Qualitative Analysis Of Solutions By The Two Types Of Ecological Models And A Class Of Differential Inequalities The Asymptotic Theory

The interference of external and the time delay and irrgular migration and com-petition between populations have more influence and effective control on population density for most species in ecology. Hopf bifurcation of a class of general logistic model and the global attractivity of feedback control logistic model with discrete and continuous delay and the oscillation for a second-order differential integral inequal-ity with muti-delay are investigated by the eigenvalue theory and using constructing Lyapunov functions and comparison principl,including the existence and uniquity of the positive solution, local asymptotic stability of the positive equlibria.the global attractivity of the positive solution,oscillation of the solutions.The population of species is affected by external factors, time delay and the number of its own.Firstly,the hopf bifurcation of a class of general logistic model with time delay, disturb and yield rate is investigated.The sufficient conditions of the existence and uniquity of the pos-itive equlibria by applying functional derivative is obtained,the condition for the existence of bifuration period solution is obtained by the eigenvalue theory,the form of approximate period solution is derived by the orthogonal condition,fitted curve figures are achieved by Matlab,when asign the differents, the effect of parameteras on the period,swing,position equilibrium of the periodic solution are discussed.In order to protect diversity of species and ecological balance,Using necessary artificial methods control the number of species.secondly, the global attractivity of feedback control logistic model with discrete and continuous delay is discussed.The boundedness of the solutions of this model has been proved through the Comparison principle.Sufficient condition of the global attractivity for this model are derived by using the method of constructing Lyapunov functional.The number of some species is affected by species itself and the continuous impact of human. The number of species maybe increase rapidly or reduce rapidly at some times, or become extinct at a moment. Changes in species density is re-lated to current time and any time before.In other words, this is continuous delay of functional differential equations. Researching continuous delay of functional differ-ential equations has important theoretical and practical significance on Differential Equations and the oscillation of inequality. Finally,the oscillation for a second-order differential integral inequality with muti-delay is discussed. Using Lebesgue's deminated convergence theorem,sufficient condition to the existence of the positive solution for inequality is gained, by the method of disproof, the sufficient condition to the inexistence of the positive solution for inequality is obtained,making use of discussion, the sufficient and necessary condition to oscillation of it's solution is derived.