Nonlocal Competition and Spatial Multi-peak Periodic Pattern Formation in Diffusive Holling-Tanner Predator-prey Model

In this paper, we investigate the periodic pattern formations with spatial multi-peaks in a classic diffusive Holling-Tanner predator-prey model with nonlocal intraspecific prey competition. The main innovation is that a spatial dependently kernel is considered in the nonlocal effect, which mathematically complicates the linear stability analysis. We first generate the existences of Hopf, Turing, Turing-Hopf and double-Hopf bifurcations, and determine the stability of the positive equilibrium. It turns out that the stable parameter region for the positive equilibrium decreases with α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha $$\end{document} increasing, which implies that the parameter region of pattern formation for such kernel is smaller than the spatial average case. For double-Hopf bifurcation, we calculate the normal form up to the third-order term restricted on the center manifold, which is expressed by the original parameters of the system. Via analyzing the equivalent amplitude equations, the system exhibits stable spatially nonhomogeneous periodic patterns, the bistability of such periodic solutions, as well as unstable spatially nonhomogeneous quasi-periodic solutions, all of them possess multiple spatial peaks. Interestingly, some possible strange attractors are found numerically near the double-Hopf singularity. Biologically, the emerging spatio-temporal patterns imply that such nonlocal intraspecific competition can promote the coexistence of the prey and predator species in the form of more complex periodic states.