# Dynamic Behaviors in a Droop Model for Phytoplankton Growth in a Chemostat with Nutrient Periodically Pulsed Input

Authors: Kai Wang; Zhidong Teng; Xueliang Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

The dynamic behaviors in a droop model for phytoplankton growth in a chemostat with nutrient periodically pulsed input are studied. A series of new criteria on the boundedness, permanence, extinction, existence of positive periodic solution, and global attractivity for the model are established. Finally, an example is given to demonstrate the effectiveness of the results in this paper.


---

## Body

## 1. Introduction

The chemostat is a very important apparatus used to study the growth of microorganisms in a continuous cultured environment in a laboratory (see [1]). It may be viewed as a laboratory model of a simple lake with continuous stirring. Chemostat models have attracted widely the attention of the scientific community, since they have a wide range of applications, for example, waste-water treatment, production by genetically altered organisms (like production of insulin), and so forth. The growth in a chemostat is described by the systems of ordinary differential equations or functional differential equations. Many important and interesting results can be found in articles [2–10] and the references cited therein.The droop model [11, 12] of phytoplankton growth is essential in theoretical phytoplankton ecology. The droop model takes into consideration that phytoplankton cells store nutrient and that the growth rate depends on the stored nutrient. Algae can uptake nutrient in excess of current needs and continue to grow during nutrient poor conditions. These nutrients are supplied from an external reservoir; in several earlier works, this concentration is assumed to be constant. The droop model of phytoplankton growth in a chemostat has been widely investigated in many literatures (see [13–19]). As in [16], the classical chemostat equations modeling phytoplankton population dynamics originally related the growth rate of the cells to the nutrient concentration in the medium. It is assumed that the nutrient uptake rate is proportional to the rate of reproduction. The constant of proportionality that converts units of nutrient to units of organisms is called the yield constant. Because of the assumed constant value of the yield, the classical Monod model is referred to as the constant-yield model in [15]. In [17], both uniform persistence and the existence of periodic coexistence state are established for a periodically forced droop model on two phytoplankton species competition in a chemostat under some appropriate conditions. In [18], the authors considered the nonautonomous droop model for phytoplankton growth in a chemostat in which the nutrient input varies nonperiodically. It is assumed that growth rate varies with the internal nutrient level of the cell and the uptake rate of phytoplankton depends on both the external and the internal nutrient concentrations. A series of new criteria on the positivity, boundedness, permanence, and extinction of the population is established.This work was motivated by [19], where the droop model was presented with general functions for growth and uptake, which takes the following forms:
(1.1)dS(t)dt=D(S0(t)-S)-N(t)ρ(S,Q),dQ(t)dt=ρ(S,Q)-μ(Q)Q,dN(t)dt=N(μ(Q)-D).
The author proved that the periodically forced droop model has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the subthreshold condition hold, the phytoplankton population is washed out of the chemostat. If the superthreshold condition holds, then there is a unique periodic solution to which all solutions approach asymptotically.As well known, countless organisms live in seasonally or diurnally forced environment, in which the populations obtain food, so the effects of this forcing may be quite profound. Recently, many papers studied chemostat model with variations in the supply of nutrients or the washout. The chemostat models with impulsive input perturbation have been studied in many articles, see [20–30] and the references cited therein, where many important and interesting results on the persistence, permanence and extinction of microorganisms, global stability, the existence of periodic oscillation and dynamical complexity of the systems are discussed.However, we find that, up to now, few papers consider the droop model with nutrient periodically pulsed input. From the above discussion, this subject has broad biological significance. In pulse case, we will give some good extensions of the corresponding results given by Smith in [19]. We introduce the following droop model for phytoplankton growth in a chemostat with nutrient periodically pulsed input(1.2)dS(t)dt=-DS-N(t)ρ(S,Q),dQ(t)dt=ρ(S,Q)-μ(Q)Q,t≠nT,dN(t)dt=N(μ(Q)-D),S(t+)=S(t)+p,Q(t+)=Q(t),t=nT,n∈N,N(t+)=N(t).
For model (1.2), we will investigate the permanence and extinction of the species, and the existence of positive T-periodic solution and global attractivity of the model. We also will give an example and numerical simulations to demonstrate the effectiveness of the results in this paper.This paper is organized as follows. In the following section we will firstly introduce some basic assumption formodel (1.2). Furthermore, we also will give several useful lemmas. In Section 3, we will state and prove an ultimate boundedness theorem of solutions for model (1.2). In Section 4, we will state and prove the sufficient condition on the permanence of species for model (1.2). In Section 5, the sufficient condition on the extinction of species for model (1.2) is given. In Section 6, the existence of positive T-periodic solution and the global attractivity of solutions for model (1.2) are established. Finally, in Section 7, we will discuss an example and give some numerical simulations.


