# Competition between Plasmid-Bearing and Plasmid-Free Organisms in a Chemostat with Pulsed Input and Washout

Authors: Sanling Yuan; Yu Zhao; Anfeng Xiao
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We consider a model of competition between plasmid-bearing and plasmid-free organisms in the chemostat with pulsed input and washout. We investigate the subsystem with nutrient and plasmid-free organism and study the stability of the boundary periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields the invasion threshold of the plasmid-bearing organism. By using the standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, plasmid-free, and plasmid-bearing organisms. Numerical simulations are carried out to illustrate our results.


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## Body

## 1. Introduction

The chemostat, a laboratory apparatus used for the continuous culture of microorganisms, has played an important role in microbiology and population biology. It is the most simple idealization of a biological system where the parameters are measurable, the experiments are reasonable, and the mathematics is tractable [1]. Experimental verification of the match between theory and experiment in the chemostat can be found in [2]. For a general discussion of competition, see [3, 4] while a detailed mathematical description of competition in the chemostat may be found in [1].The ability to manufacture desired products through genetically altered organisms represents one of the major developments in biotechnology. The genetic alteration commonly takes place through the insertion of a recombinant DNA into the cell in the form of a plasmid to code for the production of the desired protein. The load imposed by production can result in the genetically altered (plasmid-bearing) organism being a less able competitor than the plasmid-free (or “wild-type’’) organism. Unfortunately, the plasmid can be lost in the reproductive process. Since commercial production can take place on a scale of many generations, it is possible for the plasmid-free organism to take over the culture. In pharmaceuticals, changes in the plasmid could cause changes in the amino acid sequence of a protein product or changes in the background from which it must be purified. It is vital to produce a uniform product if it is a drug intended for human use. Since commercial production of products manufactured by genetically altered organisms is a reality, understanding the competition between plasmid-bearing and plasmid-free organisms in a mathematically rigorous fashion seems important. The study of mathematical models for the competition between plasmid-free and plasmid-bearing populations has recently been a problem of considerable interest. We refer the readers to Ryder and BiBiasio [5], Stephanopoulos and Lapudis [6], Hsu et al. [7–12], Luo and Hsu [13], Lu and Hadeler [14], Ai [15], Yuan et al. [16–18] as well as Xiang and Song [19], Wu et al. [20], and the references therein for recent studies on this respect.To simulate the effect of perturbations such as seasonal or other variation in the chemostat, the chemostat models described in impulsive differential equations have been studied by many authors, see, for example, Funasaki and Kot [21], Xiang and Song [19], Wang et al. [22], Smith and Wolkowicz [23, 24], Fan and Wolkowicz [25], and the references therein for recent studies on this respect. Recently impulsive differential equations have been introduced in almost every domain of applied sciences. Numerous examples are given in Baĭnov et al. work [26, 27]. In this paper, we focus on a model of competition between plasmid-bearing and plasmid-free organisms in the chemostat with periodically pulsed nutrient input and washout. Assuming that the specific growth rates of the organisms take the form of Holling II type, we want to explore if some new dynamical behaviors could occur in comparison with the model with constant input and washout and under what conditions can both the plasmid-bearing and plasmid-free organisms coexist in the chemostat.The organization of this paper is as follows. In Section2, we present the model under periodic pulsed chemostat conditions. In Section 3, we investigate the existence and stability of the periodic solutions of the impulsive subsystem with nutrient and plasmid-free organism. In Section 4, we study the local stability of the boundary periodic solution of the system and obtain the threshold of the invasion of the plasmid-bearing organism. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, plasmid-bearing, and plasmid-free organisms. Finally, numerical simulations and a brief discussion are presented in Section 5.


## 2. The Model

LetS(t) be the concentration of nutrient at time t, let x1(t) be the concentration of the plasmid-free organisms at time t, and let x2(t) be the concentration of the plasmid-bearing organisms at time t. The model of competition between plasmid-bearing and plasmid-free organisms in the chemostat with pulsed input and washout can be described by the following impulsive differential equations:
(2.2)dS(t)dt=-m1δ1S(t)x1(t)a1+S(t)-m2δ2S(t)x2(t)a2+S(t),dx1(t)dt=m1S(t)x1(t)a1+S(t)+qm2S(t)x2(t)a2+S(t),dx2(t)dt=(1-q)m2S(t)x2(t)a2+S(t),t≠nτ,ΔS(t)=D(S0-S(t)),Δx1(t)=-Dx1(t),Δx2(t)=-Dx2(t),t=nτ,S(0+)=S0>0,x1(0+)=x10>0,x2(0+)=x20>0,
where n∈N, N is the set of positive integers. All parameters are positive constants. However, q, where 0<q<1, is the fraction constant of plasmid-bearing population converting into plasmid-free population during the replication; mi,ai, i=1,2 are the maximal growth rates of plasmid-free and plasmid-bearing organisms and the Michaelis-Menten (or half saturation) constants, respectively; δi, i=1,2 are the yield constants, biologically one may assume δ1=δ2=δ. The operating parameters are S(0), D and τ, where S(0) is the input concentration of the nutrient, D is the input and washout flow of the chemostat, and τ is the period of the impulsive effects.It is convenient to perform scaling for chemostat-type problems. To avoid even more complicated parameter dependence than what we will see below, we assume that the yield constants are equal, that is,δ1=δ2=δ. Without this assumption, one has an additional parameter, the ratio of the yield constants, in the system. Specifically, let
(2.3)S̅=SS0,x̅1=x1δS0,x̅2=x2δS0,m̅i=miS0ai,bi=S0ai.
After dropping the bars, (2.2) becomes
(2.4)dS(t)dt=-m1S(t)x1(t)1+b1S(t)-m2S(t)x2(t)1+b2S(t),dx1(t)dt=m1S(t)x1(t)1+b1S(t)+qm2S(t)x2(t)1+b2S(t),dx2(t)dt=(1-q)m2S(t)x2(t)1+b2S(t),t≠nτ,ΔS(t)=D(1-S(t)),Δx1(t)=-Dx1(t),Δx2(t)=-Dx2(t),t=nτ,S(0+)=S0>0,x1(0+)=x10>0,x2(0+)=x20>0.From the point of biology, we need only to consider (2.4) in the biological meaning region R+3={(S,x1,x2)∣S,x1,x2≥0}. The form of the equations in (2.4) implies the global existence and uniqueness of its solutions.


