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A Three‐Species Food Chain System with Two Types of Functional Responses

Author

Listed:
  • Younghae Do
  • Hunki Baek
  • Yongdo Lim
  • Dongkyu Lim

Abstract

In recent decades, many researchers have investigated the ecological models with three and more species to understand complex dynamical behaviors of ecological systems in nature. However, when they studied the models with three species, they have just considered the functional responses between prey and mid‐predator and between mid‐predator and top predator as the same type. However, in the paper, in order to describe more realistic ecological world, a three‐species food chain system with two types of functional response, Holling type and Beddington‐DeAngelis type, is considered. It is shown that this system is dissipative. Also, the local and global stability of equilibrium points of the system is established. In addition, conditions for the persistence of the system are found according to the existence of limit cycles. Some numerical examples are given to substantiate our theoretical results. Moreover, we provide numerical evidence of the existence of chaotic phenomena by illustrating bifurcation diagrams of system and by calculating the largest Lyapunov exponent.

Suggested Citation

  • Younghae Do & Hunki Baek & Yongdo Lim & Dongkyu Lim, 2011. "A Three‐Species Food Chain System with Two Types of Functional Responses," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    • Handle: RePEc:wly:jnlaaa:v:2011:y:2011:i:1:n:934569
    DOI: 10.1155/2011/934569
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    References listed on IDEAS

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    1. Lv, Songjuan & Zhao, Min, 2008. "The dynamic complexity of a three species food chain model," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1469-1480.
    2. Zhao, Min & Lv, Songjuan, 2009. "Chaos in a three-species food chain model with a Beddington–DeAngelis functional response," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2305-2316.
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