The chemostat is an important experimental instrument used to provide a controlled environment. Under this condition, the experimenter can adjust the parameters of system and get the final outcome. This chemostat model had been discussed by Smith and Waltman in  . In fact, the taken nutrient will not immediately absorbed by microorganism. In other words, nutrients with transformation from the substrate to microorganism have a lag time. Many scholars     make discussion about chemostat model with discrete time delay. However, the system will have some changes because of the influence of climate; these perturbations break the continuity of the system. So the impulsive differential equations are considered into the system in     . It is important for us to know more about ecology.
In recent years, some authors pay more attention to the hibernation of the plankton. The hibernation has an important sense of adaptation in ecology. Due to unfavorable environmental conditions, the plankton enters a hibernation state in advance. In order to save energy, plankton must maintain the weak life period overcoming the difficulties, such as drought stress, cold climate and temperature. The pressures elimination will restore growth. By hibernation, animals can reduce energy requirement and survive a few months in  . Some scholars also proposed that hibernation can make animals through hardship on cold environments and limited availability of food in  . However, there are many factors of plankton movements in the lakes, such as currents and river diffusion. These researches are seen in Levin and Segel  and Okubo  . Ruan discussed Turing instability and the existence of travelling wave solutions in  .
Furthermore, it is necessary to study a chemostat model with hibernation and impulsive diffusion on nutrients. In  , the author considered the dynamics of a plankton-nutrient chemostat model with hibernation and it was described by impulsive switched systems as follows
where , , is the set of all positive integers; and represent the concentration of the nutrient in the river and reservoir at time respectively. is the concentration of the plankton in the reservoir at time . is the input nutrient concentration in the river. is the dilution rate. is the yield of plankton per unit mass of substrate. is the death rate of the plankton in the intervals of hibernation. and are the concentration of the nutrient in the river and reservoir immediately after the th diffusion pulse at time respectively, while and are the concentration of the nutrient in the river and reservoir before the th diffusion pulse at time separately. Due to the effect climate, the period of system is divided into two sections. That is normal seasons and drought seasons. In the normal seasons, the plankton grow regularly. The plankton is in hibernation in the drought seasons. are moments of torrential rain, the nutrient is diffusing between rivers and reservoir in moments of torrential rain. are moments of rainy season. and are the amount of nutrients coming from surrounding soil in moments of rainy season.
Based on the above discussion, we consider the following a hibernation plankton-nutrient chemostat model with delayed response in growth
Suppose system (1.2) is connected by impulsive diffusion spread between rivers and reservoirs. There is no nutrients input in reservoir. Nutrient input is thought to come from the upper stream. Where constant represents the time delay involved in the conversion of nutrient to plankton. Due to the chemostat outflow, is the positive constant, since it is assumed that the current change in biomass depends on the amount of nutrient consumed units of time before time and that survive in the chemostat the units of time assumed necessary to complete the nutrient conversion process. Other parameters are the same as system (1.1). is continuous on
and , there exists and .
For system (1.2), we will discuss the sufficient and necessary conditions for the permanence and extinction. This paper can be summarized as follows. In Section 2, we present some preliminary results about system (1.2). Our results about extinction are stated and proven in Section 3. In Section 4, we study the permanence of system (1.2). Finally, we give a brief discussion and numerical analysis.
2. Preliminary Results
In this part, we will give some lemmas which will be useful for our main results.
Lemma 1.  Consider the following impulsive differential system
where , and is the constant. Assume the sequence satisfies , with ; and w(t) is left-continuous at . Then
Lemma 2.  Consider the following delay differential equation:
where and are all positive constants and for .
a) If , then ;
b) If , then .
Lemma 3. For any positive solution of system (1.2) satisfy , there exists a constant , such that
The proof of Lemma 3 is simple so we omit it here.
The solution of system (1.2) corresponding to is called plankton-ex- tinction periodic solution. For system (1.2), if we select , then system (1.2) becomes the following model
Integrating and solving the system (3.1) equations between pulses, we have
Consider the stroboscopic map of system (3.1), from the third, fourth, seventh and eighth equations of system (3.1) we have:
. Equation (3.3) is difference equations. The dynamical behaviors of system (3.3) with equation (3.2) have been decided to the dynamical behaviors of system (3.1). So we focus on discussing System (3.3). System (3.3) has the following unique solution
To change System (3.3) to a map, we define the map
is the map calculate at the point . According to the lemma 3.2 and 3.4 of Refer  , we obtain
Hence, system (1.2) has a positive plankton-extinction periodic solution . In what follows, we will study the globally attractive of the plankton-extinction periodic solution of system (1.2).