## 3. Dynamical Behaviors of the Nutrient and Plasmid-Free Organism Subsystem

In the absence of the plasmid-bearing organism, (2.4) reduces to
(3.1)dS(t)dt=-m1S(t)x1(t)1+b1S(t),dx1(t)dt=m1S(t)x1(t)1+b1S(t),t≠nτ,ΔS(t)=D(1-S(t)),Δx1(t)=-Dx1(t),t=nτ,S(0+)=S0>0,x1(0+)=x10>0.
This nonlinear system has simple periodic solutions. For our purpose, we present these solutions in this section.If we add the first and second equations in system (3.1), we have d(S(t)+x1(t))/dt=0. Let Σ(t)=S(t)+x1(t), then (3.1) can be written as
(3.2)dΣ(t)dt=0,t≠nτ,Σ(nτ+)=D+(1-D)Σ(nτ),t=nτ,Σ(0+)=Σ0>0.For system (3.2), we have the following lemma.Lemma 3.1.
However, (3.2) has a positive solution Σ̃(t)=1 and for every solution Σ(t) of (3.2), we have |Σ(t)-1|→0 as t→∞, where Σ̃(t)=1, t∈(nτ,(n+1)τ], n∈N.By Lemma3.1, the following lemma is obvious.Lemma 3.2.
Let (S(t),x1(t)) be any solution of system (3.1) with initial condition S(0)>0, x1(0)>0, then limt→∞|S(t)+x1(t)-1|=0.Lemma3.2 says that the periodic solution Σ̃(t)=1 is a unique invariant manifold of the system (3.1). Denote
(3.3)m1*=-(1+b1)ln(1-D)τ.Theorem 3.3.
For system (3.1), one has the following.  (1)
Ifm1<m1*, then system (3.1) has a unique globally asymptotically stable boundary τ-periodic solution (Se(t),x1e(t)), where Se(t)=1, x1e(t)=0.(2)
Ifm1>m1*, then system (3.1) has a unique globally asymptotically stable positive τ-periodic solution (Ss(t),x1s(t)) and the τ-periodic solution (Se(t),x1e(t)) is unstable. The τ-periodic solution x1s(t) satisfies
(3.4)11-D=exp(∫0τm1(1-x1(l))1+b1(1-x1(l))dl).Proof.
By Lemma3.2, we can consider (3.1) in its stable invariant manifold Σ̃(t)=1. That is,
(3.5)dx1dt=m1(1-x1)x11+b1(1-x1),t≠nτ,Δx1=-Dx1,t=nτ,0≤x10≤1.
Suppose x1(t,x10) is a solution of (3.5), with initial condition x10∈[0,1], we have
(3.6)x1(t,x10)=x1(nτ+)exp(∫nτtm1(1-x1(l,x10))1+b1(1-x1(l,x10))dl),t∈(nτ,(n+1)τ],x1(nτ+)=(1-D)x1(nτ),x1(0+)=x10,t=nτ.
For (3.6), we have the following properties: (i)
forx10∈(0,1], 0<x1(t,x10)≤1, t∈(0,∞) is a piecewise continuous function;(ii)
the functionF(x10)=x1(t,x10), x10∈(0,1] is a increasing function;(iii)
x1(t,0)=0, t∈(0,∞) is a solution.
The periodic solutions of (3.5) satisfies the following equation:
(3.7)x10=(1-D)x10exp(∫0τm1(1-x1(l,x10))1+b1(1-x1(l,x10))dl).
By (i) and (ii), we know that if 1<1/(1-D)<exp(m1τ/(1+b1)), that is, m1>m1*, then (3.7) has a unique solution x10* in (0,1); otherwise, it has no solution in (0,1].
Ifm1<m1*, then (3.5) has a periodic solution x1e(t)=0. By Lemma 3.2, we have limt→∞|S(t)-Σ̃(t)|=0. We have proved (1).
Ifm1>m1*, then (3.5) has a unique positive periodic solution x1s(t)=x1(t,x10*). The multiplier μ1 of x1s(t) is
(3.8)μ1=(1-D)exp(∫0τm1(Ss-x1s+b1Ss2)(1+b1Ss)2dl)=exp(∫0τ-m1x1s(1+b1Ss)2dl)<1,
where Ss(t)=1-x1s(t) and (3.7) has been used. Thus the periodic solution x1s(t) of (3.5) is local stable.
To prove the global attractivity of the periodic solutionx1s(t), we define a function G(x10):x10∈(0,1) as follows:
(3.9)G(x10)=(1-D)exp(∫0τm1(1-x1(l,x10))1+b1(1-x1(l,x10))dl).
Noticing (3.6), we have
(3.10)G(x10)=x1(τ,x10)x10,x10∈(0,1).
It is obvious that G(x10*)=1.
Furthermore,∂x1(t,x10)/∂x10≥0, t∈(0,τ) (otherwise, there exist t0>0, 0<x10(1)<x10(2)<1 such that x1(t0,x10(1))=x1(t0,x10(2)), a contradiction, since different flows of (3.5) do not intersect). Thus, the function G(x10) has the following properties:
(3.11)G(x10)<1,ifx10*<x10<1,G(x10)=1,ifx10=x10*,G(x10)>1,if0<x10<x10*.
Furthermore, we obtain the following inequalities:
(3.12)x10>x1(τ,x10)>⋯>x1(nτ,x10)>⋯>x10*,ifx10*<x10≤1,x10<x1(τ,x10)<⋯<x1(nτ,x10)<⋯<x10*,if0<x10<x10*.
Let x10∈(0,1). According to (3.12), we suppose that
(3.13)limn→∞x1(nτ,x10)=α.
We will prove that the solution x1(t,α) is τ-periodic. We note that the functions x1n(t)=x1(t+nτ,x10), due to the τ-periodicity of (3.5), are also its solutions and x1n(τ)→α, as n→∞. By the continuous dependence of the solutions on the initial values, we have x1(τ,α)=limn→∞x1n(τ)=α. Hence the solution x1(t,α) is τ-periodic. Since the periodic solution x1(t,x10*) is unique, thus we have α=x10*.
Letϵ>0 be given. By [27, Theorem  2.9] on the continuous dependence of the solutions on the initial values, there exist a δ>0 such that if |x10-x10*|<δ and 0≤t≤τ, then
(3.14)|x1(t,x10)-x1(t,x10*)|<ϵ.
Choose n1>0 such that |x1(nτ,x10)-x10*|<δ for n>n1. Then |x1(t,x10)-x1(t,x10*)|<ϵ for t>nτ, which implies that
(3.15)limt→∞|x1(t,x10)-x1(t,x10*)|=0.
By Lemma 3.2, for any solution (S(t),x1(t)) of system (3.1) with initial condition S(0)>0, x1(0)>0, we have
(3.16)limt→∞|x1(t)-x1s(t)|=0,limt→∞|S(t)-Ss(t)|=0.
The proof of Theorem 3.3(2) is thus completed.


