Abstract and Applied Analysis

Volume 2014 (2014), Article ID 835310, 16 pages

http://dx.doi.org/10.1155/2014/835310

Research Article

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

Yunxian Dai,¹ Yiping Lin,¹ and Huitao Zhao1,2

¹Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

²Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou, Henan 466001, China

Received 27 November 2013; Accepted 31 March 2014; Published 11 May 2014

Academic Editor: Yuming Chen

Copyright © 2014 Yunxian Dai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delays .

1..Introduction

In [1], Xu and Chaplain studied the following delayed predator-prey model with Michaelis-Menten type functional response:

with initial conditions

where , and denote the densities of the prey, predator, and top predator population, respectively. are positive constants. , and are nonnegative constants. denote the delay in the negative feedback of the prey, predator, and top predator crowding, respectively. , are constant delays due to gestation; that is, mature adult predators can only contribute to the production of predator biomass. . are continuous bounded functions in the interval . 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 [2–12], 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 is a more interesting and difficult topic. A great deal of research has been devoted to the topics [13–21]. In this paper, let in (1); we consider Hopf bifurcation and global periodic solutions of the following system with two unequal and nonzero delays:

with initial conditions

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 .

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.

2. 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):

where and satisfy

where 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

(H ₁); (H ₂); (H ₃); (H ₄).

From [1, 25], we know that if , and hold, and always exist as nonnegative equilibria.

Let be the arbitrary equilibrium point, and let ; still denote by , respectively; then the linearized system of the corresponding equations at is as follows:

where

all the others of , and are .

The characteristic equation for system (8) is

where

We consider the following cases.

(1) . The characteristic equation reduces to

There are always a positive root and two negative roots of (12); hence is a saddle point.

(2) . Equation (10) takes the form

There is a positive root if ; hence, is a saddle point. If , is locally asymptotically stable.

(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.

Lemma 1. Consider the exponential polynomial

where and are constants. As vary, the sum of the order of the zeros of on the open right half plane can change only if a zero appears on or crosses the imaginary axis.

By using Lemma 1, we can easily obtain the following results.

Lemma 2. If is a nonnegative equilibrium point, then

(1) is unstable if ;(2) is locally asymptotically stable if , and .

Proof. (1) is a root of (14); if , then is unstable.

(2) Clearly, is not a root of (14); we should discuss the following equation instead of (14):

Assume that with is a solution of (16). Substituting into (16) and separating the real and imaginary parts yield

which implies

If , that is , there is no real root of (16). Hence there is no purely imaginary root of (18). When , (16) reduces to

If and , both roots of (19) have negative real parts. Thus, by using Lemma 1, when , and , is locally asymptotically stable.

(4) . The characteristic equation about is (10). In the following, we will analyse the distribution of roots of (10). We consider four cases.

Case a. Consider

.

The associated characteristic equation of system (3) is

Let

(H ₅).

By Routh-Hurwitz criterion, we have the following.

Theorem 3. For , assume that hold. Then when , the positive equilibrium of system (3) is locally asymptotically stable.

Case b. Consider

, .

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 be a root of (21); then we have

Separating the real and imaginary parts, we have

It follows that

where .

Denoting , (24) becomes

Let

we have

If , then . We can know that (25) has at least one positive root.

If , we obtain that when , (25) has no positive roots for . On the other hand, when , the following equation

has two real roots: . Because of and are the local minimum and the local maximum of , respectively. By the above analysis, we immediately obtain the following.

Lemma 4. (1) If and , (25) has no positive root for .

(2) If and , (25) has at least one positive root if and only if and .

(3) If  , (25) has at least one positive root.

Without loss of generality, we assume that (25) has three positive roots, defined by , respectively. Then (24) has three positive roots:

From (23) we have

Thus, if we denote

where ; then is a pair of purely imaginary roots of (21) corresponding to . Define

Let be the root of (21) near satisfying

Substituting into (21) and taking the derivative with respect to , we have

Therefore,

When ,       +  .

According to (35), we have

where . Notice that ,

then we have the following lemma.

Lemma 5. Suppose that and , where is defined by (26); then has the same sign with .

From Lemmas 1, 4, and 5 and Theorem 3, we can easily obtain the following theorem.

