Effect of Kernels on Spatio-Temporal Patterns of a Non-Local Prey-Predator Model.

Non-local interaction describes the effects of mobility of a population species in their spatial locations. Non-local interaction is incorporated into a prey-predator model by introducing an integral term with appropriate kernel function. The kernel function determines the nature and range of the accessibility of the resources. We consider a nonlocal spatio-temporal prey-predator model with additive weak Allee effect in prey growth and density-dependent predator mortality. The dynamics of the model is examined in presence of a parabolic and a triangular kernel functions. Comparisons are made between the resulting dynamics of the nonlocal model with these kernels for the same range of nonlocal interaction. A linear stability analysis is performed for both the kernels to derive Turing and spatial-Hopf bifurcation conditions. In general, Turing bifurcation curves are different for the two kernels and the same is true for spatial-Hopf bifurcation curves. This results in different dynamics for the two kernels for some parameter values. However, for a fixed range of nonlocal interaction, the dynamics of the model with these two kernels are almost similar for most of the parameter values explored. Increase in the range of nonlocal interaction up to a certain value stabilizes the system dynamics, and then, it destabilizes. Extensive numerical simulations are performed to illustrate the system dynamics.