## 4. Existence of the Positiveτ-Periodic Solution

In this section, we investigate the invasion of the plasmid-bearing organism of system (2.4). Denote Σ(t)=S(t)+x1(t)+x2(t), it follows from (2.4) that
(4.1)dΣ(t)dt=0,t≠nτ,Σ(nτ+)=D+(1-D)Σ(nτ),t=nτ,Σ(0+)>0.By Lemma3.1, the following lemma is obvious.Lemma 4.1.
Let(S(t),x1(t),x2(t)) be any solution of system (2.4) with initial value S(0)>0, xi(0)>0, i=1,2, then
(4.2)limt→∞|S(t)+x1(t)+x2(t)-1|=0.Lemma4.1 says that the periodic solution Σ̃(t)=1 of (4.1) is an invariant manifold of (2.4).By Theorem3.3, we know that (2.4) has two nonnegative boundary τ-periodic solutions:
(4.3)(Se(t),x1e(t),x2e(t))=(1,0,0),(Ss(t),x1s(t),0)(ifm1>m1*).


### 4.1. Stability of the Boundary Periodic Solutions

For convenience, in the following discussion, ifm1>m1*, we denote
(4.4)m2*=-ln(1-D)∫0τ((1-q)Ss(l)/(1+b2Ss(l)))dl.Theorem 4.2.
For (2.4), one has the following. (1)
Ifm1<m1*, then (2.4) has a unique globally asymptotically stable boundary τ-periodic solution (1,0,0).(2)
Ifm1>m1* and m2<m2*, then (2.4) has a unique globally asymptotically stable boundary τ-periodic solution (Ss(t),x1s(t),0).(3)
Ifm1>m1* and m2>m2*, then the periodic boundary solution (Ss(t),x1s(t),0) of system (2.4) is unstable.Proof.
The proof of (1) is easy, we want to prove (2) and (3). The local stability of periodic solution(Ss(t),x1s(t),0) may be determined by considering the behavior of small amplitude perturbations of the solution. Define
(4.5)S(t)=u(t)+Ss(t),x1(t)=v(t)+x1s(t),x2(t)=w(t),
then we have
(4.6)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),0≤t<τ,
where Φ(t) satisfies
(4.7)dΦ(t)dt=(-m1x1s(1+b1Ss)2-m1Ss1+b1Ss-m2Ss1+b2Ssm1x1s(1+b1Ss)2m1Ss1+b1Ssqm2Ss1+b2Ss00(1-q)m2Ss1+b2Ss)Φ(t),
and Φ(0)=I, the identity matrix. Hence, the fundamental solution matrix is
(4.8)Φ(τ)=(ϕ11(τ)ϕ12(τ)*ϕ21(τ)ϕ22(τ)**00exp(∫0τ(1-q)m2Ss1+b2Ssdl)).
The linearization of impulsive subsystem (2.4) is written as
(4.9)(u(nτ+)v(nτ+)w(nτ+))=(1-D0001-D0001-D)(u(nτ)v(nτ)w(nτ)).
The stability of the periodic solution (Ss(t),x1s(t),0) is determined by the eigenvalues of
(4.10)M=(1-D0001-D0001-D)Φ(τ),
which are μ3=(1-D)exp(∫0τ((1-q)m2Ss/(1+b2Ss))dl) and the eigenvalues μ1, μ2 of the following matrix:
(4.11)(1-D)(ϕ11(τ)ϕ12(τ)ϕ21(τ)ϕ22(τ)).
The μ1, μ2 are also the multipliers of the locally linearization system of (3.1) provided with D∈(0,D0) at the asymptotically stable periodic solution (Ss(t),x1s(t)), where D0=1-exp((1+b1)/m1*τ). According to Theorem 3.3, we have that μ1<1, μ2<1.
Ifm2<m2*, then μ3=(1-D)exp(∫0τ((1-q)m2Ss/(1+b2Ss))dl)<1, the boundary periodic solution (Ss(t),x1s(t),0) of system (2.4) is locally stable. We obtain that
(4.12)x2(t)=x2(0)(1-D)nexp(∫0t(1-q)m2Ss(l)1+b2Ss(l)dl),t∈(nτ,(n+1)τ].
Hence, we obtain that for any solution (S(t),x1(t),x2(t)) with initial value S(0)>0, xi(0)>0, i=1,2, x2(t)→0 as t→∞. By Lemma 4.1, we have limt→∞|S(t)+x1(t)-1|=0. Now using Theorem 3.3, we have limt→∞|S(t)-Ss(t)|=0 and limt→∞|x1(t)-x1s(t)|=0.
Ifm2>m2*, then μ3>1, the boundary periodic solution (Ss(t),x1s(t),0) of (2.4) is unstable. The proof of Theorem 4.2 is completed.


