Abstract and Applied Analysis

Nonlinear Dynamic in an Ecological System with Impulsive Effect and Optimal Foraging

Min Zhao and Chuanjun Dai

Full-text: Open access

Abstract

The population dynamics of a three-species ecological system with impulsive effect are investigated. Using the theories of impulsive equations and small-amplitude perturbation scales, the conditions for the system to be permanent when the number of predators released is less than some critical value can be obtained. Furthermore, because the predator in the system follows the predictions of optimal foraging theory, it follows that optimal foraging promotes species coexistence. In particular, the less beneficial prey can support the predator alone when the more beneficial prey goes extinct. Moreover, the influences of the impulsive effect and optimal foraging on inherent oscillations are studied using simulation, which reveals rich dynamic behaviors such as period-halving bifurcations, a chaotic band, a periodic window, and chaotic crises. In addition, the largest Lyapunov exponent and the power spectra of the strange attractor, which can help analyze the chaotic dynamic behavior of the model, are investigated. This information will be useful for studying the dynamic complexity of ecosystems.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 169609, 12 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412607525

Digital Object Identifier
doi:10.1155/2014/169609

Mathematical Reviews number (MathSciNet)
MR3226179

Zentralblatt MATH identifier
07021854

Citation

Zhao, Min; Dai, Chuanjun. Nonlinear Dynamic in an Ecological System with Impulsive Effect and Optimal Foraging. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 169609, 12 pages. doi:10.1155/2014/169609. https://projecteuclid.org/euclid.aaa/1412607525


Export citation

References

  • R. M. May, “Will a large complex system be stable?” Nature, vol. 238, no. 5364, pp. 413–414, 1972.
  • R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459–467, 1976.
  • R. M. May, “Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos,” Science, vol. 186, no. 4164, pp. 645–647, 1974.
  • H. Zhu, S. A. Campbell, and G. S. K. Wolkowicz, “Bifurcation analysis of a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 63, no. 2, pp. 636–682, 2002.
  • S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 42, no. 6, pp. 489–506, 2001.
  • Y. Do, H. Baek, Y. Lim, and D. Lim, “A three-species food chain system with two types of functional responses,” Abstract and Applied Analysis, vol. 2011, Article ID 934569, 16 pages, 2011.
  • S. Lv and M. Zhao, “The dynamic complexity of a three species food chain model,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1469–1480, 2008.
  • W. Wang, J. Shen, and J. J. Nieto, “Permanence and periodic solution of predator-prey system with Holling type functional response and impulses,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 81756, 15 pages, 2007.
  • L. Zhang and M. Zhao, “Dynamic complexities in a hyperparasitic system with prolonged diapause for host,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1136–1142, 2009.
  • R. K. Upadhyay and R. K. Naji, “Dynamics of a three species food chain model with Crowley-Martin type functional response,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1337–1346, 2009.
  • M. Zhao, L. Zhang, and J. Zhu, “Dynamics of a host-parasitoid model with prolonged diapause for parasitoid,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 455–462, 2011.
  • C. Wei and L. Chen, “A delayed epidemic model with pulse vaccination,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 746951, 12 pages, 2008.
  • J. Jiao, S. Cai, and L. Chen, “Analysis of a stage-structured predatory-prey system with birth pulse and impulsive harvesting at different moments,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 2232–2244, 2011.
  • R. Shi and L. Chen, “Stage-structured impulsive $SI$ model for pest management,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 97608, 11 pages, 2007.
  • H.-F. Huo, Z.-P. Ma, and C.-Y. Liu, “Persistence and stability for a generalized Leslie-Gower model with stage structure and dispersal,” Abstract and Applied Analysis, vol. 2009, Article ID 135843, 17 pages, 2009.
  • H. Yu, S. Zhong, and R. P. Agarwal, “Mathematics analysis and chaos in an ecological model with an impulsive control strategy,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 776–786, 2011.
  • L. Wang, L. Chen, and J. J. Nieto, “The dynamics of an epidemic model for pest control with impulsive effect,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1374–1386, 2010.
  • H. Yu, S. Zhong, R. P. Agarwal, and S. K. Sen, “Effect of seasonality on the dynamical behavior of an ecological system with impulsive control strategy,” Journal of the Franklin Institute: Engineering and Applied Mathematics, vol. 348, no. 4, pp. 652–670, 2011.
  • J. Luo, “Permanence and extinction of a generalized Gause-type predator-prey system with periodic coefficients,” Abstract and Applied Analysis, vol. 2010, Article ID 845606, 24 pages, 2010.
  • X. Wang, W. Wang, and X. Lin, “Dynamics of a two-prey one-predator system with Watt-type functional response and impulsive control strategy,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2392–2404, 2009.
  • R. D. Holt, “Predation, apparent competition and the structure of prey communities,” Theoretical Population Biology, vol. 12, no. 2, pp. 197–229, 1977.
  • H. Werner, “Optimal foraging and the size selection of prey by the bluegill sunfish (Lepomis macrochirus),” Ecology, vol. 55, pp. 1042–1052, 1974.
  • E. L. Charnov, “Optimal foraging: attack strategy of a mantid,” The American Naturalist, vol. 110, no. 971, pp. 141–151, 1976.
  • D. W. Stephens and J. R. Krebs, Foraging Theory, Princeton University Press, Princeton, NJ, USA, 1986.
  • V. Křivan and J. Eisner, “Optimal foraging and predator-prey dynamics III,” Theoretical Population Biology, vol. 63, no. 4, pp. 269–279, 2003.
  • V. Křivan, “Optimal foraging and predator-prey dynamics,” Theoretical Population Biology, vol. 49, no. 3, pp. 265–290, 1996.
  • V. Křivan and A. Sikder, “Optimal foraging and predator-prey dynamics, II,” Theoretical Population Biology, vol. 55, no. 2, pp. 111–126, 1999.
  • V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific, Singapore, 1989.
  • D. D. Baĭnov and P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific, Singapore, 1993.
  • D. D. Bainov and V. C. Covachev, Impulsive Differential Equations with a Small Parameter, vol. 24, World Scientific, Singapore, 1994.
  • S. Lv and M. Zhao, “The dynamic complexity of a host-parasitoid model with a lower bound for the host,” Chaos, Solitons and Fractals, vol. 36, no. 4, pp. 911–919, 2008.
  • M. Zhao and S. Lv, “Chaos in a three-species food chain model with a Beddington-DeAngelis functional response,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2305–2316, 2009.
  • M. Zhao and L. Zhang, “Permanence and chaos in a host-parasitoid model with prolonged diapause for the host,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 12, pp. 4197–4203, 2009.
  • L. Zhu and M. Zhao, “Dynamic complexity of a host-parasitoid ecological model with the Hassell growth function for the host,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1259–1269, 2009.
  • H. Yu, S. Zhong, R. P. Agarwal, and S. K. Sen, “Three-species food web model with impulsive control strategy and chaos,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 1002–1013, 2011.
  • H. Yu, S. Zhong, and M. Ye, “Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay,” Mathematics and Computers in Simulation, vol. 80, no. 3, pp. 619–632, 2009. \endinput