Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 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
Haiyun Bai and Yanhui Zhai
School of Science, Tianjin Polytechnic University, Tianjin 300387, China
Correspondence should be addressed to Yanhui Zhai; y h zhai@yahoo.com.cn
Received 19 November 2012; Revised 3 March 2013; Accepted 3 March 2013
Academic Editor: Abdelaziz Rhandi
Copyright © 2013 H. Bai and Y. 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.
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]):
𝑥̇ (𝑡) = − (𝑑 − 𝑝 (𝑦 (𝑡 − 𝜏))) 𝑥 (𝑡) ,
1
𝑦̇ (𝑡) = (𝑦0 − 𝑦 (𝑡)) 𝑑 − 𝑝 (𝑦 (𝑡)) 𝑥 (𝑡) ,
𝛾
(1)
where 𝑥(𝑡) is the concentration of the organism at time 𝑡,
𝑦(𝑡) is the concentration of the nutrient at time 𝑡, 𝑝(𝑦) =
𝑐𝑦/(𝑘+𝑦) (𝑐, 𝑘 are positive constants) is the growing rate of 𝑦,
𝛾 > 0 is the ratio of the mass of organism formed and the mass
of substrate used, 𝑦0 > 0 is the concentration of the input
nutrient, 𝜏 > 0 is time lag of digestion, and 𝑑 > 0 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 𝜏 = 0 and the
characteristic equation of (1) has no purely imaginary roots
for any 𝜏 > 0, it is also stable for any 𝜏 > 0. On the other
hand, if the equilibrium of system (1) is stable when 𝜏 = 0 and
there exist some positive values 𝜏 such that the characteristic
equation of (1) has a pair of purely imaginary roots, there
exists a domain concerning 𝜏 such that the equilibrium of
system (1) is table in the domain.
When 𝑦(𝑡) < 𝑦0 for ∀𝑡 > 0 and 𝑐 > 𝑑, the system (1) has
a unique interior equilibrium. We denote this unique interior
equilibrium by (𝑥∗ , 𝑦∗ ).
Then it satisfies
(𝑑 −
𝑐𝑦∗
) 𝑥∗ = 0,
𝑘 + 𝑦∗
(𝑦0 − 𝑦∗ ) 𝑑 −
𝑐𝑥∗ 𝑦∗
= 0.
𝛾 (𝑘 + 𝑦∗ )
(2)
2
Abstract and Applied Analysis
is satisfied. Then (7) has a pair of purely imaginary roots ±𝑖𝜔0
when 𝜏 = 𝜏𝑗 , where
That is
(𝑥∗ , 𝑦∗ ) = (
𝛾𝑐𝑦0 − 𝑑𝑦0 − 𝑑𝑘 𝑑𝑘
,
).
𝑐−𝑑
𝑐−𝑑
(3)
Let 𝑥1 (𝑡) = 𝑥(𝑡)−𝑥∗ and 𝑦1 (𝑡) = 𝑦(𝑡)−𝑦∗ and still denote
𝑥(𝑡), 𝑦(𝑡), respectively.
Then system (1) becomes
𝑥̇ (𝑡) = −𝑑 (𝑥 (𝑡) + 𝑥∗ ) +
∗
∗
∗
∗
𝑐 (𝑦 (𝑡 − 𝜏) + 𝑦 ) (𝑥 (𝑡) + 𝑥 )
,
𝑘 + 𝑦 (𝑡 − 𝜏) + 𝑦∗
𝑦̇ (𝑡) = 𝑑 (𝑦0 − 𝑦 (𝑡) − 𝑦∗ ) −
𝑐 (𝑦 (𝑡) + 𝑦 ) (𝑥 (𝑡) + 𝑥 )
.
𝛾 (𝑘 + 𝑦 (𝑡) + 𝑦∗ )
(4)
𝜔0 = (
𝜏𝑗 =
1/2
2
𝜔 + 𝐴𝐵
1
[arccos 0
+ 2𝑗𝜋] ,
𝜔0
𝐾
Proof. Let 𝑖𝜔 (𝜔 > 0) be a root of (7). Then
−𝜔2 + 𝑖 (𝐴 − 𝐵) 𝜔 − 𝐴𝐵 + 𝐾 (cos 𝜔𝜏 − 𝑖 sin 𝜔𝜏) = 0.
(11)
−𝜔2 − 𝐴𝐵 + 𝐾 cos 𝜔𝜏,
(12)
(𝐴 − 𝐵) 𝜔 − 𝐾 sin 𝜔𝜏.
(5)
𝑚
𝑛
𝑦̇ (𝑡) = − 𝑥 (𝑡) − ( + 𝑑) 𝑦 (𝑡) ,
𝛾
𝛾
Hence
where 𝑚 = 𝑘𝑐𝑥∗ /(𝑘 + 𝑦∗ )2 and 𝑛 = 𝑐𝑦∗ /(𝑘 + 𝑦∗ ), whose
characteristic equation is
𝑚
𝑚
𝑚𝑛 −𝜆𝜏
= 0.