### 4.2. Bifurcation Analysis of the Boundary Periodic Solution(Ss(t),x1s(t),0)

Letℬ denote the Banach space of piecewise continuous, τ-periodic function 𝒩:[0,τ]→R2, and have points of discontinuity τ, where they continuous from the left. In the set ℬ introduce the norm ∥𝒩∥0=sup0≤t≤τ∥𝒩(t)∥ with which ℬ becomes a Banach space with the uniform convergence topology.For convenience, we introduce Lemmas4.3 and 4.5 from Cushing [28] with small modifications.Lemma 4.3.
Supposeaij∈ℬ and 0≤di<1(i=1,2).
(a) If(1-d2)exp(∫0τa22(s)ds)≠1, (1-d1)exp(∫0τa11(s)ds)≠1, then the linear impulsive homogenous system
(4.13)dy1dt=a11y1+a12y2,dy2dt=a22y2,t≠nτ,Δy1=-d1y1,Δy2=-d2y2,t=nτ,
has no nontrivial solution in ℬ×ℬ. In this case, the nonhomogeneous system
(4.14)dx1dt=a11x1+a12x2+f1,dx2dt=a22x2+f2,t≠nτ,Δx1=-d1x1,Δx2=-d2x2,t=nτ,
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.
(b) If(1-d2)exp(∫0τa22(s)ds)=1, (1-d1)exp(∫0τa11(s)ds)≠1, then (4.13) has exactly one independent solution in ℬ×ℬ.Remark 4.4.
In fact, under the conditions of Lemma4.1(a),
(4.15)dx2dt=a22x2+f2,t≠nτ,Δx2=-d2x2,t=nτ,
has a unique solution x2∈ℬ and the operator L2:ℬ→ℬ defined by x2=L2f2 is linear and compact. Furthermore,
(4.16)dx1dt=a11x1+f3,t≠nτ,Δx1=-d1y1,t=nτ,
for f3∈ℬ has a unique solution (since (1-d1)exp(∫0τa11(s)ds)≠1) in ℬ, and x1=L1f3 defines a linear, compact operator L1:ℬ→ℬ. Then, we have
(4.17)L(f1,f2)=(L1(a12L2f2+f1),L2f2).Lemma 4.5.
Suppose thata∈ℬ, 0≤d<1, (1-d)exp(∫0τa(s)ds)=1 and f∈ℬ. Then, the impulsive equation
(4.18)dxdt=ax+f,t≠nτ,Δx=-dx,t=nτ,
has a solution x∈ℬ if and only if ∫0τf(l)exp(-∫0la(s)ds)dl=0.By Lemma4.1, in its invariant manifold Σ̃(t)=S(t)+x1(t)+x2(t)=1, (2.4) reduces to an equivalently nonautonomous system as follows:
(4.19)dx1dt=m1x1(1-x1-x2)1+b1(1-x1-x2)+qm2x2(1-x1-x2)1+b2(1-x1-x2),dx2dt=(1-q)m2x2(1-x1-x2)1+b2(1-x1-x2),t≠nτ,Δx1=-Dx1,Δx2=-Dx2,t=nτ,x10>0,x20>0,x10+x20≤1.
If m1>m1*, for (4.19), by Theorem 4.2, the boundary periodic solution (x1s(t),0) is locally asymptotically stable provided with m2<m2*, hence the value m2* plays an important role as a bifurcation threshold.For system (4.19), we have the following results.Theorem 4.6.
For system (4.19), assume m1>m1* holds, then there exists a constant λ0>0, such that for each m2∈(m2*,m2*+λ0), there exists a solution (x1,x2)∈ℬ×ℬ of (4.19) satisfying 0<x1<x1s, x2>0 and S=1-x1-x2>0 for all t>0. Hence, (2.4) has a positive τ-periodic solution (S,x1,x2).Proof.
Lety1=x1-x1s, y2=x2 in (4.19), then
(4.20)dy1dt=F11(Ss,x1s)y1+F12(m2,Ss,x1s)y2+g1(y1,y2),dy2dt=F22(m2,Ss)y2+g2(y1,y2),t≠nτ,Δy1=-Dy1,Δy2=-Dy2,t=nτ,x10>0,x20>0,x10+x20≤1,
where
(4.21)F11(Ss,x1s)=m1Ss1+b1Ss-m1x1s(1+b1Ss)2,F12(m2,Ss,x1s)=-m1x1s(1+b1Ss)2+qm2Ss1+b2Ss,F22(m2,Ss)=(1-q)m2Ss1+b2Ss.
Since (1-D)exp(∫0τ((1-q)m2Ss(l)/(1+b2Ss(l)))dl)≠1, by Lemma 4.3, using L, we can equivalently write (4.20) as the operator equation:
(4.22)(y1,y2)=L*(y1,y2)+G(y1,y2),
where
(4.23)G(y1,y2)=(L1(F12(m2,Ss,x1s)L2g2(y1,y2)+g1(y1,y2)),L2g2(y1,y2)).
Here L*:ℬ×ℬ→ℬ×ℬ is linear and compact (since L1 and L2 are compact) and satisfies G=o(|(y1,y2)|0) near (0,0). A nontrivial solution (y1,y2)≠(0,0) for some m2>m2* yields a solution (x1,x2)=(x1s+y1,y2) of (4.19). The solution (x1,x2)≠(x1s,0) is called a nontrivial solution of (4.19).
We apply the well-known local bifurcation techniques to (4.22). As is well known, bifurcation can occur only at the nontrivial solution of the linearized problem:
(4.24)(y1,y2)=L*(y1,y2).
If (y1,y2)∈ℬ×ℬ is a solution of (4.24) for some m2>0, then by the very manner in which L* was defined, (y1,y2) solves the system
(4.25)dy1dt=F11(Ss,x1s)y1+F12(m2,Ss,x1s)y2,dy2dt=F22(m2,Ss)y2,t≠nτ,Δy1=-Dy1,Δy2=-Dy2,t=nτ,
and conversely. Using Lemma 4.3(b), we see that (4.25) and hence (4.24) has one nontrivial solution in ℬ×ℬ if and only if m2=m2*. Hence, there exists a continuum C=(m2;y1,y2)⊆(0,∞)×ℬ×ℬ nontrivial solutions of (4.24) such that the closure C¯ contains (m2*;0,0). This continuum gives rise to a continuum C1=(m2;x1,x2)⊆(0,∞)×ℬ×ℬ of the solutions of (4.19) whose closure C¯1 contains the bifurcation point (m2*;x1s,0).
To see that solutions inC1 correspond to solutions (x1,x2) of (4.19), we investigate the nature of the continuum C near the bifurcation point (m2*;0,0) by expending m2 and (y1,y2) in Lyapunov-Schmidt series:
(4.26)m2=m2*+λϵ+⋯,y1=y11ϵ+y12ϵ2+⋯,y2=y21ϵ+y22ϵ2+⋯,
for yij∈ℬ where ϵ is a small parameter. If we substitute these series into the differential system (4.19) and equate coefficients of ϵ and ϵ2, we find that
(4.27)dy11dt=F11(Ss,x1s)y11+F12(m2*,Ss,x1s)y21,dy21dt=F22(m2*,Ss)y21,t≠nτ,Δy11=-Dy11,Δy21=-Dy21,t=nτ,(4.28)dy12dt=F11(Ss,x1s)y12+F12(m2*,Ss,x1s)y22+G12(y11,y12,λ),dy22dt=F22(m2*,Ss)y22+(1-q)y211+b2Ss(λSs-m2*(y11+y21)1+b2Ss),t≠nτ,Δy21=-Dy21,Δy22=-Dy22,t=nτ,
respectively. Thus, (y11,y21)∈ℬ×ℬ must be a solution of (4.22). We choose the specific solution satisfying the initial conditions y21(0)=1. Then
(4.29)y21=exp(∫nτt(1-q)m2*Ss1+b2Ssdl)>0,nτ<t≤(n+1)τ,y21(0)=1.
Moreover, y11<0 for all t (since m1>m1* and (3.8), hence -∫0τ(m1x1s(l)/(1+b1Ss(l))2)dl<0, which implies that the Green's function for first equation in (4.24) is positive). Using Lemma 4.5, we find that
(4.30)λ=∫0τ((1-q)y21m2*(y11+y21)/(1+b2Ss)2)exp(-∫0l((1-q)m2*Ss/(1+b2Ss))dl)dt∫0τ((1-q)y21Ss/(1+b2Ss))exp(-∫0l((1-q)m2*Ss/(1+b2Ss))dl)dt>0.
Thus we see that near the bifurcation point(m2*,0,0) the continuum C has two branches corresponding to ϵ<0, ϵ>0, respectively,
(4.31)C+=(m2;x1,x2):m2*<m2<m2*+λ0,x1<0,x2>0,C-=(m2;x1,x2):m2*-λ0<m2<m2*,x1>0,x2<0.
The solution is on C+ which prove the theorem, since λ>0 is equivalent to m2>m2*. We have left only to show that x1=y1+x1s for all t. This is easy, for if λ0 is small, then x1 is near x1s in the sup norm of B; thus since x1s is bounded away from zero, so is x1. At the same time, by Theorem 4.2, for (2.4), x2 is near x2s means that x1 is near x1s; thus S=1-x1-x2>0. We notice that the periodic solution (x1,x2) is τ-periodic. So S=1-x1-x2 is piecewise continuous and τ-periodic. The proof is thus completed.


