Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays

A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions areobtained fortheglobal attractiveness ofthepositiveequilibrium. Bycomparisonarguments, theglobal stabilityofthesemitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.


Introduction
The main purpose of this paper is to investigate the bifurcation phenomena from the delays for the following predatorprey system: where ( ) and ( ) stand for the population (or density) of the prey and the predator at time , respectively. From the biological sense, we assume that 2 + 2 ̸ = 0.
1 , 2 , 11 , 12 , 21 , and are positive constants, in which 1 denotes the intrinsic growth rate of the prey, 11 is the intraspecific competition rate of the prey, 12 is the capturing rate of the predator, 21 / 12 describes the efficiency of the predator in converting consumed prey into predator offspring, is the interference coefficient of the predators, and 2 is the predator mortality rate. The delay 1 ≥ 0 denotes the gestation period of the predator; 2 ≥ 0 is the hunting delay of the predator to prey.
This model is labeled "ratio-dependent, " which means that the functional and numerical responses depend on the densities of both prey and predators, especially when predator has to search for food. Such a functional response is called a ratio-dependent response function (see [1] for more details). In system (1), the ratio-dependent response function is of the form ( / ) = ( / ) 2 /( + ( / ) 2 ) = 2 /( 2 + 2 ).
The ratio-dependent predator-prey model has been studied by several researchers recently and very rich dynamics have been observed [2][3][4][5]. For example, Xu et al. [4] studied a delayed ratio-dependent predator-prey model with the same ratio-dependent response function of system (1). By means of an iteration technique, they obtained the sufficient conditions for the global attractiveness of the positive equilibrium. By comparison arguments, they proved the global stability of the semitrivial equilibrium. Finally using the theory of functional equation and Hopf bifurcation, they gave the condition on which positive equilibrium exists and the formulae to determine the quality of Hopf bifurcation. But in their work, the global continuation of local Hopf bifurcation was not mentioned.
In general, periodic solutions through the Hopf bifurcation in delay differential equations are local for the values 2 Abstract and Applied Analysis of parameters which are only in a small neighborhood of the critical values (see, e.g., [6,7]). Therefore we would like to know if these nonconstant periodic solutions obtained through local bifurcation can continue for a large range of parameter values. Recently, a great deal of research has been devoted to the topics [8][9][10][11][12]. One of the methods used in them is the global Hopf bifurcation theorem by Wu [13]. For example, Song et al. [12] studied a predator-prey system with two delays, and using the methods in [13], they get the global existence of periodic solutions.
Motivated by [12], we will study the system (1); special attention is paid to the global continuation of local Hopf bifurcation. We suppose that the initial condition for system (1) takes the form where By the fundamental theory of functional differential equations [14], system (1) has a unique solution ( ( ), ( )) satisfying initial condition (2).
The rest of the paper is organized as follows. In Section 2, we show the positivity and the boundedness of solutions of system (1) with initial condition (2). In Section 3, we study the existence of Hopf bifurcation for system (1) at the positive equilibrium. In Section 4, using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcation and the stability and other properties of bifurcating periodic solutions. In Section 5, by means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. In Section 6, we consider the global existence of bifurcating periodic solutions and give some numerical simulations. In Section 7, a brief discussion is given.

Positivity and Boundedness
In this section, we study the positivity and boundedness of solutions of system (1) with initial conditions (2). Theorem 1. Solutions of system (1) with initial condition (2) are positive for all ≥ 0.
Proof. Assume ( ( ), ( )) to be a solution of system (1) with initial condition (2). Let us consider ( ) for ≥ 0. It follows from the second equation of system (1) that then, from initial condition (2), we have ( ) > 0, for ≥ 0. We derive from the first equation of system (1) that that is, ( ) > 0 for ≥ 0. This ends the proof.
For the following discussion of boundedness, we first consider the following ordinary differential equation: where 21 , 2 , 1 , and are positive constants. From Lemma 2.1 in [5], it is easy to verify the following result.
Proof. Let ( ( ), ( )) be a positive solution of system (1) with initial condition (2). From the first equation of system (1), we havė( which yields lim sup Then, we can get at which (15) admits a pair of purely imaginary roots of the form ± 0 . Let 1 − 2 < 0 and 0 be defined above. Denote the root of (15) satisfying It is not difficult to verify that the following result holds. Proof. Differentiating (15) with respect , we obtain that it follows that from (15) and (31), we have We therefore derive that Noting that 2 0 − 2 1 > 0, hence, if (H1) and 1 − 2 < 0 hold, we have ( (Re )/ )| = > 0. Accordingly, the transversal condition holds and a Hopf bifurcation occurs at = .
By Lemma B in [5], we have the following results. Theorem 5. Suppose (H1) holds and let ℎ be defined in (12), for system (13), one has the following.

