Li et al. Advances in Difference Equations (2016) 2016:249
DOI 10.1186/s13662-016-0955-7
RESEARCH
Open Access
Dynamical analysis of a competition
model in the turbidostat with discrete delay
Zuxiong Li1,2* , Yong Yao2 , Hailing Wang2 and Zhijun Liu2
*
Correspondence:
[email protected]
1
School of Mathematics and
Statistics, Huazhong University of
Science and Technology, Wuhan,
Hubei 430074, P.R. China
2
Department of Mathematics,
Hubei University for Nationalities,
Enshi, Hubei 445000, P.R. China
Abstract
In this paper, the dynamic behaviors of a competition model in the turbidostat with
discrete delay are investigated. The stability of the positive equilibrium and the
existence of a Hopf bifurcation are discussed by choosing the delay of digestion as a
bifurcation parameter. Furthermore, we determine the direction and stability of the
bifurcating periodic solutions by the normal form and the center manifold theorem.
Moreover, some examples are given to illustrate our main results.
Keywords: turbidostat; bifurcating periodic solutions; digestion delay; stability;
competition
1 Introduction
Chemostat models have been fruitful as a source of mathematical problems (see [–]).
Generally, chemostat models lead to competitive exclusion results, which means that only
one organism can survive in the competition at last. The phenomenon of coexistence of
the organisms is a common thing in reality [, , –]. In a sense, coexistence reflects
the ecological balance. The turbidostat as well as the chemostat is an important laboratory
set for continuous cultivation of microorganisms, and also a very important medium between principle and application. Turbidostat model is one of the most important models
in mathematical biology. Figure  [] is the schematic diagram of the competition in the
turbidostat. However, little work has been done in the study of mathematical models on
the turbidostat, and the existing main studies can be listed as follows. Flegr [] showed
the coexistence of two organisms in the turbidostat by numerical analysis, De Leenheer
and Smith [] also verified Flegr’s results by theoretical analysis. Li [] and Cammarota
and Miccio [], respectively, established a mathematical model of competition in a turbidostat for an inhibitory nutrient, and one obtained sufficient conditions for coexistence
solutions. Recently, to ensure the coexistence of the species, some scholars have considered turbidostat models by controlling the dilution rate of the turbidostat (see [–]).
De Leenheer and Smith [] have considered the following system:
⎧
dS(t)
x (t)

⎪
⎪
⎨ dt = D(x(t))(S – S(t)) – γ f (S(t)) –
dx (t)
= x (t)[f (S(t)) – D(x(t))],
dt
⎪
⎪
⎩ dx (t) = x (t)[f (S(t)) – D(x(t))].
dt

x (t)
f (S(t)),
γ 
(.)

© 2016 Li et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Li et al. Advances in Difference Equations (2016) 2016:249
Page 2 of 20
Figure 1 The schematic diagram of competition in the turbidostat.
Here S(t) is the nutrient concentration and xi (t) (i = , ) is the density of the ith organism
at time t, respectively. S >  represents the input concentration of the nutrient. γi > 
i S(t)
(i = , ) stands for the yield constant. fi (S(t)) = ami +S(t)
(mi > , ai > , i = , ) are uptakes
functions. D(x(t)) = d + k x (t) + k x (t) (d > , ki > , i = , ) is the dilution of the turbidostat. ki is the gain of the ith organism. System (.) is called ‘turbidostat’ by Li [].
It was shown in [] that a turbidostat with two organisms can be made coexistent if the
dilution rate depends on the concentrations of two competing organisms. The authors
showed that system (.) possessed a unique coexistence equilibrium under the specific
conditions, and it is globally asymptotically stable. Yet people have recognized that time
delay have a complex impact on the dynamics of a system in reality (see [–] and []).
Yuan et al. [] considered the effect of delay on the dilution rate D(x(t)) of system (.).
The authors have investigated the stability of the positive equilibrium, the existence, and
the stability of Hopf bifurcation. Further we note that the chemostat models with discrete
delay due to the possibility that the organism digests (stores) the nutrient are natural and
reasonable (see Bush and Cook [], Freedman et al. [] and Zhao []). Generally, different microorganisms have the different delays of digestion, but a single time delay is
common in reality (see Lin et al. [], Bender et al. [] and Kharitonov []), so we assume that the two competitive microorganisms are homogeneous species and they have
the same delay of digestion. Based on motivation from the work of Bush and Cook [],
Freedman et al. [] and Zhao [], in this study we focus on the dynamics of a turbidostat
model with time delay of digestion which follows:
⎧
x (t)
dS(t)

⎪
⎪
⎨ dt = D(x(t))(S – S(t)) – γ f (S(t)) –
dx (t)
= x (t)[f (S(t – τ )) – D(x(t))],
dt
⎪
⎪
⎩ dx (t) = x (t)[f (S(t – τ )) – D(x(t))].
dt

x (t)
f (S(t)),
γ 
(.)

