Dynamics of a predator–prey model incorporating fear effects and linear harvesting

This work investigates a predator–prey system that incorporates fear effects and linear harvesting under Neumann boundary conditions. First, the stability of the equilibrium associated with the normal state is rigorously verified using linear operator theory and the comparison principle, with a priori estimates obtained via the maximum principle. Next, the presence or absence of spatially non-uniform positive steady states is explored through Poincaré’s inequality combined with topological degree theory, followed by an examination of Turing instability within the reaction–diffusion context. Furthermore, applying bifurcation theory alongside the center manifold approach, the direction of Hopf bifurcation and the stability of the resulting periodic solutions in the PDE system are determined. Finally, numerical simulations are conducted to validate the theoretical analysis and to clarify how critical parameters affect the formation of spatial patterns, providing deeper understanding of the mechanisms driving spatial structure evolution in the system.