Bifurcation analysis of coexistent state in a delayed two-species predator-prey model

ABSTRACT

In this paper, we consider a delayed two-species predator-prey model with general functional response under the homogeneous Neumann boundary condition. We discuss the stability of the trivial and semi-trivial solutions and obtain the spatially nonhomogeneous bifurcation solutions stemming from the semi-trivial trivial solutions $({\theta }_a,0)$ and $(0,{\theta }_b)$. Besides, the stability and some results of Hopf bifurcation at the spatially nonhomogeneous bifurcation steady-state solutions are investigated by analyzing the distribution of the eigenvalues. The method we applied here is mainly based on spectral analysis, comparison principle, Lyapunov-Schmidt reduction, and bifurcation theory.

KEYWORDS:

• Simple eigenvalue
• time delay
• comparison principle
• bifurcation theory
• Lyapunov-Schmidt reduction
• Hopf bifurcation

2000 MSC:

• 35B40
• 35K57
• 92D25

Additional information

Funding

Research supported by the NSFC (grant numbers 11671123, 11801089) and Jiangxi science and technology project (grant number GJJ170844).