## 2. Preliminaries

Inmodel (1.2), N(t) denotes the phytoplankton cells at biomass concentration, S(t) denotes the nutrient at concentration in a growth medium. Each phytoplankton cell is assumed to possess an internal pool of stored nutrient, also called quota Q(t). D>0 is the input and output flow, and its referred to as the dilution rate; p>0 is the amount of the substrate concentration pulsed each nT, where T>0 is a constant. Function μ(Q) denotes the phytoplankton growth rate, function ρ(S,Q) denotes the nutrient uptake rate. In this paper, for model (1.2), we always assume that the following condition holds.(H1)
Functionμ(Q) is continuously differentiable and dμ(Q)/dQ>0 for all Q≥Qmin, μ(Qmin)=0, where Qmin is some given positive constant.(H2)
Functionρ(S,Q) is continuously differentiable and ∂ρ(S,Q)/∂S>0, ∂ρ(S,Q)/∂Q≤0 for all S>0 and Q>Qmin. Further, ρ(0,Q)≡0 for all Q≥Qmin.The initial condition in model (1.2) is given in the following form:(2.1)S(0+)=S0>0,Q(0+)=Q0≥Qmin⁡,N(0+)=N0>0.Firstly, on the positivity of solutions formodel (1.2), we have the following result.Lemma 2.1.
Any solution(S(t),Q(t),N(t)) of  model (1.2) with initial condition (2.1) is positive, that is,  S(t)>0,  Q(t)≥Qmin and  N(t)>0 for all t>0.Proof.
Obviously, for anyt>0, we have
(2.2)N(t)=N(0+)exp⁡∫0t∂ρ(ξ(t),Q(t))∂S(μ(Q(s))-D)ds>0.
We considerS(t). If the conclusion is not true, then from S(0+)>0 and p>0 we can obtain that there is a t1>0 such that S(t1)=0 and S(t)>0 for all t∈(0,t1). From the mean-value theorem and assumption (H2), for any t∈(0,t1), there is a ξ(t)∈(0,S(t)) such that
(2.3)ρ(S(t),Q(t))=∂ρ(ξ(t),Q(t))∂SS(t).
Hence, from the first equation of model (1.2), we obtain
(2.4)dS(t)dt=-(D+N(t)∂ρ(ξ(t),Q(t))∂S)S(t)∀t∈(0,t1],t≠nT.
Choose an integer n0≥0 such that t1∈(n0T,(n0+1)T]. Obviously, S(n0T+)>0 and integrating (2.4) from n0T to t1,
(2.5)S(t1)=S(n0T+)exp⁡(-∫n0Tt1∂ρ(ξ(t),Q(t))∂S(D+N(u)∂ρ(ξ(u),Q(u))∂S))du>0,
which leads to a contradiction.
Now, we considerQ(t). We have Q(0+)≥Qmin and μ(Qmin)=0. If Q(0+)=Qmin, then since right derivative
(2.6)dQ(t)dt|t=0+=ρ(S(0+),Qmin⁡)>0,
there is a t1>0 such that Q(t)>Qmin for all t∈(0,t1]. If there is a t2>t1 such that Q(t2)=Qmin and Q(t)>Qmin for all t∈(0,t2), then left derivative dQ(t)/dt|t=t2-. But from assumption (H2), we have
(2.7)dQ(t)dt|t=t2-=ρ(S(t2),Qmin⁡)>0,
which leads to a contradiction. If Q(0+)>Qmin, then similarly to the above, we also can obtain a contradiction. This completes the proof of Lemma 2.1.Now, we consider the following linear impulsive differential equation(2.8)dω(t)dt=-Dω(t)-η,t≠nT,ω(t+)=ω(t)+p,t=nT,n∈N,
where D and p are defined in model (1.2) and η is a constant. Clearly,(2.9)ω*(t)=-ηD+S*(t),t∈(nT,(n+1)T],n∈Nis theT-periodic solution of (2.8), where(2.10)S*(t)=pe-D(t-nT)1-e-DT,t∈(nT,(n+1)T],n∈N.We say thatω*(t) is globally uniformly attractive if for any constants M>0 and ɛ>0 there is a constant T(M,ɛ)>0 such that for any initial time t0≥0 and initial value ω0 with |ω0|≤M, one has(2.11)|ω(t,t0,ω0)-ω*(t)|<ɛ∀t≥t0+T(M,ɛ),
where ω(t,t0,ω0) is the solution of (2.8) with initial condition ω(t0)=ω0. We have following result.Lemma 2.2 (see [27]).
T-periodic solution  ω*(t) of (2.8) is globally uniformly attractive.For (2.8), when η=0, we obtain the subsystem of model (1.2) with N(t)=0 as follows:
(2.12)dS(t)dt=-DS(t),t≠nT,S(t+)=S(t)+p,t=nT,n∈N.
Clearly, S*(t) is the positive T-periodic solution of (2.12). From Lemma 2.2, we obtain that for any solution S(t) of (2.12), one has(2.13)S(t)-S*(t)⟶0ast⟶∞.PuttingS(t)=S*(t) in the second equation of model (1.2), we obtain(2.14)dQ(t)dt=ρ(S*(t),Q(t))-μ(Q(t))Q(t).
From the expression of S*(t), we can choose a constant 0<α0<pe-DT/(1-e-DT) such that(2.15)inf⁡t≥0⁡S*(t)-α0≥pe-DT1-e-DT-α0>0.For anyα∈[-α0,α0], we consider the following equation:(2.16)dQ(t)dt=ρ(S*(t)+α,Q(t))-μ(Q(t))Q(t).
For any initial points t0≥0 and Q0≥Qmin, let Q0(t) and Qα(t) be the solutions of (2.14) and (2.16) satisfying initial conditions Q0(t0)=Q0 and Qα(t0)=Q0, respectively. We have the following result.Lemma 2.3.
