bifurcation analysis of a chemostat model of plasmid bearing and plasmid free competition with pulsed input

Research ArticleBifurcation Analysis of a Chemostat Model of Plasmid-Bearing and Plasmid-Free Competition with Pulsed Input Zhong Zhao, Baozhen Wang, Liuyong Pang, and Ying Chen Departme

Research Article

Bifurcation Analysis of a Chemostat Model of Plasmid-Bearing and Plasmid-Free Competition with Pulsed Input

Zhong Zhao, Baozhen Wang, Liuyong Pang, and Ying Chen

Department of Mathematics, Huanghuai University, Zhumadian, Henan 463000, China

Correspondence should be addressed to Zhong Zhao; zhaozhong8899@163.com

Received 18 April 2014; Revised 17 May 2014; Accepted 24 May 2014; Published 15 June 2014

Academic Editor: Mohamad Alwash

Copyright © 2014 Zhong Zhao 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

A chemostat model of plasmid-bearing and plasmid-free competition with pulsed input is proposed The invasion threshold of the plasmid-bearing and plasmid-free organisms is obtained according to the stability of the boundary periodic solution By use

of standard techniques of bifurcation theory, the periodic oscillations in substrate, plasmid-bearing, and plasmid-free organisms are shown when some conditions are satisfied Our results can be applied to control bioreactor aimed at producing commercial producers through genetically altered organisms

1 Introduction

Biofermentation has become an active area of research on the

continuous cultivation of microorganism in recent years [1–

3] The chemostat is an important laboratory apparatus used

to continuously culturing microorganisms [2–8] It can be

used to investigate microbial growth because the parameters

are measurable, the experiments are reasonable, and the

mathematics is tractable [9]

Fermentations using genetically modified

(recombi-nant) microorganisms typically contain two kinds of

cells-recombinant cells and wild-type cells The former contains

a genetically inserted plasmid (a foreign DNA molecule

that can exist independent of the host chromosome and

can replicate autonomously) which is responsible for the

coding functions that result in the synthesis of a desired

protein Wild-type or plasmid-free cells do not contain this

plasmid and therefore cannot generate the protein

Never-theless, they consume nutrients, grow, and multiply From

this perspective, plasmid-free cells may thus be considered

undesirable, and different methods are employed to check

their proliferation As recombinant (or plasmid-bearing) cells

have to support a larger metabolic load than plasmid-free

cells, their growth rates are smaller In addition, these cells

lose their plasmids during the fermentation process

With the scientific technology, the importance of the genetically altered technology is widely recognized Therefore, it is necessary to understand the dynamic behavior of the fermentation process Xiang and Song [10] analyze a simple chemostat model for plasmid-bearing and plasmid-free organisms with the pulsed substrate and linear functional response They prove that system

is permanent if the impulsive period is less than some critical value Shi et al [11] consider a new Monod type chemostat model with delayed growth response and pulsed input in the polluted environment Normally, the velocity of the enzyme reaction increases with the increase in substrate concentration Some enzymes, however, display the phenomenon of excess substrate inhibition, which means that large amounts of substrate can have the adverse effect and actually slow the reaction down The Monod function does not account for any inhibitory effect at high substrate concentration Therefore, it is crucial to choose a response function showing the excess substrate inhibition Pal et al [12] introduce the Monod-Haldene functional responses into a three-tier model of phytoplankton, zooplankton, and nutrient in order to investigate the phenomenon of excess substrate inhibition Therefore, we introduce the Monod-Haldene functional

Journal of Applied Mathematics

Volume 2014, Article ID 343719, 9 pages

http://dx.doi.org/10.1155/2014/343719

response into the following chemostat model with pulsed

input:

̇𝑆(𝑡) = −𝑄𝑆 − 𝜇1𝑥1𝑆

𝛿 (𝐴 + 𝑆 + 𝐵𝑆2)−

𝜇2𝑥2𝑆

𝛿 (𝐴 + 𝑆 + 𝐵𝑆2),

1(𝑡) = 𝑥1(𝜇1(1 − 𝑞) 𝑆

𝐴 + 𝑆 + 𝐵𝑆2 − 𝑄) ,

2(𝑡) = 𝑥2(𝐴 + 𝑆 + 𝐵𝑆𝜇2𝑆 2 − 𝑄) +𝐴 + 𝑆 + 𝐵𝑆𝑞𝜇1𝑆𝑥1 2,

𝑡 ̸=𝑛𝜏𝑄,

Δ𝑆 = 𝑆0𝜏,

Δ𝑥1= 0, 𝑡 = 𝑛𝜏

𝑄,

Δ𝑥2= 0,

(1)

where Δ𝑆 = 𝑆(𝑛𝜏+/𝑄) − 𝑆(𝑛𝜏/𝑄), Δ𝑥1 = 𝑥1(𝑛𝜏+/𝑄) −

𝑥1(𝑛𝜏/𝑄), Δ𝑥2 = 𝑥2(𝑛𝜏+/𝑄) − 𝑥2(𝑛𝜏/𝑄), and 𝑛 ∈ 𝑍+, 𝑍+ =

{1, 2, } 𝐴 (𝐴 > 0) can be interpreted as the

half-saturation constant in the absence of any inhibitory effect

𝐵 (𝐵 > 0) is the measure of inhibitory effect 𝑆(𝑡) denotes

the nutrient concentration at time𝑡, and 𝑥1(𝑡) denotes the

concentration of plasmid-bearing organism at time𝑡 𝑥2(𝑡)

is the concentration of plasmid-free organism at time𝑡 𝜇1

and 𝜇2 are the uptake constant of the microorganism.𝛿 is

the constant yield (It is reasonable to assume that the yield

constants for two organisms are the same since they are

the same organism just with or without the plasmid) 𝑆0

represents the input concentration of the nutrient each time,

and the probability that a plasmid is lost in reproduction is

represented by 𝑞 (0 < 𝑞 < 1) 𝑄 (0 < 𝑄 < 1) is the washout

proportion of the chemostat each time.𝑛𝜏/𝑄 is the period of

the pulse

The variables in the above system may be rescaled by

measuring𝑥(𝑡) ≡ 𝑆(𝑡)/𝑆0,𝑦(𝑡) ≡ 𝑥1(𝑡)/𝛿𝑆0,𝑧(𝑡) ≡ 𝑥2(𝑡)/𝑆0𝛿,

and𝑡 = 𝑄𝑡, and then we have the following system:

̇𝑥 (𝑡) = −𝑥 − 𝑎 𝑚1𝑥𝑦

1+ 𝑥 + 𝑎2𝑥2 − 𝑚2𝑥𝑧

𝑎1+ 𝑥 + 𝑎2𝑥2,

̇𝑦 (𝑡) = 𝑦 (𝑎𝑚1(1 − 𝑞) 𝑥

1+ 𝑥 + 𝑎2𝑥2 − 1) ,

̇𝑧 (𝑡) = 𝑧 ( 𝑚2𝑥

𝑎1+ 𝑥 + 𝑎2𝑥2 − 1) +𝑎 𝑚1𝑞𝑥𝑦

1+ 𝑥 + 𝑎2𝑥2,

𝑡 ̸= 𝑛𝜏,

Δ𝑥 = 𝜏,

Δ𝑦 = 0, 𝑡 = 𝑛𝜏,

Δ𝑧 = 0,

(2)

where𝑚1= 𝜇1/𝑄, 𝑚2= 𝜇2/𝑄, 𝑎1= 𝐴/𝑆0, 𝑎2= 𝐵𝑆0

2 The Behavior of the Substrate and Plasmid-Free Organism Subsystem

In the absence of the plasmid-bearing organism, system (2) is reduced to

̇𝑥 (𝑡) = −𝑥 − 𝑚2𝑥𝑧

𝑎1+ 𝑥 + 𝑎2𝑥2,

̇𝑧 (𝑡) = 𝑧 ( 𝑚2𝑥

𝑎1+ 𝑥 + 𝑎2𝑥2 − 1) ,

𝑡 ̸= 𝑛𝜏,

Δ𝑥 = 𝜏,

Δ𝑧 = 0, 𝑡 = 𝑛𝜏

(3)

This nonlinear system has a simple periodic solution For our purpose, we present the solution in this section

We add the first and the second equations of (3) We have

𝑑 (𝑥 + 𝑧)

Taking the variable change𝑠 = 𝑥 + 𝑦, the system (3) can be rewritten as

𝑑𝑠

𝑑𝑡 = −𝑠 , 𝑡 ̸= 𝑛𝜏, 𝑠(𝑛𝜏+) = 𝑠 (𝑛𝜏) + 𝜏, 𝑡 = 𝑛𝜏

(5)

Thus, we have the following

Lemma 1 System (5) has a positive periodic solution ̃𝑠(𝑡) and for any solution 𝑠(𝑡) of system (5)|𝑠(𝑡) − ̃𝑠(𝑡)| → 0 as 𝑡 → ∞, where ̃𝑠(𝑡) = (𝜏 exp(−(𝑡−𝑛𝜏))/(1−exp(−𝜏))), 𝑡 ∈ (𝑛𝜏, (𝑛+1)𝜏], and̃𝑠(0+) = 𝜏/(1 − 𝑒−𝜏).

Proof Clearly,̃𝑠(𝑡) is a positive periodic solution of the system (5) Any solution𝑠(𝑡) of system (5) is𝑠(𝑡) = (𝑠(0) − (𝜏/(1 −

𝑒−𝜏)))𝑒−𝑡+̃𝑠(𝑡), 𝑡 ∈ (𝑛𝜏, (𝑛+1)𝜏], 𝑛 ∈ 𝑍+ Hence,|𝑠(𝑡)−̃𝑠(𝑡)| →

0 as 𝑡 → ∞

Lemma 2 Let (𝑥(𝑡), 𝑧(𝑡)) be any solution of system (5) with initial condition 𝑥(0) ≥ 0, 𝑧(0) > 0, and then lim𝑡 → ∞|𝑥(𝑡) +

𝑧(𝑡) − ̃𝑠(𝑡)| = 0.

Lemma (5) shows that the periodic solution ̃𝑠(𝑡) is uniquely invariant manifold of system (3) Therefore, one has0 < 𝑥(𝑡) ≤

𝑀, 0 < 𝑧(𝑡) ≤ 𝑀, where 𝑀 = 𝜏/(1 − 𝑒−𝑇).

Theorem 3 For system (3), one has the following.

(1) If(1/𝜏) ∫0𝜏(𝑚2̃𝑠(𝑙)/(𝑎1+ ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2))𝑑𝑙 < 1, system

(3) has a uniquely globally stable boundary 𝜏-periodic solution(𝑥𝑒(𝑡), 𝑧𝑒(𝑡)), where 𝑥𝑒(𝑡) = ̃𝑠, 𝑧𝑒(𝑡) = 0 (2) If(1/𝜏) ∫0𝜏(𝑚2̃𝑠(𝑙)/(𝑎1+̃𝑠(𝑙)+𝑎2̃𝑠(𝑙)2))𝑑𝑙 > 1, system (3)

has a globally asymptotically stable positive 𝜏-periodic solution(𝑥𝑠(𝑡), 𝑧𝑠(𝑡)), and one has

∫𝜏

0

𝑚2(̃𝑠(𝑡) − 𝑧𝑠(𝑡))

𝑎1+ ̃𝑠(𝑡) − 𝑧𝑠(𝑡) + 𝑎2(̃𝑠(𝑡) − 𝑧𝑠(𝑡))2𝑑𝑡 = 1. (6)

Proof (1) If(1/𝜏) ∫0𝜏(𝑚2̃𝑠(𝑙)/(𝑎1+ ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2))𝑑𝑙 < 1, it is

obvious that

𝑧 (𝑡) ≤ 𝑧 (0) exp ((1𝜏∫𝜏

0

𝑚2̃𝑠(𝑙)

𝑎1+ ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2𝑑𝑙 − 1) 𝑡)

× exp (∫𝑡

0𝑝1(𝑙) 𝑑𝑙) ,

(7)

where 𝑝1(𝑡) = (𝑚2̃𝑠(𝑙)/(𝑎1 + ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2)) −

