Stochastic Characteristics and Optimal Control for a Stochastic Chemostat Model with Variable Yield

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coeﬃcient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itˆo’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of microorganism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the inﬂuence of the variable yield on the microorganism.


Introduction
e chemostat occupies a central place in mathematical ecology. It has been widely used in modeling natural ecosystems such as lakes and establishing waste-water treatment mathematical model. Many works related to the chemostat model have been published in the journal of mathematical, biological, and chemical engineering (see [1][2][3][4][5], etc.).
A large number of scholars are attracted to yield coefficient reflecting the conversion of nutrient to microorganism. Most of the models assume that the yield coefficient is a constant (see [1,2]). Smith and Waltman [1] have described a chemostat model with constant yield as follows: x ′ (t) � x(t)(p(S, x) − D), where all the parameters are positive constants. S(t) and x(t) stand for the concentrations of the nutrient and the microorganism at time t in culture vessel, respectively. S 0 is the input nutrient concentration. D is the common dilution rate. c is a "yield" constant. p(S, x) represents the growth function. Sun and Yin [2] and Wang et al. [3] have studied the chemostat model with Monod growth function. Bayen et al. [4] and Sun and Chen [5] have investigated the chemostat model with Contois growth function. Based on model (1), choosing Contois growth function p(S, x) � (mS(t)/kx(t) + S(t)), the model takes the following form: where m > 0 represents the maximal growth rate and k > 0 stands for the growth coefficient of the Contois function. However, experimental data indicate that a constant yield may fail to explain the observed oscillatory behavior in the vessel (see [6]).
is leads to the formulation of the variable yield model, for example [7][8][9]. e chemostat model with variable yield takes the following form:

mS(t) kx(t) + S(t) x(t) A + BS(t) , x ′ (t) � x(t) mS(t) kx(t) + S(t)
− D , where c(S) � A + BS is the variable yield which is considered in [9]. Both A and B are positive constants. Actually, the above models are all deterministic models, but almost all ecosystems are inevitably perturbed by various types of environmental noises. Taking into account the realistic and biological significance, many scholars have investigated stochastic chemostat models (see [3,[10][11][12][13][14]) and stochastic population models (see [15][16][17][18][19]). In the chemostat model, the micro-organism may be affected by distribution of nutrient, temperature, humidity, etc, which can be described by continuous white noise.
ere are different possible approaches to include random effects in the model, both from a biological and mathematical perspective. Some authors have superimposed white noise processes on the dilution rate [20]; others have considered the linear white noise [11,21]. Our basic approach is analogous to that of Beddington and May [21]. By this method, the white noise are directly proportional to S(t) and x(t), influenced on S′(t) and x ′ (t). A stochastic chemostat model with variable yield and Contois growth function will be investigated, and the model (3) will be rewritten as the form: where σ i > 0 (i � 1, 2) are intensities of the white noise. B i (t) (i � 1, 2) are independent standard Brownian motions defined on a complete probability space (Ω, F, F t t≥0 , P) with the filtration F t t≥0 satisfying the usual conditions. e model (4) turns into the corresponding deterministic model (3) if the noise intensities σ i � 0 (i � 1, 2). e initial conditions of (4) are given as where S 0 , x 0 are positive random variables, and denote R 2 Because the solutions of the deterministic model (3) are no longer the solutions of the stochastic model (4), the dynamic behavior of the deterministic model (3) is distinct from that of the stochastic model (4). For further details on the stochastic model, please see [10][11][12][13][14], they have investigated the stochastic characteristics of these stochastic models. However, the nonlinear term of the model (4) is different from that of their models. erefore, it is valuable to investigate the stochastic characteristics of the stochastic chemostat model with variable yield and Contois growth function. On the other hand, the optimal control plays an important role in practical application [4,22]. Many scholars have paid more attention to the optimal stochastic control problem, which covers all aspects of physics, biology, economics, etc. Ding et al. [23] have solved the distributed H ∞ state estimation problem for a class of discrete time-varying nonlinear system with both stochastic parameters and stochastic nonlinearities. Guo et al. [24] have studied the near-optimal control of a stochastic SIRS epidemic model that includes a nonmonotone incidence rate. Framstad et al. [25] have proved a sufficient maximum principle for the optimal control of jump diffusions and showed its connections to dynamic programming and given applications to financial optimization problems in a market described by such processes. us, it is necessary to study the optimal stochastic control of a stochastic chemostat model. roughout the whole paper, Lyapunov function, Itô formula, and other basic methods can be referred to these monographs [26][27][28][29][30][31][32].
is paper is organized as follows. In Section 2, the existence of the unique global positive solution for the model (4) is studied. In Section 3, the stochastic characteristics of the stochastic model (4) are investigated, including the extinction at an exponential rate of the microorganism, the strong persistence in the mean of the microorganism, and the stationary distribution of the model (4). In Section 4, the necessary condition of an optimal stochastic control for the stochastic model (4) is investigated, and the near-optimal stochastic control is mentioned. In Section 5, the numerical simulations conclude the paper, and the influence of the variable yield on the microorganism is explained by taking the different parameters.