(a)  There are constantsM1>1 and such that for any  t0≥0,  Q0≥Qmin and  α∈[-α0,α0](2.17)Qmin⁡≤liminf⁡t→∞Qα(t)≤liminf⁡t→∞Qα(t)≤M1.
(b) For eachα∈[-α0,α0], (2.16) has a positive and globally uniformly attractive T-periodic solution Qα*(t).
(c)Qα*(t) converges to Q0*(t) uniformly for t∈[0,T] as α→0.Proof.
Firstly, inequalityQ(t)≥Qmin for all t∈[0,∞) can be proved by using the similar argument as in the proof of Lemma 2.1.
From assumptions (H1) and (H2), there are positive constants K and δ such that for any α∈[-α0,α0](2.18)∫0T[ρ(S*(u)+α,K)K-μ(K)]du<-δ.
If Qα(t)≥K for all t≥t0, then integrating (2.16), for any t≥t0, we have
(2.19)Qα(t)=Qα(t0)exp⁡∫t0t[ρ(S*(u)+α,Qα(u))Qα(u)-μ(Qα(u))]du≤Qα(t0)exp⁡∫t0t[ρ(S*(u)+α,K)K-μ(K)]du≤Qα(t0)exp⁡⁡[-δ(t-t0)].
Hence, Qα(t)→0 as t→∞, which leads to a contradiction. Therefore, there is a t1>t0 such that Qα(t1)<K. Further, if there are t2>t3>t1 such that Qα(t2)>Kexp(M*T), Qα(t3)=K, and Qα(t)>K for all t∈(t3,t2], where
(2.20)M*=sup⁡t∈R+⁡{ρ(S*(t)+α0,K)K+μ(K)},
then we can choose an integer q≥0 such that t2∈(t3+qT,t3+(q+1)T] and integrating (2.16) to obtain
(2.21)Kexp⁡(M*T)<Qα(t2)=Qα(t3)exp⁡⁡∫t3t2[ρ(S*(u)+α,Qα(u))Qα(u)-μ(Qα(u))]du≤Kexp⁡⁡(∫t3t2+qT+∫t3+qTt2)[ρ(S*(u)+α,K)K-μ(K)]du≤Kexp⁡⁡∫t3+qTt2[ρ(S*(u)+α,K)K-μ(K)]du≤Kexp⁡⁡(M*T),
which also leads to a contradiction. Therefore, we finally have
(2.22)Qα(t)≤Kexp⁡⁡(M*T)∀t≥t1.
Choose constant M1=Kexp(M*T), then we have limsupt→∞Qα(t)≤M1 for all α∈[-α0,α0]. Thus, conclusion (a) is proved.
From conclusion (a), directly using the main results given by Teng and Chen in [31], we can obtain that (2.16) for each α∈[-α0,α0] has a positive T-periodic solution Qα*(t).
For any constantη0>1 and t0≥0, let Qα(t) be a solution of (2.16) with initial value Qα(t0)∈[η0-1,η0]. By conclusion (a), there is a constant M1>1 such that
(2.23)M1-1≤Qα*(t)≤M1,M1-1≤Qα(t)≤M1∀t≥t0.
Consider Liapunov function
(2.24)V(t)=|ln⁡⁡Qα(t)-ln⁡⁡Qα*(t)|.
Calculating the Dini derivative D+V(t), from
(2.25)sign⁡(Qα(t)-Qα*(t))[ρ(S*(t)+α,Qα(t))Qα(t)-ρ(S*(t)+α,Qα*(t))Qα*(t)]≤0,
we can obtain for any t≥t0 and t≠nT(2.26)D+V(t)≤sign⁡(Qα(t)-Qα*(t))[μ(Qα*(t))-μ(Qα(t))].
Using the mean-value theorem, we further obtain
(2.27)D+V(t)≤-dμ(ξ(t))dQ|Qα*(t)-Qα(t)|,
where ξ(t) is situated between Qα*(t) and Qα(t). When t=nT and n∈N, we obviously have V(t+)=V(t). Hence
(2.28)V(t)≤V(t0)∀t≥t0.
Consequently, by (2.23) we have
(2.29)|ln⁡⁡Qα(t)|≤|ln⁡⁡Qα*(t)|+V(t0)≤ln⁡⁡(η0M12),
for all t≥t0. Hence
(2.30)η0-1M1-2≤Qα(t)≤η0M12forallt≥t0.
Further, from (2.23) we obtain
(2.31)η0-1M1-2V(t)≤|Qα*(t)-Qα(t)|≤η0M12V(t),
for all t≥t0. Consequently, from (2.27) it follows that
(2.32)D+V(t)≤-η0-1M1-2dμ(ξ(t))dQV(t)≤-M0V(t),
where
(2.33)M0=η0-1M1-2min⁡⁡{dμ(ξ(t))dQ:η0-1M1-2≤Q≤η0M12}.
Hence, we further have
(2.34)V(t)≤V(t0)e-M0(t-t0)≤ln⁡⁡(η0M1)e-M0(t-t0),∀t≥t0.
For any constant ɛ>0, from (2.34), choose
(2.35)T(η0,ɛ)=-1M0ln⁡⁡ɛη0M12ln⁡⁡(η0M1),
then for any t≥t0+T(η0,ɛ) we can obtain
(2.36)V(t)<ɛη0M12∀t≥t0+T(η0,ɛ).
Therefore,
(2.37)|Qα*(t)-Qα(t)|<ɛ∀t≥t0+T(η0,ɛ).
This shows solution Qα*(t) is globally uniformly attractive. Thus, conclusion (b) is proved.
Finally, we prove conclusion (c). From assumptions (H1) and (H2), we can easily prove that the right hand of model (1.2) satisfies the uniform Lipschitz condition with respect to α∈[-α0,α0]. Hence, by the continuity of solutions with respect to parameter α, we can obtain that Qα*(t)→Q0*(t) uniformly for t∈[0,T] as α→0. Thus, conclusion (c) is proved. This completes the proof of Lemma 2.3.Whenα=0, (2.16) degenerates into (2.14), from Lemma 2.3 we have the following result.Corollary 2.4.
Equation (2.14) has a unique positive  T-periodic solution  Q0*(t) which is globally uniformly attractive.The following lemma will be used in the proof of the result on the global attractivity ofmodel (1.2).Lemma 2.5.
Let functionf(t) be continuous on  [0,+∞) and differentiable for any  t≠nT ( n∈N), where  T>0 is a constant. If there exist a constant  M>0, such that  |df(t)/dt|<M for any  t≠nT(n∈N), then  f(t) is uniformly continuous on  [0,+∞).The proof of Lemma2.1 is simple, we hence omit it here.