(1/𝜏) ∫0𝜏(𝑚2̃𝑠(𝑙)/(𝑎1 + ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2))𝑑𝑙 𝑝1(𝑡) is 𝜏-periodic

piecewise continuous function in view of(1/𝜏) ∫0𝜏𝑝1(𝑙)𝑑𝑙 = 0

For(1/𝜏) ∫0𝜏(𝑚2̃𝑠(𝑙)/(𝑎1+ ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2))𝑑𝑙 − 1 < 0, we obtain

𝑧(𝑡) that tends exponentially to zero as 𝑡 → +∞ From

Lemma 2and system (5), we have lim𝑡 → ∞|𝑥(𝑡) − ̃𝑠(𝑡)| = 0

(2) If (1/𝜏) ∫0𝜏(𝑚2̃𝑠(𝑙)/(𝑎1 + ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2))𝑑𝑙 > 1, we

consider system (3) in its stable invariant manifold̃𝑠(𝑡); that

is,

𝑑𝑧

𝑑𝑡 =

𝑚2(̃𝑠(𝑡) − 𝑧 (𝑡)) 𝑧 (𝑡)

𝑎1+ ̃𝑠(𝑡) − 𝑧 (𝑡) + 𝑎2(̃𝑠(𝑡) − 𝑧 (𝑡))2 − 𝑧,

0 ≤ 𝑧0≤ ̃𝑠(0)

(8)

Suppose𝑧(𝑡, 𝑧0) is a solution of (8) with initial condition𝑧0∈

(0, ̃𝑠(0)], and we obtain

𝑧 (𝑡, 𝑧0)

= 𝑧 (𝑛𝜏) exp (∫𝑡

𝑛𝜏( 𝑚2(̃𝑠(𝑡) − 𝑧 (𝑡))

𝑎1+ ̃𝑠(𝑡) − 𝑧 (𝑡) + 𝑎2(̃𝑠(𝑡) − 𝑧 (𝑡))2

−1) 𝑑𝑙) ,

𝑧 (𝑛𝜏) = 𝑧0, 𝑡 ∈ (𝑛𝜏, (𝑛 + 1) 𝜏]

(9) For (9), we have the following properties:

(i) the function𝐹(𝑧0) = 𝑧(𝑡, 𝑧0), 𝑧0 ∈ (0, ̃𝑠(0)] is an

increasing function;

(ii)0 < 𝑧(𝑡, 𝑧0)𝑧 < ̃𝑠(0), 𝑡 ∈ [0, ∞) is a continuous

function;

(iii)𝑧(𝑡, 0) = 0, 𝑡 ∈ [0, ∞) is a solution

The periodic solution of (9) satisfies the following equation:

𝑧0

= 𝑧0exp(∫𝜏

0 ( 𝑚2(̃𝑠(𝑙) − 𝑧 (𝑙))

𝑎1+ ̃𝑠(𝑙) − 𝑧 (𝑙) + 𝑎2(̃𝑠(𝑙) − 𝑧 (𝑙))2 − 1) 𝑑𝑙)

(10) And we denote𝑚∗

2 = (𝜏/ ∫0𝜏(̃𝑠(𝑙)/(𝑎1+ ̃𝑠(𝑙) + 𝑎2̃𝑠(𝑙)2))𝑑𝑙) By (i) and (ii), system (8) has a stable solution𝑧𝑒(𝑡) = 0 for 𝑚2<

𝑚∗2 ByLemma 2, we have lim𝑡 → ∞|𝑥(𝑡) − ̃𝑠(𝑡)| = 0

If𝑚2 > 𝑚∗

2, system (8) has a uniquely positive periodic solution We denote the positive periodic solution

𝑧𝑠(𝑡) = 𝑧 (𝑡, 𝑧∗0) , 𝑥𝑠(𝑡) = ̃𝑠(𝑡) − 𝑧 (𝑡, 𝑧∗0) (11) From (9), we obtain

∫𝜏

0

𝑚2(̃𝑠(𝑡) − 𝑧𝑠(𝑡))

𝑎1+ ̃𝑠(𝑡) − 𝑧𝑠(𝑡) + 𝑎2(̃𝑠(𝑡) − 𝑧𝑠(𝑡))2𝑑𝑡 = 1, (12) where𝑧∗0 = 𝑧𝑠(0)

In order to prove the stability of the periodic solution

𝑧𝑠(𝑡), we define a function 𝐹 : (𝑡, 𝑧0) → 𝑅 ∈ [0, ∞)×[0, ̃𝑠(0)]

as the following:

𝐹 (𝑧 (𝑡, 𝑧0)) = ∫𝑡

0

𝑚2(̃𝑠(𝑡) − 𝑧𝑠(𝑡))

𝑎1+ ̃𝑠(𝑡) − 𝑧𝑠(𝑡) + 𝑎2(̃𝑠(𝑡) − 𝑧𝑠(𝑡))2𝑑𝑡 − 𝑡.

(13) Noticing (8), we have

𝐹 (𝑧 (𝜏, 𝑧0)) = ln (𝑧 (𝜏, 𝑧𝑧 0)

0 ) , 𝑧0∈ (0, ̃𝑠(0)] , (14) and it is obvious that𝐹(𝑧(𝑛𝜏), 𝑧0∗) = 0 For any 𝑧0∈ (0, ̃𝑠(0)],

by the theorem on the differentiability of the solutions on the initial values,𝜕𝑧(𝑡, 𝑧0)/𝜕𝑧0exists Furthermore,𝜕𝑧(𝑡, 𝑧0)/𝜕𝑧0 holds for 𝑡 ∈ (0, ∞) (otherwise, there exists 0 < 𝑧1 <

𝑧2 < ̃𝑠(0) such that 𝑧(𝑡0, 𝑧1) = 𝑧(𝑡0, 𝑧2) for 𝑡1 > 0, which

is a contradiction to the different flows of system (8) not to intersect, and we havẽ𝑠(𝑙) > 𝑧(𝑙, 𝑧0) for 𝑙 ∈ [0, 𝜏] So, we obtain that

𝑑𝐹 (𝑧 (𝜏, 𝑧0))

