Impulsive Differential Model Of Integrated Pest Management

As we all know that pest is the enemy of forest system. Each year,countless trees suffer serious damage by pests. Using the mathematical model can help to quantitatively analyze how to better implement integrated pest control. Using the impulsive control approach,such as artificial or mechanical physical control, chemical control of spraying insecticide, biological protection of releasing natural enermies and so on,many articles set up models of impulsive spraying pesticide and obtain a lot of meaningful results.however many of them neglect the fact that the transmission rate of plant insect pest and the total number of plants are change over time.This paper mainly include:In the third chapter, considering that the infection rate evolves periodically with time and the total number of forest trees remain unchanged, we dicuss the SIRS model of impulsive spraying pesticide with vertical transmission and single species. Firstly, according to ’the Monodromy operator and Bohl- Brouwer fixed point theory’, we demonstrate the existence of a disease-free periodic solution of the system. Secondly,taking advantage of ’Monodromy matrix and Floquet theory’, we obtain a basic reproductive rate of the model. Lastly, we work out the conditions of locally asymptotic stability of disease-free periodic solution of the model.In the forth chapter, we discuss the SIRS model of pulse spraying pesticide with vertical transmission and single species.Suppose that the incidence rate periodically change over time and the total number of forest trees also vary with time. Firstly,according to ‘the monodromy operator and Bohl- Brouwer fixed point theory’,we demonstrate the existence of disease-free periodic solution and taking advantage of ‘monodromy matrix and floquet theory’,we obtain the condition of local asymptotical stability(LAS) for the disease-free periodic solution.In addition, we study the impulsive model stability by means of piecewise continuous Lyapunov function and differential inequality theories. As the results,we obtain conditions of global asymptotic stability(GAS)for the periodic solution of pests being relatively or totally eradicated.Lasty, we select appropriate datas and use numerical examples shows the correct of the conclusions.In the fifth chapter, Dicuss the SIRS model of pulse spraying pesticide with vertical transmission and single species.Suppose that the incidence rate periodically change over time and the total number of forest trees also vary with time. Firstly,according to ‘the monodromy operator and Bohl- Brouwer fixed point theory’,we demonstrate the existence of disease-free periodic solution and taking advantage of ‘monodromy matrix and floquet theory’,we obtain the condition of local asymptotical stability(LAS) for the disease-free periodic solution.In addition, we study the impulsive model stability by means of piecewise continuous Lyapunov function and differential inequality theories. As the results,we obtain conditions of global asymptotic stability(GAS) for the periodic solution of pests being relatively or totally eradicated.Lasty, we select appropriate datas and use numerical examples shows the correct of the conclusions.