Study On The Permanence And Extinction Of Some Mathematical Ecological Models

In this paper, we consider three kinds of mathematical ecological models:Firstly, we consider a stage-structured three-species predator-prey system with two type functional response. By using the comparison theorem, some sufficient and necessary conditions are derived for the permanence of the system. Lastly, examples together with numerical simulation are presented to illustrate the application of our main results.Secondly, a linear harvested predator-prey system with monotonic functional responses is considered. Using general qualitative analysis methods, we study the boundedness of solutions, the properties of equilibrium and the existence of limit cycles. From the analysis, we get that if the positive equilibrium point of the system is stable, then it is globally stable under some conditions, which means that the predator and the prey are permanence. If the positive equilibrium point is not stable, then the system has one limit cycle at least.Thirdly, we consider a discrete n-species non-autonomous Lotka-Volterra com-petitive system with delays and feedback controls. Using analytical and inequalities method, properly zooming the right function of the model, we obtain the sufficient conditions for the permanence of the system. Lastly, examples together with nu-merical simulation are presented to illustrate that suitable controls can be chosen to make the species coexistence in the long run.