State Feedback Control Of Disease And Insect Pests Of The Banana Leaves

This paper studies state impulsive feedback control system of disease and insect pests of the banana leaves of Logistic growth model with continuous time delay, then using the linear chain trick, we transform the system to nonlinear differential equations, by means of Lyapunov method, it is proved that positive equilibrium is globally stability under weak delay kernel function Finally, we obtain sufficient conditions for the existence of system’s periodic solution of order one by using the means of successor function, and orbitally asymptotic stability of periodic solution is proved.In Chapter l,we introduce the background and development of disease and insect pests of the banana leaves.In Chapter2,first of all,we research on equations and gain the solution of equation,we also get Logistic curve chart after simulating the solution curve of real equation;moreover,we consider theoretical research on state impulsive feedback system,then we get the period;finally,using the linear chain trick, we transform the system to nonlinear differential equations.The main conclusions in chapter2are as following:(a) Consider equation of disease and insect pests of the banana leaves here x denotes the density of pests of banana leaves, k denotes environmental carrying capacity, x0is normal number. the solution of equation is given by (b) Consider theoretical research on state impulsive feedback system here x denotes the density of pests of banana leaves, k denotes environmental carrying capacity, h denotes the ET(Economic Threshold, i.e., pest population density at which control measures should be instigated to prevent an increasing pest population from reaching the economic injury level),0<β<1is the ratio of killing pests by spraying pesticide, x0is initial value. the period is given by Next, we show that, if A1B1is the periodic solution, also A2B2is the periodic solution.(c) Study on state impulsive feedback control system of Logistic growth model with continuous time delay here x denotes the density of pests of banana leaves, r is the intrinsic rate of increase, h is the ET, a, c, w are positive constants,0<β<1is the ratio of killing pests by spraying pesticide. Then we can change the system (3) into the following by using chain transformation y=∫exp(a(t-s)x(s))ds, here y denotes the density of natural enemy, a,c,w are positive constants, satisfying In Chapter3,we define successor function of this system,by means of Lyapunov method, it is proved that positive equilibrium is globally stability under weak delay kernel function.In addition, we obtain sufficient conditions for the existence of system’s periodic solution of order one by using the means of successor function.The main conclusions in chapter3are as following:Theorem3.1Positive equilibrium E(x*,y*)=E(1/c+w,1/c+w) of system (4) is global stability.Definition2Successor function G:If the point of intersection of trajectory L and the phase set x=(1-β)h is N, the point of intersection of L and impulsive set x=h is E which generates impulses to the phase setx=(1-β)h, the phase point is N1, then the y-coordinate difference ofN and N1, is G(N)=N1y-NyTheorem3.2(1)If impulsive set0<h≤1/c+w, there exists a point M of phase set x=(1-β)h then we have G(M)=0, that is to say, there exists an order-one periodic solution of system (4).(2) If impulsive set h>1/c+w, there exists a point P of phase set x=(1-β)h, then we have G(M)=0, that is to say, there exists an order-one periodic solution of system (4), or for any t, we get y(t)≤yt+chβ/w.In Chapter4,we prove periodic solution is orbitally asymptotic stability.The main conclusions in chapter3are as following:Theorem4If1-ch-wφ0<0, then the order-one periodic orbitI0passing the point (h,φ0) of system (4) is asymptotically stable.In Chapter5,we summarize all essay,and proposed some questions that worth further studying.