A mathematical model is proposed that describes the adaptive spatial movement of prey towards higher population density to reduce predation risk. The model admits the increased nonlinearity and the global existence of solutions of the system is established in Sobolev space through analytical estimates. The conditions for the Turing instability from a coexistence steady state are obtained, and sharp conditions for the asymptotical stability of the positive equilibrium in a large region are established with the help of a Lyapunov function. Numerical simulations are presented to support the theoretical results and demonstrate the versatility of spatial models.