Skip to main content
Log in

Complex Dynamics of an Impulsive Control System in which Predator Species Share a Common Prey

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

In an ecosystem, multiple predator species often share a common prey and the interactions between the predators are neutral. In view of this fact, we propose a three-species prey-predator system with the functional responses and impulsive controls to model the process of pest management. It is proved that the system has a locally stable pest-eradication periodic solution under the assumption that the impulsive period is less than some critical value. In particular, two single control strategies (biological control alone or chemical control alone) are proposed. Finally, we compare three pest control strategies and find that if we choose narrow-spectrum pesticides that are targeted to a specific pest’s life cycle to kill the pest, then the combined strategy is preferable. Numerical results show that our system has complex dynamics including period-doubling bifurcation, quasi-periodic oscillation, chaos, intermittency and crises.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10, 707–723 (1968)

    Article  Google Scholar 

  • Bainov, D., Simeonov, P.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, New York (1993)

    MATH  Google Scholar 

  • Bainov, D., Simeonov, P.: Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore (1995)

    MATH  Google Scholar 

  • Barclay, J.H.: Combining methods of insect pest control: partitioning mortality and predicting complementarity. Res. Popul. Ecol. 34, 91–107 (1992)

    Article  Google Scholar 

  • DeBach, P., Rosen, D.: Biological Control by Natural Enemies, 2nd edn. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  • Donofrio, A.: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci. 179, 57–72 (2002)

    Article  MathSciNet  Google Scholar 

  • Ferry, N., Edwards, M.G., Gatehouse, J., Capell, T., Christou, P.A.M.R.: Gatehouse transgenic plants for insect pest control: a forward looking scientific perspective. Transgenic Res. 15, 13–19 (2006)

    Article  Google Scholar 

  • Gao, S.J., Chen, L.S.: Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete Contin. Dyn. Syst. Ser. B 7(1), 77–86 (2007)

    MATH  MathSciNet  Google Scholar 

  • Hassell, M.P.: The Dynamics of Competition and Predation. Edward Arnod, London (1976)

    Google Scholar 

  • Holling, C.S.: The functional response of predator to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45, 1–60 (1965)

    Google Scholar 

  • Kellogg, R.L., Nehring, R., Grube, A., Goss, D.W., Plotkin, S.: Environmental indicators of pesticide leaching and runoff from farm fields. United States Department of Agriculture Natural Resources Conservation Service (February 2000)

  • Kuniuki, S.: Effects of organic fertilization and pesticide application on growth and yield of field-grown rice for 10 years. Jpn. J. Crop Sci. 70(4), 530–540 (2001)

    Google Scholar 

  • Lakmeche, A., Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dyn. Contin. Discrete Impuls. Syst. 7, 165–187 (2000)

    MathSciNet  Google Scholar 

  • Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    MATH  Google Scholar 

  • Liu, X.N., Chen, L.S.: Complex dynamics of Holling type II Lotka–Volterra predator-prey system with impulsive perturbations on the predator. Chaos Solitons Fractals 16, 311–320 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Liu, B., Chen, L.S., Zhang, Y.J.: The dynamics of a prey-dependent consumption model concerning impulsive control strategy. Appl. Math. Comput. 169, 305–320 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Panetta, J.C.: A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. Bull. Math. Biol. 58, 425–447 (1996)

    Article  MATH  Google Scholar 

  • Roberts, M.G., Kao, R.R.: The dynamics of an infectious disease in a population with birth pulses. Math. Biosci. 149, 23–36 (1998)

    Article  MATH  Google Scholar 

  • Sugie, J., Howell, J.A.: Kinetics of phenol oxidation by washed cell. Biotechnol. Bioeng. 23, 2039–2049 (1980)

    Google Scholar 

  • Tener, J.S.: Muskoxen. Queens Printer, Ottawa (1965)

    Google Scholar 

  • Zhang, S.W., Dong, L.Z., Chen, L.S.: The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator. Chaos, Solitons Fractals 23, 631–643 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang, S.W., Tan, D.J., Chen, L.S.: Dynamic complexities of a food chain model with impulsive perturbations and Beddington–DeAngelis functional response. Chaos Solitons Fractals 27(3), 768–777 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yongzhen Pei.

Additional information

Communicated by Jacques Bélair.

This work is supported by National Natural Science Foundation of China (10171106).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pei, Y., Liu, S. & Li, C. Complex Dynamics of an Impulsive Control System in which Predator Species Share a Common Prey. J Nonlinear Sci 19, 249–266 (2009). https://doi.org/10.1007/s00332-008-9034-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-008-9034-x

Keywords

Mathematics Subject Classification (2000)

Navigation