Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 843735, 25 pagesdoi:10.1155/2012/843735

Research Article

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

Kai Wang,¹ Zhidong Teng,² and Xueliang Zhang¹

¹Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830054, China²College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received 23 December 2011; Accepted 20 March 2012

Academic Editor: Xue He

Copyright © 2012 Kai Wang 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.

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.

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: 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 For (1.2), we will investigate the permanence and extinction of the species, and the existence of positive -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 for (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 (1.2). In Section 4, we will state and prove the sufficient condition on the permanence of species for (1.2). In Section 5, the sufficient condition on the extinction of species for (1.2) is given. In Section 6, the existence of positive -periodic solution and the global attractivity of solutions for (1.2) are established. Finally, in Section 7, we will discuss an example and give some numerical simulations.

2. Preliminaries

In (1.2), denotes the phytoplankton cells at biomass concentration, 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 . is the input and output flow, and its referred to as the dilution rate; is the amount of the substrate concentration pulsed each , where is a constant. Function denotes the phytoplankton growth rate, function denotes the nutrient uptake rate. In this paper, for model (1.2), we always assume that the following condition holds.(H₁)Function is continuously differentiable and for all , , where is some given positive constant.(H₂)Function is continuously differentiable and , for all and . Further, for all .

The initial condition in model (1.2) is given in the following form:

Firstly, on the positivity of solutions for (1.2), we have the following result.

Lemma 2.1. Any solution of (1.2) with initial condition (2.1) is positive, that is, , and for all .

Proof. Obviously, for any , we have We consider . If the conclusion is not true, then from and we can obtain that there is a such that and for all . From the mean-value theorem and assumption , for any , there is a such that Hence, from the first equation of (1.2), we obtain Choose an integer such that . Obviously, and integrating (2.4) from to , which leads to a contradiction.Now, we consider . We have and . If , then since right derivative there is a such that for all . If there is a such that and for all , then left derivative . But from assumption (), we have which leads to a contradiction. If , 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 where and are defined in (1.2) and is a constant. Clearly,

is the -periodic solution of (2.8), where

We say that is globally uniformly attractive if for any constants and there is a constant such that for any initial time and initial value with , one has where is the solution of (2.8) with initial condition . We have following result.

Lemma 2.2 (see [27]). -periodic solution of (2.8) is globally uniformly attractive.

For (2.8), when , we obtain the subsystem of (1.2) with as follows: Clearly, is the positive -periodic solution of (2.12). From Lemma 2.2, we obtain that for any solution of (2.12), one has

Putting in the second equation of (1.2), we obtain From the expression of , we can choose a constant such that

For any , we consider the following equation: For any initial points and , let and be the solutions of (2.14) and (2.16) satisfying initial conditions and , respectively. We have the following result.

Lemma 2.3. (a)  There are constants and such that for any , and (b) For each , (2.16) has a positive and globally uniformly attractive -periodic solution . (c) converges to uniformly for as .

Proof. Firstly, inequality for all can be proved by using the similar argument as in the proof of Lemma 2.1.From assumptions () and (), there are positive constants and such that for any If for all , then integrating (2.16), for any , we have Hence, as , which leads to a contradiction. Therefore, there is a such that . Further, if there are such that , , and for all , where then we can choose an integer such that and integrating (2.16) to obtain which also leads to a contradiction. Therefore, we finally have Choose constant , then we have for all . Thus, conclusion () is proved.From conclusion (), directly using the main results given by Teng and Chen in [31], we can obtain that (2.16) for each has a positive -periodic solution .For any constant and , let be a solution of (2.16) with initial value . By conclusion (), there is a constant such that Consider Liapunov function Calculating the Dini derivative , from we can obtain for any and Using the mean-value theorem, we further obtain where is situated between and . When and , we obviously have . Hence Consequently, by (2.23) we have for all . Hence Further, from (2.23) we obtain for all . Consequently, from (2.27) it follows that where Hence, we further have For any constant , from (2.34), choose then for any we can obtain Therefore, This shows solution is globally uniformly attractive. Thus, conclusion () is proved.Finally, we prove conclusion (). From assumptions and , we can easily prove that the right hand of (1.2) satisfies the uniform Lipschitz condition with respect to . Hence, by the continuity of solutions with respect to parameter , we can obtain that uniformly for as . Thus, conclusion () is proved. This completes the proof of Lemma 2.3.