Theorem 6. For , suppose that hold.

(i)If and , then all roots of (10) have negative real parts for all , and the positive equilibrium is locally asymptotically stable for all .(ii)If either or , and , then has at least one positive roots, and all roots of (23) have negative real parts for , and the positive equilibrium is locally asymptotically stable for .(iii)If (ii) holds and , then system (3)undergoes Hopf bifurcations at the positive equilibrium for .

Case c. Consider

, .

The associated characteristic equation of system (3) is

Similar to the analysis of Case , we get the following theorem.

Theorem 7. For , suppose that hold.

(i)If and , then all roots of (38) have negative real parts for all , and the positive equilibrium is locally asymptotically stable for all .(ii)If either or , , and , then has at least one positive root , and all roots of (38) have negative real parts for , and the positive equilibrium is locally asymptotically stable for .(iii)If (ii) holds and , then system (3) undergoes Hopf bifurcations at the positive equilibrium for ,

where

where ; ; then is a pair of purely imaginary roots of (38) corresponding to . Define

Case d. Consider

, .

The associated characteristic equation of system (3) is

We consider (41) with   in its stable interval . Regard as a parameter.

Let be a root of (41); then we have

Separating the real and imaginary parts, we have

It follows that

where

Denote . If , then

We can obtain that (44) has at most six positive roots . For every fixed ,  , there exists a sequence , such that (43) holds.

Let

When , (41) has a pair of purely imaginary roots for .

In the following, we assume that

(H ₆).

Thus, by the general Hopf bifurcation theorem for FDEs in Hale [22], we have the following result on the stability and Hopf bifurcation in system (3).

Theorem 8. For , , , suppose that is satisfied. If and , then the positive equilibrium is locally asymptotically stable for . System (3) undergoes Hopf bifurcations at the positive equilibrium for .

3. 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 and regard 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 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 by , at which (43) has a pair of purely imaginary roots and system (3) undergoes Hopf bifurcation from .

Throughout this section, we always assume that . Let and . Then is the Hopf bifurcation value of system (3). System (3) may be written as a functional differential equation in

where , and

where , and

Obviously, is a continuous linear function mapping into . By the Riesz representation theorem, there exists a matrix function   , whose elements are of bounded variation such that

In fact, we can choose

where is Dirac-delta function. For , define

Then when , the system is equivalent to

where ,  . For , define

and a bilinear inner product

where . Let ; then and are adjoint operators. By the discussion in Section 2, we know that are eigenvalues of . Thus, they are also eigenvalues of . We first need to compute the eigenvector of and corresponding to and , respectively. Suppose that is the eigenvector of corresponding to . Then . From the definition of , and we can easily obtain , where

and . Similarly, let be the eigenvector of corresponding to . By the definition of , we can compute

From (57), we have

Thus, we can choose

such that .

In the remainder of this section, we follow the ideas in Hassard et al. [23] and use the same notations as there to compute the coordinates describing the center manifold at . Let be the solution of (48) when . Define

On the center manifold , we have

where and are local coordinates for center manifold in the direction of and . Note that is real if is real. We consider only real solutions. For the solution of (48), since , we have

where

By (62), we have , and then

It follows together with (50) that

Comparing the coefficients with (65), we have

where

Thus, we can determine and . Furthermore, we can determine each by the parameters and delay in (3). Thus, we can compute the following values:

which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value . Suppose . determines the directions of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcation exists for ; determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if ; and determines the period of the bifurcating periodic solutions: the period increases (decreases) if .

4. Numerical Simulation

We consider system (3) by taking the following coefficients:                                       . We have the unique positive equilibrium .

By computation, we get , , , , . From Theorem 6, we know that when , the positive equilibrium is locally asymptotically stable for . When crosses , the equilibrium loses its stability and Hopf bifurcation occurs. From the algorithm in Section 3, we have , which means that the bifurcation is supercritical and periodic solution is stable. The trajectories and the phase graphs are shown in Figures 1 and 2.

Figure 1: The trajectories and the phase graph with ; is locally asymptotically stable.

Figure 2: The trajectories and the phase graph with ; a periodic orbit bifurcate from .

