Stability And Hopf Bifurcation Of Two Classes Of Biomathematics Models

As a branch of ecology, population ecology has been developed very extensively and systemat-ically by utilizing mathematical theory. In recent years, the dynamics of prey-predator system have received a great deal of attention by mathematicians and biologists for its widely applications. The research on stability and other properties of population has important theoretical and practical significance by utilizing mathematical models. It can be applied to describe, predict and control the species as well as regulation and development trend of the development process, building a harmonious environment for development.This dissertation mainly studies two classes of population ecology models by means of the qualitative and stability theories and bifurcation method of ordinary differential equation:A stage-structured predator-prey system with time delay and Holling typeⅢfunctional response and a Leslie-Gower predator-prey system with multiple delays. This two systems are development and generalization of the according systems in recent literature. A series of results are obtained, which of them improve or extend the results in the literature. The full text is divided into four parts.The first part is the introduction. The background and the current research status of predator-prey system and the motivations of this work are briefly addressed.In part 2, some elementary tools are listed. Some fundamental theories and lemmas about the stability and Hopf bifurcation that can be used in this paper are given.In part 3, we study a class of stage-structured predator-prey system with time delay and Holling typeⅢfunctional response which is more practical based on the known model. Firstly, the conditions for the stability of the positive equilibrium point is obtained; furthermore, the delay bound and the conditions of the existence of Hopf bifurcation are obtained. A formula for determining the direction of Hopf bifurcation and stability of the bifurcating periodic solutions is given by using Hassard's method. The results can be considered as the promotion of the results obtained by Xu Rui et al.In part 4, the sufficient conditions of to ensure the stability of a Leslie-Gower predator-prey system with multiple delays have been obtained. Furthermore, the critical value of the Hopf bifurcation occurring in near positive equilibrium point is obtained when the sum of delaysτ1+τ2= r is bifurcation parameter and a formula for determining the direction of Hopf bifurcation and stability of the bifurcating periodic solutions is given by using the normal form method and center manifold theory. Due to the global Hopf bifurcation theorem by Wu, the conditions to guarantee the global existence of periodic solutions are given. Numerical simulations are presented to support the theoretical results. We generalize the results about existence of Hopf bifurcation of a Leslie-Gower predator-prey system with only one delay obtained by Liu Qiming and Zhang Jianming. Finally, some problems are concluded and improved.