## 3. Boundedness

On the ultimate boundedness of all positive solutions ofmodel (1.2), we have the following result.Theorem 3.1.
Let(S(t),Q(t),N(t)) be any positive solution of  model (1.2), then we have that there exists a constant  M3>0, which is independent of any positive solution of  model (1.2), such that
(3.1)lim⁡⁡sup⁡t→∞⁡S(t)<M3,lim⁡⁡sup⁡t→∞⁡Q(t)<M3,lim⁡⁡sup⁡t→∞⁡N(t)<M3.Proof.
Let(S(t),Q(t),N(t)) be any positive solution of model (1.2) with initial condition (2.1). Since
(3.2)dS(t)dt≤-DS(t),t≠nT,S(t+)=S(t)+p,t=nT,n∈N,
by Lemma 2.2 we directly have
(3.3)lim⁡⁡sup⁡t→∞⁡S(t)≤lim⁡⁡sup⁡t→∞⁡S*(t)≤p1-e-DT.
For any constantɛ>0, there is a T0>0 such that
(3.4)S(t)≤p1-e-DT+ɛ∀t≥T0.
From the second equation of model (1.2), we have
(3.5)dQ(t)dt≤ρ(p1-e-DT+ɛ,Q(t))-μ(Q(t))Q(t)
for all t≥T0. Using the similar argument as in the proof of Lemma 2.3, we have that there is a positive constant MQ>0, and MQ is independent of any solution (S(t),Q(t),N(t)) of model (1.2), such that
(3.6)lim⁡⁡sup⁡t→∞⁡Q(t)≤MQ.
Define a functionU(t) as follows:
(3.7)U(t)=S(t)+Q(t)N(t).
Calculating the derivative of U(t) along solution (S(t),Q(t),N(t)) of model (1.2),  we have
(3.8)dU(t)dt=-DU(t),t≠nT,U(t+)=U(t)+p,t=nT,n∈N.
From Lemma 2.2, we obtain
(3.9)lim⁡⁡sup⁡t→∞⁡U(t)≤lim⁡⁡sup⁡t→∞⁡S*(t)≤p1-e-DT.
From Lemma 2.1, we have Q(t)≥Qmin for all t≥0. Hence, by (3.9) we can obtain that N(t) is ultimately bound. This completes the proof of Theorem 3.1.Remark 3.2.
Obviously, Theorem3.1 is new and can serve as an extension of corresponding result given by Smith in [19].


