Stability And Bifurcation Analysis Of Two Delayed Prey-predator System With B-D Functional Response

The relationship between prey and predator is an important issue in the biomathematics'study. After the establishment of Lotka-Volterra system, to study and establish a dynamical model of the population of prey-predator system have received a great deal of attention by mathematicians and biologists for widely applications. The most important element in a prey-predator model is the func-tional response which mainly has two forms: prey dependent and ratio dependent.Beddington and DeAngilis established the prey-predator model with Bed-dington-DeAngilis functional response which not only has the same quality with ratio dependent functional response, but also avoid some simple dynamical be-havior the latter presents when the population is low. It has been widely applied well in biomathematics.Considering the time delay that the predator takes to convert food into its growth and the negative feedback of prey into its own growth, this paper focuses on the discussion of two delayed prey-predator system with Beddington-DeAngi- lis functional response. We consider the stability of the equilibria and the exis-tence of Hopf bifurcation by linearizing stability method. We first linearize the nonlinear system at the equilibrium, and then by analyzing the eigenvalues, we study the stability of the equilibrium and the existence of Hopf bifurcation. Sec-ond, we derive the formulae for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady state by using the normal form method and center manifold theorem.