  We explore a diffusive predator-prey system that incorporates the fear effect
in advective environments. Firstly, we analyze the eigenvalue problem and the
adjoint operator, considering Constant-Flux and Dirichlet (CF/D) boundary
conditions, as well as Free-Flow (FF) boundary conditions. Our investigation
focuses 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 functional
responses on the dynamics of the model. Through these analyses, we aim to gain
a better understanding of the intricate relationship between advection,
predation, and prey response in this system.