𝜆 + ( − 𝑛 + 2𝑑) 𝜆 − (𝑛 − 𝑑) ( + 𝑑) +
𝑒
𝛾
𝛾
𝛾
(6)
2
By setting (𝑚/𝛾) + 𝑑 = 𝐴, 𝑛 − 𝑑 = 𝐵 and (𝑚𝑛/𝛾) = 𝐾, the
characteristic equation (6) can be rewritten as
𝜆2 + (𝐴 − 𝐵) 𝜆 − 𝐴𝐵 + 𝐾𝑒−𝜆𝜏 = 0.
2
2
𝜔 =
− (𝐴2 + 𝐵2 ) ± √(𝐴2 + 𝐵2 ) − 4 (𝐴2 𝐵2 − 𝐾2 )
2
.
(13)
Obviously, (∗∗) implies that
2
𝜔0 = (
− (𝐴2 + 𝐵2 ) + √(𝐴2 + 𝐵2 ) − 4 (𝐴2 𝐵2 − 𝐾2 )
2
1/2
)
,
(14)
(7)
and, hence,
Lemma 1. When 𝜏 = 0 and
𝐴−𝐵>0
𝜏0 =
(∗)
are met, the equilibrium (𝑥∗ , 𝑦∗ ) of system (1) is asymptotically
stable.
Proof. When 𝜏 = 0, (7) becomes
𝜆2 + (𝐴 − 𝐵) 𝜆 − 𝐴𝐵 + 𝐾 = 0,
(8)
whose characteristic value is
2
𝑗 = 0, 1, 2, . . . .
(10)
𝑥̇ (𝑡) = (𝑛 − 𝑑) 𝑥 (𝑡) + 𝑚𝑦 (𝑡 − 𝜏) ,
− (𝐴 − 𝐵) ± √(𝐴 − 𝐵)2 + 4 (𝐴𝐵 − 𝐾)
,
)
The separation of the real and imaginary parts yields
The linearization of (4) around (𝑥∗ , 𝑦∗ ) is
𝜆 1,2 =
1/2
2
− (𝐴2 + 𝐵2 ) + √(𝐴2 + 𝐵2 ) − 4 (𝐴2 𝐵2 − 𝐾2 )
.
(9)
Obviously, when (∗) holds, the real parts of 𝜆 1,2 are
negative.
This completes the proof.
In the following, we investigate the distribution of the
eigenvalues of the characteristic equation (7).
2
𝜔 + 𝐴𝐵
1
arccos 0
.
𝜔0
𝐾
(15)
Define 𝜏𝑗 = 𝜏0 + (2𝑗𝜋/𝜔0 ), 𝑗 = 0, 1, 2, . . .. Then (𝜏𝑗 , 𝜔0 )
solves (12).
This means that 𝑖𝜔0 is a root of (7) when 𝜏 = 𝜏𝑗 , 𝑗 =
0, 1, 2 . . ..
This completes the proof.
Lemma 2 shows that there exist some positive values 𝜏𝑗
such that the characteristic equation (7) has a pair of purely
imaginary roots.
Lemma 3. Let 𝜆(𝜏) = 𝑎(𝜏) + 𝑖𝑏(𝜏) be the root of (7) with
𝑎(𝜏𝑗 ) = 0 and 𝑏(𝜏𝑗 ) = 𝜔0 . When (∗∗) holds, 𝑎󸀠 (𝜏𝑗 ) > 0.
Proof. By differentiating both sides of (7) with respect to 𝜏,
we obtain
2𝜆
𝑑𝜆
𝑑𝜆
𝑑𝜆
+ (𝐴 − 𝐵)
− 𝐾𝑒−𝜏𝜆 (𝜆 + 𝜏 ) = 0.
𝑑𝜏
𝑑𝜏
𝑑𝜏
(16)
Then
Lemma 2. Assume that
𝐾 > 𝐴𝐵
(∗∗)
𝑑𝜆
𝐾𝜆𝑒−𝜏𝜆
.
=
𝑑𝜏 2𝜆 + (𝐴 − 𝐵) − 𝐾𝜏𝑒−𝜏𝜆
(17)
Abstract and Applied Analysis
3
Substituting 𝜏 = 𝜏𝑗 and 𝜆 = 𝑖𝜔0 into (17), we obtain
𝐾𝑖𝜔0 𝑒−𝜏𝑗 𝜔0 𝑖
𝑑𝜆 󵄨󵄨󵄨󵄨
=
.
