Nonlinear Dynamics

, Volume 64, Issue 1–2, pp 1–12

The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and harvesting prey at different fixed moments

Original Paper
  • 162 Downloads

Abstract

In this paper, a delayed pest control model with stage-structure for pests by introducing a constant periodic pesticide input and harvesting prey (Crops) at two different fixed moments is proposed and analyzed. We assume only the pests are affected by pesticide. We prove that the conditions for global asymptotically attractive ‘predator-extinction’ periodic solution and permanence of the population of the model depend on time delay, pulse pesticide input, and pulse harvesting prey. By numerical analysis, we also show that constant maturation time delay, pulse pesticide input, and pulse harvesting prey can bring obvious effects on the dynamics of system, which also corroborates our theoretical results. We believe that the results will provide reliable tactic basis for the practical pest management. One of the features of present paper is to investigate the high-dimensional delayed system with impulsive effects at different fixed impulsive moments.

Keywords

Permanence High-dimensional delayed system with impulse Stage-structure Maturation time delay Pest management 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hastings, A.: Age-dependent predation is not a simple process, I. Continuous time models. Theor. Popul. Biol. 23, 47–62 (1983) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hastings, A.: Delay in recruitment at different trophic levels, effects on stability. J. Math. Biol. 21, 35–44 (1984) MathSciNetMATHGoogle Scholar
  3. 3.
    Qiu, Z., Wang, K.: The asymptotic behavior of a single population model with space-limited and stage-structure. Nonlinear Anal. Real World Appl. 6, 155–173 (2005) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Hui, J., Zhu, D.: Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects. Chaos Solitons Fractals 29, 233–251 (2006) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Gourley, S.A., Kuang, Y.: A stage structured predator-prey model and its dependenceon through-stage delay and death rate. J. Math. Biol. 49, 188–200 (2004) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Wei, C., Chen, L.: Eco-epidemiology model with age structure and prey-dependent consumption for pest management. Appl. Math. Model. 33, 4354–4363 (2009) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Jiao, J., Chen, L.: Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators. Int. J. Biomath. 1, 197–208 (2008) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Song, X., Cui, J.: The stage-structured predator-prey system with delay and harvesting. Appl. Anal. 81, 1127–1142 (2002) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ou, L., et al.: The asymptotic behaviors of a stage-structured autonomous predator-prey system with time delay. J. Math. Appl. 283, 534–548 (2003) MathSciNetMATHGoogle Scholar
  10. 10.
    Shi, R., Chen, L.: The study of a ratio-dependent predator–prey model with stage structure in the prey. Nonlinear Dyn. 58, 443–451 (2009) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Aiello, W.G., Freedman, H.I.: A time-delay model of single-species growth with stage structure. Math. Biosci. 101, 139–153 (1990) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Zhang, H., Jiao, J., Chen, L.: Pest management through continuous and impulsive control strategies. Biosystems 90, 350–361 (2007) CrossRefGoogle Scholar
  13. 13.
    Meng, X., Li, Z., Wang, X.: Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects. Nonlinear Dyn. 59, 503–513 (2009) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Liu, B., Chen, L.: The periodic competing Lotka–Volterra model with impulsive effect. IMA J. Math. Med. Biol. 21, 129–145 (2004) MATHCrossRefGoogle Scholar
  15. 15.
    Roberts, M.G., Kao, R.R.: The dynamics of an infectious disease in a population with birth pulse. Math. Biosci. 149, 23–36 (1998) MATHCrossRefGoogle Scholar
  16. 16.
    Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58(3), 525–538 (2009) MATHCrossRefGoogle Scholar
  17. 17.
    Liu, B., Zhang, Y., Chen, L.: The dynamical behaviors of a Lotka–Volterra predator-prey model concerning integrated pest management. Nonlinear Anal. Real World Appl. 6, 227–243 (2005) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Zhang, S., Tan, D., Chen, L.: Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations. Chaos Solitons Fractals 28, 367–376 (2006) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Yan, J.: Stability for impulsive delay differential equations. Nonlinear Anal. 63, 66–80 (2005) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Leonid, B., Elena, B.: Linearized oscillation theory for a nonlinear delay impulsive equation. J. Comput. Appl. Math. 161, 477–495 (2003) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Liu, X., Ballinger, G.: Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear Anal. 53, 1041–1062 (2003) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Gao, S., Chen, L., Teng, Z.: Impulsive vaccination of an SEIRS model with time delay and varying total population size. Bull. Math. Biol. 69, 731–745 (2007) MATHCrossRefGoogle Scholar
  23. 23.
    Meng, X., Jiao, J., Chen, L.: The dynamics of an age structured predator-prey model with disturbing pulse and time delays. Nonlinear Anal. Real World Appl. 9, 547–561 (2008) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Jiao, J., Long, W., Chen, L.: A single stage-structured population model with mature individuals in a polluted environment and pulse input of environmental toxin. Nonlinear Anal. Real World Appl. 10(5), 3073–3081 (2009) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bainov, D., Simeonov, P.: System with Impulsive Effect: Stability, Theory and Applications. Wiley, New York (1989) Google Scholar
  26. 26.
    Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) MATHGoogle Scholar
  27. 27.
    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego, California (1993) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.College of Information Science and EngineeringShandong University of Science and TechnologyQingdaoPR China
  2. 2.College of ScienceShandong University of Science and TechnologyQingdaoPR China
  3. 3.Department of Mathematics and Mathematical StatisticsUmeå UniversityUmeåSweden

Personalised recommendations