Turing Patterns Induced by Cross-Diffusion in a Predator–Prey System with Functional Response of Holling-II Type

• Published in: Qualitative theory of dynamical systems 2024-09, Vol.23 (4), Article 168
• Main Authors: Yan, Xiang-Ping, Yang, Tong-Jie, Zhang, Cun-Hua
• Format: Article
• Language: English
• Subjects: Amplitudes; Boundary conditions; Difference and Functional Equations; Dynamical Systems and Ergodic Theory; Hopf bifurcation; Mathematics; Mathematics and Statistics; Perturbation methods; Predators; Stability analysis
• ISSN: 1575-5460; 1662-3592
• DOI: 10.1007/s12346-024-01031-x

In this article, a classical predator–prey system with linear cross-diffusion and Holling-II type functional response and subject to homogeneous Neuamnn boundary condition is considered. The spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of the constant coexistence equilibrium are established with the help of the linearized analysis. When the bifurcation parameters are restricted to the Turing instability region and near the Turing bifurcation curve, the associated amplitude equations of the original system near the constant coexistence equilibrium are obtained by means of multiple-scale time perturbation analysis. According to the obtained amplitude equations, the stability and classification of spatiotemporal patterns of the original system near the constant coexistence equilibrium are determined. It is shown that the cross-diffusion in the classical predator–prey system plays an important role in formation of spatiotemporal patterns. Also, the theoretical results are verified numerically.