Nonconstant Positive Steady States for a Predator–Prey System With Signal‐Dependent Diffusion

ABSTRACT This paper investigates a highly nonlinear predator–prey reaction–diffusion system featuring signal‐dependent prey diffusion, where the motility of the prey is modulated by a chemical signal released by predators. For unbounded functional responses, we establish the global‐in‐time existence and uniform‐in‐time boundedness of classical solutions in two‐dimensional bounded domains without any parametric restrictions, and extend these results to three dimensions under a sufficiently strong quadratic damping mechanism. For bounded functional responses, global well‐posedness is verified in an arbitrary spatial dimension. We further rigorously analyze the associated steady‐state profiles and oscillatory dynamics. In spatially homogeneous environments, we demonstrate that while signal‐dependent diffusion cannot trigger stationary bifurcations, it nevertheless induces spatially inhomogeneous Hopf bifurcations. In spatially heterogeneous settings, where the underlying linearized operator is non–self‐adjoint and lacks a variational framework, we derive precise stability thresholds for the semi‐trivial steady states. By applying the cone fixed‐point index theory to compute the associated topological degree, we rigorously prove the emergence of nonconstant positive steady states as the semi‐trivial states lose their stability. Finally, brief numerical simulations are provided to complement the theoretical findings and illustrate the formation of stable spatial patterns. Through this framework, we extend and refine previously known theorems, providing new structural insights.