Neimark-Sacker bifurcation analysis in a discrete neutral Nicholson’s blowflies system with delay

Abstract

In this paper, we investigate the Neimark-Sacker bifurcation in a delayed Nicholson’s blowflies system of neutral type, which is obtained by using Euler discretization method. Choosing time delay $\mathit{\tau }$ as the bifurcation parameter and analysing the associated characteristic equation, we study the stability of the positive equilibrium and verify the existence of the Neimark-Sacker bifurcation. Then, an explicit formula which determines the stability and direction of the bifurcation is given by using centre manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate our theoretical results.

Keywords:

• Discrete neutral Nicholson’s blowflies system
• delay
• Neimark-Sacker bifurcation

Acknowledgements

The authors greatly appreciate the anonymous referees careful reading and valuable comments. Their critical comments and helpful suggestions greatly improve the presentation of the manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by National Natural Science Foundation of China [grant number 11371111]; Research Fund for the Doctoral Program of Higher Education of China [grant number 20122302110044]; Natural Science Foundation of Shandong Province, China [grant number ZR2015AQ005].