## 4.1. Stability of the Boundary Periodic Solutions

For convenience, in the following discussion, ifm1>m1*, we denote
(4.4)m2*=-ln(1-D)∫0τ((1-q)Ss(l)/(1+b2Ss(l)))dl.Theorem 4.2.
For (2.4), one has the following. (1)
Ifm1<m1*, then (2.4) has a unique globally asymptotically stable boundary τ-periodic solution (1,0,0).(2)
Ifm1>m1* and m2<m2*, then (2.4) has a unique globally asymptotically stable boundary τ-periodic solution (Ss(t),x1s(t),0).(3)
Ifm1>m1* and m2>m2*, then the periodic boundary solution (Ss(t),x1s(t),0) of system (2.4) is unstable.Proof.
The proof of (1) is easy, we want to prove (2) and (3). The local stability of periodic solution(Ss(t),x1s(t),0) may be determined by considering the behavior of small amplitude perturbations of the solution. Define
(4.5)S(t)=u(t)+Ss(t),x1(t)=v(t)+x1s(t),x2(t)=w(t),
then we have
(4.6)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),0≤t<τ,
where Φ(t) satisfies
(4.7)dΦ(t)dt=(-m1x1s(1+b1Ss)2-m1Ss1+b1Ss-m2Ss1+b2Ssm1x1s(1+b1Ss)2m1Ss1+b1Ssqm2Ss1+b2Ss00(1-q)m2Ss1+b2Ss)Φ(t),
and Φ(0)=I, the identity matrix. Hence, the fundamental solution matrix is
(4.8)Φ(τ)=(ϕ11(τ)ϕ12(τ)*ϕ21(τ)ϕ22(τ)**00exp(∫0τ(1-q)m2Ss1+b2Ssdl)).
The linearization of impulsive subsystem (2.4) is written as
(4.9)(u(nτ+)v(nτ+)w(nτ+))=(1-D0001-D0001-D)(u(nτ)v(nτ)w(nτ)).
The stability of the periodic solution (Ss(t),x1s(t),0) is determined by the eigenvalues of
(4.10)M=(1-D0001-D0001-D)Φ(τ),
which are μ3=(1-D)exp(∫0τ((1-q)m2Ss/(1+b2Ss))dl) and the eigenvalues μ1, μ2 of the following matrix:
(4.11)(1-D)(ϕ11(τ)ϕ12(τ)ϕ21(τ)ϕ22(τ)).
The μ1, μ2 are also the multipliers of the locally linearization system of (3.1) provided with D∈(0,D0) at the asymptotically stable periodic solution (Ss(t),x1s(t)), where D0=1-exp((1+b1)/m1*τ). According to Theorem 3.3, we have that μ1<1, μ2<1.
Ifm2<m2*, then μ3=(1-D)exp(∫0τ((1-q)m2Ss/(1+b2Ss))dl)<1, the boundary periodic solution (Ss(t),x1s(t),0) of system (2.4) is locally stable. We obtain that
(4.12)x2(t)=x2(0)(1-D)nexp(∫0t(1-q)m2Ss(l)1+b2Ss(l)dl),t∈(nτ,(n+1)τ].
Hence, we obtain that for any solution (S(t),x1(t),x2(t)) with initial value S(0)>0, xi(0)>0, i=1,2, x2(t)→0 as t→∞. By Lemma 4.1, we have limt→∞|S(t)+x1(t)-1|=0. Now using Theorem 3.3, we have limt→∞|S(t)-Ss(t)|=0 and limt→∞|x1(t)-x1s(t)|=0.
Ifm2>m2*, then μ3>1, the boundary periodic solution (Ss(t),x1s(t),0) of (2.4) is unstable. The proof of Theorem 4.2 is completed.


