Diffusive spatial movement with memory in an advective environment

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 reaction–diffusion–advection model memory-based diffusion homogeneous Dirichlet boundary conditions. existence nonconstant positive steady state is proven. We obtain the linear stability analysing eigenvalues associated operator: can always be linearly stable regardless memory delay, while also possess Hopf bifurcation as delay varies. Moreover, theoretical numerical results show that large advection annihilates oscillation patterns drives to concentrate downstream.