##
**Stage-structured impulsive \(SI\) model for pest management.**
*(English)*
Zbl 1146.37368

Summary: An \(SI\) epidemic model with stage structure is investigated. In the model, impulsive biological control is taken, that is, we release infected pests to the field at a fixed time periodically. We get a sufficient condition for the global asymptotical stability of the pest-eradication periodic solution \((0,0,\tilde I(t))\), and a condition for the permanence of the system. At last, a brief discussion shows that our results will be helpful for pest management.

### MSC:

37N25 | Dynamical systems in biology |

92D30 | Epidemiology |

34K45 | Functional-differential equations with impulses |

34K20 | Stability theory of functional-differential equations |

### Keywords:

epidemic model; impulsive biological control; asymptotical stability; periodic solution; permanence; pest management
PDF
BibTeX
XML
Cite

\textit{R. Shi} and \textit{L. Chen}, Discrete Dyn. Nat. Soc. 2007, Article ID 97608, 11 p. (2007; Zbl 1146.37368)

### References:

[1] | H. D. Burges and N. W. Hussey, Eds., Microbial Control of Insections and Mites, Academic Press, New York, NY, USA, 1971. |

[2] | P. E. Davis, K. Myers, and J. B. Hoy, “Biological control among vertebrates,” in Theory and Practice of Biological Control, C. B. Huffaker and P. S. Messenger, Eds., pp. 501-519, Plenum Press, New York, NY, USA, 1976. |

[3] | L. A. Falcon, “Problems associated with the use of arthropod viruses in pest control,” Annual Review of Entomology, vol. 21, pp. 305-324, 1976. |

[4] | K. Wickwire, “Mathematical models for the control of pests and infectious diseases: a survey,” Theoretical Population Biology, vol. 11, no. 2, pp. 182-238, 1977. · Zbl 0356.92001 |

[5] | W. Wang, J. Shen, and 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. · Zbl 1146.37370 |

[6] | H. Zhang, R. Zhu, and L. Chen, “On a periodic time-dependent impulsive system of strategies for controlling the apple snail in paddy fields,” Journal of Biological Systems, vol. 15, no. 3, pp. 397-408, 2007. · Zbl 1195.92068 |

[7] | H. Zhang, L. Chen, and J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,” to appear in Nonlinear Analysis: Real World Applications. · Zbl 1154.34394 |

[8] | D. Baĭnov and P. Simeonov, Eds., Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK, 1993. · Zbl 0815.34001 |

[9] | V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Eds., Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. · Zbl 0719.34002 |

[10] | S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems: Theory and Applications, vol. 394 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1997. · Zbl 0880.46031 |

[11] | A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995. |

[12] | J. J. Nieto and R. Rodríguez-López, “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 593-610, 2006. · Zbl 1101.34051 |

[13] | J. J. Nieto, “Periodic boundary value problems for first-order impulsive ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 51, no. 7, pp. 1223-1232, 2002. · Zbl 1015.34010 |

[14] | L. Chen and J. Sun, “Nonlinear boundary value problem of first order impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 726-741, 2006. · Zbl 1102.34052 |

[15] | Y. Liu, “Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 435-452, 2007. · Zbl 1119.34062 |

[16] | J. Li, J. J. Nieto, and J. Shen, “Impulsive periodic boundary value problems of first-order differential equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 226-236, 2007. · Zbl 1110.34019 |

[17] | P. Georgescu and G. Moro\csanu, “Pest regulation by means of impulsive controls,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 790-803, 2007. · Zbl 1117.93006 |

[18] | S. Gao, L. Chen, J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037-6045, 2006. |

[19] | M. Choisy, J.-F. Guégan, and P. Rohani, “Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects,” Physica D, vol. 223, no. 1, pp. 26-35, 2006. · Zbl 1110.34031 |

[20] | J. Yan, A. Zhao, and J. J. Nieto, “Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems,” Mathematical and Computer Modelling, vol. 40, no. 5-6, pp. 509-518, 2004. · Zbl 1112.34052 |

[21] | W.-T. Li and H.-F. Huo, “Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics,” Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 227-238, 2005. · Zbl 1070.34089 |

[22] | W. Zhang and M. Fan, “Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays,” Mathematical and Computer Modelling, vol. 39, no. 4-5, pp. 479-493, 2004. · Zbl 1065.92066 |

[23] | X. Zhang, Z. Shuai, and K. Wang, “Optimal impulsive harvesting policy for single population,” Nonlinear Analysis. Real World Applications, vol. 4, no. 4, pp. 639-651, 2003. · Zbl 1011.92052 |

[24] | A. d’Onofrio, “A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences,” Physica D, vol. 208, no. 3-4, pp. 220-235, 2005. · Zbl 1087.34028 |

[25] | S. Gao, Z. Teng, J. J. Nieto, and A. Torres, “Analysis of an SIR epidemic model with pulse vaccination and distributed time delay,” Journal of Biomedicine and Biotechnology, vol. 2007, Article ID 64870, 10 pages, 2007. |

[26] | W. G. Aiello and H. I. Freedman, “A time-delay model of single-species growth with stage structure,” Mathematical Biosciences, vol. 101, no. 2, pp. 139-153, 1990. · Zbl 0719.92017 |

[27] | X. Song and L. Chen, “Modelling and analysis of a single-species system with stage structure and harvesting,” Mathematical and Computer Modelling, vol. 36, no. 1-2, pp. 67-82, 2002. · Zbl 1024.92015 |

[28] | S. Liu, L. Chen, and Z. Liu, “Extinction and permanence in nonautonomous competitive system with stage structure,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 667-684, 2002. · Zbl 1039.34068 |

[29] | X. Song and L. Chen, “Optimal harvesting and stability for a predator-prey system with stage structure,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 3, pp. 423-430, 2002. · Zbl 1054.34125 |

[30] | W. Wang and L. Chen, “A predator-prey system with stage-structure for predator,” Computers & Mathematics with Applications, vol. 33, no. 8, pp. 83-91, 1997. |

[31] | Y. Xiao and L. Chen, “Stabilizing effect of cannibalism on a structured competitive system,” Acta Mathematica Scientia Series A, vol. 22, no. 2, pp. 210-216, 2002 (Chinese). · Zbl 1041.34033 |

[32] | W. G. Aiello, H. I. Freedman, and J. Wu, “Analysis of a model representing stage-structured population growth with state-dependent time delay,” SIAM Journal on Applied Mathematics, vol. 52, no. 3, pp. 855-869, 1992. · Zbl 0760.92018 |

[33] | S. Liu, L. Chen, and R. Agarwal, “Recent progress on stage-structured population dynamics,” Mathematical and Computer Modelling, vol. 36, no. 11-13, pp. 1319-1360, 2002. · Zbl 1077.92516 |

[34] | Y. Xiao and L. Chen, “An SIS epidemic model with stage structure and a delay,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 4, pp. 607-618, 2002. · Zbl 1035.34054 |

[35] | Y. Xiao, L. Chen, and F. ven den Bosch, “Dynamical behavior for a stage-structured SIR infectious disease model,” Nonlinear Analysis: Real World Applications, vol. 3, no. 2, pp. 175-190, 2002. · Zbl 1007.92032 |

[36] | Y. Xiao and L. Chen, “On an SIS epidemic model with stage structure,” Journal of Systems Science and Complexity, vol. 16, no. 2, pp. 275-288, 2003. · Zbl 1138.92369 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.