## 4.2. Bifurcation Analysis of the Boundary Periodic Solution(Ss(t),x1s(t),0)

Letℬ denote the Banach space of piecewise continuous, τ-periodic function 𝒩:[0,τ]→R2, and have points of discontinuity τ, where they continuous from the left. In the set ℬ introduce the norm ∥𝒩∥0=sup0≤t≤τ∥𝒩(t)∥ with which ℬ becomes a Banach space with the uniform convergence topology.For convenience, we introduce Lemmas4.3 and 4.5 from Cushing [28] with small modifications.Lemma 4.3.
Supposeaij∈ℬ and 0≤di<1(i=1,2).
(a) If(1-d2)exp(∫0τa22(s)ds)≠1, (1-d1)exp(∫0τa11(s)ds)≠1, then the linear impulsive homogenous system
(4.13)dy1dt=a11y1+a12y2,dy2dt=a22y2,t≠nτ,Δy1=-d1y1,Δy2=-d2y2,t=nτ,
has no nontrivial solution in ℬ×ℬ. In this case, the nonhomogeneous system
(4.14)dx1dt=a11x1+a12x2+f1,dx2dt=a22x2+f2,t≠nτ,Δx1=-d1x1,Δx2=-d2x2,t=nτ,
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.
(b) If(1-d2)exp(∫0τa22(s)ds)=1, (1-d1)exp(∫0τa11(s)ds)≠1, then (4.13) has exactly one independent solution in ℬ×ℬ.Remark 4.4.
In fact, under the conditions of Lemma4.1(a),
(4.15)dx2dt=a22x2+f2,t≠nτ,Δx2=-d2x2,t=nτ,
has a unique solution x2∈ℬ and the operator L2:ℬ→ℬ defined by x2=L2f2 is linear and compact. Furthermore,
(4.16)dx1dt=a11x1+f3,t≠nτ,Δx1=-d1y1,t=nτ,
for f3∈ℬ has a unique solution (since (1-d1)exp(∫0τa11(s)ds)≠1) in ℬ, and x1=L1f3 defines a linear, compact operator L1:ℬ→ℬ. Then, we have
(4.17)L(f1,f2)=(L1(a12L2f2+f1),L2f2).Lemma 4.5.
Suppose thata∈ℬ, 0≤d<1, (1-d)exp(∫0τa(s)ds)=1 and f∈ℬ. Then, the impulsive equation
(4.18)dxdt=ax+f,t≠nτ,Δx=-dx,t=nτ,
has a solution x∈ℬ if and only if ∫0τf(l)exp(-∫0la(s)ds)dl=0.By Lemma4.1, in its invariant manifold Σ̃(t)=S(t)+x1(t)+x2(t)=1, (2.4) reduces to an equivalently nonautonomous system as follows:
(4.19)dx1dt=m1x1(1-x1-x2)1+b1(1-x1-x2)+qm2x2(1-x1-x2)1+b2(1-x1-x2),dx2dt=(1-q)m2x2(1-x1-x2)1+b2(1-x1-x2),t≠nτ,Δx1=-Dx1,Δx2=-Dx2,t=nτ,x10>0,x20>0,x10+x20≤1.
If m1>m1*, for (4.19), by Theorem 4.2, the boundary periodic solution (x1s(t),0) is locally asymptotically stable provided with m2<m2*, hence the value m2* plays an important role as a bifurcation threshold.For system (4.19), we have the following results.Theorem 4.6.
For system (4.19), assume m1>m1* holds, then there exists a constant λ0>0, such that for each m2∈(m2*,m2*+λ0), there exists a solution (x1,x2)∈ℬ×ℬ of (4.19) satisfying 0<x1<x1s, x2>0 and S=1-x1-x2>0 for all t>0. Hence, (2.4) has a positive τ-periodic solution (S,x1,x2).Proof.
Lety1=x1-x1s, y2=x2 in (4.19), then
(4.20)dy1dt=F11(Ss,x1s)y1+F12(m2,Ss,x1s)y2+g1(y1,y2),dy2dt=F22(m2,Ss)y2+g2(y1,y2),t≠nτ,Δy1=-Dy1,Δy2=-Dy2,t=nτ,x10>0,x20>0,x10+x20≤1,
where
(4.21)F11(Ss,x1s)=m1Ss1+b1Ss-m1x1s(1+b1Ss)2,F12(m2,Ss,x1s)=-m1x1s(1+b1Ss)2+qm2Ss1+b2Ss,F22(m2,Ss)=(1-q)m2Ss1+b2Ss.
Since (1-D)exp(∫0τ((1-q)m2Ss(l)/(1+b2Ss(l)))dl)≠1, by Lemma 4.3, using L, we can equivalently write (4.20) as the operator equation:
(4.22)(y1,y2)=L*(y1,y2)+G(y1,y2),
where
(4.23)G(y1,y2)=(L1(F12(m2,Ss,x1s)L2g2(y1,y2)+g1(y1,y2)),L2g2(y1,y2)).
