Global bifurcation for a predator-prey system with nonlinear boundary-mediated dispersal

This paper presents a complete bifurcation analysis of steady-state solutions for a predator-prey reaction-diffusion-advection system featuring density-dependent boundary conditions. By developing a novel synthesis of local and global bifurcation theories, we characterize the full solution structure emerging from semi-trivial states, revealing how nonlinear boundary terms qualitatively alter system dynamics. Our main results establish: (i) precise conditions for coexistence state bifurcations that coincide with stability transitions of semi-trivial solutions; (ii) complete global continuation of solution branches showing either unbounded persistence or novel connection patterns; and (iii) quantitative criteria for stability thresholds governed by boundary-mediated feedback. The analysis overcomes significant technical challenges through sharp a priori estimates and careful treatment of non-self-adjoint operators arising from the nonlinear boundary conditions. These theoretical advances provide mechanistic explanations for edge effects in ecological systems while developing mathematical tools applicable to broader pattern formation problems. The work opens new directions for studying nonlinear boundary phenomena in biological systems, with natural extensions to multi-species communities and chemotaxis models.