Therefore,𝐹(𝑧(𝜏, 𝑧0)) is monotonously decreasing con-tinuous function for𝑧0 ∈ [0, ̃𝑠(0)] Now, we choose 𝜀 such

as0 < 𝜀 < 𝑧∗0 < ̃𝑠(0), and we have the following cases:

ln𝑧 (𝜏, 𝑧0) − ln 𝑧0< 0, if 𝑧0∗< 𝑧0< ̃𝑠(0) ,

ln𝑧 (𝜏, 𝑧0) − ln 𝑧0 = 0, if 𝑧∗0 = 𝑧0,

ln𝑧 (𝜏, 𝑧0) − ln 𝑧0> 0, if 𝜀 < 𝑧0< 𝑧∗0

(16)

Furthermore, we obtain the following equations:

𝑧0> 𝑧 (𝜏, 𝑧0) > ⋅ ⋅ ⋅ 𝑧 (𝑛𝜏, 𝑧0) > 𝑧0∗, if 𝑧0∗< 𝑧0≤ ̃𝑠(0) ,

𝑧0< 𝑧 (𝜏, 𝑧0) < ⋅ ⋅ ⋅ 𝑧 (𝑛𝜏, 𝑧0) < 𝑧0∗, if 𝜀 < 𝑧0< 𝑧∗0

(17) Suppose that

lim

𝑛 → ∞𝑧 (𝑛𝜏, 𝑧0) = 𝑎 (18)

We will prove that the solution𝑧(𝑡, 𝑎) is 𝜏-periodic Because the function 𝑧𝑛(𝑡) = 𝑧(𝑡 + 𝑛𝜏, 𝑧0) is also a solution of system (8) and 𝑧𝑛(0) → 𝑎 as 𝑛 → ∞, we have that 𝑧(𝜏, 𝑎) = lim𝑛 → ∞𝑧𝑛(𝜏) = 𝑎 by the continuous dependence of

the solutions on the initial values Hence, the solution𝑧(𝑡, 𝑎)

is 𝜏-periodic The periodic solution 𝑧(𝑡, 𝑧∗

0) is unique and

𝑎 = 𝑧∗

0 Let𝜀 > 0 be given, by the theorem on the continuous

dependence of the solutions on the initial values, there exists

a𝛿 > 0 such that

󵄨󵄨󵄨󵄨𝑧(𝑡,𝑧0) − 𝑧 (𝑡, 𝑧∗0)󵄨󵄨󵄨󵄨 < 𝜀, (19)

if|𝑧0 − 𝑧∗

0| < 𝛿 and 0 ≤ 𝑡 ≤ 𝜏 Choose 𝑛1 > 0 such as

|𝑧(𝑛𝜏, 𝑧0) − 𝑧0∗| < 𝛿 for 𝑛 > 𝑛1 Then,|𝑧(𝑡, 𝑧0) − 𝑧(𝑡, 𝑧0∗)| < 𝜀

for𝑡 > 𝑛𝜏, which shows that

lim

𝑛 → ∞󵄨󵄨󵄨󵄨𝑧(𝑡,𝑧0) − 𝑧 (𝑡, 𝑧0∗)󵄨󵄨󵄨󵄨 = 0, 𝑧0∈ (0, ̃𝑠(0)] (20)

For system (3), we obtain lim𝑡 → ∞|𝑥−𝑥𝑠| = 0 and lim𝑡 → ∞|𝑧−

𝑧𝑠| = 0 for any solution (𝑥(𝑡), 𝑧(𝑡)) with the initial condition

𝑥(0) ≥ 0, 𝑧(0) > 0 From the 𝜏-periodic solution 𝑧𝑠 being

asymptotically stable, we obtain the multiplier𝜇 of 𝑧𝑠, which

satisfies

𝜇 = exp (∫𝜏

0

𝑚2𝑥𝑠(𝑙) (𝑎1− 𝑎2𝑥2𝑠(𝑙)) (𝑎1+ 𝑥𝑠(𝑙) + 𝑎2𝑥2

𝑠(𝑙))2𝑑𝑙) < 1. (21) This conclusion will be used inSection 3

3 The Bifurcation of the System

In order to investigate the properties of system (2), we add

the first, second, and third equations of system (2) and take

variable change𝑠 = 𝑥 + 𝑦 + 𝑧, and the following lemma is

obvious

Lemma 4 Let 𝑋(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) be any solution of

system (2) with 𝑋(0) > 0, and then

lim

𝑡 → ∞󵄨󵄨󵄨󵄨𝑥(𝑡) + 𝑦(𝑡) + 𝑧(𝑡) − ̃𝑠(𝑡)󵄨󵄨󵄨󵄨 = 0 (22)

For convenience, we suppose𝑚2> 𝑚∗

2 and denote that

(1 − 𝑞) ∫0𝜏(𝑧𝑠(𝑙) / (𝑎1+ 𝑧 (𝑙) + 𝑎2𝑧2

𝑠(𝑙))) 𝑑𝑙. (23)

Theorem 5 Let (𝑥(𝑡), 𝑦(𝑡)𝑧(𝑡)) be any solution of system (2)

with 𝑋(0) > 0, and we obtain the following.

(1) If 𝑚2 < 𝑚∗

2, then system (2) has a unique glob-ally asymptoticglob-ally stable boundary 𝜏-periodic solution (̃𝑠(𝑡), 0, 0).

(2) If 𝑚2 > 𝑚∗

2 and 𝑚1 < 𝑚∗

1, then system (2)

has a unique globally asymptotically stable boundary 𝜏-periodic solution (𝑥𝑠(𝑡)), 0, 𝑧𝑠(𝑡).

(3) If𝑚2 > 𝑚∗2 and𝑚1 > 𝑚∗1, then boundary 𝜏-periodic solution(𝑥𝑠(𝑡)), 0, 𝑧𝑠(𝑡) of system (2) is unstable Proof The proof of (1) is easy; we want to prove (2) and (3).

