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Stability analysis and optimal harvesting control of a cross-diffusion prey-predator system

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  • Ma, Tingting
  • Meng, Xinzhu
  • Hayat, Tasawar
  • Hobiny, Aatef

Abstract

In this paper, we consider a cross-diffusion prey-predator system with fear effect and prey refuge. The upper and lower bounds of the system are obtained by using priori estimates and Harnack Inequality. Then sufficient conditions for the local stability and global stability of the system are established. We obtain that the cross-diffusion coefficients can affect the stability of the original system, meanwhile the fear effect and prey refuge suppress the formation of Turing instability. By using the Leray-Schauder degree theory, we study the existence and nonexistence of the non-constant steady states. Moreover, we discuss the effects of the fear effect and prey refuge on the optimal harvesting. Finally, we obtain the optimal harvesting strategies under different fear effect values and prey refuge values, the different maximum sustainable yields (MSY) are correspondingly given. Numerical simulations are carried out to verify and illustrate these theoretical results.

Suggested Citation

  • Ma, Tingting & Meng, Xinzhu & Hayat, Tasawar & Hobiny, Aatef, 2021. "Stability analysis and optimal harvesting control of a cross-diffusion prey-predator system," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    • Handle: RePEc:eee:chsofr:v:152:y:2021:i:c:s0960077921007724
    DOI: 10.1016/j.chaos.2021.111418
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    References listed on IDEAS

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    1. Wang, Zhen & Xie, Yingkang & Lu, Junwei & Li, Yuxia, 2019. "Stability and bifurcation of a delayed generalized fractional-order prey–predator model with interspecific competition," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 360-369.
    2. Zhao, Yu & Yuan, Sanling, 2017. "Optimal harvesting policy of a stochastic two-species competitive model with Lévy noise in a polluted environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 477(C), pages 20-33.
    3. Duan, Daifeng & Niu, Ben & Wei, Junjie, 2019. "Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 206-216.
    4. Liu, Guodong & Chang, Zhengbo & Meng, Xinzhu & Liu, Siyu, 2020. "Optimality for a diffusive predator-prey system in a spatially heterogeneous environment incorporating a prey refuge," Applied Mathematics and Computation, Elsevier, vol. 384(C).
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    Cited by:

    1. Meng Zhu & Jing Li & Xinze Lian, 2022. "Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation," Mathematics, MDPI, vol. 10(17), pages 1-20, September.
    2. Li, Peiluan & Gao, Rong & Xu, Changjin & Li, Ying & Akgül, Ali & Baleanu, Dumitru, 2023. "Dynamics exploration for a fractional-order delayed zooplankton–phytoplankton system," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

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