GLOBAL DYNAMICS OF A MICROORGANISM FLOCCULATION MODEL WITH TIME DELAY

. In this paper, we consider a microorganism ﬂocculation model with time delay. In this model, there may exist a forward bifurcation/backward bi- furcation. By constructing suitable positively invariant sets and using Lyapunov-LaSalle theorem, we study the global stability of the equilibria of the model under certain conditions. Furthermore, we also investigate the permanence of the model, and an explicit expression of the eventual lower bound of microor- ganism concentration is given.


1.
Introduction. Microorganisms are widely used in many fields such as environmental protection industry, pharmaceutical industry, food industry, and energy industry, etc (see, e.g., [2,26]). Hence, the study about continuous cultivation, collection and extraction, degradation of microorganisms has been drawn much attention (see, e.g., [9,17,18,20,21,24,28,29]). Generally speaking, the physical method of flocculation to collect and extract microorganisms, which is one of the most efficient methods in the world, is playing an important role in production and application in microbial industry.
Recently, models of continuous cultivation of microorganisms have been proposed by many authors (see, e.g., the models with time delay [1,3,4,15,14,24], ones without time delay [6,11,12,16]). They investigated the stability or permanence of these models by applying some theories in delay differential equations [5,7,10,13,22,27]. In this paper, we consider the permanence and the global stability for the following time-delayed model of microorganism flocculation which is proposed in [24].
where x(t), y(t), z(t) represent the concentrations of nutrient, microorganisms and flocculant at time t, respectively. The positive constant d represents the same velocities of inflow and outflow of nutrient and flocculant as well as the velocity of outflow of microorganisms. The constants x 0 > 0, z 0 > 0 indicate the input concentrations of nutrient for cultivating microorganisms and flocculant for precipitating microorganisms, respectively.h 1 , h,h 2 andh 3 are positive constants standing for the consumption rate of nutrient, the growth rate of microorganisms, the flocculating rate of microorganisms, and the consumption rate of flocculant, respectively. The nonnegative constant τ is the time delay.
(ii) If w < R < 1 (i.e. (2) undergoes a backward bifurcation, see Fig. 2), then there exist two positive equilibria E * = (x * , y * , z * ) T and E * * = (x * * , y * * , z * * ) T , where (iii) If R = w, then there exists a unique positive equilibrium E * = (x * , y * , z * ) T . (III) Otherwise, in any other cases, (2) has no positive equilibria.   The purpose of this paper is to study the global stability of the equilibria of (2) under some conditions by constructing suitable positively invariant sets and using Lyapunov-LaSalle theorem. In addition, we also investigate the permanence of (2), and give a specific estimation of the eventual lower bound of microorganism concentration by using a thorough analysis (also see [8]) which differs from traditional methods.
2. Permanence of the system. We will consider the permanence of (2) in this section. We have where q ∈ (0, r − 1 − h 2 ) and Proof. We can first obtain that X is positively invariant for (2). Let u t = (x t , y t , z t ) T be the solution of (2) with any φ ∈ X, and let Thus, we only need to show that (4) holds. Define the functional V as follows, The derivative of V along this solution u t is taken aṡ By (5), there exists T = T (φ) > 0 such that x(t) ≥ 1/2 (r + 1) for all t ≥ T. Let Now, we claim that for any t 0 ≥ T > 0, it is impossible to satisfy y(t) ≤ y 1 (∀t ≥ t 0 ). Otherwise, there exists some t 0 ≥ T such that y(t) ≤ y 1 for all t ≥ t 0 . Then it follows from the first equation of (2) that, for t ≥ t 0 , which implies that Thus, it follows that for t ≥ t 0 + T 0 , where 1/2 (r + 1) − A < 0 is used. Then from (8), we obtain rρ − 1 − h 2 > 0 and we also haveV then we get from the second equation of (2) and (8) Clearly, this is a contradiction. Hence, y(t) ≥ y m for all t ≥ t 0 +T 0 . In consequence, we have that for all t ≥ t 0 + T 0 , which implies that V (u t ) → ∞ as t → ∞. This contradicts with the boundedness of V (u t ). The claim is proved.