󵄨󵄨
𝑑𝜏 󵄨󵄨𝜏=𝜏𝑗 2𝑖𝜔0 + (𝐴 − 𝐵) − 𝐾𝜏𝑗 𝑒−𝜏𝑗 𝜔0 𝑖
By using the Taylor series and letting 𝜏 = 𝜏0 + 𝜇, we have
(18)
𝑥̇ (𝑡) = (𝜏0 + 𝜇) [ (𝑛 − 𝑑) 𝑥 (𝑡) + 𝑚𝑦 (𝑡 − 1)
According to (12),
+
𝑖𝜔0 (𝜔02 − 𝑖 (𝐴 − 𝐵) 𝜔0 − 𝐴𝐵)
𝑑𝜆 󵄨󵄨󵄨󵄨
=
.
󵄨󵄨
𝑑𝜏 󵄨󵄨𝜏=𝜏𝑗 2𝑖𝜔0 + (𝐴 − 𝐵) − 𝜏𝑗 (𝜔02 − 𝑖 (𝐴 − 𝐵) 𝜔0 − 𝐴𝐵)
(19)
Hence
+𝑂 (√𝑥(𝑡)2 + 𝑦(𝑡 − 1)2 ) ] ,
(22)
𝑛
𝑚
𝑦̇ (𝑡) = (𝜏0 + 𝜇) [ − 𝑥 (𝑡) − ( + 𝑑) 𝑦 (𝑡)
𝛾
𝛾
𝑑𝜆 󵄨󵄨󵄨󵄨
𝑎 (𝜏𝑗 ) = Re
󵄨
𝑑𝜏 󵄨󵄨󵄨𝜏=𝜏𝑗
󸀠
=
𝑘𝑐
𝑥 (𝑡) 𝑦 (𝑡 − 1)
(𝑘 + 𝑦∗ )2
+
𝜔02 (2𝜔02 + 𝐴2 + 𝐵2 )
2
[𝐴 + 𝐵 − 𝜏𝑗 𝜔02 + 𝜏𝑗 𝐴𝐵] + [2𝜔0 + (𝐴 + 𝐵) 𝜏𝑗 𝜔0 ]
2
> 0.
(20)
The conclusion is completed.
Lemma 3 explains that the real parts 𝑎󸀠 (𝜏) are monotonously increased in a small neighbourhood concerning 𝜏𝑗 .
In other words, the root of (7) crosses the imaginary axis from
the left to the right as 𝜏 continuously varies from a number
less than 𝜏𝑗 to one greater than 𝜏𝑗 .
Lemma 4. When (∗) and (∗∗) hold, then there exist 𝜏0 < 𝜏1 <
𝜏2 < ⋅ ⋅ ⋅ such that all the roots of (7) have negative real parts
when 𝜏 ∈ [0, 𝜏0 ) and (7) has at least one root with positive real
parts when 𝜏 ∈ (𝜏𝑗 , 𝜏𝑗+1 ), 𝑗 = 1, 2, 3, . . ., where 𝜏𝑗 is defined as
in (10).
𝑘𝑐
2
𝛾(𝑘 + 𝑦∗ )
𝑥 (𝑡) 𝑦 (𝑡)
+𝑂√𝑥(𝑡)2 + 𝑦(𝑡)2 ] .
Clearly, 𝜇 = 0 is the Hopf bifurcation value of system (21).
Let 𝐶1 = 𝐶([−1, 0], 𝑅+2 ). For ∀𝜙 = (𝜙1 , 𝜙2 )𝑇 ∈ 𝐶1 , let
𝐿 𝜇 (𝜙) = (𝜏0 + 𝜇) 𝐴𝜙 (0) + (𝜏0 + 𝜇) 𝐵𝜙 (−1) ,
(23)
𝑓 (𝜇, 𝜙)
= (𝜏0 + 𝜇)
𝑘𝑐
(𝑘 + 𝑦∗ )
×(
𝑘𝑐
2
𝜙2 (−1) 𝜙1 (0) + 𝑂 (√𝜙22 (−1) + 𝜙12 (0))
𝜙 (0) 𝜙2 (0) + 𝑂 (√𝜙12 (0) + 𝜙22 (0))
2 1
𝛾(𝑘 + 𝑦∗ )
),
(24)
𝑛−𝑑
0
( −(𝑛/𝛾)
−((𝑚/𝛾)+𝑑) ),
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.
where 𝐴 =
𝐵 = ( 00 𝑚0 ).
By Riesz’s representation theorem, there exists a matrix
whose components are bounded variation functions 𝜂(𝜃, 𝜇)
in 𝜃 ∈ [−1, 0), such that
Theorem 5. If (∗) and (∗∗) are satisfied, then the equilibrium
(𝑥∗ , 𝑦∗ ) is asymptotically stable for 𝜏 ∈ [0, 𝜏0 ) and unstable for
𝜏 > 𝜏0 . System (4) undergoes a Hopf bifurcation at (𝑥∗ , 𝑦∗ )
when 𝜏 = 𝜏𝑗 , 𝑗 = 0, 1, 2 . . ., where 𝜏𝑗 is defined as in (10).
