Oscillation Of Three Kinds Of Ecological Models And Hope-Bifurcation For The Gerneral Logistic Biological Model With Delays

By investigating asymptotic properties of solutions for differential equations, suchas local or global attractivity, stability, periodicity, oscillation and persistence etc, ecological systemcould be understood and controlled by people in further extent, and its balance could be keptin some degree. In this paper there are four parts. Firstly, oscillation of solutions for threekinds of ecological models are investigated separately, including the existence of positive solutions,oscillation of the solutions, linearized oscillation of the solutions; secondly, the hopf-bifurcation fora general logistic biological model with discret delays and disturbs is studied.A family of nonautonomous linear third-order delay differential equation hayed been studiedin the chapter 2. The relation between generalized characteristic equation and existence of positivesolutions is investigated, and simplified sufficient conditions for the existence of eventually positivesolutions are achieved in details. Firstly necessary and sufficient condition for the existence ofpositive solutions is obtained by using the theory of functional analysis and fixed point theorem;secondly by the method of creating function, some sufficient results to existence of the eventuallypositive solution are derived on the basis of different characters of the equation; in the end therealization of theorem's condition is checked through examples.Adding to the ceaseless influence of people, nature has its own rule of movement and devel-opment, some population density is caused to increase or reduce lastly in some time, or extictionin some instant in the ecosystem. The above diversification could be relate to any instant andsome arbitrary instant before current, that is to say it has continuous delay in function-differentialequations. And then the study of oscillation for intergro-differential equation or inequality withcontinuous delays is important to theory and application. In the chapter 3, the oscillatory prob-lem for a second-order nonlinear integro-differential inequality with continuous delay haved beenstudied. Firstly, taking advantage of Lebesgue's deminated convergence theorem, some sufficientcondition to the existence of the positive solution for inequality is obtained; then by the methodof disproof, the sufficient condition to inexistence of the positive solution for inequality is gained;finally, by using of discussion and analysis, the sufficient and necessary condition to oscillation ofit's solution is derived. For differential equation, the study of the nonlinear equation's property generally is madeuse of linear equation. Such as oscillation, if some nonlinear equation has the same oscillatoryproperty.with its corresponding linear equation, and so its oscillatory property could be decribedby its corresponding linear equation. Then the study of oscillatory property would be simplifiedin great extent to the nonlinear equation. Because the change of population density is under theinfluence of time directly, which often represents that the coefficients is variable in the differentialequation. In the chapter 4, the linearized oscillation for a second-order differential equation withvariable coefficients and constant delay is studied. Firstly, by using the Knaster-Tarski fixed-pointtheorem, the linearized oscillation criterion of the equation is obtained as the variable coefficientsin some range; then the condition as the variable coefficients in the different range is discussed;finally, the sufficient and necessary condition of it is derived, and so the oscillatin of the equation'ssolution is simplified.The change of population density is often complicated in the population dynamical system.Generally, which is relate to some current instant and some instant before current. Furthermore,some outside factor would worked on the density. For example, the population would be disturbedby some outside factor, and the bifurcation would be appear. In the chapter 5, the hopf-bifurcationfor a general logistic biological model with discret delays and disturbs are discussed. First of all,according to the eigenvalue theory, the condition for the existence of bifurcation period solution isobtained; after that, the form of approximate period solution is derived by the conditions of thefunction orthogonal; in the end, through examples, fitted curve figures are achieved by using Matlabwhen asign the different values to the parameters, and then the influence to swing, period, positiveequilibrium of period solution are discussed when parameters in the process of diversification.