Nonlocal perception as a mechanism for stability and pattern formation in predator–prey interaction

In this paper, we formulate a diffusive predator–prey system with nonlocal perception in both species. The main focus is to investigate the joint influence of both the perception strength on the spatiotemporal dynamics of the system. First, the temporal system is inspected, and stability conditions for the coexisting equilibrium are produced. The parametric conditions for Hopf bifurcation are also demonstrated when spatial components are absent. The nonlocal detection function is characterized by a top-hat kernel, in which species can detect resources up to an equal fixed distance but not beyond. Moreover, such a form of kernel generally induces a sinusoidal function, and hence, they can instigate richer pattern selection from kernels that are purely monotone. We perform stability analysis and find the sufficient condition for the coexisting equilibrium state to undergo Turing bifurcation by taking the perceptual radius as the bifurcating parameter. A thorough numerical simulations are conducted to validate the theoretical findings. Bifurcation regions are plotted, and it is observed that increasing the nonlocal radius generally leads to stability in the system. Further, it is also shown how variation in both the perception strength plays a significant role in the class of patterning within the Turing region. Finally, we demonstrate that combined changes in the temporal parameter and the nonlocal radius can give rise to additional spatial instability, like the pure Hopf and Turing–Hopf instabilities. The spatial evolution of both prey and predator species is shown in all the bifurcation regimes. The results highlight the importance of nonlocal perception strength and its influence in generating various spatial instabilities.