Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

and Applied Analysis 3 Proof. Differentiating both sides of (10) with respect to τ, we obtain (2λ + α 1 − r 2 − α 2 r 1 τe −λτ ) dλ dτ = α 2 r 1 λe −λτ . (19) Therefore, sign{d (Re (λ))


Introduction
The theoretical study of predator-prey systems in mathematical ecology has a long history beginning with the famous Lotka-Volterra equations because of their universal existence and importance.One of the ecological models proposed and analyzed by Bazykin [1] is where , , and  are positive constants and  1 and  2 are functions of time representing population densities of prey and predator, respectively.This system can be used to describe the dynamics of the prey-predator system when the nonlinearity of predator reproduction and prey competitive are both taken into account.Bazykin [1] pointed out that for the system (1) the degenerate Bogdanov-Takens bifurcation exists when  = 4/3,  = 1/3, and  = 1/4 and conjectured that it is a nondegenerate codim 3 bifurcation.Kuznetsov [2] proved the conjecture is correct by using critical (generalized) eigenvectors of the linearized matrix and its transpose.However, time delays commonly exist in biological system, information transfer system, and so on.Therefore, time delays of one type or another have been incorporated into mathematical models of population dynamics due to maturation time, capturing time, or other reasons.In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay may lead to changes of stability of equilibrium and the fluctuation of the populations.So far, a great deal of research has been devoted to the delayed predator-prey system.See, for example, the monographs of Cushing [3], Gopalsamy [4], and Kuang [5] for general delayed biological systems and Beretta and Kuang [6,7], Faria [8], Gopalsamy [9,10], May [11], Song et al. [12][13][14], Xiao and Ruan [15], and Liu and Yuan [16] and the references cited therein for studies on delayed prey-predator systems.In the above references, normal form and center manifold theory were one of important methods to study the stability and Hopf bifurcation of the delayed predator-prey systems.Considering the maturation time of the predator, Bazykin [1] becomes the following delayed model: In this paper, we first discuss the effect of the time  on the stability of the positive equilibrium of the system (2).Then we investigate the existence of the Hopf bifurcation, the bifurcating direction, and the stability of the bifurcation

The Existence of Hopf Bifurcations
In this section, we study the existence of the Hopf bifurcations of system (2).Clearly, when −1 <  < 0, system (2) has only one positive equilibrium, that is, ) . ( Let then system (2) becomes By introducing the new variables 2 and denoting ( 1 ,  2 ) = − 2 +( 2 /(+ 2 )) 1  2 , system (5) can be rewritten in a simpler form as where and Then the linearization of system (2) at  is The associated characteristic equation of ( 8) is given by That is, The equilibrium  is stable if all roots of (10) have negative real parts.Clearly, when  = 0, the characteristic equation ( 10) becomes By directly computing, we known that  2 < 0 when −1 <  < 0. Therefore all roots of (11) have negative real parts.
Obviously,  =  ( > 0) is a root of ( 10) if and only if  satisfies Separating the real and imaginary parts, we have which leads to has only one positive root Substituting ( 15) into (13), we obtain Thus, when  =   , the characteristic equation ( 10) has a pair of purely imaginary roots ± * .
Furthermore, from Lemma 2, the following theorem holds.
By the Reiz representation theorem, there exists a function (, ) of bounded variation for  ∈ [−1, 0], such that In fact, we can choose where For  ∈ where where () = (, 0).Then  * and (0) are adjoint operators.By the discussion of Section 2, we known that ± 0  0 are eigenvalues of (0).Thus, they are also eigenvalues of  * .
Using the same notations as in Hassard et al. [18] and Song et al. [19], we first compute the center manifold  0 at  = 0. Let   be the solution of (21) when  = 0. Define On the center manifold  0 , we have where where  and  are local coordinates for center manifold  0 in the direction of  * and  * .Note that  is real if   is real.