Permanence On Predator-prey System And Backward Bifurcation For SIS Model

In this dissertation we study two systems, one is a predaor-prey system and the other is a dynamical system about infectious disease.In the first model we mainly discuss the effect of the dispersal on the positive equilibrium and permanence. We obtain that the dispersal system has at most one positive equilibrium, then discuss the permanence and obtain some new conditions which are better than [11].In the second part we propose an SIS model with saturation recovery from infective individuals to understand the effect of limited resources for treatment of infectives on the emergency disease control. It is shown that saturation recovery from infective individuals lead to vital dynamics, such as bistability and backward bifurcation, when the basic reproduction number R₀ is less than unity, which raises many new challenges to effective infection control. We also shown the model exibits Bogdanov-Takens i.e. there are saddle-node bifurcation, subcritical Hopf bifurcation and homoclinic bifurcation.