Global Dynamics of a Predator–Prey System With Indirect Prey‐Taxis and Nonlinear Boundary Fluxes

ABSTRACT We investigate a reaction–diffusion predator–prey model incorporating indirect prey‐taxis, where predators respond to chemical cues released by prey rather than directly tracking prey density. The model incorporates two biologically relevant boundary conditions: (i) homogeneous Dirichlet conditions, representing lethal habitat edges, and (ii) nonlinear boundary fluxes accounting for density‐dependent emigration. We establish the global existence and boundedness of solutions under biologically reasonable assumptions, proving that indirect prey‐taxis prevents finite‐time blowup, even in higher‐dimensional settings. Using spectral theory and bifurcation analysis, we characterize the existence and stability of semitrivial and coexistence steady states, identifying critical thresholds for predator conversion efficiency and prey growth rate that determine long‐term dynamics. Finally, we classify the long‐term dynamics into distinct ecological regimes (extinction, prey dominance, and coexistence) and quantify how boundary conditions modify these regimes through their influence on stability thresholds. These results rigorously demonstrate the stabilizing role of indirect taxis in spatial ecological systems while advancing the mathematical theory of nonlinear boundary value problems in biological contexts.