Here L*:ℬ×ℬ→ℬ×ℬ is linear and compact (since L1 and L2 are compact) and satisfies G=o(|(y1,y2)|0) near (0,0). A nontrivial solution (y1,y2)≠(0,0) for some m2>m2* yields a solution (x1,x2)=(x1s+y1,y2) of (4.19). The solution (x1,x2)≠(x1s,0) is called a nontrivial solution of (4.19).
We apply the well-known local bifurcation techniques to (4.22). As is well known, bifurcation can occur only at the nontrivial solution of the linearized problem:
(4.24)(y1,y2)=L*(y1,y2).
If (y1,y2)∈ℬ×ℬ is a solution of (4.24) for some m2>0, then by the very manner in which L* was defined, (y1,y2) solves the system
(4.25)dy1dt=F11(Ss,x1s)y1+F12(m2,Ss,x1s)y2,dy2dt=F22(m2,Ss)y2,t≠nτ,Δy1=-Dy1,Δy2=-Dy2,t=nτ,
and conversely. Using Lemma 4.3(b), we see that (4.25) and hence (4.24) has one nontrivial solution in ℬ×ℬ if and only if m2=m2*. Hence, there exists a continuum C=(m2;y1,y2)⊆(0,∞)×ℬ×ℬ nontrivial solutions of (4.24) such that the closure C¯ contains (m2*;0,0). This continuum gives rise to a continuum C1=(m2;x1,x2)⊆(0,∞)×ℬ×ℬ of the solutions of (4.19) whose closure C¯1 contains the bifurcation point (m2*;x1s,0).
To see that solutions inC1 correspond to solutions (x1,x2) of (4.19), we investigate the nature of the continuum C near the bifurcation point (m2*;0,0) by expending m2 and (y1,y2) in Lyapunov-Schmidt series:
(4.26)m2=m2*+λϵ+⋯,y1=y11ϵ+y12ϵ2+⋯,y2=y21ϵ+y22ϵ2+⋯,
for yij∈ℬ where ϵ is a small parameter. If we substitute these series into the differential system (4.19) and equate coefficients of ϵ and ϵ2, we find that
(4.27)dy11dt=F11(Ss,x1s)y11+F12(m2*,Ss,x1s)y21,dy21dt=F22(m2*,Ss)y21,t≠nτ,Δy11=-Dy11,Δy21=-Dy21,t=nτ,(4.28)dy12dt=F11(Ss,x1s)y12+F12(m2*,Ss,x1s)y22+G12(y11,y12,λ),dy22dt=F22(m2*,Ss)y22+(1-q)y211+b2Ss(λSs-m2*(y11+y21)1+b2Ss),t≠nτ,Δy21=-Dy21,Δy22=-Dy22,t=nτ,
respectively. Thus, (y11,y21)∈ℬ×ℬ must be a solution of (4.22). We choose the specific solution satisfying the initial conditions y21(0)=1. Then
(4.29)y21=exp(∫nτt(1-q)m2*Ss1+b2Ssdl)>0,nτ<t≤(n+1)τ,y21(0)=1.
Moreover, y11<0 for all t (since m1>m1* and (3.8), hence -∫0τ(m1x1s(l)/(1+b1Ss(l))2)dl<0, which implies that the Green's function for first equation in (4.24) is positive). Using Lemma 4.5, we find that
(4.30)λ=∫0τ((1-q)y21m2*(y11+y21)/(1+b2Ss)2)exp(-∫0l((1-q)m2*Ss/(1+b2Ss))dl)dt∫0τ((1-q)y21Ss/(1+b2Ss))exp(-∫0l((1-q)m2*Ss/(1+b2Ss))dl)dt>0.
Thus we see that near the bifurcation point(m2*,0,0) the continuum C has two branches corresponding to ϵ<0, ϵ>0, respectively,
(4.31)C+=(m2;x1,x2):m2*<m2<m2*+λ0,x1<0,x2>0,C-=(m2;x1,x2):m2*-λ0<m2<m2*,x1>0,x2<0.
The solution is on C+ which prove the theorem, since λ>0 is equivalent to m2>m2*. We have left only to show that x1=y1+x1s for all t. This is easy, for if λ0 is small, then x1 is near x1s in the sup norm of B; thus since x1s is bounded away from zero, so is x1. At the same time, by Theorem 4.2, for (2.4), x2 is near x2s means that x1 is near x1s; thus S=1-x1-x2>0. We notice that the periodic solution (x1,x2) is τ-periodic. So S=1-x1-x2 is piecewise continuous and τ-periodic. The proof is thus completed.


