Analysis Of The Dynamic Behaviors Of A Special Kind Of Predator-prey Model

In recent years, population ecology models has been received great concern dueto there contensive applications. Using the method of establishing mathematicalmodels to study the dynamical system of the population, not only can we reveal theinherent law of the system, and predict the development of the population, but alsocan better control or adjust the ecological development of the population. On theother hand, the introduction of time delay in the system can better describe thebehavior of the system and conform to the real word. This paper studies the Hopfbifurcation of a class of predator-prey models with time delay and capture effect. Inthis thesis, we investigate this retarted differential equation, use the theory andmethods including local bifurcation theorem, the center manifold theorem, normform method, topological degree theory and global bifurcation theorem, we studythe stability of the equation, the existence of periodic solutions and global Hopfbifurcation of this sistem. Details as follow:First, chapter one introduces the background of the predator-prey model, andsimply states the research status at home and abroad and the task of this paper.Chapter two introduce the theory of the stability of the equilibrium and the Hopfbifurcation. The theory of global Hopf bifurcation is also stated here.Moreover, we studies the dynamic behavior of predator-prey model with timedelay, capture effect and the effect of outside toxins. Through the analysis of thedistribution of characteristic roots of equation for the linearzation of the system atthe equilibrium points，we discuss the stability of several equilibrium points,especially the stability of positive equilibrium points and conditions causing localHopf bifurcation. Then by using Hassard’s center manifold theorem and normal formtheory, we give the expression related to the direction and stability of Hopfbifurcation and stability of periodic solution for bifurcation. Finally, under theconditions that cause local Hopf bifurcation, we prove that the connectedcomponents through the isolated centers are unbounded. We make use of the globalHopf bifurcation theorem established by Krawcewicz et al, showing that theexistence a periodic solution in the system on a large scale.Finally is about the numerical simulation. An example is given by the toolkit ofSimulink in Matlab to verify the theoretical results.