The Properties Of Three Classes Differential Equations

The properties of the solution of differential equation include the stability of the solution , oscillation and periodicity and so on ,these properties reveal the long-term behavior of the dynamical system, since a lot of biological regular and phenomenon in the biological were described by using differential equations, which drew the attention of a lot of experts and scholars ,and formed many new topics that have strong practical background .we know that the study of the co-exist ,stability , oscillation of the spices has very important practical meaning to keep ecological equilibration and protect ecological environment , even to save valuable and rare biologics which are on the verge of becoming extinct.In this paper ,we discuss the properties of three classes of differential equations.The population dynamics system which Applied can describe,calculate in order to regulate and control the developmental processes and the developmental trend of species,which is one of theoretical basis for people exploiting resources,making use of resources and protecting resources.ln the population dynamics system .persistence and globally asymptotically stability is a subject of great topical interest,in the document formerly, generally,condition of the persistence is obtained by applying the comparison theorem.we obtain the globally asymptotically stability by using Liapunov function. In the chapter 2. we consider a predator-prey system with time delay,mathematical analyses of system with regard to the positive invariant set, nature of boundary equilibrium by using characteristic equation is analyzed, the sufficient condition is obtained for locally asymptotically stability of positive equilibrium when time delay r is small,and Hopf bifurcation can occur as the delay increases to (?)₀.Last we obtain globally asymptotically stability of boundary equilibrium by applying locally asymptotically stability and attractiveness and persistence by using uniform repeller theorem. The correctness of the conditions of the theorem is shown by example and using matlab software.Generally,for a abstract functional differential equation,if it satisfies some conditions, then we can get a periodic solution.In the chapter 3,we base on the existence of periodic solution ,using the method of linearization and applying the mean value theorem and the variation-of-constants formula, we get the sufficient condition of the exponential stability of periodic solution of a class of nonlinear functional differential system . we get the exponential stability of a class of periodic Lotka-Volterra type n-species competitive systems.In the last 50 years the oscillation theory of ordinary,functional,neutral,partial and impulsive differential equations has attracted many researchers.Theoretically,we know that the oscillation...