Analysis Of The Stability Of Several Types Of Eco-mathematical Model

In the recent years, the study of mathematical ecology models has widely attracted the attention of researchers and many good results have been obtained. Some results are on stability, an important feature in describing mathematical ecology models. The study on stability of populations can provide better guidance in exploiting and transforming the nature.In this paper, using the theory and methods from ordinary differential equations, we propose and carefully study four dynamical models.Firstly, we build a competitive L-V model with two delays and a stage structure. By using the Hurwitz criterion, we first obtain conditions on the locally asymptotic stability of the equilibria. Then, from the point of view of bifurcation, we find that delays do not affect the stability of the positive equilibrium. Moreover, with the help of the theory of monotonic systems and the iteration method, we establish sufficient conditions on the global stability of the positive equilibrium.Secondly, we consider a Leslie type predator-prey epidemic model. We provide condi-tions on the persistence of the system and on the locally asymptotic stability of the equilibria. When there is a delay, we find the conditions on Hopf bifurcation at the positive equilibrium.Thirdly, we propose a rat model of sterility control with density-dependent birth and death. We study the stability of the equilibria and analyze the effect of the density-dependence on the stability of equlibria and the population size.Finally, we establish a rat model with infertility control and a stage structure. As above, we consider the stability of equilibria and study the impact of the infertility ratio on the population size.