Abstract.
A state-dependent impulsive model is proposed for integrated pest management (IPM). IPM involves combining biological, mechanical, and chemical tactics to reduce pest numbers to tolerable levels after a pest population has reached its economic threshold (ET). The complete expression of an orbitally asymptotically stable periodic solution to the model with a maximum value no larger than the given ET is presented, the existence of which implies that pests can be controlled at or below their ET levels. We also prove that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the LambertW function and Poincaré map. Moreover, we show that the existence of an order two periodic solution implies the existence of an order one periodic solution. Various positive invariant sets and attractors of this impulsive semi-dynamical system are described and discussed. In particular, several horseshoe-like attractors, whose interiors can simultaneously contain stable order 1 periodic solutions and order 2 periodic solutions, are found and the interior structure of the horseshoe-like attractors is discussed. Finally, the largest invariant set and the sufficient conditions which guarantee the global orbital and asymptotic stability of the order 1 periodic solution in the meaningful domain for the system are given using the Lyapunov function. Our results show that, in theory, a pest can be controlled such that its population size is no larger than its ET by applying effects impulsively once, twice, or at most, a finite number of times, or according to a periodic regime. Moreover, our theoretical work suggests how IPM strategies could be used to alter the levels of the ET in the farmers’ favour.
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Mathematics Subject Classifications (2000): 34A36, 34D45, 92B05, 34C25
Acknowledgement The authors thank Professor L.S. Chen for support and encouragement in this research, as well as Y.N. Xiao for fruitful discussions. We would also like to thank the referees for their careful reading of the original manuscript and many valuable comments and suggestions, including the new method of proving proposition 2.1 attached in Appendix, that greatly improved the presentation of this paper.
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Tang, S., Cheke, R. State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50, 257–292 (2005). https://doi.org/10.1007/s00285-004-0290-6
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DOI: https://doi.org/10.1007/s00285-004-0290-6