The Nature Of The Solution Of Two Types Of Ecological Models And A Class Of Second-order Non-autonomous Differential Equations Of Vibration Research

Ecological mathematics traslates complex biological problems into a mathemat-ical problem by the establishment of mathematical models.Then ecological mathe-matics studies all kinds of natural phenomenon through the mathematical theory and way.The properties of two ecological systems and the oscillation of a second-order differential equation are investigated.Here,the behavior includes the existence and uniqueness of the positive equilibrium state,the existence and approximate expression of the periodic solution,the boundedness of the solutions,the global attractivity of the solutions and the oscillations of solutions.The research of the time delay leads to the periodic solutions' has developed rapidly in recent years,and the hopf bifurcation produced by the time delay and the periodic solutions similar expressions of the bifurcation have been studied in lots of papers. Frist,the hopf bifurcation of a class of general Logistic model with a discrete time delay and stocking item is investigated.The part stability of the positive equilibrium state and existence of the hopf branch are discussed through the eigenvalue theory. The form of the approximate periodic solution is obtained by orthogonal conditions.The paper points out the example to confirm the theorem's realizability, and fitted curve figures with different values,in which the influence toperiod, swing, positive equilibrium of period solution are discussed, are achieved by using matlab.In nature,the relationship between one group and another is restriced and de-pendent,such as predator-prey,competition and reciprocity coexistence. Sceond,the global attractivity of the solutions of a two-species competitive system with feedback controls and mutl-delay is investigated.The boundedness of the solutions of this model has been proved through the oscillation theory and the thinking of limit.Sufficient condition of the global attractivity for this model are derived by using the method of constructing Liapunov functional and inequality valuation.Because the differential equation's oscillation theory is one of the important issues about the power system study. Third,the linearized oscillation for a second-order differential equation with mutl-delay is studied.Through the first-order coefficient s in the different range was discussed.The linearized oscillation criterion of the equation was obtained by using the Knaster-Tarski fixed- point theorem and the value theorem.The whole sufficient and neces-sary condition of it was derived,and so the oscillatin of the equation's solution was simplified.