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Dynamical analysis in a hybrid bioeconomic system with multiple time delays and strong Allee effect

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  • Liu, Chao
  • Lu, Na
  • Zhang, Qingling

Abstract

In this paper, a multiple delayed differential–algebraic prey–predator system is established, where commercial harvesting on predator and strong Allee effect in prey growth are considered. Three time delays are introduced to represent maturation delay for prey (τ1), reaction delay to changes of prey surviving environment due to strong Allee effect (τ2) and gestation delay for predator (τ3), respectively. Positivity of solutions and uniform persistence of system are discussed. In absence of time delay, existence of singularity induced bifurcation and local stability analysis are investigated due to variation of economic interest of commercial harvesting. In presence of multiple time delays, existence of Hopf bifurcation and local stability analysis are discussed by analyzing associated characteristic equation. By using new normal form of multiple delayed differential–algebraic system and center manifold theorem, properties of Hopf bifurcation are studied. Furthermore, existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem.

Suggested Citation

  • Liu, Chao & Lu, Na & Zhang, Qingling, 2017. "Dynamical analysis in a hybrid bioeconomic system with multiple time delays and strong Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 136(C), pages 104-131.
    • Handle: RePEc:eee:matcom:v:136:y:2017:i:c:p:104-131
    DOI: 10.1016/j.matcom.2016.12.005
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    References listed on IDEAS

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    1. H. Scott Gordon, 1954. "The Economic Theory of a Common-Property Resource: The Fishery," Journal of Political Economy, University of Chicago Press, vol. 62(2), pages 124-124.
    2. H. Scott Gordon, 1954. "The Economic Theory of a Common-Property Resource: The Fishery," Palgrave Macmillan Books, in: Chennat Gopalakrishnan (ed.), Classic Papers in Natural Resource Economics, chapter 9, pages 178-203, Palgrave Macmillan.
    3. Aulisa, Eugenio & Jang, Sophia R.-J., 2014. "Continuous-time predator–prey systems with Allee effects in the prey," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 105(C), pages 1-16.
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    1. Liu, Chao & Wang, Luping & Zhang, Qingling & Li, Yuanke, 2018. "Modeling and dynamical analysis of a triple delayed prey–predator–scavenger system with Lévy jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 1216-1239.

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