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A complete analysis of the global dynamics of a diffusive predator and toxic prey model

Author

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  • Lv, Yun-fei
  • Li, Tongtong
  • Pei, Yongzhen
  • Yuan, Rong

Abstract

Considering many species can release toxic substances to protect themselves against predators, a diffusive predator and toxic prey system with spatial heterogeneity in predator and prey populations has been investigated. For this system, we give a complete and rigorous analysis of the global dynamics with the boundedness, globally asymptotical stability, transcritical bifurcation, Hopf bifurcation and its direction, and the stability of the bifurcating periodic solutions. Meanwhile, we consider the effects of toxins produced by the prey on the dynamic behavior. The consequence of the global stability of the coexistence equilibrium is that the toxin’s intrinsic characteristic will not change the stability of the system irreversibly. Our results show that the toxin-produced by the prey (phytoplankton) may be used as a bio-control agent for the Harmful Algal Bloom problems.

Suggested Citation

  • Lv, Yun-fei & Li, Tongtong & Pei, Yongzhen & Yuan, Rong, 2016. "A complete analysis of the global dynamics of a diffusive predator and toxic prey model," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 182-196.
    • Handle: RePEc:eee:apmaco:v:291:y:2016:i:c:p:182-196
    DOI: 10.1016/j.amc.2016.06.040
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    References listed on IDEAS

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    1. Tang, Xiaosong & Song, Yongli, 2015. "Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator–prey model with herd behavior," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 375-391.
    2. Roy, Shovonlal, 2009. "The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy," Theoretical Population Biology, Elsevier, vol. 75(1), pages 68-75.
    3. Shi, Hong-Bo & Li, Yan, 2015. "Global asymptotic stability of a diffusive predator–prey model with ratio-dependent functional response," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 71-77.
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