3. Global stability of equilibria. In this section, we shall consider the global stability of equilibria under some conditions. According to Figs. 1 and 2, for the global stability of the boundary equilibrium, we only need to consider two cases: for the global stability of the positive equilibrium, we only need to consider R > 1. First, for the boundary equilibrium, we have Proof. Obviously, 1 <r < 1+h 2 , and then r <r implies R < 1. If h 2 h 3 > h 1 +h 1 h 2 , then r <r also implies R < w < 1. Consider the functional V as defined in (6) on G. It can be seen that V is continuous on G. Since r <r if and only if Let u t = (x t , y t , z t ) T be the solution of (2) with any φ ∈ G. By (5), there is a T = T (φ) > 0 such that z(t) ≥ h 1 /ε (h 1 + rh 3 ) for all t ≥ T. The derivative of V along this solution of (2) for t ≥ T is given aṡ Let u t = (x t , y t , z t ) T be the solution of (2) with any ψ ∈ M . Note that M is invariant for system (2), consequently, by contradiction, if there exists some t 1 ∈ R such that y(t 1 ) > 0, then from the second equation of (2), it follows that for t ≥ t 1 , (1+h2z(s))ds > 0.
This contradicts y(t) = 0 for t ≥ T (ψ). Thus, we have y(t) = 0 for t ∈ R. The first and the third equations of (2) together with the invariance of M yield that x(t) = z(t) = 1 for t ∈ R. Hence, M = {E 0 } and then E 0 is gloabally attractive.
With the local stability of E 0 established in [24, Theorem 3.1], we thus prove the global stability of E 0 .
Proof. It is not difficult to show that X is positively invariant for (2). Let (x t , y t , z t ) T be the solution of (2) with any φ ∈ X and define p(t) : Then it follows that (x t , y t ) T is a solution of the following nonautonomous system: By (2), it is clear to find that for t ≥ τ, Hence, we obtain lim t→∞ p(t) = r/h 1 . This implies that (9) has the following limiting system: (10) Accordingly, the nonautonomous solution semiflow of (9) is asymptotic to the autonomous solution semiflow of (10) on [19,25]).
From the continuous-time version of [27, Lemma 1.2.2] (also see [19, Theorem 1.8]), it follows that ω ⊂ X 1 is an internally chain transitive set for the solution semiflow of (10). It holds that W s (E 1 ) = X 1 , where W s (E 1 ) is the stable set of E 1 for the solution semiflow of (10). Consequently, ω ∩ W s (E 1 ) = ∅. Whence, [27, Theorem 1.2.1 and Remark 1.3.2] with A = E 1 imply that ω = {E 1 }, and then lim t→∞ (x(t), y(t)) T = E 1 . Thus, Therefore, E * is globally attractive in X, which is together with the local stability of E * established from [24, Theorems 3.2 and 3.3], ensures the global stability of E * .

4.
Conclusions. Tai, Ma and Guo et al. [24] proposed (1) (i.e., (2)) on the basis of some practical problems about microorganism continuous culture and flocculation, and they gave a complete analysis on the local stability of the equilibria of (2). In this paper, we consider the global dynamics of (2) based on the construction of suitable positively invariant sets and the application of Lyapunov-LaSalle theorem. Some detailed analyses on the global stability of equilibria of (2) under certain conditions are carried out. It is shown that, when r <r (it implies two cases: (i) R < 1; (ii) h 2 h 3 > h 1 + h 1 h 2 , R < w), the boundary equilibrium E 0 is globally asymptotically stable for any τ ≥ 0. Furthermore, it is also shown that, if R > 1 and 4h 1 ≥ rh 3 hold, the positive equilibrium E * is globally asymptotically stable for any τ ≥ 0. The two results for global stability show that, for (2), the time delay has no effect on both global asymptotic properties of the boundary equilibrium E 0 and the positive equilibrium E * . For permanence of (2), it is shown that, when R > 1, (2) is permanent, which also ensures the global stability of E * . In addition, the explicit lower bound ν of lim inf t→∞ y(t) is given in Theorem 2.1 by using some analysis techniques.