𝐿 𝜇 (𝜃) = ∫ 𝑑 (𝜂 (𝜃, 𝜇) 𝜙 (𝜃))
3. Direction and Stability of the Bifurcating
Periodic Solutions
Throughout the following section, 𝐶([−1, 0]; 𝑅+2 ) is a phase
space, and 𝐴 stands for an operator, which is different from
𝐴 in Section 2.
In Section 3, we will research the stability and direction
of the bifurcating periodic solutions of system (1). For
convenience, let 𝑡 = 𝑠𝜏 and still denote 𝑡. Then the system
(5) can be rewritten as
𝑥̇ (𝑡) = 𝜏 (𝑛 − 𝑑) 𝑥 (𝑡) + 𝜏𝑚𝑦 (𝑡 − 1) ,
𝑦̇ (𝑡) = −
𝑚
𝜏𝑛
𝑥 (𝑡) − 𝜏 ( + 𝑑) 𝑦 (𝑡) .
𝛾
𝛾
(21)
0
−1
(25)
for 𝜙 ∈ 𝐶.
For 𝜙 ∈ 𝐶([−1, 0], 𝑅2 ), we define the operators 𝐴 and 𝑅
as
𝑑𝜙 (𝜃)
{
,
{
{
{ 𝑑𝜃
𝐴 (𝜇) 𝜙 (𝜃) = { 0
{
{∫ 𝑑 (𝜂 (𝑡, 𝜇) 𝜙 (𝑡)) ,
{
{ −1
𝑅 (𝜇) 𝜙 (𝜃) = {
0,
𝑓 (𝜇, 𝜃) ,
𝜃 ∈ [−1, 0) ,
𝜃 = 0,
(26)
𝜃 ∈ [−1, 0) ,
𝜃 = 0.
Then the system (21) is equivalent to the following
abstract differential equation [5]:
𝑢̇𝑡 = 𝐴 (𝜇) 𝑢𝑡 + 𝑅 (𝜇) 𝑢𝑡 ,
𝑇
where 𝑢 = (𝑢1 , 𝑢2 ) , 𝑢𝑡 (𝜃) = 𝑢(𝑡 + 𝜃) for 𝜃 ∈ [−1, 0).
(27)
4
Abstract and Applied Analysis
As in [6], the bifurcating periodic solutions 𝑥(𝑡, 𝜇) of
system (21) are indexed by a small parameter 𝜀. A solution
𝑥(𝑡, 𝜇(𝜀)) has amplitude 𝑂(𝜀), period 𝑇(𝜀), and nonzero
Floquet exponent 𝛽(𝜀) with 𝛽(0) = 0. Under the present
assumptions, 𝜇, 𝑇, and 𝛽 have expansions
𝜇 = 𝜇2 𝜀2 + 𝜇4 𝜀4 + ⋅ ⋅ ⋅ ,
𝑇=
2𝜋
(1 + 𝑇2 𝜀2 + 𝑇4 𝜀4 + ⋅ ⋅ ⋅ ) ,
𝜔
2
(28)
𝛽 = 𝛽2 𝜀 + 𝛽4 𝜀 + ⋅ ⋅ ⋅ .
The sign of 𝜇2 determines the direction of bifurcation: if
𝜇2 > 0 (< 0), then the Hopf bifurcation is forward (backward). 𝛽2 determines the stability of the bifurcating periodic
solutions: asymptotically orbitally stable (unstable) if 𝛽2 <
0 (> 0). And 𝑇2 determines the period of the bifurcating
periodic solutions: the period increases (decreases) if 𝑇2 >
0 (< 0).
Next, we only compute the coefficients 𝜇2 , 𝑇2 , 𝛽2 in these
expansions.
We define the adjoint operator 𝐴∗ of 𝐴 as
(29)
For 𝜙 ∈ 𝐶[−1, 0] and 𝜓 ∈ 𝐶[0, 1], define a bilinear form
𝑇
0
𝜃
𝑧 (𝑡) = ⟨𝑞∗ , 𝑢𝑡 ⟩ ,
where 𝜂(𝜃) = 𝜂(𝜃, 0).
To determine the normal form of operator 𝐴, we need
to calculate the eigenvectors 𝑞(𝜃) and 𝑞1∗ (𝑠) of 𝐴 and 𝐴∗
corresponding to 𝑖𝜏0 𝜔0 and −𝑖𝜏0 𝜔0 , respectively.
Proposition 6. Assume that 𝑞(𝜃) and 𝑞∗ (𝑠) are the eigenvectors of 𝐴 and 𝐴∗ corresponding to 𝑖𝜏0 𝜔0 and −𝑖𝜏0 𝜔0 ,
respectively, satisfying ⟨𝑞∗ , 𝑞⟩ = 1 and ⟨𝑞∗ , 𝑞⟩ = 0.
