Stability and bifurcation of a reaction–diffusion predator–prey model with non-local delay and Michaelis–Menten-type prey-harvesting

Abstract

We investigate a reaction–diffusion predator–prey system with homogeneous Neumann boundary conditions and non-local delay due to predator gestation. By analysing the corresponding characteristic equations, we establish the local stability of the positive steady state. We also discuss the existence of Hopf bifurcations at the positive steady state. We derive sufficient conditions for the global stability of the positive steady state of the proposed problem using the Lyapunov functional. Numerical simulations illustrate the results and reveal that as the discrete delay τ increases, the species may tend to extinction. Changes in the harvesting effort E and non-local delay β can transform an unstable system into a stable one.

Keywords:

• non-local delay
• stability
• Hopf bifurcation
• harvesting
• global stability

2010 AMS Subject Classifications:

• 34K20
• 34K18

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under grants [61174155] and by the Qing Lan Project of Jiangsu.