Two Types Of Three-dimensional System Stability And Hopf Bifurcation

In this paper, by using the qualitative and stability theories and bifurcation method of ordinary differential equations, two population models in ecology are studied .The whole paper consists of three chapters. Details are as follows:1.The first chapter is the introduction. The development of ecology and the main works of the thesis are introduced, then some fundamental theories and lemmas about ecology and stability that can be used in this paper are given.2.A mathematical model of two prey and one predator system which has the switching property of predation is considered. By using the Routh-Hurwitz criteria and Lya-punov method ,the condition of the stable positive equilibrium is obtained. In the special case that two prey species have the same intrinsic growth rates, it is shown that the system asymptotically settles a Volterra's oscillation in three-dimensional space.Then the Hopf bifurcation of the system is discussed.The conditions of the Hopf bifurcation are obtained when k₂ is parameters.Further ,the necessary conditions of the Hopf bifurcation are obtained when k₂ = f(k₁)(k₁ > 0) ,and several examples are given when k₂ = lk₁, k₂ = k₁^α(α> 1), k₂ = k₁^α(0 <α< 1).3.A prey-predator system of two species with stage structure and time delay is investigated . The boundary of the solution to the system is obtained by using summary theory. By using the stability theory of the differential equation ,the invariance of non-negativity, nature of the boundary equilibrium ,the local stability and Hopf bifurcation of the positive equilibrium are analyzed.The sufficient conditions for local asymptotic stability of the positive equilibrium are given when time delayτ= 0.Furthermore, it shows that positive equilibrium is locally asymptotically stable when time delayτ=τ₁ +τ₂ is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases .That is ,a family of periodic solutions bifurcates from positive equilibrium as r passes through the critical value .