## 4. Permanence

On the permanence of species formodel (1.2), we have the following result.Theorem 4.1.
Suppose(4.1)∫0T[μ(Q0*(t))-D]dt>0.
Then model (1.2) is permanent.Proof.
From Lemma2.1 we directly have liminft→∞Q(t)≥Qmin>0. Hence, Q(t) is obviously permanent. We now prove the permanence of N(t). From (4.1), we can choose a constant ɛ0>0 small enough such that
(4.2)∫0T[μ(Q0*(t)-ɛ0)-D]dt>ɛ0.
Let (S(t),Q(t),N(t)) be any solution of model (1.2) with initial condition (2.1). From Theorem 3.1, for any constant ɛ>0 there is a T0>0 such that
(4.3)S(t)≤M3+ɛ,Q(t)≤M3+ɛ,N(t)≤M3+ɛ
for all t≥T0. For any α∈[-α0,α0], we consider the following assistant equation:
(4.4)dQ(t)dt=ρ(S*(t)+α,Q(t))-μ(Q(t))Q(t).
By Lemma 2.3, we obtain that (4.4) has a unique positive T-periodic solution Qα*(t), which is globally uniformly attractive and Qα*(t) converges to Q0*(t) uniformly for t∈R+ as α→0. Hence, there is an α*>0 and α*<ɛ0 such that
(4.5)Qα*(t)>Q0*(t)-ɛ02∀t∈R+,|α|<α*.
Choose positive constantsɛ1 and ɛ2 such that
(4.6)inf⁡t∈R+⁡S*(t)-ɛ2-ɛ1ρ(M3+ɛ,Qmin⁡)D>0,(4.7)ɛ2+ɛ1ρ(M3+ɛ,Qmin⁡)D<α*.
We first proof that
(4.8)lim⁡t→∞⁡sup⁡⁡N(t)>ɛ1
for any solution (S(t),Q(t),N(t)) of model (1.2) with initial condition (2.1).
In fact, if (4.8) is not true, then there is a solution (S(t),Q(t),N(t)) of model (1.2) such that limsupt→∞N(t)≤ɛ1. Hence, there is a T1>T0 such that N(t)<ɛ1, for all t≥T1. From the fist equation of model (1.2) and (4.3), we have
(4.9)dS(t)dt≥-DS(t)-ɛ1ρ(M3+ɛ,Qmin⁡),t≥T1,t≠nT,S(t+)=S(t)+p,t=nT,n∈N.
From the comparison theorem of impulsive differential equations and Lemma 2.2, for above ɛ2>0, there is an n1∈N and n1T>T1 such that
(4.10)S(t)≥-ɛ1ρ(M3+ɛ,Qmin⁡)D+S*(t)-ɛ2∀t>n1T.
From the second equation of model (1.2), we have
(4.11)dQ(t)dt≥ρ(S*(t)-ɛ1ρ(M3+ɛ,Qmin⁡)D-ɛ2,Q(t))-μ(Q(t))Q(t)
for all t>n1T. Let α1=-ɛ1ρ(M3+ɛ,Qmin)/D-ɛ2, from Lemma 2.3 and the comparison theorem, there exists an n2>n1 such that
(4.12)Q(t)≥Qα1*(t)-ɛ02≥Q0*(t)-ɛ0∀t>n2T.
Then, from the third equation of model (1.2), we have
(4.13)dN(t)dt≥N(t)(μ(Q0*(t)-ɛ0)-D)∀t>n2T.
Integrating (4.13) from n2T to t>n2T, we obtain
(4.14)N(t)≥N(n2T)exp⁡⁡∫n2Tt(μ(Q0*(s)-ɛ0)-D)ds.
Obviously, from (4.2) and (4.14), we obtain N(t)→∞ as t→∞, which leads to a contradiction. Therefore, (4.8) is true.
Now, we prove that there is a constantm1>0 such that
(4.15)lim⁡⁡inf⁡t→∞⁡N(t)≥m1
for any solution (S(t),Q(t),N(t)) of model (1.2) with initial condition (2.1).
Assume that (4.15) is not true, then there exists a sequence of initial values zk=(Sk,Qk,Nk), which satisfy Sk>0, Qk≥Qmin, and Nk>0 such that for solution (S(t,zk),Q(t,zk),N(t,zk)) of model (1.2) with initial condition S(0+,zk), Q(0+,zk), and N(0+,zk)(4.16)lim⁡⁡inf⁡t→∞⁡N(t,zk)<ɛ1k2∀k=1,2,….
From (4.8) and (4.16), we obtain that there exist two sequences {vq(k)} and {tq(k)} such that for each k=1,2,…(4.17)0<v1(k)<t1(k)<v2(k)<t2(k)<⋯<vq(k)<tq(k)<⋯,vq(k)⟶∞,vq(k)⟶∞asq⟶∞,(4.18)N(tq(k),zk)=ɛ1k2,N(vq(k),zk)=ɛ1k,(4.19)ηk2<N(t,zk)<ɛ1k∀t∈(vq(k),tq(k)).
From Theorem 3.1, there is a T(k)≥T0 such that
(4.20)S(t,zk)≤M3+ɛ,Q(t,zk)≤M3+ɛ,N(t,zk)≤M3+ɛ
for all t≥T(k). Further for each k, from (4.17) there is an integer K(k)>0 such that vq(k)>T(k) for all q>K(k). Hence for any t∈[vq(k),tq(k)] and q>K(k), we have
(4.21)dN(t,zk)dt=N(t,zk)(μ(Q(t,zk))-D)≥-γN(t,zk),
where γ=D+μ(M3+ɛ). Therefore, for any q≥K(k) and k=1,2,…, integrating (4.21) on [vq(k),tq(k)], we obtain from (4.18)
(4.22)ɛ1k2=N(tq(k),zk)≥N(vq(k),zk)exp⁡⁡[-γ(tq(k)-vq(k))]=ɛ1kexp⁡⁡[-γ(tq(k)-vq(k))].
Consequently,
(4.23)tq(k)-vq(k)≥ln⁡⁡kγ∀q≥K(k),k=1,2,….
Consider the following equation:(4.24)du(t)dt=-Du(t)-ɛ1ρ(M3+ɛ0,Qmin⁡),t≠nT,u(t+)=u(t)+p,t=nT.
From Lemma 2.2, there is a constant T*>0 such that for any t0≥0 and 0≤u0≤M3+ɛ(4.25)u(t)≥S*(t)-ɛ1ρ(M3+ɛ,Qmin⁡)D-ɛ2∀t≥t0+T*,
where u(t) is the solution of (4.24) with initial value u(t0)=u0.
Further consider the following equation:(4.26)dv(t)dt=ρ(S*(t)+α1,v(t))-μ(v(t))v(t).
From Lemma 2.3, there is a constant T**>0 such that for any t0≥0 and 0≤v0≤M3+ɛ(4.27)v(t)≥Qα1*(t)-ɛ02∀t≥t0+T**,
where v(t) is the solution of (4.26) with initial value v(t0)=v0.
From (4.23), we can choose an integer N0>0 such that
(4.28)tq(k)-vq(k)>T*+T**+T∀k>N0,q>K(k).
Since for any integer k, q≥K(k), t∈[vq(k),tq(k)] and q≥K(k) we have
(4.29)dS(t,zk)dt=-DS(t,zk)-N(t,zk)ρ(S(t,zk),Q(t,zk))≥-DS(t,zk)-ɛ1ρ(M3+ɛ,Qmin⁡),t≠nT,S(t+,zk)=S(t,zk)+p,t=nT,
from the comparison theorem and inequality (4.25), we have
(4.30)S(t,zk)≥u(t)≥-ɛ1ρ(M3+ɛ,Qmin⁡)D+S*(t)-ɛ2=S*(t)+α1
for all t∈[vq(k)+T*,tq(k)] and q≥K(k). Further, since
(4.31)dQ(t,zk)dt≥ρ(S*(t)+α1,Q(t,zk))-μ(Q(t,zk))Q(t,zk)
for all t∈[vq(k)+T*,tq(k)] and q≥K(k). From the comparison theorem and inequality (4.27),
(4.32)Q(t,zk)≥Qα1*(t)-ɛ02≥Q0*(t)-ɛ0,
for all t∈[vq(k)+T*+T**,tq(k)] and q≥K(k). Hence, for any q≥K(k), k≥N0 and t∈[tq(k)-T,tq(k)], we have
(4.33)dN(t,zk)dt=N(t,zk)(μ(Q(t,zk)-D))≥N(t,zk)(μ(Q0*(t)-ɛ0)-D).
Integrating (4.33) from tq(k)-T to tq(k), by (4.19) we obtain
(4.34)ɛ1k2=N(tq(k),zk)≥N(tq(k)-T,zk)exp⁡⁡∫tq(k)-Ttq(k)(μ(Q0*(t)-ɛ0)-D)dt≥ɛ1k2exp⁡⁡(ɛ0)>ɛ1k2,
which is contradictory. This contradiction shows that (4.15) is true for any positive solution (S(t),Q(t),N(t)) of model (1.2).
Next, we prove thatS(t) in model (1.2) is permanent. Let (S(t),Q(t),N(t)) be any solution of model (1.2) with initial condition (2.1). From the assumption of ρ(S,Q) and the mean-value theorem, for any t≥0, there exists a ξ(t)∈(0,S(t)) such that
(4.35)ρ(S(t),Q(t))S(t)=ρ(S(t),Q(t))-ρ(0,Q(t))S(t)=∂ρ(ξ(t),Q(t))∂S.
Further, from assumption (H2) and the boundedness of solution (S(t),Q(t),N(t)), there exists a constant Ms>0 such that
(4.36)ρ(S(t),Q(t))S(t)=∂ρ(ξ(t),Q(t))∂S<Ms∀t≥0.
From the first equation of model (1.2), we obtain
(4.37)dS(t)dt=-DS(t)-N(t)ρ(S(t),Q(t))≥-(D+(M3+ɛ)Ms)S(t),
for all t≥T0 and t≠nT, and
(4.38)S(t+)=S(t)+p,t=nT,n∈N.
Using the comparison theorem and Lemma 2.2, we can obtain
(4.39)lim⁡⁡inf⁡t→∞⁡S(t)≥pexp⁡⁡(-(D+(M3+ɛ)Ms)T)-(D+(M3+ɛ)Ms)T.
This shows that S(t) is permanent. This completes the proof of Theorem 4.1.