Here S(t), x (t), x (t), D(x(t)), fi (S(t – τ )) (i = , ) and the parameters play similar roles to
system (.), τ >  is the time delay of digestion.
The organization of this paper is as follows. We investigate the local stability and Hopf
bifurcation of the positive equilibrium of system (.) in the next section. In Section ,
by using the normal form method and the center manifold theory introduced by Hassard
et al. [], we analyze the direction of Hopf bifurcation and the stability of bifurcating
periodic solutions. In Section , some examples are given to illustrate our results.
Li et al. Advances in Difference Equations (2016) 2016:249
Page 3 of 20
2 Local stability and Hopf bifurcation
In the section, we focus on investigating the local stability of the positive equilibrium and
the existence of local Hopf bifurcations for system (.).
For the sake of simplicity, we let
S̄(t) =
S(t)
,
S
k̄i = γi S ki ,
x̄i (t) =
xi (t)
,
γi S
f¯i S̄(t) = fi S̄(t)S ,
i = , .
The bars dropped, system (.) becomes
⎧
dS(t)
⎪
⎪ dt = (d + k x (t) + k x (t))( – S(t)) – x (t)f (S(t)) – x (t)f (S(t)),
⎨
dx (t)
= x (t)[f (S(t – τ )) – (d + k x (t) + k x (t))],
dt
⎪
⎪
⎩ dx (t) = x (t)[f (S(t – τ )) – (d + k x (t) + k x (t))].

dt

 
(.)
 
As in [], we introduce the same hypothesis:
(H) The graphs of the functions f and f intersect once at S∗ :
f S∗ = f S∗ = D∗ ,
where S∗ ∈ (, ) and f (S∗ ) = f (S∗ ).
Assume that d ∈ (, D∗ ) and
(H ) k <
D∗ – d
< k
 – S∗
or k <
D∗ – d
< k ,
 – S∗
then system (.) has a unique positive equilibrium E∗ = (S∗ , x∗ , x∗ ), where
S∗ =
m a  – m a 
,
m – m
x∗ =
D∗ – k ( – S∗ ) – d
,
k – k
x∗ =
k ( – S∗ ) – D∗ + d
.
k – k
In the following, we will investigate the local stability of E∗ and the existence of Hopf
bifurcations induced by the time delay.
Set y (t) = S(t) – S∗ , y (t) = x (t) – x∗ , y (t) = x (t) – x∗ . Then system (.) can be written
as
⎧ dy (t)
= –ay (t) + (k – k S∗ – D∗ )y (t) + (k – k S∗ – D∗ )y (t)
⎪
dt
⎪
⎪
⎪
⎪
+ (–x∗ b – x∗ b )y (t) + (–k – b )y (t)y (t) + (–k – b )y (t)y (t)
⎪
⎪
⎪
⎪
⎪
– b y (t)y (t) – b y (t)y (t) + (–b x∗ – b x∗ )y (t) + · · · ,
⎪
⎪
⎪
⎪
⎪ dydt (t) = –k x∗ y (t) – k x∗ y (t) + x∗ b y (t – τ ) – k y (t) – k y (t)y (t)
⎨
+ x∗ b y (t – τ ) + b y (t – τ )y (t) + b y (t – τ )y (t)
⎪
⎪
⎪
+ b x∗ y (t – τ ) + · · · ,
⎪
⎪
⎪
⎪
dy
(t)

⎪
= –k x∗ y (t) – k x∗ y (t) + x∗ b y (t – τ ) – k y (t) – k y (t)y (t)
⎪
dt
⎪
⎪
⎪
⎪
+ x∗ b y (t – τ ) + b y (t – τ )y (t) + b y (t – τ )y (t)
⎪
⎪
⎩
+ b x∗ y (t – τ ) + · · · .
(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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Here
a = d + k x∗ + k x∗ + b x∗ + b x∗
= D∗ + b x∗ + b x∗ ,
(j)
bij =
fi (S∗ )
,
j!
i = , , j = , , , . . . .
So the linearized system of (.) at E∗ is
⎧
dy (t)
∗
∗
∗
∗
⎪
⎪
⎨ dt = –ay (t) + (k – k S – D )y (t) + (k – k S – D )y (t),
dy (t)
= –k x∗ y (t) – k x∗ y (t) + x∗ b y (t – τ ),
dt
⎪
⎪
⎩ dy (t) = –k x∗ y (t) – k x∗ y (t) + x∗ b y (t – τ ).
dt
  
  
(.)
  
We obtain the characteristic equation from (.)
λ + pλ + qλ + rλe–λτ + ke–λτ = ,
(.)
where
p = k x∗ + k x∗ + a,
q = a k x∗ + k x∗ ,
r = D∗ x∗ b + D∗ x∗ b – x∗ + x∗ b k x∗ + b k x∗ ,
k = x∗ x∗ D∗ (k – k )(b – b ).
In order to study the distribution of the roots of (.), we consider the following two
cases: τ =  and τ > .
For τ = , (.) becomes
λ + pλ + (q + r)λ + k = .
(.)
It is obvious that p > . According to the Routh-Hurwitz criterion, we immediately have
the following lemma.
Lemma . If (H ) k > , p(q + r) > k, then all roots of (.) have negative real parts.
For τ > , Beretta and Kuang [] have studied the general characteristic equation with
delay dependent parameters:
Pn (λ; τ ) + Qm (λ; τ )e–λτ = ,
where Pn and Qm are, respectively, n-degree and m-degree polynomials in λ, with n >
m and with delay dependent polynomial coefficients. In (.), the parameters are delay
independent, so the stability switch of the (.) can be obtained as a particular case of the
results in [].
Li et al. Advances in Difference Equations (2016) 2016:249
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We set
P(λ) = Pn (λ; τ ) = λ + pλ + qλ,
Q(λ) = Qn (λ; τ ) = rλ + k.
(.)
We assume that Pn (λ; τ ) and Qm (λ; τ ) cannot have imaginary roots. That is, for any real
number w,
Pn (iw, τ ) + Qm (iw, τ ) = .
We have

