In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

A conceptual minimal model demonstrating spatially heterogeneous wave regimesinterpreted as pursuit-evasion in predator-prey system is constructed and investigated. Themodel is based on the earlier proposed hypothesis that taxis accelerations of prey and predators are proportional to the density gradient of another population playing a role of taxis stimulus. Considering acceleration rather than immediate velocity allows obtaining realistic solutions even while ignoring variations of total abundances of both modelled populations. Linear analysis of the model shows that stationary homogeneous regime becomes oscillatory unstable with respect to small heterogeneous perturbations if either taxis activities or total population abundances are high enough. The ability for active directed movement of both prey and predators is the necessary condition for spatial self-organization. Numerical simulations illustrate analytical results. The relation between the proposed model and conventional two-component systems with cross-diffusion is discussed.