## 5. Extinction

On the extinction of species ofmodel (1.2), we have the following result.Theorem 5.1.
Suppose(5.1)∫0T[μ(Q0*(t))-D]dt<0.
Then for any positive solution (S(t),Q(t),N(t)) of model (1.2)
(5.2)lim⁡t→∞⁡S(t)=S*(t),lim⁡t→∞⁡Q(t)=Q0*(t),lim⁡t→∞⁡N(t)=0.Proof.
From (5.1), there exists a constant ɛ0>0 such that
(5.3)∫0T[μ(Q0*(t)+ɛ0)-D]dt<0.
For any α∈[-α0,α0], we consider the following equation:
(5.4)dQ(t)dt=ρ(S*(t)+α,Q(t))-μ(Q(t))Q(t).
By Lemma 2.3, we obtain that positive T-periodic solution Qα*(t) of (5.4) is globally uniformly attractive and Qα*(t) converges to Q0*(t) uniformly for t∈R+ as α→0. Hence, there is an α*>0 and α*<ɛ0 such that
(5.5)Qα*(t)<Q0*(t)+ɛ0∀t∈R+,|α|<α*.
Let (S(t),Q(t),N(t)) be any solution of model (1.2) with initial condition (2.1). Since
(5.6)dS(t)dt≤-DS(t),t≠nT,S(t+)=S(t)+p,t=nT,
for any constant 0<ɛ1<α*, from Lemma 2.2, there is a T0>0 such that
(5.7)S(t)≤S*(t)+ɛ1,t≥T0.
From the second equation of model (1.2), we have
(5.8)dQ(t)dt≤ρ(S*(t)+ɛ1,Q(t))-μ(Q(t))Q(t)∀t≥T0.
From Lemma 2.3 and the comparison theorem, there exists a function β(t):R+→R satisfying β(t)→0 as t→∞ such that
(5.9)Q(t)≤Qɛ1*(t)+β(t)≤Q0*(t)+β(t)+ɛ0∀t≥T0.
Since limt→∞β(t)=0, we can obtain
(5.10)lim⁡t→∞⁡∫tt+T[μ(Q0*(t)+β(t)+ɛ0)-D]dt=∫0T[μ(Q0*(t)+ɛ0)-D]dt<0.
Hence, there exist constants η¯>0 and T1>T0 such that when t≥T1(5.11)∫tt+T[μ(Q0*(t)+β(t)+ɛ0)-D]dt≤-η¯
and |β(t)|<1.
For anyt≥T1, from the third equation of model (1.2), we have
(5.12)dN(t)dt≤N(t)[μ(Q0*(t)+β(t)+ɛ0)-D]∀t≥T1.
We choose an integer rt≥0 such that t∈(T1+rtT,T1+(rt+1)T]. Then, integrating (5.12) from T1 to t, we obtain
(5.13)N(t)≤N(T1)exp⁡⁡∫T1t[μ(Q0*(t)+β(t)+ɛ0)-D]dt≤N(T1)exp⁡⁡{(∫T1T1+rtT+∫T1+rtTt)[μ(Q0*(t)+β(t)+ɛ0)-D]dt≤N(T1)exp⁡⁡(-η¯rt)exp⁡⁡([μ(M1+1+ɛ0)-D]T).
Therefore, by (5.11) we have N(t)→0 as t→∞.
For any constantɛ>0, there is a T*>0 such that S(t)≤M3+ɛ and N(t)≤ɛ for all t>T*. When t>T*, from model (1.2) we obtain
(5.14)dS(t)dt≤-DS(t),t≠nT,S(t+)=S(t)+p,t=nT,n∈N,dS(t)dt≥-DS(t)-ɛρ(M3+ɛ,Qmin⁡),t≠nT,S(t+)=S(t)+p,t=nT,n∈N.
Using the comparison theorem and Lemma 2.2, we can obtain that there is a T>T* such that
(5.15)-ɛρ(M3+ɛ,Qmin⁡)D+S*(t)-ɛ≤S(t)≤S*(t)+ɛ
for all t≥T. From the arbitrariness of ɛ, we finally obtain
(5.16)S(t)⟶S*(t)ast⟶∞.
Finally, from Lemma 2.3, we can easily obtain
(5.17)Q(t)⟶Q0*(t)ast⟶∞.
This completes the proof of Theorem 5.1.Remark 5.2.
Obviously, system (1.2) has a semitrivial T-periodic solution (S*(t),Q0*(t),0) at which microorganism culture fails. Theorem 5.1 shows that T-periodic solution (S*(t),Q0*(t),0) of system (1.2) is global attractivity.Remark 5.3.
The biological meaning of Theorems3.1–5.1 is very significant, which can be seen in [18] (see Remarks 3 and  5 in [18]).Remark 5.4.
We notice that it is quite difficult to obtain the sufficient and necessary conditions on the extinction and permanence of speciesN. So, we take this problem in the future.