Then
𝑚
𝑛 𝑇
+ 𝑑, − ) 𝑒𝑖𝜏0 𝜔0 𝜃 ,
𝛾
𝛾
𝑇
𝑛
𝑇
𝑞∗ (𝑠) = 𝐷(𝛼∗ , 𝛽∗ ) 𝑒𝑖𝜏0 𝜔0 𝑠 = 𝐷(− , 𝑖𝜔0 − 𝑛 + 𝑑) 𝑒𝑖𝜏0 𝜔0 𝑠 ,
𝛾
(31)
where 𝐷 = 1/((𝑚/𝛾) + 2𝑑 − 𝑛 − 𝑚𝜏0 𝑒
).
Proof. Without loss of generality, we just consider the eigenvector 𝑞(𝜃).
Firstly, when 𝜃 ∈ [−1, 0), by the definition of 𝐴 and
𝑞(𝜃), we obtain the form 𝑞(𝜃) = (𝛼, 𝛽)𝑇 𝑒𝑖𝜔0 𝜏0 (here, 𝛼, 𝛽 are
unknown parameters).
𝑊 (𝑡, 𝜃) = 𝑢𝑡 (𝜃) − 2 Re {𝑧 (𝑡) 𝑞 (𝜃)} .
(33)
𝑊 (𝑡, 𝜃) = 𝑊 (𝑧 (𝑡) , 𝑧(𝑡), 𝜃) ,
(34)
where
𝑊 (𝑧, 𝑧, 𝜃) = 𝑊20 (𝜃)
𝑧2
𝑧3
𝑧2
+ 𝑊11 (𝜃) 𝑧𝑧 + 𝑊02 + 𝑊30 ⋅ ⋅ ⋅ .
2
2
6
(35)
𝑧 and 𝑧 are local coordinates for the center manifold 𝐶0
in the direction of 𝑞 and 𝑞∗ , respectively. Since 𝜇 = 0, we have
𝑧󸀠 (𝑡) = 𝑖𝜏0 𝜔0 𝑧 (𝑡) + ⟨𝑞∗ (𝜃) , 𝑓 (𝑊 + 2 Re {𝑧 (𝑡) 𝑞 (𝜃)})⟩
= 𝑖𝜏0 𝜔0 𝑧 (𝑡) + 𝑞∗ (0)𝑓 (𝑊 (𝑧, 𝑧, 0) + 2 Re {𝑧 (𝑡) 𝑞 (0)})
≜ 𝑖𝜏0 𝜔0 𝑧 (𝑡) + 𝑞∗ (0)𝑓0 (𝑧, 𝑧) ,
(30)
−𝑖𝜏0 𝜔0
(32)
On the center manifold 𝐶0 , we have
𝑇
𝑞 (𝜃) = (𝛼, 𝛽)𝑇 𝑒𝑖𝜏0 𝜔0 𝜃 = (𝑖𝜔0 +
𝑛
𝛽=− .
𝛾
Now we construct the coordinates of the center manifold
𝐶0 at 𝜇 = 0.
Let
⟨𝜓, 𝜙⟩ = 𝜓 (0) 𝜙 (0) − ∫ ∫ 𝜓 (𝜉 − 𝜃) 𝑑𝜂 (𝜃) 𝜙 (𝜉) 𝑑𝜉,
−1 0
𝑚
+ 𝑑,
𝛾
𝛼 = 𝑖𝜔0 +
Finally, by ⟨𝑞∗ , 𝑞⟩ = 1, we obtain the parameter 𝐷.
The proof is completed.
4
𝑑𝜓 (𝑠)
{
,
𝑠 ∈ (0, 1] ,
−
{
{
{
𝑑𝑠
𝐴∗ 𝜓 (𝑠) = { 0
{
{
{
∫ 𝑑 (𝜂𝑇 (𝑡, 0) 𝜓 (−𝑡)) , 𝑠 = 0.
{ −1
In what follows, notice that 𝑞(0) = (𝛼, 𝛽)𝑇 , and 𝐴𝑞(0) =
∫−1 𝑑(𝜂(𝑡, 𝜇)𝜙(𝑡)) = 𝑖𝜔0 𝜏0 𝑞(0). We have
0
(36)
where
𝑓0 (𝑧, 𝑧) = 𝑓𝑧2
𝑧2
𝑧2
+ 𝑓𝑧2 + 𝑓𝑧𝑧 𝑧𝑧 + ⋅ ⋅ ⋅ .
2
2
(37)
We rewrite this as
𝑧󸀠 (𝑡) = 𝑖𝜏0 𝜔0 𝑧 + 𝑔 (𝑧, 𝑧) ,
(38)
with
𝑔 (𝑧, 𝑧) = 𝑞∗ (0) 𝑓0 (𝑧, 𝑧)
= 𝑔20
𝑧2
𝑧2
𝑧2 𝑧
+ 𝑔11 𝑧𝑧 + 𝑔02 + 𝑔21
+ ⋅⋅⋅ .
