State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences.

*(English)*Zbl 1080.92067Summary: 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 Lambert W function and Poincaré maps. 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 sufficient conditions which guarantee the global orbital and asymptotic stability of order 1 periodic solutions 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.

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 sufficient conditions which guarantee the global orbital and asymptotic stability of order 1 periodic solutions 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.

##### MSC:

92D40 | Ecology |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

37N25 | Dynamical systems in biology |

92D30 | Epidemiology |

34A37 | Ordinary differential equations with impulses |

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\textit{S. Tang} and \textit{R. A. Cheke}, J. Math. Biol. 50, No. 3, 257--292 (2005; Zbl 1080.92067)

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##### References:

[1] | Andronov, A.A., Leontovich, E.A., Gordan, L.L., Maier, A.G.: Qualitative theory of second-order dynamic systems. Translated from Russian by D. Louvish, John Wiley & Sons, New York, 1973 · Zbl 0282.34022 |

[2] | Bainov, D.D., Simeonov, P.S.: Impulsive differential equations: periodic solutions and applications, Pitman Monographs and Surveys in Pure and Appl. Math. 66, (1993) · Zbl 0815.34001 |

[3] | Bainov, D.D., Simeonov, P.S.: Systems with impulse effect, theory and applications, Ellis Hardwood series in Mathematics and its Applications, Ellis Hardwood, Chichester, 1989 · Zbl 0671.34052 |

[4] | Barclay, H.J.: Models for pest control using predator release, habitat management and pesticide release in combination. J. Appl. Ecol. 19, 337-348 (1982) |

[5] | Chellaboina, V.S., Bhat, S.P., Haddad, W.M.: An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlinear Anal. TMA 53, 527-550 (2003) · Zbl 1082.37018 |

[6] | Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On The Lambert W Function. Adv. Comput. Math. 5, 329-359 (1996) · Zbl 0863.65008 |

[7] | Flint, M.L., ed.: Integrated Pest Management for Walnuts, University of California Statewide Integrated Pest Management Project, Division of Agriculture and Natural Resources, Second Edition, University of California, Oakland, CA, publication 3270, 1987, pp. 3641 |

[8] | Grasman, J., Van Herwaarden, O.A., Hemerik, L., Van Lenteren, J.C.: A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci. 169, 207-216 (2001) · Zbl 0966.92026 |

[9] | Greathead, D.J.: Natural enemies of tropical locusts and grasshoppers: their impact and potential as biological control agents. In: C.J. Lomer, C. Prior (eds.), Biological control of locusts and grasshoppers. Wallingford, UK: C.A.B. International, 1992, pp. 105-121 |

[10] | Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl. Math. Sci. 42, (1983) · Zbl 0515.34001 |

[11] | Kaul, S.: On impulsive semidynamical systems. J. Math. Anal. Appl. 150, 120-128 (1990) · Zbl 0711.34015 |

[12] | Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of impulsive differential equations. World Scientific series in Modern Mathematics, Vol. 6, Singapore, 1989 · Zbl 0719.34002 |

[13] | Matveev, A.S., Savkin, A.V.: Qualitative theory of hybrid dynamical systems. Birkhäuser, 2000 · Zbl 1052.93004 |

[14] | Pedigo, L.P., Higley, L.G.: A new perspective of the economic injury level concept and environmental quality. Am. Entomologist 38, 12-20 (1992) |

[15] | Pedigo, L.P.: Entomology and Pest Management. Second Edition. Prentice-Hall Pub., Englewood Cliffs, NJ, 1996, p. 679 |

[16] | Qi, J.G., Fu, X.L.: Existence of limit cycles of impulsive differential equations with impulses as variable times. Nonl. Anal. TMA 44, 345-353 (2001) · Zbl 0993.34043 |

[17] | Simeonov, P.S., Bainov, D.D.: Orbital stability of periodic solutions of autonomous systems with impulsive effect. INT. J. Syst. SCI 19, 2561-2585 (1988) · Zbl 0669.34044 |

[18] | Stern, V.M., Smith, R.F., van den Bosch, R., Hagen, K.S.: The integrated control concept. Hilgardia 29, 81-93 (1959) |

[19] | Stern, V.M.: Economic Thresholds. Ann. Rev. Entomol., 259-280 (1973) |

[20] | Tang, S.Y., Chen, L.S.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 44, 185-199 (2002) · Zbl 0990.92033 |

[21] | Tang, S.Y., Chen, L.S.: Multiple attractors in stage-structured population models with birth pulses. Bull. Math. Biol. 65, 479-495 (2003) · Zbl 1334.92371 |

[22] | Tang, S.Y., Chen, L.S.: The effect of seasonal harvesting on stage-structured population models. J. Math. Biol. 48, 357-374 (2004a) · Zbl 1058.92051 |

[23] | Tang, S.Y., Chen, L.S.: Modelling and analysis of integrated pest management strategy. Disc. Continuous Dyn. Syst. B 4, 759-768 (2004b) · Zbl 1114.92074 |

[24] | Tang, S.Y., Chen, L.S.: Global attractivity in a ?food limited? population model with impulsive effects. J. Math. Anal. Appl. 292, 211-221 (2004c) · Zbl 1062.34055 |

[25] | Van Lenteren, J.C.: Integrated pest management in protected crops. In: Dent, D., (ed.), Integrated pest management, Chapman & Hall, London, 1995, pp. 311-320 |

[26] | Van Lenteren, J.C.: Measures of success in biological control of arthropods by augmentation of natural enemies. In: Wratten, S., Gurr G. (eds.), Measures of Success in Biological Control, Kluwer Academic Publishers, Dordrecht, 2000, pp. 77-89 |

[27] | Van Lenteren, J.C., Woets, J.: Biological and integrated pest control in greenhouses. Ann. Rev. Ent. 33, 239-250 (1988) |

[28] | Volterra, V.: Variations and fluctuations of a number of individuals in animal species living together. Translation In: R.N. Chapman: Animal Ecology, New York: McGraw Hill, 1931, pp. 409-448 |

[29] | Xiao, Y.N., Van Den Bosch, F.: The dynamics of an eco-epidemic model with biological control. Ecol. Modelling 168, 203-214 (2003) |

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