When , (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 -periodic solution which is globally uniformly attractive.

The following lemma will be used in the proof of the result on the global attractivity of (1.2).

Lemma 2.5. Let function be continuous on and differentiable for any ( ), where is a constant. If there exist a constant , such that for any , then is uniformly continuous on .

The proof of Lemma 2.1 is simple, we hence omit it here.

3. Boundedness

On the ultimate boundedness of all positive solutions of (1.2), we have the following result.

Theorem 3.1. Let be any positive solution of (1.2), then we have that there exists a constant , which is independent of any positive solution of (1.2), such that

Proof. Let be any positive solution of (1.2) with initial condition (2.1). Since by Lemma 2.2 we directly have For any constant , there is a such that From the second equation of (1.2), we have for all . Using the similar argument as in the proof of Lemma 2.3, we have that there is a positive constant , and is independent of any solution of (1.2), such that Define a function as follows: Calculating the derivative of along solution of (1.2), we have From Lemma 2.2, we obtain From Lemma 2.1, we have for all . Hence, by (3.9) we can obtain that is ultimately bound. This completes the proof of Theorem 3.1.

Remark 3.2. Obviously, Theorem 3.1 is new and can serve as an extension of corresponding result given by Smith in [19].

4. Permanence

On the permanence of species for (1.2), we have the following result.

Theorem 4.1. Suppose Then (1.2) is permanent.

Proof. From Lemma 2.1 we directly have . Hence, is obviously permanent. We now prove the permanence of . From (4.1), we can choose a constant small enough such that Let be any solution of (1.2) with initial condition (2.1). From Theorem 3.1, for any constant there is a such that for all . For any , we consider the following assistant equation: By Lemma 2.3, we obtain that (4.4) has a unique positive -periodic solution , which is globally uniformly attractive and converges to uniformly for as . Hence, there is an and such that Choose positive constants and such that We first proof that for any solution of (1.2) with initial condition (2.1).In fact, if (4.8) is not true, then there is a solution of (1.2) such that . Hence, there is a such that , for all . From the fist equation of (1.2) and (4.3), we have From the comparison theorem of impulsive differential equations and Lemma 2.2, for above , there is an and such that From the second equation of (1.2), we have for all . Let , from Lemma 2.3 and the comparison theorem, there exists an such that Then, from the third equation of (1.2), we have Integrating (4.13) from to , we obtain Obviously, from (4.2) and (4.14), we obtain as , which leads to a contradiction. Therefore, (4.8) is true.Now, we prove that there is a constant such that for any solution of (1.2) with initial condition (2.1).Assume that (4.15) is not true, then there exists a sequence of initial values , which satisfy , , and such that for solution of (1.2) with initial condition , , and From (4.8) and (4.16), we obtain that there exist two sequences and such that for each From Theorem 3.1, there is a such that for all . Further for each , from (4.17) there is an integer such that for all . Hence for any and , we have where . Therefore, for any and , integrating (4.21) on , we obtain from (4.18) Consequently, Consider the following equation: From Lemma 2.2, there is a constant such that for any and where is the solution of (4.24) with initial value .Further consider the following equation: From Lemma 2.3, there is a constant such that for any and where is the solution of (4.26) with initial value .From (4.23), we can choose an integer such that Since for any integer , , and we have from the comparison theorem and inequality (4.25), we have for all and . Further, since for all and . From the comparison theorem and inequality (4.27), for all and . Hence, for any , and , we have Integrating (4.33) from to , by (4.19) we obtain which is contradictory. This contradiction shows that (4.15) is true for any positive solution of (1.2).Next, we prove that in (1.2) is permanent. Let be any solution of (1.2) with initial condition (2.1). From the assumption of and the mean-value theorem, for any , there exists a such that Further, from assumption () and the boundedness of solution , there exists a constant such that From the first equation of (1.2), we obtain for all and , and Using the comparison theorem and Lemma 2.2, we can obtain This shows that is permanent. This completes the proof of Theorem 4.1.

5. Extinction

On the extinction of species of (1.2), we have the following result.

