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We explore a diffusive predator-prey system that incorporates the fear effectin advective environments. Firstly, we analyze the eigenvalue problem and theadjoint operator, considering Constant-Flux and Dirichlet (CF/D) boundaryconditions, as well as Free-Flow (FF) boundary conditions. Our investigationfocuses on determining the direction and stability of spatial Hopf bifurcation,with the generation delay $\tau$ serving as the bifurcation parameter.Additionally, we examine the influence of both linear and Holling-II functionalresponses on the dynamics of the model. Through these analyses, we aim to gaina better understanding of the intricate relationship between advection,predation, and prey response in this system.

Dynamics of a diffusive predator-prey system with fear effect in advective environments