## 5. Simulations and Discussion

The ability to manufacture desired products through genetically altered organisms represents one of the major developments in biotechnology. Competition between plasmid-bearing and plasmid-free organisms is a subject of considerable interest [29]. In this paper, we have considered a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with pulsed nutrient input and washout. Our research shows that the dynamical behaviors of model (2.4) are completely determined by two thresholds: m1*, the invasion threshold of the plasmid-free organism and m2*, the invasion threshold of the plasmid-bearing organism. If m1<m1*, the periodic solution (1,0,0) is globally asymptotically stable, both the plasmid-bearing and the plasmid-free organisms are eradicated from the chemostat except the nutrient; if m1>m1* and m2<m2*, the periodic solution (Ss(t),x1s(t),0) is globally asymptotically stable, then the nutrient, plasmid-free organisms coexist periodically in the chemostat and the plasmid-bearing organism tends to extinction; if m1>m1* and m2>m2*, (2.4) has a positive periodic solution (S(t),x1(t),x2(t)) and therefore, both the plasmid-bearing and the plasmid-free organisms coexist periodically in the chemostat.To justify the theoretic results, we obtained in Section4, we give two examples which concern the results in Theorems 4.2 and 4.6, respectively.Example 5.1.
In (2.4), set m2=1.2, a1=0.3, a2=0.25, D=0.7, q=0.0015, we can compute m1*=̇1.4448. By Theorem 4.2, we know that if m1<m1*, the periodic solution (1,0,0) is globally asymptotically stable; if m1>m1*, the plasmid-free organism begins to invade the system and the periodic solution (Ss(t),x1s(t),0) is globally asymptotically stable. Our simulations support these results (see Figures 1 and 2, where m1=1.2 and m1=1.8, resp.).Time-series of (2.4) with periodically input and washout with initial value S0=1, x10=0.5, x20=0.5: m1=1.2, m2=1.2, a1=0.3, a2=0.25, D=0.7, q=0.0015, τ=1.
(a)(b)(c)Time-series of system (2.4) with periodically input and washout with initial value S0=1, x10=0.5, x20=0.5: m1=1.8, m2, a1, a2, D, q, and τ have similar values as in Figure 1.
(a)(b)(c)Example 5.2.
In (2.4), set m1=1.8(>m1*) and a1, a2, D, q have similar values as in Example 5.1. We can estimate numerically that m2*=̇1.74 (since we cannot compute the exact value from the expression of m2* in (4.4)). By Theorem 4.6, we know that if m2>m2*, the plasmid-bearing organism begins to invade the system and, in this case, (2.4) has a positive periodic solution (S(t),x1(t),x2(t)). Our simulations support this result (see Figure 3, where m2=1.745).Time-series of (2.4) with periodically input and washout with initial value S0=1, x10=0.5, x20=0.5: m1=1.8(>m1*), m2=1.745, a1, a2, D, q, τ have similar values as in Figures 1 and 2.
(a)(b)(c)From the chemical engineering standpoint, the consumption of nutrient by the plasmid-free organism represents a loss of production in the bioreactor. Moreover, if it is a sufficiently better competitor, the plasmid-bearing organism (the production) may be eliminated from the chemostat (in this situation, the periodic solution(Ss(t),x1s(t),0) exists and is globally stable). Then no product is manufactured (and nutrient is consumed). This is an undesirable situation. Note also that the plasmid-bearing organism can always lose the plasmid. Thus, if the plasmid-bearing organism does not go to extinction, then neither does the plasmid-free organism. This can also be seen from the form of the equations in system (2.4) (as a consequence, there is no nonnegative periodic solution (Ss(t),0,x2s(t)) in the space x1=0). Thus, the best situation for the manufacture by genetically altered organism is that both the plasmid-free and the plasmid-bearing organisms coexist in the chemostat. Therefore, the study of the existence of the positive periodic solution (S(t),x1(t),x2(t)) of the system is paramount. By Theorem 4.6 we can see from the expressions of m1*, m2* given, respectively, by (3.3) and (4.4) and the scaling formulations in (2.3) that the controllers of the bioreactor can do this by only control the parameters S0,D and τ such that the conditions m1>m1* and m2>m2* are satisfied.For the situation when both the plasmid-free and the plasmid-bearing organisms coexist in the culture, we further want to know the optimum values ofS0,D, and τ under which the product manufacturer can obtain the maximum production by genetically altered organism. This is a challenging question to answer. We leave this for future consideration.



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*Source: 204632-2009-12-01.xml*