Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion

• Main Authors: Qiushuang Shi, Ming Liu, Xiaofeng Xu
• Format: Article
• Language: English
• Published: SpringerOpen 2019-08-01
• Series: Advances in Difference Equations
• Subjects: Ratio-dependent; Reaction-diffusion; Turing–Hopf bifurcation; Predator-prey model
• Online Access: http://link.springer.com/article/10.1186/s13662-019-2123-3

Abstract In this paper, the Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion and Neumann boundary condition is considered. Firstly, we present a kind of double parameters selection method, which can be used to analyze the Turing–Hopf bifurcation of a general reaction-diffusion equation under Neumann boundary condition. By analyzing the distribution of eigenvalues, the stable region, the unstable region (including Turing unstable region), and Turing–Hopf bifurcation point are derived in a double parameters plane. Secondly, by applying this method, the Turing–Hopf bifurcation of a ratio-dependent predator-prey model with diffusion is investigated. Finally, we compute normal forms near Turing–Hopf singularity and verify the theoretical analysis by numerical simulations.