The Local Bifurcation and the Hopf Bifurcation for Eco-Epidemiological System with One Infectious Disease

In this paper, we established the conditions of the occurrence of local bifurcation (such as saddle-node, transcritical and pitchfork) with particular emphasis on the Hopf bifurcation near of the positive equilibrium point of eco-epidemiological mathematical model consisting of prey-predator model involving SIS infectious disease in prey population are established. After the study and analysis, of the observed incidence transcritical bifurcation near equilibrium point E_0 as well as the occurrence of saddle-node bifurcation at equilibrium points E_1, E_2. It is worth mentioning, there are no possibility occurrence of the pitchfork bifurcation at each point E_(i)i= 0,1,2. Finally, some numerical simulations are used to illustration the occurrence of local bifurcation of this model.