Stability Analysis Of The Ecological Model With Time Delay

Population ecology is an important branch of mathematical ecology which can explain, predict and control population developmental tendency through the analysis and study of population model. Ecological system is influenced by present factor as well as by past factor, that is to say the effect of delay, so the dynamical model with time delay can be more close to ecological background. Stability is the most important basic feature of the ecological system. Through the study of this stability, it can guide people to make full use of and remold the nature. The research has great theoretical and practical significance for the sustainable development of the ecological environment.Based on the results, time delay and ratio function response is considered, a predator-prey model with discrete and distributed delay is further studied. Sufficient conditions for the boundedness and persistence of the solution are established by using the comparison principle theorem, the existence of the positive equilibrium is obtained by algebraic equations, and global stability of the positive equilibrium is discussed by constructing a suitable Liapunov functional. Second, fully considering the factors of delay, the general functional response is introduced to the model. A predator-prey model of the two groups with functional response and time delay is considered. The persistence and bound of the positive equilibrium are obtained by using differential inequality. A criterion of global stability in the positive equilibrium is obtained by using a Liapunov functional method. And then, the conclusion of the two population model is extended to the n-species model with time delay. Sufficient condition for the bound is established by using the differential inequality theorem, and global stability of the positive equilibrium is investigated by making use of quadratic theory. In the research of global stability about two species with time delay, the method and structure of the Liapunov functional are attempted from different perspective. Finally, we find that the small change of time delay affect the stability of population model severely from the numerical simulation result of population model which is obtained by using the Matlab software. Phase spaces which are at the different delay and the curve of population which changes by time are given, moreover, the development of population is intuitively observed, which are very important to reveal the change of population.