 F(w) = P(w; τ ) – Q(w; τ ) = w + p – q w + q – r w – k  .
(.)
Hence, F(w) =  implies
w + p – q w + q – r w – k  = .
(.)
Set v = w , then (.) becomes
v + p – q v + q – r v – k  = .
(.)
Denote
f (v) = v + p – q v + q – r v – k  .
(.)
Since f () = –k  < , limv→+∞ f (v) = +∞, it is obvious that (.) has at least one positive
root. By (.), we get
f (v) = v +  p – q v + q – r .
(.)
Denote = (p – q) – (q – r ). If ≤ , then the function f (v) is monotone increasing in v ∈ [, ∞). Thus, (.) has only a positive real root; on the other hand,
when
√
–(p –q)+ 



and
> , the equation v + (p – q)v + q – r =  has two real roots v =

√

v = –(p –q)–
. Since p – q > , one can get v < . We immediately know that f (v) has

only a positive real root too, and f (v) is monotone increasing in v ∈ (v , ∞). Thus, (.)
has only a positive root denoted by v . Furthermore, we have the fact that v > v is true
√
when > . Then (.) has unique positive root w = v .
Furthermore, PR (iw , τ ) = –pw , PI (iw , τ ) = –w + qw , QR (iw , τ ) = k, QI (iw , τ ) = rw .
Hence, we have
sin θ =
=
–PR (iw , τ )QI (iw , τ ) + PI (iw , τ )QR (iw , τ )
|Q(iw , τ )|
prw + (–w + q)kw
,
k  + r w
cos θ = –
=–
PR (iw , τ )QR (iw , τ ) + PI (iw , τ )QI (iw , τ )
|Q(iw , τ )|
–pkw + (–w + q)rw
.
k  + r w
(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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We denote the corresponding critical value of time delay that is satisfied τ ∗ ,
Sn τ ∗ = τ ∗ – τn τ ∗ = ,
n ∈ N .
(.)
Thus,
τ ∗ = τn =
θ + nπ
,
w
n ∈ N .
(.)
From (.), we have
Fw (w) = w w +  p – q w + q – r = wf (v).
(.)
Differentiating (.) with respect to τ , we have
dλ
dτ
–
=
λ + pλ + q
r
τ
+
– .
λ(–λ – pλ – qλ) (rλ + k)λ λ
(.)
Hence, a direct calculation shows that
Re
dλ
dτ
–
= Re
λ=iw
=
w
λ + pλ + q
λ(–λ – pλ – qλ)

)w
+ Re
λ=iw

r
(rλ + k)λ
λ=iw

r
+ (–q + p
+q
–
.
(rw ) + k 
(–w + qw ) + (pw )
(.)
Also we have
–w + qw


+ pw = (rw ) + k  .
(.)
Hence,
– dλ
d(Re λ)
= sign Re
δ τ ∗ = sign
dτ
dτ
λ=iw
λ=iw





w + (–q + p )w + q – r
= sign
(rw ) + k 
f (w )
= sign
(rw ) + k 
= sign f w .
(.)
λ)
We conclude that the sign of ( d(Re
)λ=iw is determined by that of f (w ).
dτ
According to the above discussion, we know f (v ) > . Thus, δ(τ ∗ ) > .
From Theorem . in [], we have the following result.
Lemma . The characteristic equation (.) has a pair of simple and conjugate pure
imaginary roots λ = ±iw at τ ∗ if Sn (τ ∗ ) = τ ∗ – τn (τ ∗ ) =  for some n ∈ N . Since δ(τ ∗ ) > ,
this pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to
right.
Li et al. Advances in Difference Equations (2016) 2016:249
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From Lemmas . and ., we have the following lemma.
Lemma . If (H ) holds, then (.) has a pair of simple imaginary roots ±iw when τ = τn ,
n ∈ N . Furthermore, when τ ∈ [, τ ), all roots of (.) have negative real parts; when
τ = τ , all roots of (.) except ±iw have negative real parts; when τ ∈ (τn , τn+ ], (.) has
(n + ) roots with positive real parts.
Thus, from Lemmas .-. and Theorem . in [], we have the following theorem.
Theorem . For system (.), assume that (H ) and (H ) hold, there exists a positive number τ such that the positive equilibrium E∗ is asymptotically stable when τ ∈ [, τ ) and
unstable when τ > τ . Furthermore, system (.) undergoes a Hopf bifurcation at E∗ when
τ = τ .
3 Direction and stability of the bifurcating periodic solutions
In Section , we have obtained the conditions under which system (.) undergoes Hopf
bifurcation when τ = τ . In this section, by using the normal form and the center manifold
theory that introduced by Hassard et al. [], we will consider the direction of the Hopf
bifurcation and the stability of bifurcating periodic solutions of system (.).
Set ȳi (t) = yi (τ t), τ = τ + μ, where τ is defined by (.), and drop the bars for convenience, then system (.) can be written as a FDE in C = C([–, ], R ),
ẏ(t) = Lμ (yt ) + h(μ, yt ),
(.)
where y(t) = (y (t), y (t), y (t))T ∈ R , and Lμ : C → R , h : R × C → R are, respectively,
given by
⎞⎛
⎞
ϕ ()
k  – k S ∗ – D ∗ k – k S ∗ – D ∗
⎟⎜
⎟
–k x∗
–k x∗
⎠ ⎝ϕ ()⎠
–k x∗
–k x∗
ϕ ()
⎞⎛
⎞
ϕ (–)