2
2
2
(39)
By (27) and (36), we obtain
̇
̇ − 𝑧𝑞
𝑊̇ = 𝑢̇𝑡 − 𝑧𝑞
∗
𝜃 ∈ [−1, 0)
{𝐴𝑊 − 2 Re {𝑞 (𝜃) 𝑓0 𝑞 (𝜃)} ,
(40)
={
∗
𝐴𝑊 − 2 Re {𝑞 (0) 𝑓0 𝑞 (0)} + 𝑓0 , 𝜃 = 0
{
≜ 𝐴𝑊 + 𝐻 (𝑧, 𝑧, 𝜃) ,
Abstract and Applied Analysis
5
where
and by comparing coefficients with (41),we obtain
𝐻 (𝑧, 𝑧, 𝜃) = 𝐻20
2
2
𝑧
𝑧
+ 𝐻11 𝑧𝑧 + 𝐻02 + ⋅ ⋅ ⋅ .
2
2
(41)
𝐻20 (𝜃) = −𝑔20 𝑞 (𝜃) − 𝑔02 𝑞 (𝜃) ,
By comparing the coefficients of the previous series, we obtain
(𝐴 − 2𝑖𝜏0 𝜔0 𝐼) 𝑊20 (𝜃) = −𝐻20 (𝜃) ,
𝐴𝑊11 (𝜃) = −𝐻11 (𝜃) , . . . .
(42)
Noticing 𝑢𝑡 = (𝑢1𝑡 , 𝑢2𝑡 ) = 𝑧𝑞(𝜃) + 𝑧𝑞(𝜃) + 𝑊(𝑡, 𝜃), it
follows that
𝑧2
(2)
𝑦 (𝑡 − 1) = 𝛽𝑒−𝑖𝜏0 𝜔0 𝑧 + 𝛽𝑒𝑖𝜏0 𝜔0 𝑧 + 𝑊20
(−1)
2
(2)
𝑢2𝑡 (0) = 𝛽𝑧 + 𝛽𝑧 + 𝑊20
(0)
𝑊20 (𝜃) =
(43)
𝑔11 = 𝐷
𝑘𝑐𝜏0
𝑧2
(2)
+ 𝑊11
(0) 𝑧𝑧
2
𝑊11 (𝜃) = −
(𝑘 + 𝑦∗ )2
𝑖𝑔
𝑖𝑔11
𝑞 (0) 𝑒𝑖𝜏0 𝜔0 𝜃 + 11 𝑞 (0) 𝑒−𝑖𝜏0 𝜔0 𝜃 + 𝐸2 ,
𝜏0 𝜔0
𝜏0 𝜔0
0
(𝛼𝛼∗ 𝛽𝑒−𝑖𝜏0 𝜔0 + 𝛼𝛼∗ 𝛽𝑒𝑖𝜏0 𝜔0
𝑔21 = 2𝐷
𝑘𝑐𝜏0
(𝑘 +
𝑘𝑐𝜏0
(𝑘 + 𝑦∗ )2
𝑦∗ )2
−1
=
1
(𝛼𝛼∗ 𝛽𝑒𝑖𝜏0 𝜔0 + 𝛼𝛽𝛽∗ ) ,
𝛾
𝐸1(1) =
1
(1)
(2)
+ 𝛼∗ 𝛽𝑒𝑖𝜏0 𝜔0 𝑊20
(0) + 𝛼𝛼∗ 𝑊20
(−1)
2
2𝑘𝑐𝛼𝛽
(𝑘 +
1
1
(1)
(2)
+ 𝛽𝛽∗ 𝑊11
(0) + 𝛼𝛽∗ 𝑊20
(0) .
𝛾
2𝛾
(44)
Since there are 𝑊20 , 𝑊11 in 𝑔21 , we still need to compute
them.
For 𝜃 ∈ [−1, 0),
(45)
2
𝑦∗ )
(𝑒−𝑖𝜔0 𝜏0 (2𝑖𝜔0 +
× ((2𝑖𝜔0 − 𝑛 + 𝑑) (2𝑖𝜔0 +
1
1
(2)
(1)
+ 𝛼𝛽∗ 𝑊11
(0)) + 𝛽𝛽∗ 𝑊20
(0)
𝛾
2𝛾
= − 𝑔 (𝑧, 𝑧) 𝑞 (𝜃) − 𝑔 (𝑧, 𝑧) 𝑞 (𝜃) ,
𝑇
1
(𝛼𝛽𝑒−𝑖𝜔0 𝜏0 , 𝛼𝛽) .
