  In this paper, we investigate the global boundedness, asymptotic stability
and pattern formation of predator-prey systems with density-dependent preytaxis
in a two-dimensional bounded domain with Neumann boundary conditions, where the
coefficients of motility (diffusion) and mobility (preytaxis) 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 linearized system and nonlinear systems are quite
different, and the prey density-dependent motility function can trigger the
pattern formation.
