On a Hibernation Plankton-Nutrient Chemostat Model with Delayed Response in Growth ()
1. Introduction
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 [1] . 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 [2] [3] [4] [5] 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 [6] [7] [8] [9] . 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 [10] . Some scholars also proposed that hibernation can make animals through hardship on cold environments and limited availability of food in [11] . 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 [12] and Okubo [13] . Ruan discussed Turing instability and the existence of travelling wave solutions in [14] .
Furthermore, it is necessary to study a chemostat model with hibernation and impulsive diffusion on nutrients. In [15] , the author considered the dynamics of a plankton-nutrient chemostat model with hibernation and it was described by impulsive switched systems as follows
(1.1)
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
(1.2)
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. [16] 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. [17] 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
where
.
The proof of Lemma 3 is simple so we omit it here.
3. Extinction
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
(3.1)
Integrating and solving the system (3.1) equations between pulses, we have
(3.2)
Consider the stroboscopic map of system (3.1), from the third, fourth, seventh and eighth equations of system (3.1) we have:
(3.3)
where
,
. 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
(3.4)
To change System (3.3) to a map, we define the map
(3.5)
is the map calculate at the point
. According to the lemma 3.2 and 3.4 of Refer [15] , we obtain
(3.6)
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
(3.7)
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
(3.8)
where
. From the second equation of (1.2) we have
.
Consider the following equations with pulse
(3.9)
From(3.6) and (3.9) we have that
and
as
. Therefore, there exist a integer
and an arbitrary positive parameter
, such that
(3.10)
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
(3.11)
The system (3.11) has a positive solution
, where
are expressed as follows
(3.12)
In which
(3.13)
and
,
. For arbitrary
, there exists a constant
, such that, for all
,
Let
, we obtain
(3.14)
For
and
,
and
. The proof of Theorem 1 is completed.
4. Permanence
In this section we shall study the permanence of system (1.2).
Theorem 2. System (1.2) is permanent, if
(4.1)
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
(4.2)
Define
Derive
along with solution of system (1.2), we obtain
(4.3)
Set
From (4.4) we obtain
. We may select a positive integer
small enough, such that
(4.4)
where
For
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
(4.5)
Consider the following impulsive differential equation
(4.6)
The system (4.6) has a unique globally asymptotically stable positive solution as follows
(4.7)
where
(4.8)
There exists
, such that
(4.9)
where
.
Take
. For any
, from (4.3) and (4.9) we obtain
(4.10)
Let
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
.
Let
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
and
So
and
are permanent. The proof of Theorem 2 is completed.
According to Theorem 1 and 2, we may derive the following conclusion.
Corollary
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
where
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.