2
𝛾
(𝑘 + 𝑦∗ )
2𝑘𝑐𝜏0
(49)
Thus
(1)
(2)
(𝛼∗ 𝛽𝑒−𝑖𝜏0 𝜔0 𝑊11
(0) + 𝛼𝛼∗ 𝑊11
(−1)
𝐻 (𝑧, 𝑧, 𝜃) = − 𝑞∗ (0) 𝑓0 𝑞 (𝜃) − 𝑞∗ (0) 𝑓0 𝑞 (𝜃)
(48)
(2𝑖𝜏0 𝜔0 𝐼 − ∫ 𝑒2𝑖𝜏0 𝜔0 𝜃 𝑑𝜂 (𝜃)) 𝐸1
1
1
+ 𝛼𝛽∗ 𝛽 + 𝛼𝛽𝛽∗ ) ,
𝛾
𝛾
𝑔02 = 2𝐷
𝑖𝑔02
𝑞 (0) 𝑒−𝑖𝜏0 𝜔0 𝜃 + 𝐸1 𝑒2𝑖𝜏0 𝜔0 𝜃 ,
3𝜏0 𝜔0
where 𝐸1 = (𝐸1(1) , 𝐸1(2) ), 𝐸2 = (𝐸2(1) , 𝐸2(2) ).
From the definition of 𝐴 and (46), we obtain
1
(𝛼𝛼∗ 𝛽𝑒−𝑖𝜏0 𝜔0 + 𝛼𝛼∗ 𝛽) ,
𝛾
(𝑘 + 𝑦∗ )2
(47)
𝑖𝑔20
𝑞 (0) 𝑒𝑖𝜏0 𝜔0 𝜃
𝜏0 𝜔0
+
Thus, from (39), we have
𝑘𝑐𝜏0
󸀠
𝑊20
(𝜃) = 2𝑖𝜏0 𝜔0 𝑊20 (𝜃) + 𝑔20 𝑞 (𝜃) + 𝑔02 𝑞 (𝜃) .
𝑧2
+ ⋅⋅⋅ .
2
(2)
+ 𝑊02
(0)
𝑔20 = 2𝐷
By substituting these relations into (42), we can derive the
following equation:
𝑧2
(1)
+ 𝑊11
(0) 𝑧𝑧
2
𝑧2
+ ⋅⋅⋅ ,
2
(1)
+ 𝑊02
(0)
(46)
By solving for 𝑊20 (𝜃), 𝑊11 (𝜃), we obtain
(2)
+ 𝑊11
(−1) 𝑧𝑧 + ⋅ ⋅ ⋅ ,
(1)
𝑢1𝑡 (0) = 𝛼𝑧 + 𝛼 𝑧 + 𝑊20
(0)
𝐻11 (𝜃) = −𝑔11 𝑞 (𝜃) − 𝑔11 𝑞 (𝜃) .
+
𝐸1(2) =
2
𝑦∗ )
𝑛
(2𝑖𝜔0 − 𝑛 + 𝑑 − 𝑒−𝑖𝜔0 𝜏0 )
𝛾
× ((2𝑖𝜔0 − 𝑛 + 𝑑) (2𝑖𝜔0 +
+
𝑚
+ 𝑑)
𝛾
𝑚𝑛 −2𝑖𝜔0 𝜏0 −1
) ,
𝑒
𝛾
2𝑘𝑐𝛼𝛽
(𝑘 +
𝑚
𝑚
+ 𝑑) + 𝑒−2𝑖𝜔0 𝜏0 )
𝛾
𝛾
𝑚
+ 𝑑)
𝛾
𝑚𝑛 −2𝑖𝜔0 𝜏0 −1
) .
𝑒
𝛾
(50)
6
Abstract and Applied Analysis
6
Similarly, we have
𝐸2(1) =
4𝑘𝑐
2
(𝑘 + 𝑦∗ )
(Re {𝛼𝛽𝑒−𝑖𝜔0 𝜏0 } (
+ Re {𝛼𝛽}
× ((𝑑 − 𝑛) (
𝐸2(2)
5.5
𝑚
+ 𝑑)
𝛾
5
4.5
𝑚 −2𝑖𝜔0 𝜏0
)
𝑒
𝛾
4
−1
𝑚
𝑚𝑛 −2𝑖𝜔0 𝜏0
) ,
+ 𝑑) +
𝑒
𝛾
𝛾
4𝑘𝑐
1
=
( (𝑛 − 𝑑) Re {𝛼𝛽}
2
∗
(𝑘 + 𝑦 ) 𝛾
3.5
2.5
2
−0.2
𝑛
+ Re {𝛼𝛽𝑒−𝑖𝜔0 𝜏0 })
𝛾
× ((𝑑 − 𝑛) (
𝑔
1 󵄨󵄨 󵄨󵄨2
󵄨󵄨𝑔02 󵄨󵄨 ) + 21 ,
3
2
𝜔0
(52)
4. Numerical Simulation
In this section, we give a particular example to illustrate the
effectiveness of our results. We take the coefficients 𝑦0 = 0.65,
𝑐 = 0.7, 𝑑 = 0.56, 𝛾 = 2, 𝑘 = 0.1 in (1). By simple
computing, we have the equilibrium (𝑥∗ , 𝑦∗ ) = (0.4, 5.4),
𝜔0 ≐ 0.2540, 𝜏0 ≐ 2.018. Further, we obtain the numerical
results directly by means of the software Matlab:
𝑔11 ≐ 0.0629 + 0.0164𝑖
𝑔21 ≐ −0.0271 + 0.0643𝑖.