 
⎟⎜
⎟
⎜
+ (τ + μ) ⎝ x∗ b  ⎠ ⎝ϕ (–)⎠
x∗ b  
ϕ (–)
⎛
–a
⎜
Lμ ϕ = (τ + μ) ⎝ 

⎛
(.)
and
⎛ ⎞
h
⎜ ⎟
h(μ, ϕ) = (τ + μ) ⎝h ⎠ .
h
Here
h = –x∗ b – x∗ b ϕ () + (–k – b )ϕ ()ϕ () + (–k – b )ϕ ()ϕ ()
– b ϕ ()ϕ () – b ϕ ()ϕ () + –b x∗ – b x∗ ϕ () + · · · ,
h = –k ϕ () – k ϕ ()ϕ () + x∗ b ϕ (–) + b ϕ (–)ϕ () + b ϕ (–)ϕ ()
+ b x∗ ϕ (–) + · · · ,
(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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h = –k ϕ () – k ϕ ()ϕ () + x∗ b ϕ (–) + b ϕ (–)ϕ () + b ϕ (–)ϕ ()
+ b x∗ ϕ (–) + · · · ,
and ϕ = (ϕ , ϕ , ϕ )T ∈ C.
By the Riesz representation theorem, there exists a function η(θ , μ) of bounded variation
for θ ∈ [–, ] such that

Lμ ϕ =
dη(θ , μ)ϕ(θ ),
for ϕ ∈ C.
(.)
–
In fact, we can choose
⎞
k – k S ∗ – D ∗ k – k S ∗ – D ∗
⎟
–k x∗
–k x∗
⎠ δ(θ )
∗
∗
–k x
–k x
⎞

 
⎜
⎟
– (τ + μ) ⎝ x∗ b  ⎠ δ(θ + ),
x∗ b  
⎛
–a
⎜
η(θ , μ) = (τ + μ) ⎝ 

⎛
(.)
where δ is the Dirac delta function.
For ϕ ∈ C  ([–, ], R ), we define the operators A and R as
dϕ(θ)
dθ

A(μ)ϕ =
,
– dη(θ , μ)ϕ(θ ),
θ ∈ [–, ),
θ = ,
(.)
and
R(μ)ϕ =
,
θ ∈ [–, ),
h(μ, ϕ), θ = .
(.)
Then system (.) becomes
ẏt = A(μ)yt + R(μ)yt ,
(.)
where yt (θ ) = y(t + θ ) for θ ∈ [–, ].
As in [], the bifurcating periodic solutions y(t, μ) of system (.) are indexed by a small
parameter ε. The solution y(t, μ(ε)) has an amplitude O(ε), a nonzero Floquet exponent
β(ε) with β() =  and a period T(ε). Under the assumptions, μ, T, and β have expansions
⎧


⎪
⎨ μ = μ ε + μ ε + · · · ,
π

T = ω ( + T ε + T ε + · · · ),
⎪
⎩
β = β ε  + β ε  + · · · .
(.)
Here μ determines the directions of the bifurcation: if μ <  (> ), then the Hopf bifurcation is subcritical (supercritical); β determines the stability of the bifurcating periodic
solutions: the bifurcating periodic solutions are stable (unstable) if β <  (> ); and the
period of the bifurcating periodic solutions is determined by T : if T >  (< ), the period
increases (decreases).
Li et al. Advances in Difference Equations (2016) 2016:249
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Next, we only need to compute the coefficients μ , T , β in the above expansions.
For ψ ∈ C  ([, ], (R )∗ ), we define the adjoint operator A∗ of A as
∗
–dψ(s)
A ψ=
,
s ∈ (, ],
ds T
– dη (t, )ψ(–t), s = .
(.)
Meanwhile, we define a bilinear inner product as follows:
ψ(s), ϕ(θ ) = ψ̄ T ()ϕ() –
 θ
ψ̄ T (ξ – θ ) dη(θ )ϕ(ξ ) dξ ,
–
(.)
ξ =
where η(θ ) = η(θ , ).
From the discussion in Section , we know that ±iw τ are eigenvalues of A(). Thus,
they are also eigenvalues of A∗ , we first need to calculate the eigenvectors of A() and A∗
corresponding to iw τ and –iw τ , respectively.
Assume that q(θ ) = (, α , β )T eiw τ θ is the eigenvector of A() corresponding to iw τ ,
then A()q(θ ) = iw τ q(θ ). By the definition of A() and (.), (.), and (.), we obtain
⎛
–k + k S∗ + D∗
iw + k x∗
k x∗
iw + a
⎜ ∗
τ ⎝ –x b e–iw τ
–x∗ b e–iw τ
⎞⎛ ⎞
–k + k S∗ + D∗

