Abstract and Applied Analysis

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

Younghae Do, Hunki Baek, Yongdo Lim, and Dongkyu Lim

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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.

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Abstr. Appl. Anal., Volume 2011 (2011), Article ID 934569, 16 pages.

First available in Project Euclid: 12 August 2011

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Do, Younghae; Baek, Hunki; Lim, Yongdo; Lim, Dongkyu. A Three-Species Food Chain System with Two Types of Functional Responses. Abstr. Appl. Anal. 2011 (2011), Article ID 934569, 16 pages. doi:10.1155/2011/934569. https://projecteuclid.org/euclid.aaa/1313171123

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  • P. A. Braza, “The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing,” SIAM Journal on Applied Mathematics, vol. 63, no. 3, pp. 889–904, 2003.
  • C. Cosner, D. L. Deangelis, J. S. Ault, and D. B. Olson, “Effects of spatial grouping on the functional response of predators,” Theoretical Population Biology, vol. 56, no. 1, pp. 65–75, 1999.
  • S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445–1472, 2001.
  • M. Fan and Y. Kuang, “Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 15–39, 2004.
  • S. Gakkhar and R. K. Naji, “Seasonally perturbed prey-predator system with predator-dependent functional response,” Chaos, Solitons & Fractals, vol. 18, no. 5, pp. 1075–1083, 2003.
  • R. Arditi and L. R. Ginzburg, “Coupling in predator-prey dynamics: ratio-dependence,” Journal of Theoretical Biology, vol. 139, no. 3, pp. 311–326, 1989.
  • H. I. Freedman and R. M. Mathsen, “Persistence in predator-prey systems with ratio-dependent predator influence,” Bulletin of Mathematical Biology, vol. 55, no. 4, pp. 817–827, 1993.
  • L. A. Segel, Modeling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge University Press, Cambridge, UK, 1984.
  • S. Gakkhar and R. K. Naji, “Order and chaos in predator to prey ratio-dependent food chain,” Chaos, Solitons & Fractals, vol. 18, no. 2, pp. 229–239, 2003.
  • A. Hastings and T. Powell, “Chaos in a three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991.
  • S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “A ratio-dependent food chain model and its applications to biological control,” Mathematical Biosciences, vol. 181, no. 1, pp. 55–83, 2003.
  • A. Klebanoff and A. Hastings, “Chaos in three-species food chains,” Journal of Mathematical Biology, vol. 32, no. 5, pp. 427–451, 1994.
  • S. Lv and M. Zhao, “The dynamic complexity of a three species food chain model,” Chaos, Solitons & Fractals, vol. 37, no. 5, pp. 1469–1480, 2008.
  • R. K. Naji and A. T. Balasim, “Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1853–1866, 2007.
  • C. Shen, “Permanence and global attractivity of the food-chain system with Holling IV type functional response,” Applied Mathematics and Computation, vol. 194, no. 1, pp. 179–185, 2007.
  • J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel, and J. C. Sprott, “Chaos in low-dimensional Lotka-Volterra models of competition,” Nonlinearity, vol. 19, no. 10, pp. 2391–2404, 2006.
  • M. Zhao and S. Lv, “Chaos in a three-species food chain model with a Beddington-DeAngelis functional response,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2305–2316, 2009.
  • B. Liu, Z. Teng, and L. Chen, “Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 347–362, 2006.
  • F. Cao and L. Chen, “Asymptotic behavior of nonautonomous diffusive Lotka-Volterra model,” Systems Science and Mathematical Sciences, vol. 11, no. 2, pp. 107–111, 1998.
  • H. I. Freedman and P. Waltman, “Persistence in models of three interacting predator-prey populations,” Mathematical Biosciences, vol. 68, no. 2, pp. 213–231, 1984.
  • R. M. May, Stablitiy and Complexity in Model Ecosystem, Princeton University Press, Princeton, NJ, USA, 1973.
  • M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D, vol. 65, no. 1-2, pp. 117–134, 1993.
  • J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, New York, NY, USA, 2003.