Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

and Applied Analysis 3

,   (,  = 1, 2, 3) are positive constants. 11 ,  21 ,  22 ,  32 , and  33 are nonnegative constants. 11 ,  22 ,  33 denote the delay in the negative feedback of the prey, predator, and top predator crowding, respectively. 21 ,  32 , are constant delays due to gestation; that is, mature adult predators can only contribute to the production of predator biomass. = max{ 11 ,  21 ,  22 ,  32 ,  33 }.  () ( = 1, 2, 3) are continuous bounded functions in the interval [−, 0].The authors proved that the system is uniformly persistent under some appropriate conditions.By means of constructing suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium of the system.Time delays of one type or another have been incorporated into systems by many researchers since a time delay could cause a stable equilibrium to become unstable and fluctuation.In [2][3][4][5][6][7][8][9][10][11][12], authors showed effects of two delays on dynamical behaviors of system.
It is well known that periodic solutions can arise through the Hopf bifurcation in delay differential equations.However, these periodic solutions bifurcating from Hopf bifurcations are generally local.Under some circumstances, periodic solutions exist when the parameter is far away from the critical value.Therefore, global existence of Hopf bifurcation with initial conditions   () =   () ,  ∈ [−, 0] ,   (0) > 0, Our goal is to investigate the possible stability switches of the positive equilibrium and stability of periodic orbits arising due to a Hopf bifurcation when one of the delays is treated as a bifurcation parameter.Special attention is paid to the global continuation of local Hopf bifurcation when the delays  1 ̸ =  2 .This paper is organized as follows.In Section 2, by analyzing the characteristic equation of the linearized system of system (3) at positive equilibrium, the sufficient conditions ensuring the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained [22].Some explicit formulas determining the direction and stability of periodic solutions bifurcating from Hopf bifurcations are demonstrated by applying the normal form method and center manifold theory due to Hassard et al. [23] in Section 3. In Section 4, we consider the global existence of these bifurcating periodic solutions [24] with two different delays.Some numerical simulation results are included in Section 5.

Stability of the Positive Equilibrium and Local Hopf Bifurcations
In this section, we first study the existence and local stability of the positive equilibrium and then investigate the effect of delay and the conditions for existence of Hopf bifurcations.There are at most four nonnegative equilibria for system (3):  1 = (0, 0, 0) ,  2 = (  1  11 , 0, 0) , where ( x1 , x2 , 0) and ( * 1 ,  * 2 ,  * 3 ) satisfy where  3 is a nonnegative equilibrium point if there is a positive solution of ( 6), and  * is a nonnegative equilibrium point if there is a positive solution of (7).Let From [1,25], we know that if ( 1 ), ( 2 ), ( 3 ), and ( 4 ) hold,  3 and  * always exist as nonnegative equilibria.
The characteristic equation for system (8) is where We consider the following cases.
(1)  =  1 .The characteristic equation reduces to There are always a positive root  1 and two negative roots  2 ,  3 of (12); hence  1 is a saddle point.
(2)  =  2 .Equation ( 10) takes the form There is a positive root  = ( (3)  =  3 .The characteristic equation is We will analyse the distribution of the characteristic root of ( 14) from Ruan and Wei [26], which is stated as follows.

Case a.
Consider The associated characteristic equation of system (3) is By Routh-Hurwitz criterion, we have the following.
The associated characteristic equation of system (3) is We want to determine if the real part of some root increases to reach zero and eventually becomes positive as  varies.Let  =  ( > 0) be a root of (21); then we have Separating the real and imaginary parts, we have It follows that where Let we have If  10 = ( 0 +  0 ) 2 −  2 0 < 0, then ℎ 1 (0) < 0, lim  → +∞ ℎ 1 () = +∞.We can know that (25) has at least one positive root.
The associated characteristic equation of system (3) is Similar to the analysis of Case , we get the following theorem.