⎟⎜ ⎟
∗
k x 
⎠ ⎝α ⎠ = ,
iw + k x∗
β
which yields
α =
e–iw τ [k x∗ x∗ (b – b ) + iw x∗ b ]
,
–w + iw (k x∗ + k x∗ )
e–iw τ [k x∗ x∗ (b – b ) + iw x∗ b ]
.
β =
–w + iw (k x∗ + k x∗ )
(.)
Similarly, set q∗ (s) = D(, α , β )eiw τ s is the eigenvector of A∗ corresponding to –iw τ . It
follows from the definition of A∗ and (.), (.), and (.) that
⎛
iw – a
⎜
τ ⎝ k – k S ∗ – D ∗
k – k S ∗ – D ∗
x∗ b eiw τ
iw – k x∗
–k x∗
⎞⎛ ⎞
x∗ b eiw τ

⎟⎜ ⎟
–k x∗ ⎠ ⎝α ⎠ = ,
iw – k x∗
β
we can easily obtain
α =
(ak – iw k )e–iw τ + (D∗ + k S∗ – k )b
,
x∗ k (b – b ) + iw b
(ak – iw k )e–iw τ + (D∗ + k S∗ – k )b
β =
.
x∗ k (b – b ) + iw b
From (.), we have
∗
q (s), q(θ ) = D̄(, ᾱ , β̄ )(, α , β )T
 θ
–
D̄(, ᾱ , β̄ )e–iw τ (ξ –θ) dη(θ )(, α , β )T eiξ w τ dξ
–
ξ =
(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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
= D̄  + α ᾱ + β β̄ –
(, ᾱ , β̄ )θ eiθw τ dη(θ )(, α , β )T
–
= D̄  + α ᾱ + β β̄ + τ ᾱ x∗ b + β̄ x∗ b e–iw τ .
Since q∗ (s), q(θ ) = , we have
D=

.
 + ᾱ α + β̄ β + τ (α x∗ b + β x∗ b )eiw τ
(.)
In the following, we follow the ideas in Hassard et al. and use the same notations to
compute the coordinates describing the center manifold C at μ = . Set yt be the solution
of (.) when μ = . Define
z(t) = q∗ (s), yt (θ ) ,
W (t, θ ) = yt (θ ) –  Re z(t)q(θ ) .
(.)
On the center manifold C , we have
z
z̄
W (t, θ ) = W z(t), z̄(t), θ = W (θ ) + W (θ )zz̄ + W (θ ) + · · · ,


(.)
where z and z̄ are local coordinates for center manifold C in the direction of q∗ and q̄∗ .
Note that W is real if yt is real. We consider only real solutions. For the solution yt ∈ C
of (.), since μ = , we have
ż(t) = iw τ z + q̄∗ (θ ), h , W (z, z̄, θ ) +  Re zq(θ )
= iw τ z + q̄∗ ()h , W (z, z̄, ) +  Re zq()
= iw τ z + q̄∗ ()h (z, z̄).
def
(.)
We rewrite the equation as
ż(t) = iw τ z(t) + g(z, z̄),
(.)
where
g(z, z̄) = q̄∗ ()h (z, z̄)
= g
z̄
z
z z̄
+ g zz̄ + g + g
+ ··· .



(.)
The expressions of μ , T , and β include the coefficients g , g , g , and g . Next, we
need to compute g , g , g , and g .
By (.), we have yt (θ ) = (yt (θ ), yt (θ ), yt (θ ))T = W (t, θ ) + zq(θ ) + zq(θ ), and then
⎧


()
()
() z + W() ()zz̄ + W
() z̄ + z + z̄ + O(|(z, z̄)| ),
yt () = W
⎪
⎪
⎪
⎪
()
()
()
⎪
z
z̄

