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Dynamic analysis of a plankton–herbivore state-dependent impulsive model with action threshold depending on the density and its changing rate

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Abstract

A plankton–herbivore state-dependent impulsive model with nonlinear impulsive functions and action threshold including population density and rate of change is proposed. Since the use of action threshold makes the model have complex phase set and pulse set, we adopt the Poincar\(\acute{\text{ e }}\) map as a tool to study its complex dynamics. The Poincar\(\acute{\text{ e }}\) map is defined on the phase set and its properties in different situations are analyzed. Furthermore, the periodic solution of model is discussed, including the existence and stability conditions of the order-1 periodic solution and the existence of the order-k (\(k\ge 2\)) periodic solutions. Compared with the fixed threshold in the existing literature, our results show that the use of action threshold is more practical, which is conducive to the sustainable development of population and makes people obtain more economic benefits. The analysis method used in this paper can study the complex dynamics of the model more comprehensively and deeply.

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Data availability statement

The author declares that the data supporting the results of this study are available in the article.

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Funding

This work is supported by the National Natural Science Foundation of China (11371230), the SDUST Research Fund (2014TDJH102), Research on the basic theoretical framework and symmetry theory of soft robot dynamics(11872335).

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Shandong University of Science and Technology, Qingdao, 266590, Shandong, People’s Republic of China

    Wei Li, Yufei Wang & Huidong Cheng

  2. Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia

    Tonghua Zhang

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  1. Wei Li
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  2. Tonghua Zhang
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  4. Huidong Cheng
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Correspondence to Huidong Cheng.

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Li, W., Zhang, T., Wang, Y. et al. Dynamic analysis of a plankton–herbivore state-dependent impulsive model with action threshold depending on the density and its changing rate. Nonlinear Dyn 107, 2951–2963 (2022). https://doi.org/10.1007/s11071-021-07022-w

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  • DOI: https://doi.org/10.1007/s11071-021-07022-w

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