Direction and Stability of the Hopf Bifurcation
In Section 2, we obtain the conditions under which system (3) undergoes the Hopf bifurcation at the positive equilibrium  * .In this section, we consider with  2 =  * 2 ∈ [0,  2 10 ) and regard  1 as a parameter.We will derive the explicit formulas determining the direction, stability, and period of these periodic solutions bifurcating from equilibrium  * at the critical values  1 by using the normal form and the center manifold theory developed by Hassard et al. [23].Without loss of generality, denote any one of these critical values  1 =  () 1 ( = 1, 2, . . ., 6;  = 0, 1, 2, . ..) by τ1 , at which (43) has a pair of purely imaginary roots ± and system (3) undergoes Hopf bifurcation from  * .
Throughout this section, we always assume that  * 2 <  10 .
Then  = 0 is the Hopf bifurcation value of system (3).System (3) may be written as a functional differential equation in where  = ( 1 ,  2 ,  3 )  ∈ R 3 , and where , Obviously,   () is a continuous linear function mapping . By the Riesz representation theorem, there exists a 3 × 3 matrix function (, ) (−1 ⩽  ⩽ 0), whose elements are of bounded variation such that In fact, we can choose where  is Dirac-delta function.
Regarding  1 as a parameter and let  2 = 2.9 ∈ [0, 3.2348), we can observe that with  1 increasing, the positive equilibrium  * loses its stability and Hopf bifurcation occurs (see Figures 3 and 4).

Global Continuation of Local Hopf Bifurcations
In this section, we study the global continuation of periodic solutions bifurcating from the positive equilibrium ( * , 1 ), ( = 1, 2, . . ., 6;  = 0, 1, . ..).Throughout this section, we follow closely the notations in [24] and assume that  2 =  * 2 ∈ [0,  2 10 ) regarding  1 as a parameter.For simplification of notations, setting   () = ( 1 ,  2 ,  3 )  , we may rewrite system (3) as the following functional differential equation: where and  3 () denote the densities of the prey, the predator, and the top predator, respectively; the positive solution of system (3) is of interest and its periodic solutions only arise in the first quadrant.Thus, we consider system (3) only in the domain + under the assumption ( 1 )-( 4 ).Following the work of [24], we need to define is a -periodic solution of system (71)} , 1 is defined by (43).We know that ℓ ( * , () 1 ,2/ For the benefit of readers, we first state the global Hopf bifurcation theory due to Wu [24] for functional differential equations.
From the first equation of system (3), we can get Applying the third equation of system (3), we know It follows that This shows that the nontrivial periodic solution of system ( 3) is uniformly bounded and the proof is complete.2 )] > 0 hold, then system (3) has no nontrivial  1 -periodic solution.
Proof.Suppose for a contradiction that system (3) has nontrivial periodic solution with period  1 .Then the following system (83) of ordinary differential equations has nontrivial periodic solution: which has the same equilibria to system (3); that is, Note that   -axis ( = 1, 2, 3) are the invariable manifold of system (83) and the orbits of system (83) do not intersect each other.Thus, there are no solutions crossing the coordinate axes.On the other hand, note the fact that if system (83) has a periodic solution, then there must be the equilibrium in its interior, and that  1 ,  2 ,  3 are located on the coordinate axis.Thus, we conclude that the periodic orbit of system (83) must lie in the first quadrant.If ( 7 ) holds, it is well known that the positive equilibrium  * is globally asymptotically stable in the first quadrant (see [1]).Thus, there is no periodic orbit in the first quadrant too.The above discussion means that (83) does not have any nontrivial periodic solution.It is a contradiction.Therefore, the lemma is confirmed.In following we prove that the hypotheses (A1)-(A4) in [24] hold.
(2) It follows from system (3) that Then under the assumption ( From (86), we know that the hypothesis (A2) in [24] is satisfied.

Conclusion
In this paper, we take our attention to the stability and Hopf bifurcation analysis of a predator-prey system with Michaelis-Menten type functional response and two unequal delays.We obtained some conditions for local stability and Hopf bifurcation occurring.When  1 ̸ =  2 , we derived the explicit formulas to determine the properties of periodic solutions by the normal form method and center manifold theorem.Specially, the global existence results of periodic solutions bifurcating from Hopf bifurcations are also established by using a global Hopf bifurcation result due to Wu [24].