Spatio-Temporal Pattern Formation in Holling–Tanner Type Model with Nonlocal Consumption of Resources

Abstract

A wide variety of spatio-temporal models are available in literature which are unable to generate stationary patterns through Turing bifurcation. Introduction of nonlocal terms to the same model can produce Turing patterns and this is true even for a single species population model. In this paper, we consider a prey–predator model of Holling–Tanner type with a generalist predator and a nonlocal interaction in the intra-specific competition term of the prey population. Nonmonotonic functional response is assumed to describe consumption rate of the prey by the predator. The Turing instability condition has been studied for the model without the nonlocal term around coexisting steady states. We also determine the Turing domain in the presence of nonlocal interaction term. The spatial-Hopf bifurcation has been studied and it plays an important role to find the pure Turing domain for the nonlocal model. Furthermore, in the presence of nonlocal interaction, the nonlocal model produces traveling wave solution. Using linear stability analysis, we have obtained the wave speed for the traveling wave front analytically. With the help of numerical simulation, we have verified that the speed of the traveling wave front for the complete nonlinear nonlocal model matches with the analytical approximation. The emergence of wave trains has also been established for higher range of nonlocal interaction.