Global Stability Analysis of Coexistence Steady State in a General Predator–Prey System with Double Prey-Taxis

In this paper, we consider a general two-prey one-predator reaction–diffusion system with prey competition and double prey-taxis. We first present some preliminary results, including global-in-time existence and a priori estimates of classical solutions to this system with ratio-dependent and non-ratio-dependent predator functional responses. Our main concern is the global stability of spatially homogeneous coexistence steady states. For generalized prey-dependent models, we show that the global asymptotic stability of the coexistence steady state relates to two prey-taxis coefficients provided that the predator functional response, the conversion, and diffusion rates are given. Since there is little prospect of establishing a unified Lyapunov functional for the systems with a predator-dependent functional response, we propose an explicit predator-dependent model and construct the corresponding Lyapunov functional. The theoretical results can cover most three-species chemotaxis systems.