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Bifurcation analysis and global dynamics in a predator–prey system of Leslie type with an increasing functional response

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  • Shang, Zuchong
  • Qiao, Yuanhua
  • Duan, Lijuan
  • Miao, Jun

Abstract

The dynamical behaviors of a Leslie type predator–prey system are explored when the functional response is increasing for both predator and prey. Qualitative and quantitative analysis methods based on stability theory, bifurcation theory and numerical simulation are adopted. It is showed that the system is dissipative and permanent, and its solutions are bounded. Global stability of the unique positive equilibrium is investigated by constructing Dulac function and applying Poincaré–Bendixson theorem. The bifurcation behaviors are further explored and the number of limit cycles is determined. By calculating the first Lyapunov number and the first two focus values, it is proved that the positive equilibrium is not a center but a weak focus of multiplicity at most two, so the system undergoes Hopf bifurcation and Bautin bifurcation. The normal form of Bautin bifurcation is also obtained by introducing the complex system. Moreover, numerical simulations are run to demonstrate the validity of theoretical results.

Suggested Citation

  • Shang, Zuchong & Qiao, Yuanhua & Duan, Lijuan & Miao, Jun, 2021. "Bifurcation analysis and global dynamics in a predator–prey system of Leslie type with an increasing functional response," Ecological Modelling, Elsevier, vol. 455(C).
    • Handle: RePEc:eee:ecomod:v:455:y:2021:i:c:s0304380021002192
    DOI: 10.1016/j.ecolmodel.2021.109660
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    References listed on IDEAS

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    1. Savoca, S. & Grifó, G. & Panarello, G. & Albano, M. & Giacobbe, S. & Capillo, G. & Spanó, N. & Consolo, G., 2020. "Modelling prey-predator interactions in Messina beachrock pools," Ecological Modelling, Elsevier, vol. 434(C).
    2. Lajmiri, Z. & Khoshsiar Ghaziani, R. & Orak, Iman, 2018. "Bifurcation and stability analysis of a ratio-dependent predator-prey model with predator harvesting rate," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 193-200.
    3. Colon, C. & Claessen, D. & Ghil, M., 2015. "Bifurcation analysis of an agent-based model for predator–prey interactions," Ecological Modelling, Elsevier, vol. 317(C), pages 93-106.
    4. Ghosh, Uttam & Pal, Swadesh & Banerjee, Malay, 2021. "Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    5. Negi, Kuldeep & Gakkhar, Sunita, 2007. "Dynamics in a Beddington–DeAngelis prey–predator system with impulsive harvesting," Ecological Modelling, Elsevier, vol. 206(3), pages 421-430.
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