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Zbl 1104.92054
Jiao, Jianjun; Chen, Lansun
A pest management SI model with periodic biological and chemical control concern. (English)
[J] Appl. Math. Comput. 183, No. 2, 1018-1026 (2006). ISSN 0096-3003

Summary: We consider an SI model for pest management, with concerns about impulsive releases of infective pests and pesticides sprays. We prove that all solutions of $$\cases S'(t)=rS(t)\left(1-\bigl(S(t)+\theta I(t)\bigr)/K \right)-\beta S(t) I^2(t), I'(t)=\beta S(t)I^2(t)-wI(t),\ t\ne n\tau,\\ \Delta S(t)=-\mu_1S(t), \quad \Delta I(t)=-\mu_2I(t)+\mu,\quad t=n\tau,\ n=1,2,\dots,\endcases\tag 1$$ are uniformly ultimately bounded and there exist globally asymptotically stable periodic solutions of pest-extinction when $$\ln\frac{1}{1-\mu_1}>r\tau-\frac{r \mu\theta\bigl(1-\exp(-wt)\bigr)}{Kw\bigl(1-(1-\mu_2)\exp(-w\tau)\bigr)}-\frac {\beta \mu^2\bigl(1-\exp(-2w\tau)\bigr)}{2w\bigl(1-(1-\mu_2)\exp(-w\tau) \bigr)^2}$$ is satisfied, and a condition for permanence of system (1) is also obtained. It is concluded that the approach of combining impulsive infective releasing with impulsive pesticide spraying provides a reliable tactic basis for practical pest management.

MSC 2000:: *92D30 Epidemiology
34A37 Differential equations with impulses
93C15 Control systems governed by ODE
93C95 Appl. of control theory
34C25 Periodic solutions of ODE

Keywords: impulsive; infective; chemical control; pest- extinction

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Zentralblatt MATH Berlin [Germany]