⎪
⎪
⎨ yt () = W ()  + W ()zz̄ + W ()  + α z + ᾱ z̄ + O(|(z, z̄)| ),


()
()
() z + W() ()zz̄ + W
() z̄ + β z + β̄ z̄ + O(|(z, z̄)| ),
yt () = W
⎪
⎪
⎪
()
()
()
z
z̄
⎪
⎪ yt (–) = W (–)  + W (–)zz̄ + W (–)  + ze–iw τ + z̄eiw τ
⎪
⎪
⎩
+ O(|(z, z̄)| ).
(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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From (.), we obtain
g(z, z̄) = q̄∗ ()h (z, z̄) = τ D̄(, ᾱ , β̄ )(h , h , h )T = τ D̄(h + ᾱ h + β̄ h )
= τ D̄ –x∗ b – x∗ b ϕ () + (–k – b )ϕ ()ϕ ()
+ (–k – b )ϕ ()ϕ () – b ϕ ()ϕ () – b ϕ ()ϕ ()
+ –b x∗ – b x∗ ϕ () + · · · + ᾱ –k ϕ () – k ϕ ()ϕ ()
+ x∗ b ϕ (–) + b ϕ (–)ϕ () + b ϕ (–)ϕ ()
+ b x∗ ϕ (–) + · · · + β̄ –k ϕ () – k ϕ ()ϕ () + x∗ b ϕ (–)
+ b ϕ (–)ϕ () + b ϕ (–)ϕ () + b x∗ ϕ (–) + · · · .
(.)
By substituting (.) into (.) and comparing the coefficients with (.), we obtain
⎧
⎪
g = τ D̄[(–x∗ b – x∗ b ) + α (–k – b ) + β (–k – b )
⎪
⎪ 
⎪
⎪
+ ᾱ (–k α – k α β + x∗ b e–iw τ + b α e–iw τ )
⎪
⎪
⎪
⎪
⎪
+ β̄ (–k β – k α β + x∗ b e–iw τ + b β e–iw τ )],
⎪
⎪
⎪
⎪
⎪ g = τ D̄[(–x∗ b – x∗ b ) + (–k – b )(α + ᾱ ) + (–k – b )(β + β̄ )
⎪
⎪
⎪
⎪
⎪
+ ᾱ (–k α ᾱ – k (α β̄ + ᾱ β ) + x∗ b + b (α eiw τ + ᾱ e–iw τ ))
⎪
⎪
⎪
⎪
⎪
+ β̄ (–k β β̄ – k (α β̄ + ᾱ β ) + x∗ b + b (β eiw τ + β̄ e–iw τ ))],
⎪
⎪
⎪
⎪
⎪
g = τ D̄[(–x∗ b – x∗ b ) + ᾱ (–k – b ) + β̄ (–k – b )
⎪
⎪
⎪
⎪
+ ᾱ (–k ᾱ – k ᾱ β̄ + x∗ b eiw τ + b ᾱ eiw τ )
⎪
⎪
⎪
⎪
⎪
+ β̄ (–k β̄ – k ᾱ β̄ + x∗ b eiw τ + b β̄ eiw τ )],
⎪
⎪
⎪
⎨ g = τ D̄[(–x∗ b – x∗ b )(W () () + W () ()) + (–k – b )(W () ()







 
 
()
()
()
()


⎪
W
()
+
W
()
ᾱ
+
W
()α
)
+
(–k
–
b
)(W
()
+




⎪






⎪
⎪
()
()
⎪
⎪
() +  W
()β̄ + W() ()β ) – b (α + ᾱ ) – b (β + β̄ )
+  W
⎪
⎪
⎪
()
⎪
⎪
()) – k (α W() ()
+ (–b x∗ – b x∗ ) + ᾱ (–k (α W() () + ᾱ W
⎪
⎪
⎪
()
()
()


∗
⎪
+  ᾱ W () +  W ()β̄ + W ()β ) + x b (W() (–)e–iw τ
⎪
⎪
⎪
⎪
()
()
()
⎪
+ W
(–)eiw τ ) + b (α W() (–) +  W
(–)ᾱ +  W
()eiw τ
⎪
⎪
⎪
()
⎪
∗
–iw
τ
–iw
τ
–iw
τ
⎪
+ W ()e   ) + b (α + e   ᾱ ) + b x e   )
⎪
⎪
⎪
()
()
⎪
⎪
()β̄ ) – k (α W() () +  ᾱ W
()
+ β̄ (–k (β W() () + W
⎪
⎪
⎪
()
()
()
()


⎪
⎪
β̄
β̄
W
()
+
β
W
())
+
b
(β
W
(–)
+
W
(–)
+
 
  
⎪
  
  
⎪
⎪
()
()
()

∗
iw τ
–iw τ
–iw τ
⎪
⎪
W
()e
+
W
()e
)
+
x
b
(W
+



 (–)e

⎪

⎪
⎩
()
+ W (–)eiw τ ) + b (β + β̄ e–iw τ ) + b x∗ e–iw τ )].
(.)
In order to determine g , we still need to compute W (θ ) and W (θ ). From (.) and
(.), we get
˙
Ẇ = ẏt – żq – z̄q̄
θ ∈ [–, ),
AW –  Re{q̄∗ ()h (z, z̄)q(θ )},
=
∗
AW –  Re{q̄ ()h (z, z̄)q(θ )} + h (z, z̄), θ = 
def
= AW + H(z, z̄, θ ),
(.)
where
H(z, z̄, θ ) = H (θ )
z
z̄
z
+ H (θ )zz̄ + H (θ ) + H (θ ) + · · · .



(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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By comparing the coefficients (.), we get
(A – iw τ I)W (θ ) = –H (θ ),
AW (θ ) = –H (θ ), . . . .
(.)
From (.), we know that, for θ ∈ [–, ),
H(z, z̄, θ ) = –q̄∗ ()h (z, z̄)q(θ ) – q∗ ()h̄ (z, z̄)q̄(θ )
= –g(z, z̄)q(θ ) – ḡ(z, z̄)q̄(θ )

= – g q(θ ) + ḡ q̄(θ ) z – g q(θ ) + ḡ q̄(θ ) zz̄ + · · · .