The local stability of periodic solution(𝑥𝑠(𝑡), 0, 𝑧𝑠(𝑡)) may be determined by considering the behavior of small amplitude

of the solution Define

𝑥 (𝑡) = 𝑤 (𝑡) + 𝑥𝑠(𝑡) , 𝑦 (𝑡) = V (𝑡) , 𝑧 (𝑡) = 𝑤 (𝑡) + 𝑧𝑠(𝑡)

(24) There may be written

(𝑢 (𝑡)

V (𝑡)

𝑤 (𝑡)) = Φ (𝑡) (

𝑢 (0)

V (0)

𝑤 (0)) , 0 ≤ 𝑡 ≤ 𝜏, (25) whereΦ(𝑡) = {𝜑𝑖𝑗}𝑖,𝑗=1,2,3satisfies

𝑑Φ (𝑡)

( ( ( (

−1 −𝑚2(𝑎1− 𝑎2𝑥𝑠) 𝑧𝑠 (𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2 −

𝑚1𝑥𝑠

𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠 −𝑎 𝑚2𝑥𝑠

1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠

𝑚2(𝑎1− 𝑎2𝑥𝑠) 𝑧𝑠 (𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2 −1 +

𝑚2𝑥𝑠

𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠

𝑚1𝑞𝑥𝑠

𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠

1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠

) ) ) )

andΦ(0) = 𝐼 is the identity matrix Hence, the fundament solution matrix is

Φ (𝜏) = (

0( 𝑚1(1 − 𝑞) 𝑥𝑠(𝑙)

𝑎1+ 𝑥𝑠(𝑙) + 𝑎2𝑥2

𝑠(𝑙)− 1) 𝑑𝑙)

and there is no need to calculate the exact form(∗) and ∗∗ as

it is not required in the analysis that follows

The eigenvalues of the matrix Φ(𝜏) are 𝜇3 =

exp(∫0𝜏((𝑚1(1 − 𝑞)𝑥𝑠(𝑙)/(𝑎1+ 𝑥𝑠(𝑙) + 𝑎2𝑥2

𝑠(𝑙))) − 1)𝑑𝑙), 𝜇1, and

𝜇2, where𝜇1and𝜇2are determined by the following matrix:

Φ (𝜏) = (𝜑11(𝜏) 𝜑12(𝜏)

𝜑21(𝜏) 𝜑22(𝜏)) (28)

𝜇1 and𝜇2 are also the multipliers of the local linearization

system of (3) According toTheorem 3, we have that𝜇1 < 1

and𝜇2< 1

If 𝑚2 > 𝑚∗2 and𝑚1 < 𝑚∗1 and 𝜇3 = exp(∫0𝜏((𝑚1(1 −

𝑞)𝑥𝑠(𝑙)/(𝑎1 + 𝑥𝑠(𝑙) + 𝑎2𝑥2

𝑠(𝑙))) − 1)𝑑𝑙) < 1, the boundary periodic solution (𝑥𝑠(𝑡), 0, 𝑧𝑠(𝑡)) of system (2) is locally

asymptotically stable In view of𝑦(𝑡) ≤ 𝑦(0) exp(∫0𝑡((𝑚1(1 −

𝑞)𝑥𝑠(𝑙)/(𝑎1+ 𝑥𝑠(𝑙) + 𝑎2𝑥2

𝑠(𝑙))) − 1)𝑑𝑙), we obtain 𝑦(𝑡) → 0 as

𝑡 → ∞

From lim𝑡 → ∞|𝑥(𝑡) + 𝑦(𝑡) + 𝑧(𝑡) − ̃𝑠(𝑡)| = 0, we have

lim𝑡 → ∞|𝑥(𝑡) + 𝑧(𝑡) − ̃𝑠(𝑡)| = 0 Now, usingTheorem 3, we

have lim𝑡 → ∞|𝑧(𝑡) − ̃𝑧𝑠(𝑡)| = 0 and lim𝑡 → ∞|𝑥(𝑡) − ̃𝑥𝑠(𝑡)| = 0

If 𝑚2 > 𝑚∗

2, 𝑚1 > 𝑚∗

1 and 𝜇3 = exp(∫0𝜏((𝑚1(1 − 𝑞)𝑥𝑠/(𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)) − 1)𝑑𝑙) > 1, the boundary periodic solution(𝑥𝑠(𝑡), 0, 𝑠𝑠(𝑡)) of system (2) is unstable The proof is

completed

Let𝐵 denote the Banach space of continuous, 𝜏-periodic

function 𝑁 : [0, 𝜏] → 𝑅2 In the set 𝐵, we introduce

the norm |𝑁|0 = sup0≤𝑡≤𝜏|𝑁(𝑡)| with which 𝐵 becomes

a Banach space with the uniform convergence topology

For convenience, just like [13], we introduce the following

Lemmas6and7

Lemma 6 Suppose 𝑎𝑖𝑗∈ 𝐵.

(a) If ∫0𝜏𝑎22(𝑠)𝑑𝑠 ̸= 0, ∫0𝜏𝑎11(𝑠)𝑑𝑠 ̸= 0, then the linear

homogenous system

𝑑𝑦1

𝑑𝑡 = 𝑎11𝑦1+ 𝑎12𝑦2,

𝑑𝑦2

𝑑𝑡 = 𝑎22𝑦1,

(29)

has no nontrivial solution in 𝐵 × 𝐵 In this case, the

nonhomo-geneous system

𝑑𝑦1

𝑑𝑡 = 𝑎11𝑦1+ 𝑎12𝑦2+ 𝑓1,

𝑑𝑦2

𝑑𝑡 = 𝑎22𝑦1+ 𝑓2,

(30)

has, for every(𝑓1, 𝑓2) ∈ 𝐵 × 𝐵, a unique solution (𝑥1, 𝑥2) ∈

𝐵 × 𝐵 and the operator 𝐿 : 𝐵×𝐵 → 𝐵×𝐵 defined by (𝑥1, 𝑥2) =

𝐿(𝑓1, 𝑓2) is linear and compact If we define that 𝑥󸀠2= 𝑎22𝑥2+𝑓2

has a unique solution𝑥2 ∈ 𝐵 and the operator 𝐿2 : 𝐵 → 𝐵

defined by𝑥2 = 𝐿2𝑓2 is linear and compact Further,𝑥󸀠

1 =

𝑎11𝑥1+𝑓3(𝑓3∈ 𝐵) has a unique solution (since ∫0𝜏𝑎11(𝑠)𝑑𝑠 ̸= 0)

in 𝐵 and 𝑥1= 𝐿3𝑓3define a linear, compact operator𝐿1: 𝐵 →

𝐵 Then, we have

Ł(𝑓1, 𝑓2) ≡ (𝐿1(𝑎12𝐿2𝑓2) , 𝐿2𝑓2) (31)

(b) If∫0𝜏𝑎22(𝑠)𝑑𝑠 = 0, ∫0𝜏𝑎11(𝑠)𝑑𝑠 ̸= 0, then (29) has exactly one independent solution in 𝐵 × 𝐵.

