Several Types Of Biological Models Pulse Mapping To Determine The Global Analysis Of Differential Equations

In recent years, many biological problems have been described by impulsive differential equations, including drug delivery, the physical control of plant diseases, integrated control of pests, making it more and more widely apply in drugs, medicine, agriculture and forestry, as well as other fields. Since the discontinuity of the solution described by impulsive differential equations, which brings great difficulties to the research theory. However, if the continuous part of impulsive differential equations can be solved or exists the first integral, then the existence and stability of periodic solutions of impulsive differential equations can be transformed into studying the existence and stability of their corresponding difference equations determined by the pomcare map.In this paper, global analysis of impulsive differential equations arising from pharmacokinetics, plant diseases and integrated pest management are carried out. In particular, the existence and stability of periodic solutions of proposed models are investigated through studying the difference equations determined by Lambert W function.For pharmacokinetics, the impulsive differential equations with Michaelis-Menten elimination rate can be described as follows where C(t) is the concentration of drug at time t, T is the time interval of the dose injection, Vmax represents the maximum rate of change of concentration, Km is the Michaelis-Menten constant, and impulsive input of quantity is described by D, V denotes the apparent volume of distribution.With the strategies of roguing infected plants and replanting healthy plants used for impulsive control of plant diseases, we can create the following model for integrated control of plant diseases where S(t), I(t) represent the number of healthy plants and infected plants, respec-tively, ET is the economic threshold, specific biological significance refer to the text.If we use impulsive chemical control, biological control and physical control for pest, then impulsive prey-predator model can be established as follows where x(t),y(t) denote the number of prey and predator, respectively, ET is also economic threshold, specific biological significance refer to the text.A common feature of researching the above three models is to find the critical condition that guarantee the existence and stability of periodic solutions. The study of this issue can be transformed into researching a unified form of difference equations as followsXn+1=-MLambert W[i,-NXnf(Xn)}+L(?)g(Xn), n=1,2,…,i=0,-1, where M, N, L are positive.By using the definition and properties of Lambert W function,Poincare map and globally stable theorem of difference equations, we mainly focus on the global stability of periodic solutions of above three types of impulsive differential equations, and provide the condition which guarantees the global stability and instability. The conclusions obtained in this paper have extended the classical results of the corresponding models which arise from drug administration, integrated disease control and integrated pest management. The proposed methods in this article can be extended and employed to study fixed-time impulsive differential equations and state-dependent impulsive differential equations.