Theorem 1. The periodic solution of system (1.2) is globally attractive, if
where and parametric and are given in (3.4).
Proof. Suppose is any positive solution of system (1.2) with . Based on the condition (3.7), we set
, then is strictly increasing function for arbitrary . We may select sufficiently small , such that
where . From the second equation of (1.2) we have .
Consider the following equations with pulse
From(3.6) and (3.9) we have that and as . Therefore, there exist a integer and an arbitrary positive parameter , such that
for all , where . For , from (3.10) and the second equation of (1.1), we have
Consider the following impulsive differential equation
According to lemma 2 and condition (3.8), we obtain that . Since when , by the impulsive delay differential equation and the nonnegative of the solutions, we obtain as . Without loss of generality, for all , we may suppose that . By
the second equation of the system (1.2), we have
Consider the following comparison system with pulse
The system (3.11) has a positive solution , where are expressed as follows
and , . For arbitrary , there exists a constant , such that, for all ,
Let , we obtain
For and , and . The proof of Theorem 1 is completed.
In this section we shall study the permanence of system (1.2).
Theorem 2. System (1.2) is permanent, if
where , and are given in (3.4) and (4.8) respectively.
Proof. Suppose is any positive solution of system (1.2) with . We may rewrite the second equation of the system (1.2) as follows
Derive along with solution of system (1.2), we obtain
From (4.4) we obtain . We may select a positive integer small enough, such that
any nonnegative integer , we claim that inequality is not hold for all . Otherwise, there exists a positive parameter , such that for all . From the system (1.2) we obtain
Consider the following impulsive differential equation
The system (4.6) has a unique globally asymptotically stable positive solution as follows
There exists , such that
Take . For any , from (4.3) and (4.9) we obtain
In what follows, we prove that for all . Otherwise, there exists a positive constant , for any , we have , and . Therefore, according to the third equation of (1.2) and (4.10), we further obtain
which is a contradiction. So we have that for any . From (4.4) we have that and from which we obtain
as . This contradict to . Hence, for any nonnegative constant with , the inequality is not hold.
On the one hand, if always holds for large enough, then our target is obtained. Otherwise, suppose is oscillatory about .
In what follows, we shall prove that for all . There exist two positive integer and such that
. When is large enough, is hold true for any . Since is uniformly continuous without impacted by pulse. Therefore, for any and
, we have . By this, we have
for . When , our goal is obtained. When , we have from the third equation of (1.2) that . According to , we obtain for . Then can hold true for . For and , we obtain that . By above similar argument, we can show that for . Since the interval is arbitrarily chosen by us, we get that for large enough. In view of our above arguments, the choice of is independent of any solution of (1.2) which satisfies that for large enough. So is hold true for large enough.
For all , from lemma 3 we have that . From Theorem 1, we have and as and , where
So and are permanent. The proof of Theorem 2 is completed.
According to Theorem 1 and 2, we may derive the following conclusion.
1) The plankton-extinction periodic solution is globally attractive if and only if
where these parametric are the same as the Theorem 1.
2) The plankton of System (1.2) is permanent if and only if
5. Discussions and Numerical Analysis
In this paper, we investigate the necessary and sufficient conditions for the plankton-extinction periodic solution and permanence of sys-
tem (1.2). If the time delay exceeds a certain amount of time, the plankton of system (1.2) will become extinct. If the time delay is under a certain amount of time, the plankton will be lasting survival in the system. So delay plays an important role in affecting the dynamic behavior of the system. Next, we use numerical simulation to illustrate our mathematical results.
From Theorem 1, we consider dynamical behavior of the system (1.2) with , , , , , , , , , , , , , , ,
. From (3.7) we obtain that . The plan- kton-extinction solution is globally attractive; the plankton of system (1.2) will become extinct in this case.
From Theorem 2, we consider dynamical behavior of the system (1.2) with , , , , , , , , , , , , , ,
. From (4.1) we obtain that . The
plankton of System (1.2) is permanent; the plankton will be lasting survival in the system.
It is difficult to study the global attractivity of system (1.2) analytically. From the numerical simulation (Figure 1) we see that there has a unique -period solution of system (1.2) which is globally attractive. The numerical simulation (Figure 2) also shows that system (1.2) is permanent. In
(a) (b)(c) (d)
Figure 1. The plankton-extinction solution of system (1.2) is globally attractive.
(a) (b)(c) (d)
Figure 2. The plankton of System (1.2) is permanent.
view of analytical results, we showed the possibility of establishing control strategy of system (1.2) based on impulsive diffusion and time delay.
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