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Dynamic analysis of a fractional order delayed predator–prey system with harvesting

Abstract

In the study, we consider a fractional order delayed predator–prey system with harvesting terms. Our discussion is divided into two cases. Without harvesting, we investigate the stability of the model, as well as deriving some criteria by analyzing the associated characteristic equation. With harvesting, we investigate the dynamics of the system from the aspect of local stability and analyze the influence of harvesting to prey and predator. Finally, numerical simulations are presented to verify our theoretical results. In addition, using numerical simulations, we investigate the effects of fractional order and harvesting terms on dynamic behavior. Our numerical results show that fractional order can affect not only the stability of the system without harvesting terms, but also the switching times from stability to instability and to stability. The harvesting can convert the equilibrium point, the stability and the stability switching times.

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Acknowledgments

We would like to express our gratitude to the editor and anonymous referees for their valuable comments and suggestions, which helped to improve this manuscript. The work was supported by National Natural Science Foundation of China (Grant Nos. 61174155 and 11571170), as well as being sponsored by the Qing Lan Project of Jiangsu and Jiangsu Province Science Foundation (BK20150420).

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Correspondence to Hongyong Zhao.

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Song, P., Zhao, H. & Zhang, X. Dynamic analysis of a fractional order delayed predator–prey system with harvesting. Theory Biosci. 135, 59–72 (2016). https://doi.org/10.1007/s12064-016-0223-0

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Keywords

  • Predator–prey system
  • Time delay
  • Harvesting
  • Stability
  • Fractional derivative