Pattern dynamics of a cross-diffusion predator–prey model with nonlinear harvesting term

In this paper, we investigate a cross-diffusion predator–prey model incorporating a nonlinear harvesting term, with a particular focus on the role of cross-diffusion in shaping Turing patterns of positive equilibria. The diffusion instability of the positive equilibrium of the model with Neumann boundary conditions is discussed. In order to better understand the influence of cross-diffusion on pattern formation, the pattern formation process without cross-diffusion is first given, and then the cross-diffusion coefficient is selected as the main control parameter to observe its influence on the prey pattern formation. Through a series of numerical simulations, diverse Turing structures in the parameter space are obtained, including hole, strip, and spot patterns. Finally, leveraging computer-aided analysis, we simulate spiral patterns under four distinct initial conditions. Our results collectively demonstrate that cross-diffusion significantly enriches the pattern dynamics of the predator–prey system, offering valuable insights into the spatio-temporal complexity and dynamic properties of such ecosystems.