Re {𝜆󸀠 (𝜏0 )} ≐ 1.1591 × 10−5 > 0.
Re {𝐶1 (0)} ≐ −0.0276 < 0.
1
1.2
5.5
5
4.5
3
2
.
Theorem 7. If Re{𝐶1 (0)} < 0 (> 0), the direction of the Hopf
bifurcation of the system (1) at the equilibrium (𝑥∗ , 𝑦∗ ) when
𝜏 = 𝜏0 is forward (backward) and the bifurcating periodic
solutions are orbitally asymptotically stable (unstable).
𝑔02 ≐ 0.0397 − 0.1067𝑖,
0.8
2.5
From the discussion in Section 2, we know that
Re{𝜆󸀠 (𝜏0 )} > 0. We therefore have the following result.
𝑔20 ≐ −0.1839 + 0.0979𝑖,
0.6
4
𝛽2 = 2 Re {𝐶1 (0)} ,
Im {𝐶1 (0)} + 𝜇2 (Im {𝜆󸀠 (𝜏0 )})
0.4
3.5
Re {𝐶1 (0)}
,
𝜇2 = −
Re {𝜆󸀠 (𝜏0 )}
𝑇2 = −
0.2
6
Thus, we can compute the parameters 𝑊(0) and 𝑊(−1).
In conclusion, we have computed all the coefficients in
(39): 𝑔20 , 𝑔11 , 𝑔02 , and 𝑔21 .
Next, we can compute the following quantities:
󵄨 󵄨
(𝑔 𝑔 − 2 󵄨󵄨󵄨𝑔11 󵄨󵄨󵄨 −
𝐶1 (0) =
2𝜏0 𝜔0 20 11
0
Figure 1: When 𝜏 = 2.1, the equilibrium (0.4, 5.4) is unstable and
periodic solution occurs around them.
𝑚
𝑚𝑛 −2𝑖𝜔0 𝜏0 −1
) .
+ 𝑑) +
𝑒
𝛾
𝛾
𝑖
3
(51)
(53)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure 2: When 𝜏 = 2, the equilibrium (0.4, 5.4) becomes locally
stable.
From the previous arithmetic, the equilibrium (0.4, 5.4) is
asymptotically stable when 𝜏 = 2 < 𝜏0 = 2.018 (see Figure 2).
When 𝜏 = 2.1 > 𝜏0 = 2.018 and Re{𝐶1 (0)} ≐ −0.0276 < 0, the
stable periodic solutions occur from the equilibrium (0.4, 5.4)
(see Figure 1).
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 𝜏 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 𝜏 = 𝜏0 . 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
Abstract and Applied Analysis
expected. It is interesting to describe the global dynamics of
the model by means of the local properties of the interior
equilibrium.
References
[1] X. Li, J. Pan, and Q. Huang, “Hopf bifurcation analysis of
some modified chemostat models,” Northeastern Mathematical
Journal, vol. 14, no. 4, pp. 392–400, 1998.
[2] X.-y. Li, M.-h. Qian, J.-p. Yang, and Q.-C. Huang, “Hopf bifurcations of a chemostat system with bi-parameters,” Northeastern
Mathematical Journal, vol. 20, no. 2, pp. 167–174, 2004.
[3] H. I. Freedman, J. W.-H. So, and P. Waltman, “Coexistence in
a model of competition in the chemostat incorporating discrete
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[4] J. Hale, Theory of Functional Differential Equations, Springer,
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[5] J. K. Hale and S. M. V. Lunel, Introduction to Functional
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[6] G. Mircea, M. Neamtu, and D. Opris, Dynamical Systems
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[7] J. Wei and C. Yu, “Hopf bifurcation analysis in a model of
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[8] N. A. M. Monk, “Oscillatory expression of Hes1, p53, and NF𝜅B driven by transcriptional time delays,” Current Biology, vol.
13, no. 16, pp. 1409–1413, 2003.
[9] Y. Song and J. Wei, “Bifurcation analysis for Chen’s system with
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[10] Y. Song, J. Wei, and M. Han, “Local and global Hopf bifurcation
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[11] S. Ruan and J. Wei, “On the zeros of transcendental functions
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[12] J. Wei, “Bifurcation analysis in a scalar delay differential equation,” Nonlinearity, vol. 20, no. 11, pp. 2483–2498, 2007.
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