The Stability And Hopf Bifurcation Of Two Classes Differential-algebraic Ecological System

As an important branch of ecology,population ecology has beeen developed veryextensively and systematically by utilizing mathematical theory.Because of its widelyapplications，the job of the research about the prey-predator system is more thorough, moreclose to the social reality.In this paper,we put the economic factors into account,and establishseveral classes of differential-algebraic economic system with Beddington-DeAngelis typepredator-prey model.And the stability and the Hopf bifurcation problems of the systems arediscussed.As an introduction,in Section One,the background and the current research status ofbiomathematics models are given.At the same time,the real meaning of this work is brieflyaddressed.In Section Two, in order to make the model consistent with the actual ecological andcultural background as far as possible, the establishment of a class of the differentialalgebraic system with single harvesting is done. According to the new normal form and theHopf bifurcation theorem, we study the stability and Hopf bifurcation problems of the system.Here, economic factor m as a bifurcation parameter, it is found that the periodicsolutionwill occur when the parameter m closes to a certain valuem0.Finally,thenumerical simulation results illustrate the theoretical analysis results.From the model in Section Two,we research the differential-Algebraic system withmultiple harvestings in Section Three. And, the stability and the existance conditions ofbranch are obtained.Then,we discuss the problemof the direction of Hopf bifurcation and thestability of periodic solutions.The results of numerical simulation are consistant with theresults we obtained.Finally, summarized and prospects are given.