Dynamics in a general predator–prey chemotaxis model with signal-dependent diffusion and sensitivity

We examine a predator–prey chemotaxis model by assuming that preys move away from and predators move towards areas with higher concentrations of a shared chemical substance released by both species. In this model, the diffusion and sensitivity parts of prey and predator are postulated to be signal-dependent. In a more general prey–predator interaction setting, we give the global existence and boundedness of solutions to the system within a two-dimensional spatial domain by employing the Moser-Alikakos iteration method. With a special interaction including the Beddington–DeAngelis functional response, we construct some proper Lyapunov functionals and give some conditions on the global asymptotic stability as well as the convergence rates of both the semi-trivial equilibrium and the positive equilibrium of the model. Additionally, a linearized stability analysis indicates that a high sensitivity function of predators at the signal-component of the positive equilibrium can destabilize the constant coexistence steady state via Turing bifurcation or Hopf bifurcation under various conditions. Finally, several numerical simulations are presented to corroborate our theoretical findings.