Direction and Stability of Hopf Bifurcations
In Section 3, we have shown that system (13) admits a periodic solution bifurcated from the positive equilibrium * at the critical value 0 . In this section, we derive explicit formulae to determine the direction of Hopf bifurcations and stability of periodic solutions bifurcated from the positive equilibrium * at critical value 0 by using the normal form theory and the center manifold reduction (see, e.g., [15,16]). Set = 0 + ; then = 0 is a Hopf bifurcation value of system (13). Thus we can consider the problem above in the phase space C = C([− , 0], R 2 ).
In the following, we first compute the coordinates to describe the center manifold 0 at = 0. Define On the center manifold 0 , we have where and are local coordinates for center manifold 0 in the directions of and . Note that is real if is real. We consider only real solutions. For the solution ∈ 0 , since = 0, we havė (1) 20 (1) 20 In order to assure the value of 21 , we need to compute 20 ( ) and 11 ( ). By (42) and (45), we havė where ( , , ) = 20 ( ) Notice that near the origin on the center manifold 0 , we havė thus, we have Comparing the coefficients with (51) gives that From (56), (54), and the definition of (0), we can geṫ Notice that ( ) = (0) 0 ; we have In the same way, we can also obtain In what follows, we will compute 1 and 2 . From the definition of (0) and (54), we have where ( ) = (0, ). From (51), (58), and (60) and noting that we have From (52), (59), and (61) and noting that we have Abstract and Applied Analysis 7 Thus, we can determine 20 ( ) and 11 ( ) from (58) and (59). Furthermore, we can determine each . Therefore, each is determined by the parameters and delay in (13). Thus, we can compute the following values [15]: which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value ; that is, 2 determines the directions of the Hopf bifurcation: if 2 > 0 (< 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcation exists for > 0 (< 0 ); 2 determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if 2 < 0 (> 0); and 2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if 2 > 0 (< 0).

Lemma 8.
Assume that ( * , , ) is an isolated center satisfying the hypotheses ( 1 )-( 4 ) in [13]. Denote by ℓ ( * , , ) the connected component of ( * , , ) in Γ. Then either Clearly, if (ii) in Lemma 8 is not true, then ℓ ( * , , ) is unbounded. Thus, if the projections of ℓ ( * , , ) onto -space and onto -space are bounded, then the projection onto -space is unbounded. Further, if we can show that the projection of ℓ ( * , , ) onto -space is away from zero, then the projection of ℓ ( * , , ) onto -space must include interval [ , +∞). Following this ideal, we can prove our results on the global continuation of local Hopf bifurcation.
are uniformly bounded.
Proof. Suppose that = ( ), = ( ) are nonconstant periodic solutions of system (13) and define It follows from system (13) that which implies that the solutions of system (13) cannot cross the -axis and -axis. Thus the nonconstant periodic orbits must be located in the interior of each quadrant. It follows from initial conditions of system (13) that ( ) > 0, ( ) > 0.
From system (13), we can get Since ( ) > 0, ( ) > 0, it follows from the first equation of (114) that on the other hand, by the second equation of (114) and (115), we have where ℎ is defined in (12). From the discussion above, the lemma follows immediately. (13) has no nonconstant periodic solution with period .

