Turing instability in a diffusive predator-prey model with multiple Allee effect and herd behavior

TL;DR

This paper analyzes how diffusion-driven Turing instability occurs in a predator-prey model with herd behavior, Allee effects, and quadratic mortality, revealing conditions for pattern formation and bifurcations.

Contribution

It introduces a predator-prey model incorporating multiple Allee effects and herd behavior, and provides a comprehensive analysis of Turing instability and bifurcation conditions.

Findings

Large prey diffusion rates inhibit Turing instability

Conditions for Turing instability are explicitly derived

Numerical simulations support theoretical results

Abstract

Diffusion-driven instability and bifurcation analysis are studied in a predator-prey model with herd behavior and quadratic mortality by incorporating multiple Allee effect into prey species. The existence and stability of the equilibria of the system are studied. And bifurcation behaviors of the system without diffusion are shown. The sufficient and necessary conditions for Turing instability occurring are obtained. And the stability and the direction of Hopf and steady state bifurcations are explored by using the normal form method. Furthermore, some numerical simulations are presented to support our theoretical analysis. We found that too large diffusion rate of prey prevents Turing instability from emerging. Finally, we summarize our findings in the conclusion.