Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect

Mathematics

Yangyang Shao¹  Xinyue Xu¹  Yan Meng¹ 

[1] School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China;

关键词: cross-diffusion;    Allee effect;    Turing instability;    pattern formation;    numerical simulation;   

DOI  :  10.3390/math10091500

来源: DOAJ

【 摘 要 】

The Allee effect is widespread among endangered plants and animals in ecosystems, suggesting that a minimum population density or size is necessary for population survival. This paper investigates the stability and pattern formation of a predator–prey model with nonlinear reactive cross-diffusion under Neumann boundary conditions, which introduces the Allee effect. Firstly, the ODE system is asymptotically stable for its positive equilibrium solution. In a reaction system with self-diffusion, the Allee effect can destabilize the system. Then, in a reaction system with cross-diffusion, through a linear stability analysis, the cross-diffusion coefficient is used as a bifurcation parameter, and instability conditions driven by the cross-diffusion are obtained. Furthermore, we show that the system (5) has at least one inhomogeneous stationary solution. Finally, our theoretical results are illustrated with numerical simulations.

【 授权许可】

Unknown