Bifurcation analysis in a discrete toxin-producing phytoplankton–zooplankton model with refuge

Abstract

This paper concerns a discrete-time phytoplankton–zooplankton model in which the effects of toxin produced by phytoplankton and refuges provided for phytoplankton on the interactions between phytoplankton and zooplankton are considered. We first discuss the existence and stability of fixed points. When two parameters change, it is shown that there exist some codimension-two bifurcations, including fold-flip bifurcation and strong resonance bifurcations. Moreover, we give the corresponding bifurcation diagrams, phase diagrams and maximum Lyapunov exponent diagrams. The results show that phytoplankton toxin and refuges play important roles in the occurrence and termination of algal blooms in freshwater lakes.

Keywords:

• Discrete-time model
• phytoplankton
• zooplankton
• fold-flip bifurcation
• strong resonance bifurcations

Mathematics Subject Classifications:

• 39A28
• 39A30
• 39A33

Acknowledgments

The authors wish to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that led to substantial improvements in the original manuscript.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Grant Nos: 12271421, 12371504, and 11771373], the Shaanxi Province Innovation Talent Promotion Plan Project [Grant No: 2023KJXX-056], China Postdoctoral Science Foundation [Grant No. 2019M663610], Natural Science Basic Research Program of Shaanxi Province [Grant No. 2021JM-445] and the Young Talent fund of University Association for Science and Technology in Shaanxi [Grant No. 20200510].