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

Yangyang Shao, Yan Meng and Xinyue Xu
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Yangyang Shao: School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
Yan Meng: School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
Xinyue Xu: School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Mathematics, 2022, vol. 10, issue 9, 1-15

Abstract: 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.

Keywords: cross-diffusion; Allee effect; Turing instability; pattern formation; numerical simulation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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