Bifurcation in a modified Leslie-Gower model with nonlocal competition and fear effect

A diffusive predator-prey model with nonlocal competition and the fear effect is considered in this paper. This study investigates how parameters affect the existence, multiplicity, and stability of nonhomogeneous steady-state solutions. Establish the criteria for Hopf, Turing, Turing-Hopf, and Hopf-Hopf bifurcations, and determine the stable regions of the positive equilibrium. Under some circumstances, the model with nonlocal competition can produce a Hopf-Hopf bifurcation in contrast to the model without nonlocal competition. The normal form at the Hopf-Hopf bifurcation singularity is computed using the qualitative analysis to examine the many dynamic characteristics the model displays in various parameter ranges. Numerical simulations are carried out to verify the viability of the obtained results and the dependence of the dynamic behavior on nonlocal competition. By using numerical simulations, it is demonstrated that in specific scenarios, nonlocal competition leads to both stable spatially non-homogeneous quasi-periodic solutions and stable spatially non-homogeneous periodic solutions.