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Bifurcation analysis of a non-smooth prey–predator model by a differential linear complementarity system

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  • Luo, Jianfeng
  • Zhao, Yi

Abstract

The aim of this paper is to investigate a prey–predator system with threshold harvesting for both species, whereas most researchers are devoted to studying a threshold policy for one population to avoid intractable mathematical simulation occurred by their coupling. Due to the equivalence between differential inclusion and differential complementarity system, we reformulate this ecosystem as a differential linear complementarity system. Then specialized algorithms for the complementarity problem allow us to handle such non-smooth structure, thereby performing a numerical examination of the dynamics and bifurcations of our proposed system with success. As a result, we can observe that this system exhibits many peculiar bifurcation patterns that are inherent to a non-smooth dynamic system, including boundary node bifurcation, pseudo-saddle–node bifurcation, touching bifurcation, and sliding homoclinic bifurcation. Specifically, we observe a multiple crossing bifurcation that results from the superposition of a sliding homoclinic bifurcation and a boundary saddle bifurcation. They are originated from the coupling of their own independent non-smooth structures of the two species. This system also admits some conventional bifurcations like saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, which are identified previously in a prey–predator system without threshold policy. Both the theoretical and numerical results indicate that the non-smooth structure of the threshold harvesting policy increases equilibria, as well as the dynamical complications of the system.

Suggested Citation

  • Luo, Jianfeng & Zhao, Yi, 2023. "Bifurcation analysis of a non-smooth prey–predator model by a differential linear complementarity system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 581-599.
  • Handle: RePEc:eee:matcom:v:205:y:2023:i:c:p:581-599
    DOI: 10.1016/j.matcom.2022.10.016
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    References listed on IDEAS

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    1. Chen, Hongyan & Wang, Wendi & Fu, Rui & Luo, Jianfeng, 2015. "Global analysis of a mathematical model on malaria with competitive strains and immune responses," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 132-152.
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