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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 829045, 7 pages
http://dx.doi.org/10.1155/2013/829045
Research Article

Hopf Bifurcation Analysis for the Model of the Chemostat with One Species of Organism

School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 19 November 2012; Revised 3 March 2013; Accepted 3 March 2013

Academic Editor: Abdelaziz Rhandi

Copyright © 2013 Haiyun Bai and Yanhui Zhai. 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 research the dynamics of the chemostat model with time delay. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. By using the normal form theory and center manifold method, we derive the explicit formulas determining the stability and direction of bifurcating periodic solutions. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

1. Introduction

Since late 60s, many researchers have been devoted to studying the chemostat, which is considered as an important laboratory set used for breeding microorganism and studying biological systems. In [1] Li et al. propose some new ideals by modifying the basic chemostat model with one species of organism and studying the Hopf bifurcations and stability of the modified one. In [2] Li et al. study the chemostat model with two time delays. They only research the stability of the equilibrium and the existence of the local Hopf bifurcation. However, some subtle mathematical questions on the behavior of solutions of the model are far from completely answered, for example, the bifurcating direction and stability of periodic solutions. Based on this, the main purpose of this study is to provide an insight into these unexplored aspects of the model by using the theory of the center manifold and the normal forms method.

Now we consider the basic model of the chemostat with one species of organism (see [3]): where is the concentration of the organism at time , is the concentration of the nutrient at time , ( , are positive constants) is the growing rate of , is the ratio of the mass of organism formed and the mass of substrate used, is the concentration of the input nutrient, is time lag of digestion, and is flowing rate.

2. Stability and Local Hopf Bifurcation

To consider the meaning of the biology, in the section we only focus on investigating the local stability of the interior equilibrium for the system (1). We know that if the equilibrium of system (1) is stable when and the characteristic equation of (1) has no purely imaginary roots for any , it is also stable for any . On the other hand, if the equilibrium of system (1) is stable when and there exist some positive values such that the characteristic equation of (1) has a pair of purely imaginary roots, there exists a domain concerning such that the equilibrium of system (1) is table in the domain.

When for and , the system (1) has a unique interior equilibrium. We denote this unique interior equilibrium by .

Then it satisfies

That is

Let and and still denote , , respectively.

Then system (1) becomes

The linearization of (4) around is where and , whose characteristic equation is

By setting , and , the characteristic equation (6) can be rewritten as

Lemma 1. When and are met, the equilibrium of system (1) is asymptotically stable.

Proof. When , (7) becomes whose characteristic value is
Obviously, when holds, the real parts of are negative.
This completes the proof.

In the following, we investigate the distribution of the eigenvalues of the characteristic equation (7).

Lemma 2. Assume that is satisfied. Then (7) has a pair of purely imaginary roots when , where

Proof. Let ( ) be a root of (7). Then
The separation of the real and imaginary parts yields
Hence
Obviously, implies that and, hence,
Define , . Then solves (12).
This means that is a root of (7) when , .
This completes the proof.

Lemma 2 shows that there exist some positive values such that the characteristic equation (7) has a pair of purely imaginary roots.

Lemma 3. Let be the root of (7) with and . When holds, .

Proof. By differentiating both sides of (7) with respect to , we obtain
Then
Substituting and into (17), we obtain
According to (12),
Hence
The conclusion is completed.

Lemma 3 explains that the real parts are monotonously increased in a small neighbourhood concerning . In other words, the root of (7) crosses the imaginary axis from the left to the right as continuously varies from a number less than to one greater than .

Lemma 4. When and hold, then there exist such that all the roots of (7) have negative real parts when and (7) has at least one root with positive real parts when , , where is defined as in (10).

In fact, according to Lemmas 1, 2, and 3, it is easy to obtain the results.

Applying Lemmas 2, 3, and 4 and the Hopf bifurcation theorem (see [4]), we have the following results.