Lemma 7 Suppose 𝑎 ∈ 𝐵 and (1/𝜏) ∫0𝜏𝑎(𝑙)𝑑𝑙 = 0 Then,

𝑥󸀠

1 = 𝑎𝑥 + 𝑓(𝑓 ∈ 𝐵) has a solution 𝑥 ∈ 𝐵 if and only if

(1/𝜏) ∫0𝜏𝑓(𝑙)(exp(− ∫0𝑙𝑎(𝑠)𝑑𝑠)𝑑𝑙) = 0.

By Lemma 7 , in its invariant manifold ̃𝑠= 𝑥(𝑡)+𝑦(𝑡)+𝑧(𝑡), system (2) is reduced to a equivalently nonautonomous system

as the following:

̇𝑥 (𝑡) = −𝑥 − 𝑎 𝑚1𝑥𝑦

1+ 𝑥 + 𝑎2𝑥2 −𝑚𝑎2𝑥 (̃𝑠− 𝑥 − 𝑦)

1+ 𝑥 + 𝑎2𝑥2 ,

̇𝑦 (𝑡) = 𝑦 (𝑎𝑚1(1 − 𝑞) 𝑥

1+ 𝑥 + 𝑎2𝑥2 − 1) , 𝑡 ̸= 𝑛𝜏,

Δ𝑥 = 𝜏,

Δ𝑦 = 0, 𝑡 = 𝑛𝜏,

(32)

and if𝑚2> 𝑚∗

2, the boundary periodic solution(𝑥𝑠, 0) is locally asymptotically stable provided with𝑚1< 𝑚∗

1and it is unstable provided with𝑚1 > 𝑚∗1; hence𝑚∗1 practices as a bifurcation threshold For system (32), we have the following result.

Theorem 8 If 𝑚2 > 𝑚∗

2 and𝑎1− 2𝑎2𝑥2

𝑠 ≥ 0 hold, then there exists a constant𝜆0, such that𝑚1∈ (𝑚∗

1, 𝑚∗

1 + 𝜆0), and there exists a solution (𝑥, 𝑦) satisfying 0 < 𝑥 < 𝑥𝑠, 𝑦 > 0, 𝑧 = ̃𝑠 −

𝑥 − 𝑦 > 0 Hence, system (2) has a positive 𝜏-periodic solution (𝑥, 𝑦, ̃𝑠− 𝑥 − 𝑦).

Proof Let𝑥1= 𝑥 − 𝑥𝑠, 𝑥2= 𝑦, and then system (32) becomes

𝑑𝑥1

𝑑𝑡 = 𝐹11(𝑥𝑠, ̃𝑠) 𝑥1+ 𝑥2+ 𝑔1(𝑥1, 𝑥2) ,

𝑑𝑥2

𝑑𝑡 = 𝐹22(𝑚1, 𝑥𝑠) 𝑥2+ 𝑔2(𝑥1, 𝑥2) ,

(33)

where

𝐹11(𝑥𝑠, ̃𝑠) = 𝑚𝑎2(̃𝑠− 𝑥𝑠)

1− 2𝑎2𝑥2

𝑠 −𝑚2𝑥𝑠(𝑎1− 2𝑎2𝑥

2

𝑠) (𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2

− 1 − 𝑚2̃𝑠𝑥𝑠(1 + 2𝑎2𝑥𝑠) (𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2,

𝐹12(𝑚1, 𝑥𝑠, ̃𝑠) = 𝑚2𝑥𝑠(𝑎1− 2𝑎2𝑥

2

𝑠) (𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2 +

𝑚2̃𝑠𝑥𝑠(1 + 2𝑎2𝑥𝑠) (𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2

(𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2,

𝐹22(𝑚1, 𝑥𝑠) = (1 − 𝑞) 𝑚1𝑥𝑠

(𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2 − 1.

(34)

1.2

1

0.8

x

t

(a)

t

y

6e − 13 5e − 13 4e − 13 3e − 13 2e − 13 1e − 13 0

(b)

t

z

4e − 05

3e − 05

2e − 05

1e − 05

0

(c)

z

1

0.8

0.8

0.6 0.4

0.4

0.2 0

1

0.8 0.6 0.4 0.2 0

0

1.2

(d) Figure 1: Time series of the system (2) with pulse.𝑎1= 0.2, 𝑎2= 0.2, 𝑚1= 1.3, 𝑞 = 0.16, and 𝑇 = 0.5

We know that∫0𝜏(((1 − 𝑞)𝑚1𝑥𝑠(𝑙)/(𝑎1+ 𝑥𝑠(𝑙) + 𝑎2𝑥2

𝑠)2(𝑙))) − 1)𝑑𝑙 ̸= 0, and, byLemma 7, we can equivalently write system

(33) as the operator equation

(𝑥1, 𝑥2) = 𝐿∗(𝑥1, 𝑥2) + 𝐺 (𝑥1, 𝑥2) , (35)

where

𝐺 (𝑥1, 𝑥2) = 𝐿1(𝐹12(𝑥𝑠, ̃𝑠) 𝑔2(𝑥1, 𝑥2) + 𝑔1(𝑥1, 𝑥2) ,

𝐿2𝑔2(𝑥1, 𝑥2)) (36) Here, 𝐿∗ : 𝐵 × 𝐵 → 𝐵 × 𝐵 is linear and compact 𝐺 :

𝐵 × 𝐵 → 𝐵 × 𝐵 is continuous and compact (since 𝐿1

and 𝐿2 are compact) and satisfies𝐺 = 𝑜(|(𝑥1, 𝑥2)|0) near

(0, 0) A nontrivial solution (𝑥1, 𝑥2) ̸= (0, 0) for some 𝑚1 > 1

yields a solution(𝑥, 𝑦) = (𝑥𝑠+ 𝑥1, 𝑥2) ̸= (0, 0) of system (32)

Thus, the existence of the periodic solution of system (2)

can be considered as the bifurcation problem of system (35)

Now, we apply local bifurcation technique to (35) As is well

known, bifurcation can occur only at the nontrivial solution

of linearized problem

(𝑦1, 𝑦2) = 𝐿∗(𝑦1, 𝑦2) , 𝑚1> 0 (37)

If𝑦1× 𝑦2∈ 𝐵 × 𝐵 is a solution of (37) for some𝑚1> 0, then (𝑦1, 𝑦2) satisfies

𝑑𝑦1

𝑑𝑡 = 𝐹11(𝑥𝑠, ̃𝑠) 𝑦1+ 𝐹12(𝑥𝑠, ̃𝑠) 𝑦2,

𝑑𝑦2

𝑑𝑡 = 𝐹22(𝑥𝑠) 𝑦2.