(.)
By comparing the coefficients with (.), we have
H (θ ) = –g q(θ ) – ḡ q̄(θ ),
H (θ ) = –g q(θ ) – ḡ q̄(θ ).
(.)
It follows from (.), (.), and the definition of A that
Ẇ (θ ) = iw τ W (θ ) + g q(θ ) + ḡ q̄(θ ).
Notice q(θ ) = (, α , β )T eiw τ θ , then
W (θ ) =
ig
iḡ
q̄()e–iw τ θ + eiw τ θ E ,
q()eiw τ θ +
w τ
w τ
(.)
where E = (E , E , E )T ∈ R is a constant vector.
Similarly, from (.) and (.), it follows that
W (θ ) =
–ig
iḡ
q̄()e–iw τ θ + E ,
q()eiw τ θ +
w τ
w τ
(.)
where E = (E , E , E )T ∈ R is also a constant vector.
Next, we will search E and E . From (.) and the definition of A, we can obtain

dη(θ )W (θ ) = iw τ W () – H (),
– dη(θ )W (θ ) = –H (),
–

(.)
where η(θ ) = η(θ , ).
Noting that q(θ ) is the eigenvector of A() and from (.) and the definition of A(),
we have

dη(θ )W (θ ) =
–
ig
w τ

dη(θ )q(θ ) +
–
iḡ
w τ


dη(θ )E eiw τ θ
dη(θ )q̄(θ ) +
–
ig iḡ =
iw τ q() +
–iw τ q̄() +
w τ
w τ

ḡ
= –g q() +
dη(θ )E eiw τ θ ,
q̄() +

–
–

dη(θ )E eiw τ θ
–
(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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and
iw τ W () = –g q() –
ḡ
q̄() + iw τ E .

(.)
Hence, the first equation of (.) becomes
H () = –g q() – ḡ q̄() + iw τ –

dη(θ )eiw τ θ E .
(.)
–
Similarly, from (.), it follows that


dη(θ )W (θ ) = g q() + ḡ q̄() +
–
dη(θ )E .
(.)
–
Thus, the second equation of (.) becomes

H () = –g q() – ḡ q̄() –
dη(θ )E .
(.)
–
From (.), it follows that
H () = –g q() – ḡ q̄() + τ (w , w , w )T ,
(.)
where
⎧
∗
∗
⎪
⎨ w = –x b – x b + α (–k – b ) + β (–k – b ),
w = –k α – k α β + x∗ b e–iw τ + b α e–iw τ ,
⎪
⎩
w = –k β – k α β + x∗ b e–iw τ + b β e–iw τ .
Also
H () = –g q() – ḡ q̄() – τ (v , v , v )T ,
where
⎧
∗
∗
⎪
⎨ v = x b + x b + Re{(k + b )α + (k + b )β },
v = k α ᾱ + k Re{α β̄ } – x∗ b – b Re{α eiw τ },
⎪
⎩
v = k β β̄ + k Re{α β̄ } – x∗ b – b Re{β eiw τ }.
From (.) and (.), we have
iw τ –

dη(θ )eiw τ θ E = τ (w , w , w )T ,
–
which leads to
⎛
iw + a
⎜ ∗
⎝ –x b e–iw τ
–x∗ b e–iw τ
–k + k S∗ + D∗
iw + k x∗
k x∗
⎞
⎛ ⎞
–k + k S∗ + D∗
w
⎟
⎜ ⎟
∗
k x 
⎠ E =  ⎝w ⎠ .
∗
iw + k x
w
(.)
Li et al. Advances in Difference Equations (2016) 2016:249
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Thus, we have
⎛
iw + a
⎜ ∗
E =  ⎝ –x b e–iw τ
–x∗ b e–iw τ
–k + k S∗ + D∗
iw + k x∗
k x∗
⎞– ⎛ ⎞
–k + k S∗ + D∗
w
⎟ ⎜ ⎟
∗
k x 
⎠ ⎝w ⎠ .
iw + k x∗
w
(.)
Similarly, from (.) and (.), we have
⎛
–a
⎜ ∗
⎝ x b
x∗ b
k – k S ∗ – D ∗
–k x∗
–k x∗
⎞
⎛ ⎞
k – k S ∗ – D ∗
v
⎟
⎜ ⎟
∗
–k x
⎠ E =  ⎝v ⎠ .
–k x∗
v
Then we have
⎛
–a
⎜
E =  ⎝ x∗ b
x∗ b
k – k S ∗ – D ∗
–k x∗
–k x∗
⎞– ⎛ ⎞
k – k S ∗ – D ∗
v
⎟ ⎜ ⎟
–k x∗
⎠ ⎝ v ⎠ .
–k x∗
v
(.)
Hence, we can determine W (θ ) and W (θ ) from (.) and (.). Furthermore, g can
be expressed explicitly. Next, we can compute the following values:
⎧
c () = wi τ (g g – |g | –  |g | ) +
⎪
⎪
⎪
⎪
⎨ μ = – Re(c  ()) ,
Re(λ (τ ))
⎪
β =  Re(c ()),
⎪
⎪
⎪
⎩
 Im(λ (τ ))
T = – Im(c ())+μ
.
w τ
g
,

