The Asymptotic Behavior Of Two Ecological Systems And The Existence Of Positive Solutions Of A Class Second-Order Non-Autonomous Linear Delay Differential Equations

In this paper , asymptotic behavior of two ecological systems and the existence of positive solutions of a class second-order non-autonomous linear delay differential equations are studied , including the stability of equilibrium , the existence and uniqueness of the limit cycle , conditions of the existence of Hopf bifurcation etc . People may make full use of nature by mesns of study of these properties . These will have very important practical meaning to protect ecological systems and keep the population sustainable development .In the study of ecological models , the method of characteristic root has widely been used in studying the existence of solution , stability and asymptotic behavior and so on , and the existence of positive solution has a close relation with oscillation . There are many study results about the oscillation of constant coefficient linear delay differential equations , but the studying about the nonautonomous equations is much less . In the chapter 2 , the characteristic equation of a class second-order nonautonomous linear delay differential equations . sufficient and necessary condition of existence of positive solutions and the relation between the roots of generalized characteristic equation and the positive solution of delay differential equations are abtained by using the theory of functional theory and fix point theorem . The conclusion contains the general characteristic equation's . Provide a better way for studying the existence and oscillation of sencond-order nonautonomous equation and a new way for discusing the asymptotic behavior of the solution .In the natural world , change of biological group density is extremely complex , in actual ecosystems , delay usually have a certain effect on density constraints , in other words , Growth of the group in a moment not only concern with the moment group density , but with a moment before the moment, and all in the past . Because of this , In order to make the model more accord with its ecological background and reality . In the chapter 3 , a class ecological model with the distributed and distete delay are established . Sufficient conditions of absolute stability are obtained by using the theory of characteristic roots , and it is shown that the delay Ï„ is locally harmless . Furthermore , regarding the delay as a parameter , conditions of the existence of Hopf bifurcation and the stability of model .In the establishment of ecological models , the simple linear functional response function does not have a generalization , not fully describe the complex interrelationships between groups , so how to add reasonable functional response function was widespread concerned , which interferencewith the index-type functional response function considered interference factors , thus it is more meaningful . In the chapter 4 , a class of two species predator-prey model with interference with the index-type functional response function is studied . The stability of equilibrium , the sufficient conditions of nonexistence and conditions of the existence and uniqueness of limit cycle around the positive equilibrium of the system are discussed . On the basis of the study in the predecessors the parameters' range extend .