Preliminaries
First of all, the notations are described for the whole paper as follows: (i) Ω: a set of the elementary events (ii) F: a family of the subsets of Ω (iii) F t t≥0 : a family of increasing sub− σ− algebras of F (iv) P(ω): the probability of events ω (v) EX: the expectation of X (vi) ∅: the empty set (vii) I A : the indicator function of a set A, i.e., I A (x) � 1 if x ∈ A or otherwise 0 (viii) a.s.: almost surely (ix) E n : the n− dimensional Euclidean space (x) R: the set of all real numbers (xi) R + : the set of all nonnegative real numbers, i.e., In view of biological significance and dynamical behavior, the first concerned thing is whether the solution is unique, global, and positive. Hence, to further study the stochastic chemostat model with variable yield and Contois growth function (4), the first problem to be solved is the existence of the unique global positive solution, namely, there is no explosion in a finite time under the initial value (5). If the coefficients of the equations are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [28,30]), the stochastic differential equations for any given initial value have a unique global solution. However, the stochastic model (4) may allow the solutions to explode at a finite time because the coefficients of the stochastic model (4) do not satisfy the linear growth condition. So, we need to search for the positive solutions. e following theorem assures that the solution of the model (4) with the initial value (5) is unique, global, and positive. First, a lemma and a remark are given.
Lemma 1 (see [19]). The following inequality holds: Remark 1. Based on Lemma 1, since the inequality u ≤ 2(u + 1 − ln u) − (4 − 2 ln 2), we can obtain u ≤ 2(u + 1 − ln u) holds for all u > 0. Next, the existence of the unique global positive solution for the model (4) is proved. Theorem 1. For any initial value (S 0 , x 0 ) ∈ R 2 + , there exists a unique positive solution (S(t), x(t)) to the model (4) for t ≥ 0, and the solution will remain in R 2 + with probability one (i.e., Proof. Since the coefficients of the stochastic differential equations (4) satisfy the local Lipschitz condition, the model (4) has a unique positive local solution (S(t), where τ e is the explosion time. In order to show the solution is global, we just need to prove that τ e � ∞ a.s.. Select k 0 ≥ 1 sufficiently large such that S 0 and x 0 all lie within the interval can be verified. Next, for τ ∞ � ∞, the proof process is as follows: Define a C 2 − function V: by using Lemma 1 so, where Integrating equation (11) both sides from 0 to τ k ∧T and taking expectation, By using Gronwall's inequality, we have Complexity 3 For ∀ω ∈ τ k ≤ T , it exists that S(τ k , ω) or x(τ k , ω) equals either k or (1/k), us, where so P τ ∞ � ∞ � 1 as required. e proof is thus complete.