(.)
It follows that Re(λ (τ )) >  from Lemma .. Thus, Re(c ()) determines the signs of μ
and β . We have the following theorem.
Theorem . If Re(c ()) <  (> ), then the direction of the Hopf bifurcation of the system
(.) at the positive equilibrium E∗ (S∗ , x∗ ) when τ = τ is supercritical (subcritical) and the
bifurcating periodic solutions are stable (unstable).
4 Numerical simulation and discussion
It was shown in [] that the coexistence of two organisms was achieved in the turbidostat.
While it was also shown in [] that the coexistence was achieved if we consider system
(.) with the delay on the dilution rate. In this paper, system (.) with the time delay
of digestion was investigated. By choosing the time delay as the bifurcation parameter
and analyzing the characteristic equation, we obtained sufficient conditions for the local
stability of the positive equilibrium and the existence of a Hopf bifurcation. The direction,
stability, and the other properties of the bifurcating periodic solutions were determined
by the normal form theory and the center manifold theorem. These results show that it is
possible to make the two organisms in the turbidostat coexist.
We next take an example to illustrate our main results. Setting a = ., a = .,
m = , m = , k = ., k = ., and d = ., we consider the following example of sys-
Li et al. Advances in Difference Equations (2016) 2016:249
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Figure 2 The positive equilibrium E ∗ = (0.5750, 0.2127, 0.2123) of (4.1) is asymptotically stable when
.
τ = 29.3 < τ0 = 29.85. Here (S(0), x1 (0), x2 (0)) = (0.5, 0.2, 0.3).
tem (.):
⎧
x (t)S(t)
dS(t)
⎪
⎪
⎨ dt = (. + .x (t) + .x (t))( – S(t)) – .+S(t) –
dx (t)
S(t–τ )
= x (t)[ .+S(t–τ
– . – .x (t) – .x (t)],
dt
)
⎪
⎪
⎩ dx (t) = x (t)[ S(t–τ ) – . – .x (t) – .x (t)].
dt

.+S(t–τ )

x (t)S(t)
,
.+S(t)
(.)

It is easy to verify that the conditions (H ) and (H ) hold, and then we can obtain the
positive equilibrium E∗ = (S∗ , x∗ , x∗ ) = (., ., .). By a simple calculation, we
.
.
λ)
)τ =τ = . × – > . By Theorem ., the
have w = ., τ = ., and ( d(Re
dτ
positive equilibrium E∗ is asymptotically stable when τ = . (or .) < τ (see Figure 
and Figure ). The positive equilibrium E∗ is unstable and a Hopf bifurcation occurs, i.e.,
a bifurcating periodic solution occurs from E∗ when τ = . (or ) > τ (see Figure  and
Figure ).
According to (.), we can compute
.
g = –. – .i,
.
g = . + .i,
.
g = . + .i,
.
g = –. + .i.
Li et al. Advances in Difference Equations (2016) 2016:249
Figure 3 The positive equilibrium E ∗ = (0.5750, 0.2127, 0.2123) of (4.1) is also asymptotically stable
.
when τ = 29.6 < τ0 = 29.85. Here (S(0), x1 (0), x2 (0)) = (0.4, 0.3, 0.3).
Figure 4 The positive equilibrium E ∗ = (0.5750, 0.2127, 0.2123) of (4.1) is unstable and a bifurcating
.
periodic solution occurs from E ∗ when τ = 29.9 > τ0 = 29.85. Here (S(0), x1 (0), x2 (0)) = (0.4, 0.3, 0.2).
Page 16 of 20
Li et al. Advances in Difference Equations (2016) 2016:249
Figure 5 The positive equilibrium E ∗ = (0.5750, 0.2127, 0.2123) of (4.1) is unstable and a bifurcating
.
periodic solution occurs from E ∗ when τ = 31 > τ0 = 29.85. Here (S(0), x1 (0), x2 (0)) = (0.4, 0.3, 0.3).
Figure 6 τ -S(t), τ -x1 (t), and τ -x2 (t) are simulated by numerical investigation of system (4.1). Here
(S(0), x1 (0), x2 (0)) = (0.5, 0.2, 0.3).
Page 17 of 20
Li et al. Advances in Difference Equations (2016) 2016:249
Page 18 of 20
Figure 7 Matlab simulations of system (4.1) when τ = 130 ∈ (τ1 , τ2 ). Here (S(0), x1 (0), x2 (0)) = (0.5, 0.2, 0.3).
Figure 8 Matlab simulations of system (4.1) when τ = 300 ∈ (τ2 , τ3 ). Here (S(0), x1 (0), x2 (0)) = (0.5, 0.2, 0.3).
Li et al. Advances in Difference Equations (2016) 2016:249
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Hence, from (.), we can obtain
.
c () = –. + .i,
.
μ = . > ,
.
β = –. < ,
.
T = . > .
Therefore, when τ = . (or ) ∈ (τ , τ ), μ > , and β < , then the Hopf bifurcation
for system (.) is supercritical, and the stable bifurcating periodic solutions can occur
from the positive equilibrium E∗ (., ., .). From Figure  and Figure , we
can find that the dynamics of system (.) changes when τ is located near τ . Comparing
Figure  and Figure , we can see that the size of τ affects the dynamical behaviors of the
turbidostat model. These are also shown by Figure . By Figures  and , we see that the
bifurcating periodic solutions increase.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the manuscript.
Acknowledgements
The authors thank the anonymous referees and the editor for their careful reading of the original manuscript and their
kind comments and valuable suggestions. This work is supported by the National Nature Science Foundation of China
(Grant No. 11561022 and Grant No. 11261017), the China Postdoctoral Science Foundation (Grant No. 2014M562008).
Received: 31 August 2015 Accepted: 24 August 2016
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