## 6. Global Attractivity

Now, we discuss the global attractivity of all positive solutions and the existence of positiveT-periodic solution of model (1.2); we have the following result.Theorem 6.1.
Suppose inequality (4.1) holds. Then  model (1.2) has a unique positive  T-periodic solution, which is globally attractive.Proof.
Define Liapunov functionW(t) as follows:
(6.1)W(t)=S(t)+Q(t)N(t).
From the proof of Theorem 3.1, we have W(t)→S*(t) as t→∞. Hence, W(t)=S*(t)+α(t), where α(t) is defined on R+ with limt→∞α(t)=0. Substitute S(t)=S*(t)+α(t)-Q(t)N(t) into model (1.2), we obtain
(6.2)dQ(t)dt=ρ(S*(t)+α(t)-Q(t)N(t),Q(t))-μ(Q(t))Q(t),dN(t)dt=N(t)(μ(Q(t))-D).
Further, let u(t)=Q(t)N(t); then system (6.2) changes into the following form:
(6.3)du(t)dt=u(t)[ρ(S*(t)+α(t)-u(t),Q(t))Q(t)-D],dQ(t)dt=ρ(S*(t)+α(t)-u(t),Q(t))-μ(Q(t))Q(t).
Let (u1(t),Q1(t)) and (u2(t),Q2(t)) be any two positive solutions of system (6.3), for any constant ϵ0>0 with m̲-ϵ0>0, from Theorems 4.1 and 5.1, there exits a constant T0>0 such that
(6.4)m̲-ϵ0≤Qi(t)≤M3+ϵ0,(m̲-ϵ0)2≤ui(t)≤(M3+ϵ0)2,i=1,2,
for all t≥T0. Define the Liapunov function as follows:
(6.5)V(t)=|ln⁡⁡u1(t)-ln⁡⁡u2(t)|+|ln⁡⁡Q1(t)-ln⁡⁡Q2(t)|.
Obviously, V(t+)=V(t) for all t=nT and n∈N. Calculating the Dini derivative of V(t), for any t≥T0 and t≠nT, we have
(6.6)D+V(t)=sign⁡(u1(t)-u2(t))×[ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)]+sign⁡(Q1(t)-Q2(t))×[ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)+μ(Q2(t))-μ(Q1(t))ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)]=(sign⁡(u1(t)-u2(t))+sign⁡(Q1(t)-Q2(t)))×[ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)]-dμ(ξ(t))dQ|Q1(t)-Q2(t)|,
where ξ(t) is situated between Q1(t) and Q2(t). We claim that
(6.7)(sign⁡(u1(t)-u2(t))+sign⁡(Q1(t)-Q2(t)))×[ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)]≤0.
In fact, there are the following several possible cases:(1)
u1(t)<u2(t),Q1(t)<Q2(t),(2)
u1(t)<u2(t),Q1(t)=Q2(t),(3)
u1(t)<u2(t),Q1(t)>Q2(t),(4)
u1(t)=u2(t),Q1(t)<Q2(t),(5)
u1(t)=u2(t),Q1(t)=Q2(t),(6)
u1(t)=u2(t),Q1(t)>Q2(t),(7)
u1(t)>u2(t),Q1(t)<Q2(t),(8)
u1(t)>u2(t),Q1(t)=Q2(t),(9)
u1(t)>u2(t),Q1(t)>Q2(t).We only need to prove case (1).  Cases (2)–(8) can be easily proved and case (9) can be proved similarly to case (1). For case (1), we have
(6.8)sign⁡(u1(t)-u2(t))+sign⁡(Q1(t)-Q2(t))=-2.
From assumption (H2), we have
(6.9)ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)=ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q1(t))Q1(t)+ρ(S*(t)+α(t)-u2(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)≥ρ(S*(t)+α(t)-u2(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)≥0.
Hence, (6.7) is true. Therefore,
(6.10)D+V(t)≤-dμ(ξ(t))dQ|Q1-Q2|≤-A|Q1-Q2|,
for all t≥T0 and t≠nT, where
(6.11)A=inf⁡⁡{dμ(Q(t))dQ:m̲-ϵ0≤Q(t)≤M3+ϵ0}.
For any t≥T0, integrating from T0 to t, we have
(6.12)V(t)+∫T0tA|Q1(s)-Q2(s)|ds≤V(T0).
Hence,
(6.13)∫T0∞|Q1(s)-Q2(s)|ds<∞.
From Theorem 3.1 and system (6.3), we easily see that dQi(t)/dt(i=1,2)  are bounded for t≥T0 and t≠nT. Since |Q1(t)-Q2(t)| is continuous for all t≥T0, by Lemma 2.5, |Q1(t)-Q2(t)| is uniformly continuous on [T0,∞). Therefore, from Barbalat lemma, we obtain
(6.14)lim⁡t→∞⁡(Q1(t)-Q2(t))=0.
Let
(6.15)G(S,Q)=ρ(S,Q)Q,α=inf⁡⁡{∂ρ(S,Q)∂S:S∈[m̲-ϵ0,M3+ϵ0],Q∈[m̲-ϵ0,M3+ϵ0]},β=sup⁡⁡{|∂G(S,Q)∂Q|:S∈[m̲-ϵ0,M3+ϵ0],Q∈[m̲-ϵ0,M3+ϵ0]}.
From assumption (H2), we have α>0 and β>0. Further, we consider the following Liapunov function:
(6.16)V1(t)=|ln⁡⁡u1(t)-ln⁡⁡u2(t)|.
Obviously, V1(t+)=V1(t) for all t=nT and n∈N. Calculating the Dini derivative of V1(t), for any t≥T0 and t≠nT, from the mean-value theorem, we have
(6.17)D+V1(t)=sign⁡(u1(t)-u2(t))×[ρ(S*(t)+α(t)-u1(t),Q1(t))Q1-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)]=sign⁡(u1(t)-u2(t))×[ρ(S*(t)+α(t)-u1(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q1(t))Q1(t)+ρ(S*(t)+α(t)-u2(t),Q1(t))Q1(t)-ρ(S*(t)+α(t)-u2(t),Q2(t))Q2(t)]≤-∂ρ(ξ1(t),Q1(t))Q1(t)∂S|u1(t)-u2(t)|+|∂G(S*(t)+α(t)-u2(t),ξ2(t))∂Q||Q1(t)-Q2(t)|≤-α(m̲-ϵ0)2|ln⁡⁡u1(t)-ln⁡⁡u2(t)|+δ(t)=-α(m̲-ϵ0)2V1(t)+δ(t),
where ξ1(t) is situated between S*(t)+α(t)-u1(t) and S*(t)+α(t)-u2(t), ξ2(t) is situated between Q1(t) and Q2(t) and δ(t)=β|Q1(t)-Q2(t)|. By the comparison theorem and the formula of variational constant for one-order linear differential equations, we can obtain
(6.18)V1(t)≤V(T0)e-α(m̲-ε0)2(t-T0)+∫T0tδ(s)eα(m̲-ε0)2(s-t)ds,
for all t≥T0. From limt→∞δ(t)=0, we easily obtain V1(t)→0 as t→∞. Consequently,
(6.19)lim⁡t→∞⁡(u1(t)-u2(t))=0.
For any two positive solutions (Si(t),Qi(t),Ni(t))(i=1,2), from Ni(t)=ui(t)/Qi(t) and Si(t)=S*(t)+α(t)-Qi(t)Ni(t)(i=1,2), we can further obtain
(6.20)lim⁡t→∞⁡(S1(t)-S2(t))=0,lim⁡t→∞⁡(N1(t)-N2(t))=0.
This shows that all solutions of model (1.2) is globally attractive.
LetX(t,z)=(S(t,z),Q(t,z),N(t,z)) be some positive solution of model (1.2) with initial condition (S(0),Q(0),N(0))=z, where z=(z1,z2,z3). From Theorems 3.1 and 5.1, there exist two constants M>m>0 such that
(6.21)m≤liminf⁡t→∞S(t,z)≤limsup⁡t→∞⁡S(t,z)≤M,m≤liminf⁡t→∞Q(t,z)≤limsup⁡t→∞⁡Q(t,z)≤M,m≤liminf⁡t→∞N(t,z)≤limsup⁡t→∞⁡N(t,z)≤M.
Hence, there exists an integer n0>0 such that sequence of vectors {X(nT,z)∈R3:n≥n0} is bounded. Hence, there exists a subsequence {nk} and a z*∈R3 such that
(6.22)lim⁡k→∞⁡X(nkT,z)=z*.
Let X(t,z*)=(S(t,z*),Q(t,z*),N(t,z*)) be the solution of model (1.2) with initial condition (S(0),Q(0),N(0))=z*. By the continuity of solutions of system (2.1) with respect to initial values, we obtain
(6.23)lim⁡k→∞⁡X((nk+1)T,z)=lim⁡k→∞⁡X(T,X(nkT,z))=X(T,z*).
On the other hand, we easily prove that X(t+T,z) also is the solution of model (1.2). Hence, from the global attractivity of system (1.2) we can obtain
(6.24)lim⁡t→∞⁡(X(t,z)-X(t+T,z))=0.
Consequently,
(6.25)lim⁡k→∞(X(nkT,z)-X((nk+1)T,z))=0.
Further from (6.22) and (6.23), we obtain X(T,z*)=z*. This shows that X(t,z*) is a positive T-periodic solution of model (1.2).
Finally, from above discussion we obtain thatmodel (1.2) has a unique positive T-periodic solution, which is globally attractive. This completes the proof of Theorem 6.1.Remark 6.2.
According to Theorem6.1, we can find that model (1.2) has a unique positive T-periodic solution that is globally attractive as long as it is only permanent.Remark 6.3.
We easily find that Theorem6.1 is a very good extension in pulse case of the corresponding results given by Smith in [19].


