*(English)*Zbl 1151.34037

Summary: This paper investigates the onset of nontrivial periodic solutions for an integrated pest management model which is subject to pulsed biological and chemical controls. The biological control consists in the periodic release of infective individuals, while the chemical control consists in periodic pesticide spraying. It is assumed that both controls are used with the same periodicity, although not simultaneously. To model the spread of the disease which is propagated through the release of infective individuals, an unspecified force of infection is employed.

The problem of finding nontrivial periodic solutions is reduced to showing the existence of nontrivial fixed points for the associated stroboscopic mapping of time snapshot equal to the common period of controls. The latter problem is in turn treated via a projection method. It is then shown that once a threshold condition is reached, a stable nontrivial periodic solution emerges via a supercritical bifurcation.

##### MSC:

34C60 | Qualitative investigation and simulation of models (ODE) |

34A37 | Differential equations with impulses |

92D25 | Population dynamics (general) |

34C23 | Bifurcation (ODE) |

37G15 | Bifurcations of limit cycles and periodic orbits |

34H05 | ODE in connection with control problems |

34C25 | Periodic solutions of ODE |