Neimark–Sacker bifurcation of discrete Pielou model with delay

Abstract

In this paper, we investigate the existence of the Neimark–Sacker bifurcation of discrete delay Pielou Model with positive parameters. It is shown that this model undergoes a Neimark–Sacker bifurcation when the parameter passes a critical value. Furthermore, based on the computational algorithm developed by Murakami in [The invariant curve caused by Neimark-Sacker bifurcation, Dyn. Contin. Discrete Impuls. Syst. Ser. A 9 (2002), pp. 121–132], the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcations is derived. Also, an explicit approximate expression of the invariant curve caused by Neimark–Sacker bifurcation is given. Some numerical simulations are carried out to illustrate the analytical results found.

Keywords:

• Pielou model
• equilibrium
• stability
• Neimark–Sacker bifurcation
• delay

2020 Mathematics Subject Classification:

• 39A28

Acknowledgments

The author would like to thank the referees very much for their careful reading and valuable suggestions which lead to improvement of the manuscript.

Disclosure statement

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

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (No: 12171485,12271525).