Abstract
This paper proposes a novel impulsive stochastic nonautonomous chemostat model with the saturated and bilinear growth rates in a polluted environment. Using the theory of impulsive differential equations and Lyapunov functions method, we first investigate the dynamics of the stochastic system and establish the sufficient conditions for the extinction and the permanence of the microorganisms. Then we demonstrate that the stochastic periodic system has at least one nontrivial positive periodic solution. The results show that both impulsive toxicant input and stochastic noise have great effects on the survival and extinction of the microorganisms. Furthermore, a series of numerical simulations are presented to illustrate the performance of the theoretical results.
1. Introduction
The chemostat is a continuous culture device that keeps the flow of the nutrient solution constant and makes the microorganism reproduce under the condition of its maximum growth rate. In a chemostat, the density of microorganisms is controlled by the concentration of the growth-limiting nutrient, and the growth rate is controlled by the washout rate which can be adjusted arbitrarily. The chemostat model is mainly used for laboratory theoretical research. So far, many scholars have obtained significant results for chemostat models [1–11]. While the populations in nature suffer from instantaneous discontinuous interference (for example, toxic input, seasonal harvest, and spraying pesticides), this interference phenomenon can be described as an impulse mathematically [12–16]. Furthermore, the theory and method of impulsive differential equations are widely used in many domains of biological science [17–20]. In reality, many populations are inevitably affected by stochastic disturbances [21–28], and many scholars have explored many stochastic dynamic systems and obtained some new results [29–33]. Recently, impulsive stochastic dynamics models attract the research interests of scholars [34–37]. Therefore, we also consider the influence of stochastic disturbances and impulsive toxic input on the chemostat.
In previous studies, there are almost works on research of chemostat models with one of some different growth rates. In fact, microorganisms take different forms to absorb nutrients at different times, so the growth rates of microorganisms are different. For example, some take saturated growth rate and some adopt bilinear growth rate. Consequently, a model can utilize the transfer function as , where represent the probability of occurrence of and [38]. In this paper, we consider two types of growth rates, bilinear growth rate and saturated growth rate. Hence we assume , where denotes the probability of occurrence of bilinear growth rate and is correspondingly the probability of occurrence of saturated growth rate. In addition, we also consider the impact of time on the system. Then the system becomes nonautonomous: where represents the concentration of the unconsumed nutrient at time and represents the biomass of the population of microorganism at time . is the washout rate at time , is the concentration of the growth-limiting nutrient at time , represents the nutrient intake rate of each individual in microbial population, and it also describes the conversion rate from nutrient to microorganism. is the semisaturated rate at time .
As we know, environmental pollution is one of the most important social and ecological problems at present. It affects the quality of life of human beings, the persistence of species, and the ecological vicissitude of habitat. In order to reasonably apply and control toxic substances, we must evaluate the degree of toxicant’s damage to the population. In recent years, many scholars have already studied the effects of toxicant on various ecosystems [39–44]. In this paper, we use impulsive differential equations to describe the effects of toxicant on microbial population. Besides toxicant, we also consider the effects of environmental noises. In this paper, we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in two types of growth response rate, so that . Then, a nonautonomous impulsive chemostat model with stochastic perturbation in a polluted environment takes the following form:where and denote the concentrations of the toxicant in the organism and in the environment at time , respectively. is the rate of decrease of the intrinsic growth rate at time because of toxicant, represents environmental toxicant uptake rate per unit mass organism, and are organismal net ingestion and depuration rates of toxicant, respectively, denotes the loss rate of toxicant from the environment itself by volatilization, and is the amount of pulsed input concentration of the toxicant at each . Moreover, all the above parameters are positive and , , , and are both positive functions according to its biomathematics meaning, and , , , , , , , , and are both periodic functions with .
The paper is organized as follows. Preliminaries are provided in Section 2. Existence and uniqueness of the global positive solution are demonstrated in Section 3. In Section 4, we explore the sufficient conditions for the extinction and the permanence of system (2). In Section 5, we show that the stochastic system has at least one nontrivial positive periodic solution by constructing a suitable Lyapunov function and a rectangular set. Finally, some numerical simulations are developed to illustrate the performance of the theoretical results.
2. Preliminaries
In this section, we give some notations and lemmas which will be used for the following reasoning process. Throughout this paper, we assume that , , and are continuous at and is left continuous at and , and let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). For an integrable function on , define , , and ; here is a bounded function on . Due to the properties of continuous positive periodic functions, it is obvious that the coefficients of the system (2) satisfy and .
Lemma 1. For any positive solution of system (2) with initial value , we have , .
Proof. From the first two equations of system (2), we have which implies that , . This completes the proof of Lemma 1.
Lemma 2 (see [3]). (i) The microorganism is said to be extinctive if .
(ii) The species is said to be permanent in the mean if there exists a positive constant such that .
Lemma 3 (see [45]). Assume that
(i) system has a unique global solution;
(ii) there is a function which is -periodic in and satisfies the following conditions:andwhere the operator is given byThen the system has a -periodic solution.
Now we consider the following subsystem of system (2):
Lemma 4 (see [3, 20]). System (7) has a unique positive -periodic solution and for each solution of (7), , and as . Moreover, , for all if , and , where for and .
From Lemma 4, we have
3. Existence and Uniqueness of the Global Positive Solution
This section discusses the existence and uniqueness of the global positive solution as an important prerequisite for further research. We take into account the following subsystem of system (2): where the bounded function is defined by system (7).
Theorem 5. For any initial value , there is a unique positive solution of system (2) on , and the solution will remain in with probability one.
Proof. Owing to the coefficients of the system (10) obey local Lipschitz conditions; there exists a unique solution on , where denotes the explosion time. Now, let us show that is global, i.e., .
