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

and Applied Analysis 3 Substituting τ = τ j and λ = iω 0 into (17), we obtain


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 ,  > 0 is the ratio of the mass of organism formed and the mass of substrate used,  0 > 0 is the concentration of the input nutrient,  > 0 is time lag of digestion, and  > 0 is flowing rate.

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  = 0 and the characteristic equation of (1) has no purely imaginary roots for any  > 0, it is also stable for any  > 0. On the other hand, if the equilibrium of system (1) is stable when  = 0 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.
This completes the proof.
In the following, we investigate the distribution of the eigenvalues of the characteristic equation (7).
Proof.By differentiating both sides of (7) with respect to , we obtain Then Substituting  =   and  =  0 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  0 <  1 <  2 < ⋅ ⋅ ⋅ such that all the roots of (7) have negative real parts when  ∈ [0,  0 ) and (7) has at least one root with positive real parts when  ∈ (  ,  +1 ),  = 1, 2, 3, . .., where   is defined as in (10).
In fact, according to Lemmas 1, 2, and 3, it is easy to obtain the results.

Direction and Stability of the Bifurcating Periodic Solutions
Throughout the following section, ([−1, 0];  2 + ) is a phase space, and  stands for an operator, which is different from  in Section 2.
Next, we only compute the coefficients  2 ,  2 ,  2 in these expansions.
Next, we can compute the following quantities: From the discussion in Section 2, we know that Re{  ( 0 )} > 0. We therefore have the following result.Theorem 7. If Re{ 1 (0)} < 0 (> 0), the direction of the Hopf bifurcation of the system (1) at the equilibrium ( * ,  * ) when  =  0 is forward (backward) and the bifurcating periodic solutions are orbitally asymptotically stable (unstable).
Thus, the numerical simulation clarifies the effectiveness of our results.

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  =  0 .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.