Stationary pattern and Hopf bifurcation of a diffusive predator–prey model

Abstract

This paper is concerned with a predator–prey model with prey-taxis and linear prey harvesting under the homogeneous Neumann boundary condition. The stability of the unique positive constant solution of the predator–prey model without prey-taxis is derived. Also, the emergence of Hopf bifurcation is concluded by choosing the proper Hopf bifurcation parameters. By the center manifold theorem and normal form, we compute the direction of Hopf bifurcation and the stability of the bifurcating solution. Moreover, the stationary pattern with prey-taxis is investigated. The conclusions show that prey harvesting and prey-taxis can enrich the dynamics.

Keywords:

• Hopf bifurcation
• manifold theorem
• normal form
• stationary pattern
• degree theory

2010 Mathematics Subject Classifications:

• 35B32
• 35B35
• 35B36
• 92D25

Acknowledgments

We would like to thank the anonymous referees for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Science Foundation of China (NSFC) Grant 11801566 and by Natural Science Foundation of Shandong Province (Nos. ZR2019MA006, ZR2021MA025, ZR2021MA028).