Transcritical bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with herd behaviour and square root functional response

ABSTRACT

In this paper, a discrete predator-prey model incorporating herd behaviour and square root response function is deduced from its continuous version by the semi-discretization method. Firstly, the existence and local stability of fixed points of the system are studied by applying a key lemma. Secondly, by employing the centre manifold theorem and bifurcation theory, the conditions for the occurrences of the transcritical bifurcation and Neimark-Sacker bifurcation are obtained. Not only that but the direction and stability conditions of the bifurcated closed orbits are also clearly shown. Finally, numerical simulations are also given to confirm the existence of Neimark-Sacker bifurcation.

KEYWORDS:

• Predator-prey system with herd behaviour
• square root functional response
• semi-discretization method
• transcritical bifurcation
• Neimark-Sacker bifurcation

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (grant: 61473340), the Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province (grant: F703108L02), and the Natural Science Foundation of Zhejiang University of Science and Technology (grant: F701108G14).

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Disclosure statement

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

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

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Funding

The work was supported by the National Natural Science Foundation of China [61473340]; Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province [F703108L02].