Stochastic Characteristics
In this section, the stochastic characteristics of the stochastic model (4) are discussed. As is known to all, the equilibriums of the deterministic model (3) are no longer the equilibriums of the stochastic model (4), then what happens in the stochastic model (4)? Next, the sufficient conditions of the extinction and persistence of microorganism and the stationary distribution of the model (4) are investigated.
In order to study the model (4), we introduce the following definitions and lemmas, and Lemma 2 (see [14]). For any initial value (5), the solutions S(t) and x(t) of the model (4) have the properties that at is, there are two positive constants H 1 and H 2 such that for all t ≥ 0, a.s..

(19)
Lemma 3 (see [28]). Let M � M t t≥0 be a real-valued continuous local martingale vanishing at t � 0. en Next, the extinction at an exponential rate of the microorganism is studied.

Theorem 2.
If the condition holds, then for any initial value (5), the solution (S(t), x(t)) of the model (4) satisfies which means at is, the microorganism will be extinct at an exponential rate with probability one.

Proof. Define a function
using Itô formula Integrating both sides from 0 to t, we can obtain where According to equation (27), we have which implies that us, the proof is completed.
eorem 2 shows that the condition σ 2 2 > 2(m − D) makes the microorganism extinct, but it has nothing to do with m < D or m > D, which means that large noises can lead to the extinction of the microorganism, although the microorganism is persistent in the corresponding deterministic model (3) (see Figures 1 and 2) A definition and a lemma are given before discussing the strong persistence in the mean of the microorganism.
Definition 1 (see [17]). e population x(t) is said to be strong persistence in the mean if where 〈x〉 * � lim inf t⟶∞ 〈x〉.
and lim t⟶∞ (F(t)/t) � 0, a.s., then Theorem 3. For any initial value (5), if the condition holds: then the solution (S(t), x(t)) of the model (4) satisfies at is, the microorganism is almost surely strongly persistent in the mean.
On the other hand, we will discuss the long time behavior of the microorganism x in two cases, we define so, Substituting (39) into (45), we obtain Complexity take note of the condition of eorem 3, we have λ 1 > 0. us, from (42), Lemmas 3 and 4, we can obtain that where λ 2 � (m/kH 2 + 1) − D − (1/2)σ 2 2 , similar to the above, we can obtain that so, Obviously, (54) where F 2 (t) � σ 2 B 2 (t) + ln x 0 , similarly, we have Similar to the above, we have In conclusion, no matter what S(t) chooses, we have is completes the proof of eorem 3.

Remark 3.
It follows from eorem 3 that the microorganism will be strongly persistent in the mean if the condition (1/2)σ 2 2 < ζ≜ min (m/kH 2 + 1) − D, (mS 0 /kH 2 + 1)− D} is satisfied. at is, when the intensity of the white noise is small enough, the microorganism will survive (see Figures 3  and 4).
Stationary distribution is one of the most significant dynamical characteristics of the stochastic model; that is, the stochastic model has a stationary distribution which represents the persistence of microorganism in the future. erefore, in the rest of this section, we will make a positive decision by the existence of the stationary distribution. Before proving the main result, several known results are given for the stochastic differential equations.
Assume X(t) is a regular time-homogeneous Markov Process in n− dimensional Euclidean space E n . e stochastic differential equation takes the following form: (60) e diffusion matrix of the process X(t) is defined as follows: (61) Define the differential operator L associated with (60) by Lemma 5 (see [30] [31,32]). To validate condition (2) of Lemma 5, we need to show that there exists a neighborhood U and a non-negative C 2 -function V such that for any (S, x) ∈ R 2 + ∖U, LV is negative.
8 Complexity e following theorem shows that the model (4) has a unique stationary distribution for any given initial value (5).
Proof. According to the model (4), we get So, the diffusion matrix is Let U be any bounded open domain in R 2 + , then there exists a positive constant such that 2 i,j�1 for all (S, x) ∈ U, ξ ∈ R 2 + , which implies that condition (1) of Lemma 5 is satisfied.
Next, we need to construct a non-negative C 2 -function V (S, x) and a open set U ∈ R 2 + , such that en, we select θ ∈ (0, 1) to be a sufficiently small constant such that choosing a large enough positive constant N such that where ψ u 1 � sup (S,x)∈R 2 + \U ψ 1 (S, x), and ψ 1 (S, x) is given in (A.3).

