diffusive spatial movement with memory in an advective environment

Abstract

Abstract The movements of species in a river are driven by random diffusion, unidirectional water flow, and cognitive judgement with spatial memory. In this paper, we formulate a reaction–diffusion–advection model with memory-based diffusion and homogeneous Dirichlet boundary conditions. The existence of a nonconstant positive steady state is proven. We obtain the linear stability of the steady state by analysing the eigenvalues of the associated linear operator: the nonconstant steady state can always be linearly stable regardless of the memory delay, while the model can also possess Hopf bifurcation as the memory delay varies. Moreover, theoretical and numerical results show that large advection annihilates oscillation patterns and drives the species to concentrate downstream.