(38)

By virtue ofLemma 7(b) and system (38), system (37) has one nontrivial solution in𝐵 × 𝐵 if and only if 𝑚∗

1 = 𝑚1 Therefore,

we obtain one piecewise continuous periodic solution of system (35), which is all nontrivial solutions expect for (𝑚∗

1, 0, 0) Now, we investigate the solution (𝑚1, 𝑥1, 𝑥2) near the bifurcation point(𝑚∗

1, 0, 0) by expanding 𝑚1and𝑥1, 𝑥2in

a Lyapunov-Schmidt series:

𝑚1= 𝑚∗1 + 𝜆𝜀 + ⋅ ⋅ ⋅ ,

𝑥1= 𝑥11𝜀1𝑥12𝜀2+ ⋅ ⋅ ⋅ ,

𝑥2= 𝑥21𝜀 + 𝑥22𝜀2+ ⋅ ⋅ ⋅ ,

(39)

0.8

0.6

0.4

x

t

(a)

t

y

6e − 17 7e − 17

5e − 17 4e − 17 3e − 17 2e − 17 1e − 17 0

(b)

t

0.398

0.396

0.394

0.392

0.39

z

(c)

z

1

0.8 0.6 0.4 0.2 0

0.8 0.4

0

0

1.2

(d) Figure 2: Time series of the system (2) with pulse.𝑎1= 0.2, 𝑎2= 0.2, 𝑚1= 1.3, 𝑚2= 1.5, 𝑞 = 0.2, and 𝑇 = 0.8

where𝑥𝑖𝑗 ∈ 𝐵, 𝜀 is a small parameter If we substitute those

series into the system (32) and equate coefficients of𝜀 and 𝜀2,

we find that

𝑥󸀠11= 𝐹11(𝑥𝑠, ̃𝑠) 𝑥11+ 𝐹12(𝑚∗1, 𝑥𝑠, ̃𝑠) 𝑥21,

𝑥21󸀠 = 𝐹22(𝑚∗1, 𝑥𝑠) 𝑥21,

𝑥󸀠12= 𝐹11(𝑥𝑠, ̃𝑠) 𝑥12+ 𝐹12(𝑚∗1, 𝑥𝑠, ̃𝑠) 𝑥22,

𝑥󸀠22= 𝐹22(𝑚∗1, 𝑥𝑠) 𝑥22+ 𝑥21(1 − 𝑞)

𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠

× (𝜆 + 𝑚

∗

1𝑥11(𝑎1− 2𝑎2𝑥𝑠2)

𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠 )

(40)

Thus,(𝑥1, 𝑥2) ∈ 𝐵 × 𝐵 must exist a solution of (37) Choose

the specific solution such as the initial condition𝑥21(0) = 1,

and then we have

𝑥21= exp (∫𝑡

0((1 − 𝑞) 𝑚∗1𝑥𝑠(𝑙)

𝑎1+ 𝑥𝑠(𝑙) + 𝑥2

𝑠(𝑙)− 1) 𝑑𝑙) > 0,

(41)

𝑥11< 0 for all 𝑡 (since 𝑚2> 𝑚∗

2and (14),∫0𝜏((𝑚2(̃𝑠− 𝑥𝑠)/(𝑎1− 2𝑎2𝑥2

𝑠))−(𝑚2𝑥𝑠(𝑎1−2𝑎2𝑥2

𝑠)/(𝑎1+𝑥𝑠+𝑎2𝑥2

𝑠)2) – 1−(𝑚2̃𝑠𝑥𝑠(1+ 2𝑎2𝑥𝑠)/(𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2))𝑑𝑙 = − ∫0𝜏((𝑚2𝑥𝑠(𝑎1− 2𝑎2𝑥2

𝑠)/(𝑎1+

𝑥𝑠+ 𝑎2𝑥2

𝑠)2) + (𝑚2̃𝑠𝑥𝑠(1 + 2𝑎2𝑥𝑠)/(𝑎1+ 𝑥𝑠+ 𝑎2𝑥2

𝑠)2))𝑑𝑙 < 0 implies that Green’s function for the first equation in (38) is positive) FromLemma 7, we obtain that

𝜆 = − ((∫𝜏

0

𝑚∗

1𝑥11(𝑡) 𝑥21(𝑡) (𝑎1− 𝑎2𝑥2

𝑠(𝑡)) (𝑎1+ 𝑥𝑠(𝑡) + 𝑎2𝑥2

𝑠(𝑡))2

× exp (∫𝑡

0( (1 − 𝑞) 𝑚∗1𝑥𝑠(𝑙)

𝑎1+ 𝑥𝑠(𝑙) + 𝑎2+ 𝑥2

𝑠(𝑙) − 1) 𝑑𝑙) 𝑑𝑡)

× (∫𝜏

0

𝑥21(𝑡)

𝑎1+ 𝑥𝑠(𝑡) + 𝑎2𝑥2

𝑠(𝑡)

0.6

0.4

0.2

x

t

(a)

t

0.104 0.103 0.102 0.101 0.1 y

(b)

z

0.416

0.414

0.412

0.41

0.408

0.406

0.404

0.402

0.4

0.398

t

(c)