Theorem 5. If and are satisfied, then the equilibrium is asymptotically stable for and unstable for . System (4) undergoes a Hopf bifurcation at when , , where is defined as in (10).

3. Direction and Stability of the Bifurcating Periodic Solutions

Throughout the following section, is a phase space, and stands for an operator, which is different from in Section 2.

In Section 3, we will research the stability and direction of the bifurcating periodic solutions of system (1). For convenience, let and still denote . Then the system (5) can be rewritten as

By using the Taylor series and letting , we have

Clearly, is the Hopf bifurcation value of system (21).

Let . For , let where , .

By Riesz’s representation theorem, there exists a matrix whose components are bounded variation functions in , such that for .

For , we define the operators and as

Then the system (21) is equivalent to the following abstract differential equation [5]: where , for .

As in [6], the bifurcating periodic solutions of system (21) are indexed by a small parameter . A solution has amplitude , period , and nonzero Floquet exponent with . Under the present assumptions, , , and have expansions

The sign of determines the direction of bifurcation: if , then the Hopf bifurcation is forward (backward). determines the stability of the bifurcating periodic solutions: asymptotically orbitally stable (unstable) if . And determines the period of the bifurcating periodic solutions: the period increases (decreases) if .

Next, we only compute the coefficients , , in these expansions.

We define the adjoint operator of as

For and , define a bilinear form where .

To determine the normal form of operator , we need to calculate the eigenvectors and of and corresponding to and , respectively.

Proposition 6. Assume that and are the eigenvectors of and corresponding to and , respectively, satisfying and .
Then where .

Proof. Without loss of generality, we just consider the eigenvector .
Firstly, when , by the definition of and , we obtain the form (here, , are unknown parameters).
In what follows, notice that , and . We have
Finally, by , we obtain the parameter .
The proof is completed.

Now we construct the coordinates of the center manifold at .

Let

On the center manifold , we have where

and are local coordinates for the center manifold in the direction of and , respectively. Since , we have where

We rewrite this as with

By (27) and (36), we obtain where By comparing the coefficients of the previous series, we obtain

Noticing , it follows that

Thus, from (39), we have

Since there are , in , we still need to compute them.

For , and by comparing coefficients with (41),we obtain

By substituting these relations into (42), we can derive the following equation:

By solving for , , we obtain where , .

From the definition of and (46), we obtain

Thus

Similarly, we have

Thus, we can compute the parameters and .

In conclusion, we have computed all the coefficients in (39): , , , and .

Next, we can compute the following quantities:

From the discussion in Section 2, we know that . We therefore have the following result.

Theorem 7. If , the direction of the Hopf bifurcation of the system (1) at the equilibrium when is forward (backward) and the bifurcating periodic solutions are orbitally asymptotically stable (unstable).

4. Numerical Simulation

In this section, we give a particular example to illustrate the effectiveness of our results. We take the coefficients , , , , in (1). By simple computing, we have the equilibrium , , . Further, we obtain the numerical results directly by means of the software Matlab:

From the previous arithmetic, the equilibrium is asymptotically stable when (see Figure 2). When and , the stable periodic solutions occur from the equilibrium (see Figure 1).

829045.fig.001
Figure 1: When , the equilibrium is unstable and periodic solution occurs around them.
829045.fig.002
Figure 2: When , the equilibrium becomes locally stable.

Thus, the numerical simulation clarifies the effectiveness of our results.

5. Conclusions

In this paper, we have discussed the chemostat model with one species of organism. Firstly, we get the stable domain of equilibrium, and by regarding the delays as the bifurcation parameters and applying the theorem of Hopf bifurcation, we draw the sufficient conditions of the Hopf bifurcation. Further, by using the center manifold and the normal form method, we research the Hopf bifurcating direction and the stability of the model when . Our analysis indicates that the dynamics of the model of the chemostat with one species of organism can be much more complicated than we may have expected. It is interesting to describe the global dynamics of the model by means of the local properties of the interior equilibrium.

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