Complexity
Define a nonnegative C 2 -Lyapunov function where H(S, and   H(S,x) is a C 2 -function with a unique minimum value point (S * ,x * ). Using Itô formula for V 1 , V 2 , V 3 , we estimate LV 1 firstly. So, thus, (74) erefore, according to the previous function l(S), we have Obviously, So, thus, Now, we estimate LV 2 and LV 3 , So, where the positive constant H 2 is mentioned in Lemma 2. So, where 12 Complexity rough observation, we find that Let U � (ε, (1/ε)) × (ε, (1/ε)), where ε is a small enough constant. It follows that is implies that the model (4) admits a unique stationary distribution μ(·).
us, the proof is completed. □ Remark 5. By eorem 4, we known that the model (4) has a unique stationary distribution if the condition holds, which means that the microorganism is persistent in the future.

Stochastic Maximum Principle
In this section, the existence of the optimal stochastic control for the model (4) is proved. Our aim is to find an optimal stochastic control under given the initial conditions such that the production of the microorganism is maximized at a given time T. We select the dilution rate D as stochastic control variable D(t). Several preliminaries are given.
For convenience, the model (4) can be rewritten as where e stochastic control variable D(t) is a non-negative continuous bounded function. Let the constant V denote the volume of the culture vessel, and let F(t) be the volumetric flow rate, then the dilution rate D(t) � (F(t)/V). For convenience, we define V � 1, so D(t) � F(t), the production function is expressed as where is a continuous function denoting the production of unit time of the microorganism. Since we will focus on the maximum microorganism production problem in the interval [0, T], the objective functional is given as Denote a bounded nonempty closed set A ⊂ R, and the set of admissible controls is defined as follows: For any D ∈ A a d , equation (87) is a stochastic differential equation with random coefficients, which has a unique strong solution (S(t), x(t)) called an admissible state trajectory, and (S(t), x(t), D(t)) is called an admissible triple. e objective functional (90) can be rewritten as where L(S(t), x(t), D(t)) � x(t)D(t), the optimal control condition is to select a nonanticipative decision that maximize the objective functional, that is to seek D ∈ A a d such that where D is called an optimal stochastic control. (S, x) is called the corresponding solution of the model (87) and (S, x, D) is called an optimal triple. Define a Hamiltonian function H: R + × R + × A × R × R × R × R ⟶ R by H S(t), x(t), D(t), p 1 (t), p 2 (t), q 1 (t), q 2 (t) � f 1 (S(t), x(t), D(t))p 1 (t) + f 2 (S(t), x(t), D(t))p 2 (t) + σ 1 S(t)q 1 (t) + σ 2 x(t)q 2 (t) + L(S(t), x(t), D(t)).
Furthermore, if the following conditions hold, (i) e Hamiltonian H(S, x, D, p 1 , p 2 , q 1 , q 2 ) with respect to (S, x, D) is concave for all t ∈ [0, T] (ii) H(S, x, D, p 1 , p 2 , q 1 , q 2 ) � max D∈A H(S, x, D, p 1 , p 2 , q 1 , q 2 ), for all t ∈ [0, T] then, (S, x, D) is an optimal triple and D is an optimal stochastic control. obviously, which means that when the condition of eorem 3 satisfies, then the microorganism is almost surely strongly persistent in the mean (see Figure 4). rough the above numerical simulations, we know that the parameters B rarely affect the extinction of the microorganism x(t) and have a significant impact on the persistence of the microorganism x(t), that is, the concentration of the microorganism x(t) increases with the parameter B (see Figures 3(b), 3(d), 4(b), and 4(d)). In conclusion, the intensities of white noise are disadvantageous to the growth of the microorganism, and the variable yield c(S) � A + BS is advantageous to the growth of the microorganism.

Data Availability
e "simulation" data used to support the findings of this study are included within the article. ese data are not obtained by experiments; just to satisfy the conditions of the theorem, we design them artificially.

Conflicts of Interest
e authors declare that they have no conflicts of interest.