Impact of Cross-Diffusion and Nonlocal Competition on a Predator–Prey Model With Harvesting

In this work, we examine the influence of nonlocal competition on the spatiotemporal dynamics of a predator–prey system. The nonlocal term is introduced in the intraspecific competition among the predator species. A density-dependent cross-diffusion is considered to model the movement behavior of species. First, the temporal system is examined, where the existence, boundedness, and stability of the equilibrium points are ascertained. Bifurcation analysis also reveals the presence of saddle-node, transcritical, Hopf bifurcations, and codimension-2 bifurcation such as the Bogdano–Takens bifurcation. The behavior of the system is inspected both in the presence and absence of nonlocal interaction. It is observed that the nonlocal interaction tends to destabilize the coexisting equilibrium. For lower values of diffusion coefficient, spatially nonhomogeneous solutions are detected. For higher values of diffusion coefficient, domain-induced Turing instability switching is observed. Moreover, when the domain size is small, the coexisting equilibrium is stable irrespective of the value of the diffusion coefficient. Furthermore, the stabilizing effect of harvesting effort is also seen. The formulated system is numerically solved using the operator splitting method. Extensive numerical simulations are conducted to validate all the theoretical findings.