Let be sufficiently large such that and all lie within the interval , and define the stopping time for each integer . We set ( denotes the empty set), and is distinctly increasing as . Let , and then a.s. If is true, then a.s. and a.s. for . If this hypothesis is not true, there exist a constant and an such that Hence, there is an integer such that Define a function , by where The nonnegativity of this function can be seen from Applying the Itô’s formula to leads to Here is a positive constant which is independent of and . Therefore, Then, in the same way with [34], we get which is a contradiction, and then we have . This completes the proof.
4. Extinction and Persistence in Mean
4.1. Extinction
In this section, we establish sufficient conditions for the extinction of the microorganism, which implies that microculture failed.
Theorem 6. Let be the solution of system (2) with any initial value . If one of the conditions holds(i), and ,(ii), and ,(iii), and ,(iv), and , then the microorganism will die out almost surely, i.e., , moreover, , , and .
Proof. Let . Applying Itô’s formula to system (2), we have Integrating, respectively, from to and dividing both sides of (19) by , one has where , , and by the strong law of large numbers [46] we have and .
Case 1. When and , applying (20) we have Taking the limit superior on both sides of (21), we have Thus .
Case 2. When and , applying (20) we haveTaking the limit superior on both sides of (23), we have Thus .
Case 3. When , with , applying (20) we have Taking the limit superior on both sides of (25), we have Thus .
Case 4. When , with , applying (20) we have Taking the limit superior on both sides of (27), we have This implies , a.s.
Since the limit system of (2) isby Lemma 4, it is clear that , . This completes the proof of Theorem 6.
4.2. Persistence in Mean
Theorem 7. For any initial value , system (2) is permanent in the mean if and the solution of system (2) satisfies
Proof. From system (2),Obviously, , and we obtain Applying Itô’s formula givesThenwhere and . The inequality (35) can be rewritten as where . According to Lemma 1, we see that ; thus one has , as , and .
Taking the inferior limit of both sides of (36), one can yield that By (ii) of Lemma 2, this completes the proof of Theorem 7.
5. The Existence of Nontrivial Positive Periodic Solution
In this section, based on the theory of Has’minskii [45], we testify the existence of the nontrivial positive periodic solution.
Define
Theorem 8. Assuming that , then for any initial value , system (10) has at least one positive -periodic solution.
Proof. Define a nonnegative function : as follows: where , , , is positive constant which is defined in the following discussion, is a positive constant satisfying namely, and obeys whereandIn addition, must have a minimum point as is a continuous function; then we can define a nonnegative function : as follows: Applying Itô’s formula, we can getDefine a positive -periodic function satisfyingand then we obtain We choose then Similarly,andTherefore, we get Define a bounded closed set and here we can choose sufficiently small which satisfies where , , and are both positive constants, which are defined in the following reasoning process. And for a sufficiently small , it is easy to see that if (42) holds, then (56) holds.
Divide into four domains: noting that . Next we prove that on .
Case 1. If , thenwhereAccording to (55), we get Case 2. If , then By means of (56), we can conclude that Case 3. If , thenwhere According to (57), one can get that Case 4. If , then where In view of (58), we have Thus, we have on from the above four cases. By Lemma 3, we can get that system (2) admits a positive -periodic solution. This completes the proof of Theorem 8.
6. Conclusions and Simulations
This paper investigates the dynamics of an impulsive stochastic chemostat model with two different growth rates in a polluted environment. The advantage of this paper is to take two different variable growth rates of microorganisms. We propose the novel model with the transfer function as , where represent the probability of occurrence of and . Hence we assume , and it shows that the microorganism has two types of growth rates which are bilinear and saturated growth rate. Furthermore, if , this means the growth rate of the microorganism has only one growth rate, which is saturated growth rate. On the contrary, if the growth rate of the microorganism is only bilinear growth rate, we can assume . Thus our hypothesis is more extensive. And in this paper, by using the theory of impulsive differential equations and Lyapunov functions method, we establish the conditions for the extinction and the permanence of the microorganisms. Then we demonstrate that the stochastic system has at least one positive -periodic solution. The main results suggest that
(1) If one of the four conditions of Theorem 6 is satisfied, then This implies that the microorganism goes to extinction.
(2) If , system (2) is permanent in the mean.
(3) If , system (10) exists a nontrivial positive periodic solution.
Some numerical simulations are introduced to illustrate our main results [47].
In Figure 1, we set the parameter values in system (2) as , and and ; and ; and ; and . It meets the conditions of Theorem 6 and the microorganisms in figure are all going extinct, so this is consistent with the conclusion in Theorem 6. In order to verify the condition of Theorems 7 and 8, we choose the parameter values as follows: in Figure 2. It obtains and , and the microorganism in Figure 2 is persistence in mean, which is in line with the conclusion in Theorems 7 and 8, respectively.
(a)
(b)
(c)
(d)
(a)
(b)
The results show that both impulsive toxicant input and stochastic noise have great effects on the survival and extinction of the microorganisms. High concentration of toxicant or large intensity of noises can lead to extinction of microorganisms. Appropriate toxin input and noise intensity will make the microorganisms show periodic changes.
In this paper, we considered the effects of white noises. However, there are some other random perturbations in the environment, for example, regime-switching and Markovian switching. In fact, stochastic regime-switching or Markovian switching population models were studied by several authors recently [27, 43, 48, 49]. Therefore, it is interesting to consider regime-switching and Markovian switching models, and we will leave these cases as our future work.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Research Fund for the Taishan Scholar Project of Shandong Province of China and the SDUST Research Fund (2014TDJH102).