Turing-Hopf bifurcation and spatio-temporal patterns of a ratio-dependent Holling-Tanner system with diffusion

TL;DR

This paper investigates complex bifurcations and spatio-temporal patterns in a diffusive ratio-dependent Holling-Tanner predator-prey system, revealing multiple bifurcation types and associated patterns through theoretical analysis and simulations.

Contribution

It provides a detailed analysis of Turing-Hopf bifurcation in the system using normal form methods, expanding understanding of pattern formation in ecological models.

Findings

Existence of multiple bifurcations including Turing-Hopf and Turing-Truing.

Theoretical proof of various spatio-temporal patterns such as steady states and periodic solutions.

Numerical simulations confirming theoretical predictions.

Abstract

A diffusive ratio-dependent Holling-Tanner system subject to Neumann boundary conditions is considered. The existence of multiple bifurcations, including Turing-Hopf bifurcation, Turing-Truing bifurcation, Hopf-double-Turing bifurcation and triple-Turing bifurcation, are given. Among them, the Turing-Hopf bifurcation are carried out in details by the normal form method. We theoretically prove that the system exists various spatio-temporal patterns, such as, non-constant steady state, the spatially inhomogeneous periodic or quasi-periodic solution, etc. Numerical simulations are presented to illustrate our theoretical results.