Qualitative study of cross-diffusion and pattern formation in Leslie–Gower predator–prey model with fear and Allee effects

The present study is focused on interacting prey–predator reaction–diffusion model with modified Leslie–Gower type functional response incorporating Allee and fear effects. We introduce self as well as cross-diffusion in the model. The equilibrium points and their stability along with saddle–node and Hopf bifurcations around steady states are investigated for non-spatial system. The conditions for Turing instability and the critical line of Hopf and Turing bifurcation in a spatial domain with zero-flux boundary conditions are determined. The parametric space for different regions is depicted and numerical simulation in Turing space is carried out. It is observed that cross diffusion has significant role in forming patterns such as spots, stripes, mixture of spots and stripes. The issue of spatiotemporal pattern controllability is also examined. The availability of cross diffusion results a paradox to the prey density with increasing level of Allee.