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Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay
The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.
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  is where , , and are positive constants and and 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  pointed out that for the system (1) the degenerate Bogdanov-Takens bifurcation exists when , , and and conjectured that it is a nondegenerate codim 3 bifurcation. Kuznetsov  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 , Gopalsamy , and Kuang  for general delayed biological systems and Beretta and Kuang [6, 7], Faria , Gopalsamy [9, 10], May , Song et al. [12–14], Xiao and Ruan , and Liu and Yuan  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  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 periodic solutions by the theory of normal form and center manifold. Explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are derived.
2. The Existence of Hopf Bifurcations
In this section, we study the existence of the Hopf bifurcations of system (2). Clearly, when , system (2) has only one positive equilibrium, that is, Let then system (2) becomes By introducing the new variables , and denoting , 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 , the characteristic equation (10) becomes By directly computing, we known that when . Therefore all roots of (11) have negative real parts. Obviously, is a root of (10) if and only if satisfies Separating the real and imaginary parts, we have which leads to When , (14) has only one positive root Substituting (15) into (13), we obtain Thus, when , the characteristic equation (10) has a pair of purely imaginary roots .
Lemma 1. Let be the root of (10) satisfying and then
Proof. Differentiating both sides of (10) with respect to , we obtain Therefore, Thus, the lemma follows.
Furthermore, from Lemma 2, the following theorem holds.
Theorem 3. If , then the positive equilibrium is asymptotically stable and unstable if . If , (2) undergoes a Hopf bifurcation at .
3. Stability and Direction of the Hopf Bifurcation
Let and , where . Then (2) can be written as a functional differential equation in as where , and , are given, respectively, by where denotes the higher order terms.
From the discussions above, we known that if , then system (21) undergoes a Hopf bifurcation at the zero equilibrium and the associated characteristic equation of system (21) has a pair of simple imaginary roots .
By the Reiz representation theorem, there exists a function of bounded variation for , such that In fact, we can choose where For , define Then we can rewrite (21) as where . For , define and a bilinear inner product where . Then and are adjoint operators. By the discussion of Section 2, we known that are eigenvalues of . Thus, they are also eigenvalues of .
Suppose that is the eigenvector of corresponding to . Then, . From the definition of and (25), we obtain which yields Similarly, it can be verified that is the eigenvector of corresponding to , where Let ; that is, Thus, we can choose such that , .
Using the same notations as in Hassard et al.  and Song et al. , we first compute the center manifold at . Let be the solution of (21) when . Define On the center manifold , we have where where and are local coordinates for center manifold in the direction of and . Note that is real if is real. Here we consider only real solutions. For the solution of (24), since , we have We rewrite this equation as with By (36), we have and , and then It follows, together with (23), that Comparing the coefficients with (41), we have In order to determine , we need to compute and . From (28) and (36), we have where Expanding the above series and comparing the corresponding coefficients, we obtain Following (45), we know that for , Comparing the coefficients with (46), we get Substituting these relations into (47), we obtain Solving , we obtain where is a constant vector.
Similarly, we can obtain where is also a constant vector.
In what follows, we determine the constant vectors and . From (47) and the definition of , we obtain where . From (45) and (46), we have Substituting (51) and (55) into (53) and noticing that we get that is, It follows that where .
We follow the idea in Hassard et al.  and Song et al. , which implies that the direction of the Hopf bifurcation is determined by the sign of , and the stability of the bifurcating periodic solutions is determined by the sign of and determines the period of the bifurcating periodic solution. Thus we have the following.
Theorem 4. (1) If , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for .
(2) If , then the bifurcating periodic solutions are stable (unstable).
(3) If , then the periodic of the bifurcating periodic solutions increase (decrease).
In the paper, we focused on the effect of the maturation time of the predator in Bazykin . We first discussed the effect of the time on the stability of the positive equilibrium of the system (2), and then we investigated the existence of the Hopf bifurcation, the bifurcating direction, and the stability of the bifurcating periodic solutions by the normal form and center manifold. In fact, we can also incorporate other time delays such as capturing time into the mathematical model and look at their dynamics by other methods. In this regard, we can obtain other complicated and interesting results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Natural Science Foundation of China (no. 11101371) and the Scientific Research Foundation of Zhejiang Sci-Tech University (13062176-Y). The authors would like to thank the anonymous reviewers for their suggestions and comments.
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