Sign-dependent effects of predator-taxis on spatial stability and pattern formation in a harvested predator-prey model with anti-predator cost

Predation risk affects prey populations through both demographic costs and spatial responses. We study a harvested predator-prey model that distinguishes the reproductive cost of anti-predator behavior from predator-taxis, while incorporating Holling type II predation and saturating harvesting. Local classical solvability and positive invariance are established on smooth bounded domains. For linear predator-taxis sensitivity on an interval, a weighted mass estimate, Neumann semigroup estimates and an Alikakos iteration yield global existence and uniform boundedness. The nonnegative spatially homogeneous equilibria and their temporal stability are characterized. Modal analysis reveals a sign-dependent spatial effect: predator avoidance stabilizes every nonzero stationary mode, whereas stationary spatial instability can occur only when the orientation of predator-taxis is reversed. For a simple critical Neumann mode, a local branch of nonconstant steady states is obtained by the Crandall-Rabinowitz theorem, and a Fredholm-Landau reduction characterizes its near-threshold behavior. Numerical simulations are consistent with the predicted stability threshold and display regular stripes that change little near the end of the simulated interval, together with more irregular one-dimensional dynamics as the taxis strength increases. Two-dimensional and toroidal computations further display spot and stripe-type spatial structures. The results identify the orientation of predator-taxis as a key determinant of spatial stability and pattern formation in the model.