Turing patterns induced by cross-diffusion in a predator-prey model with Smith-type prey growth and additive predation effects

We investigate a two-dimensional predator–prey model that incorporates self- and cross-diffusion, Smith-type prey growth and additive predation effects. By linearizing about the unique positive homogeneous steady state, we obtain exact criteria for the onset of Turing instability. These criteria reveal that when the prey density at the homogeneous steady state falls into a certain interval, strong predator cross-diffusion destabilizes the homogeneous state. Using the predator cross-diffusion coefficient as the bifurcation parameter, we derive the corresponding amplitude equations and, for these equations, establish sharp, explicit conditions that guarantee the existence and stability of steady-state patterns. Numerical simulations corroborate the analytical predictions and demonstrate that the prey distribution can develop a rich variety of Turing pattern, such as spots, stripes, holes, and their hybrids, depending on the strength of predator cross-diffusion.