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Species extinction and permanence in a prey–predator model with two-type functional responses and impulsive biological control

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Abstract

By introducing impulsive biological control strategy, the dynamic behaviors of the two-prey one-predator model with defensive ability and Holling type-II functional response are investigated. By using Floquet’s Theorem and the small amplitude perturbation method, we prove that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical minimum value, and permanence conditions (that is, the impulsive period is greater than some critical maximum value) are established via the method of comparison involving multiple Liapunov functions. It is shown that our impulsive control strategy is more effective than the classical one. Furthermore, the effect of impulsive perturbations on the unforced continuous system is studied. From simulations, we find that the system has more complex dynamic behaviors and is dominated by periodic, quasi-periodic, and chaotic solutions.

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References

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

    Article  Google Scholar 

  2. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics (1993)

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

    MATH  Google Scholar 

  4. Ballinger, G., Liu, X.: Permanence of population growth models with impulsive effects. Math. Comput. Model. 26, 59–72 (1997)

    Article  MathSciNet  Google Scholar 

  5. Caltagirone, L.E., Doutt, R.L.: The history of the vedalia beetle importation to California and its impact on the development of biological control. Ann. Rev. Entomol. 34, 1–16 (1989)

    Article  Google Scholar 

  6. Croft, B.A.: Arthropod Biological Control Agents and Pesticides. Wiley, New York (1990)

    Google Scholar 

  7. DeBach, P.: Biological Control of Insect Pests and Weeds. Rheinhold, New York (1964)

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Freedman, H.J.: Graphical stability, enrichment, and pest control by a natural enemy. Math. Biosci. 31, 207–225 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hassell, M.P.: The Dynamics of Competition and Predation. p. 68. Arnod, London (1976)

    Google Scholar 

  12. 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 

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

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. 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 

  17. Pei, Y.Z., Chen, L.S., Zhang, Q.R., Li, C.G.: Extinction and permanence of one-prey multi-predators of holling type II function response system with impulsive biological control. J. Theor. Biol. 235, 495–503 (2005)

    Article  MathSciNet  Google Scholar 

  18. Pei, Y.Z., Li, C.G., Chen, L.S., Wang, C.H.: Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predator. Adv. Complex. Syst. 8(41), 1–13 (2005)

    MathSciNet  Google Scholar 

  19. 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 

  20. Ruan, S., Xiao, D.: Global analysis in a predator–prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61, 1445–1472 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Shulgin, B., Stone, L., Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1–26 (1998)

    Article  Google Scholar 

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

    Google Scholar 

  23. Tang, S.Y., Chen, L.S.: Density-dependent birth rate, birth pulse and their population dynamic consequences. J. Math. Biol. 44, 185–199 (2002)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  25. Zhang, Y.J., Liu, B., Chen, L.S.: Extinction and permanence of a two-prey one-predator system with impulsive effect. Math. Med. Biol. 20, 309–325 (2003)

    Article  MATH  Google Scholar 

  26. 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 

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Authors and Affiliations

  1. School of Science, Tianjin Polytechnic University, Tianjin, 300161, China

    Yongzhen Pei

  2. Department of Mathematics, ShaoGuan University, ShaoGuan, GuangDong, 512005, China

    Guangzhao Zeng

  3. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116023, China

    Lansun Chen

Authors
  1. Yongzhen Pei
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  2. Guangzhao Zeng
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  3. Lansun Chen
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Correspondence to Yongzhen Pei.

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Pei, Y., Zeng, G. & Chen, L. Species extinction and permanence in a prey–predator model with two-type functional responses and impulsive biological control. Nonlinear Dyn 52, 71–81 (2008). https://doi.org/10.1007/s11071-007-9258-6

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  • DOI: https://doi.org/10.1007/s11071-007-9258-6

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