Regarding   as a parameter and let , we can observe that with increasing, the positive equilibrium loses its stability and Hopf bifurcation occurs (see Figures 3 and 4).

Figure 3: The trajectories and the phase graph with ; is locally asymptotically stable.

Figure 4: The trajectories and the phase graph with ; a periodic orbit bifurcate from .

5. Global Continuation of Local Hopf Bifurcations

In this section, we study the global continuation of periodic solutions bifurcating from the positive equilibrium . Throughout this section, we follow closely the notations in [24] and assume that regarding as a parameter. For simplification of notations, setting , we may rewrite system (3) as the following functional differential equation:

where for and . Since , and 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 . It is obvious that (71) has a unique positive equilibrium in under the assumption . Following the work of [24], we need to define

Let denote the connected component passing through in , where is defined by (43). We know that through is nonempty.

For the benefit of readers, we first state the global Hopf bifurcation theory due to Wu [24] for functional differential equations.

Lemma 9. Assume that is an isolated center satisfying the hypotheses (A1)–(A4) in [24]. Denote by the connected component of in . Then either

(i) is unbounded, or(ii) is bounded, is finite and

for all , where is the crossing number of if , or it is zero if otherwise.

Clearly, if (ii) in Lemma 9 is not true, then is unbounded. Thus, if the projections of onto -space and onto -space are bounded, then the projection of onto -space is unbounded. Further, if we can show that the projection of onto -space is away from zero, then the projection of onto -space must include interval . Following this ideal, we can prove our results on the global continuation of local Hopf bifurcation.

Lemma 10. If the conditions hold, then all nontrivial periodic solutions of system (71) with initial conditions

are uniformly bounded.

Proof. Suppose that are nonconstant periodic solutions of system (3) and define

It follows from system (3) that

which implies that the solutions of system (3) cannot cross the -axis . Thus, the nonconstant periodic orbits must be located in the interior of first quadrant. It follows from initial data of system (3) that for .

From the first equation of system (3), we can get

thus, we have

From the second equation of (3), we obtain

therefore, one 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.

Lemma 11. If the conditions and

(H ₇),  

hold, then system (3) has no nontrivial -periodic solution.

Proof. Suppose for a contradiction that system (3) has nontrivial periodic solution with period . 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 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 are located on the coordinate axis. Thus, we conclude that the periodic orbit of system (83) must lie in the first quadrant. If 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.

Theorem 12. Suppose the conditions of Theorem 8 and hold; let and be defined in Section 2; then when system (3) has at least periodic solutions.

Proof. It is sufficient to prove that the projection of onto -space is for each , where .

In following we prove that the hypotheses (A1)–(A4) in [24] hold.

(1)From system (3) we know easily that the following conditions hold:

(A1), where .(A3) is differential with respect to .

(2)It follows from system (3) that

 Then under the assumption , we have

 From (86), we know that the hypothesis (A2) in [24] is satisfied.(3)The characteristic matrix of (71) at a stationary solution where takes the following form:

that is,

From (88), we have

Note that (89) is the same as (20); from the discussion in Section 2 about the local Hopf bifurcation, it is easy to verify that is an isolated center, and there exist , and a smooth curve such that ,   for all and

Let

It is easy to see that on , if and only if, , , .

Therefore, the hypothesis (A4) in [24] is satisfied.

If we define

then we have the crossing number of isolated center as follows:

Thus, we have

where has all or parts of the form . It follows from Lemma 9 that the connected component through is unbounded for each center . From the discussion in Section 2, we have

where . Thus, one can get for .

Now we prove that the projection of onto -space is , where . Clearly, it follows from the proof of Lemma 11 that system (3) with has no nontrivial periodic solution. Hence, the projection of onto -space is away from zero.

For a contradiction, we suppose that the projection of onto -space is bounded; this means that the projection of onto -space is included in a interval . Noticing and applying Lemma 11 we have for belonging to . This implies that the projection of onto -space is bounded. Then, applying Lemma 10 we get that the connected component is bounded. This contradiction completes the proof.

6. 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 , 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].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to thank the anonymous referees for their careful reading and constructive suggestions which lead to truly significant improvement of the paper. This research is supported by the National Natural Science Foundation of China (no. 11061016).

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