## 7. Numerical Examples

In this section, we will give an example to demonstrate the effectiveness of our main results. We considermodel (1.2) with the following parameters:(7.1)μ(Q)=2(1-0.8Q),ρ(S,Q)=2S1+S,p=2,D=0.6,T=2.Obviously, assumptions(H1) and (H2) hold. We consider the following subsystem:(7.2)dS(t)dt=-DS(t),dQ(t)dt=ρ(S(t),Q(t))-μ(Q(t))Q(t),t≠nT,S(t+)=S(t)+p,Q(t+)=Q(t),t=nT,n∈N.
From Lemmas 2.2 and 2.3, system (7.2) has a periodic solution (S*(t),Q0*(t)). Further, let
(7.3)I=∫0T[μ(Q0*(t))-D]dt.Since the precise expression ofQ0*(t) is quite complex, it is hard to determine the value of I. Therefore, here we will use the method of numerical simulation. Applying MatLab software, we can obtain the numerical simulation of Q0*(t), see Figure 1.Figure 1
Time series ofQ0*(t).Furthermore, by numerical calculation, we can obtain(7.4)∫0T[μ(Q0*(t))-D]dt≈0.93.Therefore, we haveI>0. Thus, by Theorem 6.1 we finally obtain that model (1.2) has a unique positive T-periodic solution, which is globally attractive, see Figure 2.Figure 2
Dynamical behavior of system (1.2).



---

*Source: 843735-2012-05-20.xml*