Global bifurcation and coexistence in age-structured predator-prey models with nonlinear diffusion

This paper is concerned with positive steady-state solutions of an age-structured predator-prey system with nonlinear density-dependent diffusion. In the mortality-driven bifurcation picture, positive single-species equilibria exist and are stable below the critical thresholds λ₀ (for prey) and χ₀ (for predator), respectively. In an ordered Banach-space framework based on maximal Lq-regularity, we obtain local semi-trivial branches for the two single-species subsystems and characterize invasion thresholds via spectral radii of compact positive operators. We then reformulate the coexistence problem near the prey-only branch as a compact fixed-point problem and apply Rabinowitz's global bifurcation theorem. Under a boundedness assumption on the functional response, we derive a bounded continuum of positive coexistence steady states. Its closure contains the bifurcation point (χ₀(λ), u_λ, 0) on the prey-only branch and intersects the predator-only semi-trivial branch, thereby connecting the two semi-trivial branches. The critical value χ₀(λ) represents the predator mortality threshold below which invasion and coexistence occur. To the best of our knowledge, this is the first result establishing a bounded global continuum of positive coexistence steady states in age-structured predator-prey systems with nonlinear density-dependent diffusion.