Lemma 10. If conditions (H1) and (H2) hold, then system
Proof. Suppose for a contradiction that system (13) has nonconstant periodic solution with period . Then the following system (117) of ordinary differential equations has nonconstant periodic solution: which has the same equilibria as system (13), that is, 1 ( 1 / 11 , 0) and a positive equilibrium * ( * , * ). Note that -axis and -axis are the invariable manifold of system (13) and the orbits of system (13) do not intersect each other. Thus, there is no solution crossing the coordinate axis. On the other hand, note the fact that if system (117) has a periodic solution, then there must be the equilibrium in its interior and 1 are located on the coordinate axis. Thus, we conclude that the periodic orbit of system (117) must lie in the first quadrant. From the proof of Theorem 6, we known that if (H1) and (H2) hold, the positive equilibrium is asymptotically stable and globally attractive; thus, there is no periodic orbit in the first quadrant. This ends the proof. Proof. It is sufficient to prove that the projection of The characteristic matrix of (108) at an equilibrium = ( (1) , (2) ) ∈ R 2 takes the following form: where 0 , 1 , and 2 are defined as in Section 3. From the discussion in Section 3, each of (119) and (120) has no purely imaginary root provided that 1 > (4 12 2 2 ℎ + 12 21 2 ℎ)/ 2 21 . Thus, we conclude that (108) has no the center of the form as ( 1 , , ) and ( * , , ). On the other hand, from the discussion in Section 3 about the local Hopf bifurcation, it is easy to verify that ( * , , 2 / 0 ) is an isolated center, and there exist > 0, > 0, and a smooth curve : ( − , + ) → C such that det(Δ( ( ))) = 0, | ( ) − 0 | < for all ∈ [ − , + ] and Let Therefore, the hypotheses ( 1 )-( 4 ) in [13] are satisfied. Moreover, if we define = det (Δ ( * , ± , ) ( + 2 )) ,
Example 12. In system (1), we first choose 11 = 0.1, 12 = 1, 21 = 3/2, and = 2. As depicted in Figure 1, a bifurcation diagram is given for system (1) with respect to the parameters 1 and 2 . By the discussion in Section 3, system (1) always has a semitrivial equilibrium 1 , and if 2 > 21 , 1 is asymptotically stable; otherwise, 1 is unstable. So if we choose 0 < 2 < 21 = 3/2, as depicted in Figure 1, 1 is always unstable. In domains II, V, and VI, the positive equilibrium is not feasible. In domains I, III, and IV, system (1) has a unique positive equilibrium; it is locally asymptotically stable in domain I and is unstable in domain IV. In domain III, system (1) undergoes a Hopf bifurcation at the positive equilibrium at some 0 . Further, we choose 1 = 5/12, 2 = 1, 11 = 0.1, 12 = 1, 21 = 3/2, and = 2. In this case, system (1) has a positive equilibrium * = (5/6, 5/12). By computation, we have 0 ≈ 0.1063, 0 ≈ 10.8795, and 1 ≈ 69.9876. From Theorem 5, * is stable when < 0 as illustrated by numerical simulations (see Figure 2). When passes through the critical value 0 , the equilibrium * loses its stability and a Hopf bifurcation occurs; that is, a family of periodic solution bifurcates from * . By the algorithm derived in Section 3 and Section 4, we have ( 0 ) = 0.0053 − 0.0058 , 1 (0) = −0.4357 + 0.0265 , which implies that 2 > 0, 2 < 0, and 2 > 0. Thus, by the discussion in Section 4, the Hopf bifurcation is supercritical for > 0 , the bifurcating periodic solutions from * at 0 are asymptotically stable, and the period of these periodic solutions is increasing with the increasing of , which are depicted in Figures 3, 4, and 5. Furthermore, Figure 5 shows that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of 1 = 69.9876.

Discussion
In this paper, we have studied a ratio-dependent predatorprey model with two time delays. By analyzing the corresponding characteristic equation, the local stability of the positive equilibrium and the semitrivial equilibrium of system (1) was discussed. We have obtained the estimated length of gestation delay which would not affect the stable coexistence of both prey and predator species at their equilibrium values. The existence of Hopf bifurcation for system (1) at the positive equilibrium was also established. From theoretical analysis it was shown that the larger values of gestation time delay cause fluctuation in individual population density and hence the system becomes unstable. As the estimated length of delay to preserve stability and the critical length of time delay for Hopf bifurcation are dependent upon the parameters of system, it is possible to impose some control, which will prevent the possible abnormal oscillation in population density. The global attractiveness result in Theorem 6 implied that system (1) is permanent if the intrinsic growth rate of the prey and the conversion rate and the interference rate of the predator are high, and the death rate of the predator is low. From Theorem 7 we see that if the death rate of the predator is greater than the conversion rate of the predator, the predator population become extinct for any gestation delay. In particular, the results about boundedness and attractiveness are similar to the results of [4]. From the discussion in Sections 3 and 4, we see that if the values of 1 , 2 , 11 , 12 , 21 , and are given, we can get the Hopf bifurcation value of , and further we may determine the direction of Hopf bifurcation and the stability of periodic solutions bifurcating from the positive equilibrium * at the critical point 0 .
Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay.