THE DYNAMICAL BEHAVIOR AND PERIODIC SOLUTION IN DELAYED NONAUTONOMOUS CHEMOSTAT MODELS

In this paper, the global dynamics and existence of positive periodic solutions in a general delayed nonautonomous chemostat model are investigated. The positivity and ultimate boundedness of solutions are firstly obtained. The sufficient conditions on the uniform persistence and strong persistence of solutions are established. Furthermore, the criterion on the global attractivity of trivial solution is also established. As the applications of main results, the periodic delayed chemostat model is discussed, and the necessary and sufficient criteria on the existence of positive periodic solutions, and uniform persistence and extinction of microorganism species are obtained. Lastly, the numerical examples are presented to illustrate the main conclusions.


Introduction
The chemostat is an experimental device invented in the 1950s, first used by microbiologists to study the growth of a given species of microorganisms, its usage greatly diversified with the time going on (see [26,27,33]). It is a standard tool for microbiologists to study relationships between microbial growth and environment parameters, and it is also the focus of great interest of theoretical ecology and mathematical ecology (see [10,15,38,43] and the references cited therein). It is also used nowadays in analysis of antibiotic (see [18]) and to study recombinant problems related to genetically altered microorganisms (see [12,20]).
It is precisely because of its importance that many scholars study the dynamical properties of chemostats by establishing dynamical models of differential equations, including the nonnegative boundedness of solutions, the extinction and persistence of microorganisms (see [7,21,29]), the stability of equilibrium (see [30,31]), the existence of periodic solutions (see [1,2]), the occurrence of bifurcations (see [32]) and the dynamical complexity (see [8,9,16,41]), etc.
We all know that in autonomous chemostat models, the parameters, input and output are all constants. However, the chemostat is a dynamic system with continuous material inputs and outputs, thus simulating the open system characteristics and temporal continuity of nature (see [33]). On the other hand, in the field of ecology the chemostat is often viewed as a model of a simple lake system, of the wastewater treatment process, or of biological waste decomposition (see [6,17]). In the field of environment, various factors such as climate and environment change dynamically, making it impossible for the parameters, input and output of the model to remain constant and change over time, which are more properly characterized by nonautonomous models (see [4,14,25]). If only seasonal periodic changes are considered, then the dynamics of chemostats are studied using periodic models (see [1,2]). Moreover, when using chemostat models to study the growth of phytoplankton(see [29,40]) and the cultivation of microorganisms in natural lakes (see [5,15,43]), the results obtained in a time-fluctuating environment can better explain the population dynamics. On this account, Rehim and Teng in [28] considered the following single-species nonautonomous chemostat model: dx(t) dt =x(t)(−d(t) + Q(t, s(t))). (1.1) A series of interesting criteria on persistence, average persistence and extinction of solutions were established. It is generally agreed that time delay can have a considerable impact on the nature of the ecosystem. In fact, in population models it may well become necessary to have lags, such as a generation lag coming after a lag of the kind considered in the Volterra model (see [22]). Without exception, the influence of time delay is inevitable in the chemostat model. Because microorganisms cannot immediately convert into their own biomass after absorbing nutrients, but there is a time delay. Therefore, the chemostat model with time delay can more clearly show that the absorption of nutrients by microorganisms is not instantaneous. Moreover, considering the time delay in the chemostat model can better explain some non-stationary situations, such as periodic fluctuations and instability. In order to better simulate these actual natural phenomena, many scholars incorporated time delay into the model to study its consequences (see [1,3,11,13,19,23,24,37,39,42]). Especially, Amster et al. in [1] studied a single-species chemostat model with periodic nutrient supply and delay in the growth as follows: (1. 2) The necessary and sufficient conditions for the existence of positive periodic solution were established by constructing Poincare type mapping and using Whyburn's lemma and Leray-Schauder's degree. Further, the criterion on the extinction of microorganism species x is also obtained. Motivated by above work, in this article we propose a nonautonomous chemostat model with general delay in microorganism growth as follows:

t)s(t) − x(t)P (t, s(t))
dx(t) dt =x(t)(−d(t) + Q(t, s t )), where s(t) and x(t) are the concentrations of nutrient and microorganism when time is t, respectively, and s t = s(t + θ) with θ ∈ [−τ, 0]. P (t, s) is the per capita nutrient absorption rate of the microorganism at the concentration s and time t. Q(t, s t ) is the growth rate of the microorganism at time t, which shows that the growth of microorganisms in biomass depends on the amounts of the nutrient consumed in whole interval [t − τ, t], where τ ≥ 0 is a constant. Functions a(t), b(t) and d(t) denote, respectively, the input nutrient concentration, the dilution rate and the removal rate of microorganism. We can easily see that the following delayed nonautonomous chemostat models are special cases of model (1.3): where τ (t) is nonnegatively bounded and continuously differentiable for all t ≥ 0 and satisfies max t≥0 τ ′ (t) < 1, and where c(t, θ) is defined and nonnegative for all (t, θ) ∈ R + × [−τ, 0], and continuous for t ∈ R + and integrable for θ ∈ [−τ, 0] with ∫ 0 −τ c(t, θ)dθ ≡ 1. It is clear that model (1.2) is a special case of model (1.4).
The purpose of this paper is to investigate the global dynamic behavior and existence of positive periodic solutions of model (1.3). We will establish a series criteria on the ultimate boundedness of solutions, the uniform persistence and strong persistence of nutrient and microorganism, and the global attractivity of trivial solution. Particularly, when model (1.3) degrades into the periodic case, we will further establish the necessary and sufficient conditions for the existence of positive periodic solutions and, the uniform persistence and extinction of microorganism, respectively. The main methods used in this research are the differential inequality principle, the inequalities analysis techniques and the reduction to absurdity. This paper is organized as follows. In Section 2, as the preliminaries, we first introduce some basic assumptions for model (1.3). Then, some useful lemmas are given, and the positivity of solutions for model (1.3) with positive initial values is proved. Section 3 is devoted to demonstrate that the solutions of model (1.3) are ultimately bounded. In Section 4, the sufficient conditions on the uniform persistence and strong persistence of nutrient and microorganism are stated and proved. In Section 5, a criterion on the global attractivity of trivial solution of microorganisms vanishing is established. In allusion to time-periodic model (1.3), the necessary and sufficient conditions for the existence of positive periodic solutions, the uniform persistence and extinction of microorganism are stated in Section 6, respectively. In Section 7, the numerical experiments are presented to illustrate the main conclusions established in this paper. Finally, in Section 8, a brief conclusion is given, and some new interesting problems are proposed for the future research.
(A 4 ) For any constants H > β > 0 there exists a continuous function h(t) defined for t ≥ 0 satisfying lim inf t→∞ ∫ t+α t h(ξ)dξ > 0 for some constant α > 0 such that for any ϕ 1 , ϕ 2 ∈ C + with ϕ i ≤ H (i = 1, 2) and ϕ 1 − ϕ 2 ≥ β, one has Q(t, ϕ 1 ) − Q(t, ϕ 2 ) ≥ h(t) for any t ≥ 0. Remark 2.1. Assumptions (A 1 ) and (A 2 ) are fundamental. The Lipschitz conditions of P (t, s) and Q(t, ϕ) are given to assure the existence, uniqueness and continuability of solutions of model (1.3). For assumption (A 3 ), P (t, s) is increasing for s ≥ 0 to show that the increase of nutrient s will result in that microorganism x acquires more many nutrient. Q(t, ϕ) is increasing for ϕ ≥ 0 to show that the increase of nutrient s will make that microorganism x acquires greater growth. The condition lim inf t→∞ P (t, s) > 0 for any s > 0 will be used in the proof of ultimate boundedness of solutions (See Theorem 3.1 below). Assumption (A 4 ) will be used in the proof of extinction of microorganism x (See Theorem 5.1 below).
We see that for special models (1.4) and (1.5) assumptions (A 2 ) − (A 4 ) will degenerate into the following forms.
(A ′ 2 ) P (t, s) and Q(t, s) are continuous for any (t, s) ∈ R 2 + . P (t, 0) = Q(t, 0) ≡ 0 for any t ≥ 0. For any positive constant H there exists a K = K(H) > 0 such that For any t ≥ 0, P (t, s) and Q(t, s) are nondecreasing for s ∈ R + .
(A ′ 4 ) For any constants H > β > 0 there exists a continuous function h(t) defined for t ≥ 0 satisfying lim inf t→∞ ∫ t+α t h(ξ)dξ > 0 for some constant α > 0 such that for any t ∈ R + and s 1 , Now, we consider the following nonautonomous linear equation: where c(t) and l(t) are continuous bounded functions defined on t ≥ 0 and c(t) ≥ 0 for all t ≥ 0. We have the following lemma.  Lemma 2.1 can be proved by using the similar method given in [34], we hence omits it here.
For the convenience of narrations, we denote the functions g 1 (t, s, x) = a(t) − b(t)s − P (t, s)x and g 2 (t, s t ) = −d(t) + Q(t, s t ). Proof. Firstly, according to the fundamental theory of functional differential equations, model (1.3) has a unique solution (s(t), x(t)) satisfying initial condition The proof of positivity of solution (s(t), x(t)) is simple. In fact, integrating the second equation of model (1.3) from 0 to any t ∈ (0, T ) we directly have Suppose that there is a t 1 > 0 such that s(t 1 ) = 0 and s(t) > 0 for any t ∈ [0, t 1 ). Assumption (A 2 ) implies that P (t, s(t)) ≤ Ks(t) for any t ∈ [0, t 1 ]. From the first equation of model (1.3) it follows that for all t ∈ [0, t 1 ]. From this, we directly get From conclusion (i) of Lemma 2.1 and the comparison principle, we easily obtain the boundedness of s(t) on [−τ, T ). Then, by assumption ( The proof is completed.

Ultimate boundedness
In this section, we investigate the ultimate boundedness of solutions of model ( , if x(t) ≡ 0, i.e., there is no microorganism, then the subsystem of nutrient species is given as follows: By the comparison principle, for any constant ε > 0 there exists a T = T (ε) > 0 such that Hence, there exists a large T 0 > 0 such that s(t) ≤ H 1 for all t ≥ T 0 , where the constant H 1 > sup t≥0 s * (t). Consequently, s(t) is ultimately bounded. Now, we prove the ultimate boundedness of x(t). Assumptions (A 1 ) and (A 2 ) imply that there is a constant ε 0 > 0 small enough such that From this, it follows that for any t 0 ≥ 0 Therefore, the comparison principle implies by using the method of variation of constant, we obtain For any t > t 1 + τ integrating (3.7) one obtains 3) with initial value r n satisfies lim sup t→∞ x(t, r n ) > (2H +1)n for n = 1, 2, · · · . For every n, by the ultimate boundedness of for all t ∈ R + and a ≥ p. Then, we can choose an integer N 0 > 0 such that then from assumption (A 3 ), (3.6) and (3.11) we can obtain which leads to a contradiction. Therefore, there is a A similar argument as in the above, we further obtain Finally, from assumption (A 3 ), (3.10), (3.11) and (3.12), which is contradictory with (3.9). Therefore, (3.8) holds. That is, x(t) is also ultimately bounded. The proof is completed.
As the consequences of Theorem 3.1, for special models (1.4) and (1.5) we have the following corollaries.

Persistence
In this section, the persistence of model ( , then one says that species x is uniformly persistent. It is evident that if species x is uniformly persistent, then species x is also strongly persistent. The same concepts can be defined for nutrient s. The main results on the persistence for model (1.3) are established below.
Combining the comparison principle and conclusion (iii) of Lemma 2.1, it follows that there exists a constant m > 0 such that lim inf t→∞ s(t) ≥ m. This shows that nutrient s is uniformly persistent. This completes the proof.
We consider the following linear equation: is globally uniformly asymptotically stable. Then, conclusion (ii) of Lemma 2.1 indicates that s α (t) converges to s * (t) uniformly for t ∈ R + as α → 0. Thus, there exists an enough small constant α > 0 such that for all t ≥ T 2 . Combining the comparison principle and the global asymptotic stability of solution s α (t), one deduces that there exists a T 3 > T 2 such that Accordingly, (4.4) and (4.5) yield to For any t ≥ T 3 + τ , from assumption (A 3 ) one deduces that Finally, from (4.2) it follows that lim t→∞ x(t) = ∞, a contradiction. Therefore, lim sup t→∞ x(t) > α for any positive solution (s(t), x(t)) of model (1.3). Next, we prove that there is a constant β > 0 such that lim inf t→∞ x(t) > β for any positive solution (s(t), x(t)) of model (1.3). Suppose that the conclusion does not hold, then there exists a sequence of initial values and for q = 1, 2, · · · . By (4.2), there are positive constants p and l such that for all t ∈ R + and a ≥ p. Lets α (t) be the solution of equation (4.3) with initial valuē q , r n ). From (4.8) we can obtain that for any n, q and t ∈ [u Accordingly, the comparison principle implies (4.10) Since solution s α (t) is globally uniformly asymptotically stable, there exists a constant T ≥ p, and T does not depend on any n, such that a contradiction. This completes the proof.

(i) If x(t) is strongly persistent, then s(t) is also strongly persistent;
(ii) If x(t) is uniformly persistent, then s(t) is also uniformly persistent.
Next, we prove lim inf t→∞ s(t) > 0. Let lim sup t→∞ s(t) = β > 0. If the claim does not hold, then there exist two time sequences {u n } and {v n }, satisfying and β n 2 < s(t) < β n for all t ∈ [u n , v n ]. (4.14) Assumption (A 2 ) implies that there exists a constant c > 0 such that  This leads to a contradiction. Consequently, lim inf t→∞ s(t) > 0. Now, we prove conclusion (ii). Since x(t) is uniformly persistent and solution (s(t), x(t)) is ultimately bounded, there are constants M 1 > m 1 > 0 and T 0 > 0 such that m 1 ≤ x(t) ≤ M 1 and s(t) ≤ M 1 for all t ≥ T 0 . We first show that lim sup t→∞ s(t) > η for any positive solution (s(t), x(t)) of model (1.3), where constant η > 0 is given in (4.12). Indeed, if the claim does not hold, then there is a T * > T 0 such that s(t) ≤ η for all t ≥ T * . By assumption (A 3 ) one gets From  Then, from (4.15) we have s(v > ln n c for q = 1, 2, · · · . Accordingly, there exists an integer N 0 > 0 such that v (n) q − u (n) q ≥ 2p + τ for all n ≥ N 0 and q = 1, 2, · · · . For all n ≥ N 0 and q = 1, 2, · · · , integrating the second equation of model (1.3), by assumption (A 3 ) and (4.17) we can obtain a contradiction. Combining Theorem 3.1, it yields that s(t) is uniformly persistent. The proof is completed.

s(t), x(t)) be any positive solution of model (1.4). Then the conclusions given below hold. (i) If x(t) is strongly persistent, then s(t) is also strongly persistent;
(ii) If x(t) is uniformly persistent, then s(t) is also uniformly persistent.   [28]. In addition, we also see that the corresponding results: Theorems 2 and 3 in [7] and Theorem 5.2.2 in [43] are as the special cases of Corollaries 4.1 and 4.2.

Global attractivity of trivial solution
In this section, we investigate the global attractivity of trivial solution of which microorganism species x vanishes in model (1.3). We have the following conclusion. Proof. We first prove lim t→∞ x(t) = 0. The proof process is divided into the following two cases. Case 1. Assume lim sup t→∞ ∫ t+λ t (−d(ξ) + Q(ξ, s * ξ ))dξ < 0. Then, it follows that lim sup t→∞ t −1 ∫ t 0 (−d(ξ) + Q(ξ, s * ξ ))dξ < 0. From assumption (A 2 ), we can choose small enough constants ε > 0, δ > 0 and an enough large T > 0 such that Let (s(t), x(t)) be any positive solution of model (1.3). Due to then combining conclusion (i) of Lemma 2.1 and the comparison principle, one deduces that there exists a T 1 ≥ T such that Integrating the second equation of model (1.3), by assumption (A 3 ) we obtain be any positive solution of model (1.3) with initial condition (2.1). Firstly, Theorem 3.1 shows that there exists a constant K > 0 such that s(t) ≤ K and x(t) ≤ K for all t ≥ 0.
For any given constant ε > 0, suppose that for any t 0 ≥ 0 one has x(t) ≥ ε for all t ≥ t 0 , then lim inf t→∞ x(t) > 0, i.e., x(t) is strongly persistent. By Proposition 4.1 in Remark 4.1, then s(t) is also strongly persistent. Therefore, there exists a constant σ > 0, which is independent of any t 0 , such that s(t) ≥ σ for all t ≥ 0. Consequently, we obtain for all t ≥ t 0 . From (5.3) we further deduce that (5.4) indicates that there are constants β > 0 and T 0 > 0, which are independent of t 0 , such that for any t ≥ T 0 + t 0 , s(t) − s * (t) ≤ −β. This together with assumption (A 4 ) yields When t → ∞ we have p → ∞ and hence p(kε 0 − δ) + kλ(max t≥0 {d(t) + Q(t, s * t )} + max t≥0 |h(t)|) → −∞. Thus, from (5.7) we finally obtain x(t) → 0 as t → ∞, which leads to a contradiction with the fact that x(t) ≥ ε for all t ≥ t 0 . Therefore, there is a t 1 > 0 such that In fact, if this conclusion does not hold, then there exists a t 2 > t 1 such that Using (5.6) and (5.7), we hence have which is impossible. Therefore, x(t) ≤ εe M for all t ≥ t 1 . By the arbitrariness of ε, we finally get lim t→∞ x(t) = 0.
For any enough small constant β > 0, we consider the following equation: Let s * β (t) be the positive solution of (5.9) with initial value s * β (0) = s * (0). From conclusions (i) and (ii) of Lemma 2.1, it follows that s * β (t) is globally uniformly attractive on t ≥ 0, and for any ε > 0 there exists a β > 0 such that |s * The global uniform attractivity of s * β (t) and the comparison principle indicate that there exists a T 3 > T 2 such that s(t) > s * β (t)− ε 2 for all t ≥ T 3 . Thus, s(t) > s * (t)−ε for all t ≥ T 3 . Combining (5.2), it follows that |s(t) − s * (t)| < ε for all t ≥ T 3 . Therefore, we finally have lim t→∞ (s(t) − s * (t)) = 0. This completes the proof.
As the consequences of Theorem 5.1, some corollaries on the global attractivity of trivial solution for special models (1.4) and (1.5) are given as follows.

Remark 5.2.
It is easy to see that when τ (t) ≡ 0, Corollary 5.1 extended and improved the corresponding result given in [28], that is Theorem 4.3 in [28].
Consider the above nonautonomous linear equation (2.2), one further assumes that c(t) and l(t) denote ω-periodic continuous functions defined on t ≥ 0 and c(t) ≥ 0 for all t ∈ [0, ω]. Based on Lemma 2.1 the result is given as follows.

t) is also positive; (ii) Let v(t) be the solution of equation (2.2) andv(t) be the solution of equation (2.2) after replacing c(t) with another ω-periodic continuous functionc(t).
If v(0) = v(0), then there exists a constant L > 0 that only depends on l(t), such where the positive constants m 1 and m 2 are chosen satisfying Furthermore, we consider subsystem (3.1) of nutrient species, when assumption (B 1 ) holds, then by Lemma 6.1 the fixed solution s * (t) can be chosen as the ωperiodic solution of subsystem (3.1).

Corollary 6.7.
For ω-periodic model (1.2), assume that (P) holds. Then the following statements are equivalent, Remark 6.2. From Corollaries 6.3 and 6.7, we easily see that the main results Theorem 1 and Theorem 2 established in [1] are extended and improved to general periodic chemostat model with delayed microorganism growth. ∫ ω 0 P (ξ, s)dξ > 0 for any s > 0 to guarantee that all conclusions in Corollaries 6.1, 6.2, 6.4-6.6 and 6.8 still hold. can be removed in Corollaries 6.1, 6.2, 6.4-6.6 and 6.8, respectively, and to guarantee that the same conclusions still hold.   The numerical simulations in Figure 1 show that the open problem given in Remark 6.3 corresponding to Corollary 6.2 may be true. and

Numerical examples
From the equation ds * (t) dt = 2 + 1.9 sin 2πt − (1 + 0.7 cos 2πt)s * (t), we get the numerical simulation of s * (t), see Figure 2(a). By numerical calculation we further obtain The numerical simulations in Figure 2 indicate that the conclusions in Corollary 6.4 are right. The numerical simulations in Figure 3 imply that the open problem given in Remark 6.3 corresponding to Corollary 6.6 may be true.  From the numerical simulations in Figure 4 we see that the conclusions in Corollary 6.8 are right.

Conclusion
In this article, we investigate a nonautonomous chemostat model with general delay in microorganism growth. A series of criteria on the positivity and ultimate boundedness of solutions, uniform persistence and strong persistence of system, global attractivity of trivial solution in which microorganisms species x vanishes are established by the approaches of reductionism, comparison principle and differential inequality techniques etc. The corresponding results of uniform persistence and extinction obtained in [7,28,43] are extended to the general delayed nonautonomous case. For the two special cases of model (1.3), i.e., model (1.4) and model (1.5), the sufficient criteria for the uniform persistence and strong persistence of microorganism species x are also obtained, respectively. Furthermore, for periodic model (1.3), we obtain the necessary and sufficient criteria for the existence of positive periodic solutions, the uniform persistence of microorganism species and the global attractivity of trivial periodc solution. Additionally, similar results are obtained for ω-periodic models (1.4) and (1.5). We also see that the main result on the existence of positive periodic solution established in [1,28] is improved and extended. Finally, our main theoretical results are illustrated by some special numerical examples.
We see that only one species and one nutrient is investigated in this paper, biologically, it is more proper to extend the model (1.3) to more general chemostat models, such as delayed nonautonomous chemostat models with multiple species or multiple nutrients. In addition, some more general and complicated results, e.g., bifurcation, chaos, and the average persistence would be valuable and interesting research subjects in future.