Inhomogeneous spatial patterns in diffusive predator-prey system with spatial memory and predator-taxis

TL;DR

This paper develops a new algorithm to analyze Hopf bifurcations in a complex predator-prey model with spatial memory and predator-taxis, revealing inhomogeneous spatial patterns through theoretical and numerical methods.

Contribution

A novel algorithm for normal form calculation of Hopf bifurcation in nonlinear, delayed diffusion systems is introduced and applied to a predator-prey model with spatial memory.

Findings

Identification of stable inhomogeneous periodic solutions

Verification of theoretical results through numerical simulations

Analysis of bifurcation modes in the predator-prey system

Abstract

In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation in the reaction-diffusion system with delay can't be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation in this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also researched by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations, two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis results.