The Stability Of The Predator-prey Model With Time Delay And Integer Variables And Branch

In this paper, two types of predator-prey model for the existence, uniqueness, boundedness, stability, and various branches of analysis are studied.In biological communities, the population is usually related to the factors such as fertility, mortality and so on. These factors in the influence of the population are sustainable. In the mathematical model of this time sustained phenomenon is known as delay.In Chapter Ⅱ, a discrete time-delay and functional response function of the predator-prey model. is investigated. Sufficient conditions for the linear asymptotic stability of the positive equilibrium of the system are obtained by using the theory of characteristic value. Conditions of the existence of Hopf bifurcation and the stability of the bifurcation periodic solution are discussed by regarding the delay rasa parameter. The previous theorems and conclusions are verified by some examples. Fitted curve figures about various parameters are achieved by Matlab. The different figures are compared and analyzed.By the environment, climate, human activities such as role, partly biological populations due to seasonal breeding, migration, human beings on a regular basis to capture other factors, the number in a certain area of the periodic changes will occur, therefore, the model with piecewise constant research it is very necessary, so the third chapter, the stability and bifurcation problems were discussed in the article on a class of piecewise constant variable predator-prey model Sufficient conditions of the local asymptotically stability and non-stability of this model are derived by using Jury criterion. Furthermore, conditions of the existence of the Hopf bifurcation, the Flip bifurcation and the Neimark-Sacker bifurcation are discussed by applying the center manifold theorem. Finally, some numerical examples supporting our theoretical, and the complexity dynamic behaviors under certain conditions are also given.