0.4

0.35 0.3 0.25 0.2 0.15 0.1 0.2 0.4 0.6 0.8 1

1.2 0.04 0.06

0.08 0.1 y

z

x

(d) Figure 3: Time series of the system (2) with pulse.𝑎1= 0.2, 𝑎2= 0.2, 𝑚1= 2, 𝑚2= 1.5, 𝑞 = 0.2, and 𝑇 = 0.8

× exp (∫𝑡

0( (1 − 𝑞) 𝑚∗1𝑥𝑠(𝑙)

𝑎1+ 𝑥𝑠(𝑙) + 𝑎2+ 𝑥2

𝑠(𝑙) − 1) 𝑑𝑙) 𝑑𝑡)

−1

) > 0,

(42)

provided with𝑎1 − 2𝑎2𝑥2

𝑠 ≥ 0, which shows that near the bifurcation point(𝑚∗1, 0, 0), there exists a constant 𝜆0, such

that𝑚1 ∈ (𝑚∗1, 𝑚∗1 + 𝜆0) Thus, system (29) has a solution

(𝑥, 𝑦) ∈ (𝐵 × 𝐵), 𝑦 > 0 Next, we have only to show that

𝑥 = 𝑥1 + 𝑥𝑠 > 0 for all 𝑡 > 0; that is, if 𝜆0 is small,

then𝑥 is near 𝑥𝑠in the sup norm of𝐵 Since 𝑥𝑠is bounded

away from zero, so is𝑥 According toTheorem 5, for system

(2),𝑥 is near 𝑥𝑠, which implies that𝑧 is near 𝑧𝑠 We notice

that the period of the periodic solution(𝑥, 𝑦) is 𝜏; therefore,

𝑧 = ̃𝑠− 𝑥 − 𝑦 > 0 is piecewise continuous and 𝜏-periodic The

proof is completed

4 Numerical Simulations

In order to justify our theoretic results, we will give some numerical simulations

Let the parameters of system (2) be 𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 1.3, 𝑚2 = 1.3, 𝑞 = 0.16, and 𝑇 = 0.8 By computing,𝑚∗

2 = 1.404 > 𝑚2 = 1.3 is obtained From

Theorem 5, we know that if𝑚∗

2 > 𝑚2, system (2) has a unique globally asymptotically stable boundary𝜏-periodic solution (̃𝑠(𝑡), 0, 0), which is shown inFigure 1

Suppose that𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 1.3, 𝑚2 = 1.5,

𝑞 = 0.2, and 𝑇 = 0.8 By computing, we have 𝑚∗

2 = 1.410 <

𝑚2 = 1.5, 𝑚∗

1 = 1.763 > 𝑚1 = 1.3 According toTheorem 5, for 𝑚2 > 𝑚∗

2 and 𝑚1 < 𝑚∗

1, system (2) has a unique globally asymptotically stable boundary𝜏-periodic solution (𝑥𝑠(𝑡)), 0, 𝑧𝑠(𝑡), which is demonstrated inFigure 2

Set𝑎1= 0.2, 𝑎2= 0.2, 𝑚1= 2, 𝑚2= 1.5, 𝑞 = 0.2, and 𝑇 = 0.8 By computing, we obtain 𝑚∗1 = 1.763 < 𝑚1 = 2, 𝑚∗2 = 1.410 < 𝑚2 = 1.5 FromTheorem 5, when𝑚2 > 𝑚∗2 and

𝑚1> 𝑚∗1, system (2) exists a positive periodic solution, which

is simulated inFigure 3

5 Discussion

In this paper, we have investigated a competitive model of

plasmid-bearing and plasmid-free organisms in a pulsed

chemostat It is important in biotechnology for the study

of plasmid stability where the effects of plasmid loss in

genetically altered organisms (the plasmid-free organism is

presumably the better competitor) are investigated Firstly,

we find the invasion threshold of the plasmid-free organism,

which is𝑚∗

2 = (𝜏/ ∫0𝜏(̃𝑠(𝑙)/(𝑎1 + ̃𝑠(𝑙) + 𝑎2̃𝑠2(𝑙)))𝑑𝑙) If 𝑚2 <

𝑚∗2, the microorganism-free periodic solution(̃𝑠(0), 0, 0) is

globally asymptotically stable, which is simulated inFigure 1

If𝑚2> 𝑚∗

2, then the plasmid-free organism begins to invade

the system Furthermore, we have proved that if𝑚2 > 𝑚∗

2, there exists𝑚∗

1 = (𝜏/ ∫0𝜏((1−𝑞)𝑥𝑠(𝑙)/(𝑎1+𝑥𝑠(𝑙)+𝑎2𝑥2

𝑠(𝑙)))𝑑𝑙) as the invasion threshold of the plasmid-bearing; that is to say,

if𝑚1 < 𝑚∗

1 the boundary periodic solution(𝑥𝑠(𝑡), 0, 𝑧𝑠(𝑡))

is globally asymptotically stable (seeFigure 2) and if𝑚1 >

𝑚∗

1, the solution(𝑥𝑠(𝑡), 0, 𝑧𝑠(𝑡)) is unstable Lastly by using

standard techniques of bifurcation theory, we prove that if

𝑚2 > 𝑚∗2 and 𝑎1 − 𝑎2𝑥2𝑠 ≥ 0, the system has a positive

𝜏-periodic solution, which is shown inFigure 3

System (2) compared with other literatures is more

interesting since it can be used to model the excess substrate

inhibition in the process of the genetically modified

fermen-tation We show that the plasmid-bearing and plasmid-free

can coexist on a periodic solution The analysis indicates

that the change of one organism into another can

compli-cate the dynamics of system (2) since the plasmid-bearing

concentration is relevant to the substrate concentration and

the substrate concentration can be controlled by setting the

period of impulsive input However, how to find the optimal

period of impulsive input makes microorganisms reach the

maximum production by genetically altered biotechnology,

which is a challenging problem to solve and we will leave it

for the future

Conflict of Interests

The authors declare that there is no conflict of interests

regarding the publication of this paper

Acknowledgments

This work is supported by the National Natural Science

Foundation of China (no 11371164), NSFC-Talent Training

Fund of Henan (no U1304104), the young backbone teachers

of Henan (no 2013GGJS-214), and Henan Science and

Tech-nology Department (nos 132300410084 and 132300410250)

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