Theorem 5.1. Suppose Then for any positive solution of (1.2)

Proof. From (5.1), there exists a constant such that For any , we consider the following equation: By Lemma 2.3, we obtain that positive -periodic solution of (5.4) is globally uniformly attractive and converges to uniformly for as . Hence, there is an and such that Let be any solution of (1.2) with initial condition (2.1). Since for any constant , from Lemma 2.2, there is a such that From the second equation of (1.2), we have From Lemma 2.3 and the comparison theorem, there exists a function satisfying as such that Since , we can obtain Hence, there exist constants and such that when and .For any , from the third equation of (1.2), we have We choose an integer such that . Then, integrating (5.12) from to , we obtain Therefore, by (5.11) we have as .For any constant , there is a such that and for all . When , from (1.2) we obtain Using the comparison theorem and Lemma 2.2, we can obtain that there is a such that for all . From the arbitrariness of , we finally obtain Finally, from Lemma 2.3, we can easily obtain This completes the proof of Theorem 5.1.

Remark 5.2. Obviously, system (1.2) has a semitrivial -periodic solution at which microorganism culture fails. Theorem 5.1 shows that -periodic solution of system (1.2) is global attractivity.

Remark 5.3. The biological meaning of Theorems 3.1–5.1 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 species . 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 positive -periodic solution of (1.2); we have the following result.

Theorem 6.1. Suppose inequality (4.1) holds. Then (1.2) has a unique positive -periodic solution, which is globally attractive.

Proof. Define Liapunov function as follows: From the proof of Theorem 3.1, we have as . Hence, , where is defined on with . Substitute into (1.2), we obtain Further, let ; then system (6.2) changes into the following form: Let and be any two positive solutions of system (6.3), for any constant with , from Theorems 4.1 and 5.1, there exits a constant such that for all . Define the Liapunov function as follows: Obviously, for all and . Calculating the Dini derivative of , for any and , we have where is situated between and . We claim that In fact, there are the following several possible cases:,,,,,,,,.We only need to prove case .  Cases – can be easily proved and case can be proved similarly to case . For case , we have From assumption (), we have Hence, (6.7) is true. Therefore, for all and , where For any , integrating from to , we have Hence, From Theorem 3.1 and system (6.3), we easily see that are bounded for and . Since is continuous for all , by Lemma 2.5, is uniformly continuous on . Therefore, from Barbalat lemma, we obtain Let From assumption (), we have and . Further, we consider the following Liapunov function: Obviously, for all and . Calculating the Dini derivative of , for any and , from the mean-value theorem, we have where is situated between and , is situated between and and . By the comparison theorem and the formula of variational constant for one-order linear differential equations, we can obtain for all . From , we easily obtain as . Consequently, For any two positive solutions , from and , we can further obtain This shows that all solutions of (1.2) is globally attractive.Let be some positive solution of (1.2) with initial condition , where . From Theorems 3.1 and 5.1, there exist two constants such that Hence, there exists an integer such that sequence of vectors is bounded. Hence, there exists a subsequence and a such that Let be the solution of (1.2) with initial condition . By the continuity of solutions of system (2.1) with respect to initial values, we obtain On the other hand, we easily prove that also is the solution of (1.2). Hence, from the global attractivity of system (1.2) we can obtain Consequently, Further from (6.22) and (6.23), we obtain . This shows that is a positive -periodic solution of (1.2).Finally, from above discussion we obtain that (1.2) has a unique positive -periodic solution, which is globally attractive. This completes the proof of Theorem 6.1.

Remark 6.2. According to Theorem 6.1, we can find that (1.2) has a unique positive -periodic solution that is globally attractive as long as it is only permanent.

Remark 6.3. We easily find that Theorem 6.1 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 consider (1.2) with the following parameters:

Obviously, assumptions and hold. We consider the following subsystem: From Lemmas 2.2 and 2.3, system (7.2) has a periodic solution . Further, let

Since the precise expression of is quite complex, it is hard to determine the value of . Therefore, here we will use the method of numerical simulation. Applying MatLab software, we can obtain the numerical simulation of , see Figure 1.

Figure 1: Time series of .

Furthermore, by numerical calculation, we can obtain

Therefore, we have . Thus, by Theorem 6.1 we finally obtain that (1.2) has a unique positive -periodic solution, which is globally attractive, see Figure 2.

Figure 2: Dynamical behavior of system (1.2).

Acknowledgments

Supported by the National Natural Science Foundation of China (10961022) and Doctoral Scientific Research Fund of Xinjiang Medical University.

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