Global stability of a quasilinear predator–prey model with indirect pursuit–evasion interaction

Abstract

This paper deals with a predator–prey model with indirect prey-taxis and predator-taxis [Formula: see text] under homogeneous Neumann boundary conditions in a smoothly bounded domain [Formula: see text], where the parameters [Formula: see text] are positive, [Formula: see text] and [Formula: see text] are nonlinear diffusion functions, [Formula: see text] and [Formula: see text] are nonlinear sensitivity functions. First, under certain suitable conditions for [Formula: see text] and [Formula: see text] with [Formula: see text], the system admits a unique globally bounded classical solution, provided that [Formula: see text] and [Formula: see text]. Additionally, by constructing appropriate Lyapunov functionals, we investigate the asymptotic stability of the globally bounded solutions and provide the exact convergence rates based on the different parameter choices: When [Formula: see text], it is shown that the global bounded solution [Formula: see text] exponentially converges to [Formula: see text] as [Formula: see text]; When [Formula: see text], it is shown that the global bounded solution [Formula: see text] exponentially converges to [Formula: see text] as [Formula: see text]; When [Formula: see text], it is shown that the global bounded solution [Formula: see text] algebraically converges to [Formula: see text] as [Formula: see text].