Global qualitative analysis of new Monod type chemostat model with delayed growth response and pulsed input in polluted environment

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1 Appl. Math. Mech. -Engl. Ed., 2008, 29(1):75 87 DOI /s x c Editorial Committee of Appl. Math. Mech. and Springer-Verlag 2008 Applied Mathematics and Mechanics (English Edition) Global qualitative analysis of new Monod type chemostat model with delayed growth response and pulsed input in polluted environment MENG Xin-zhu ( ) 1,2, ZHAO Qiu-lan ( ) 1, CHEN Lan-sun ( ) 2,3 1. College of Science, Shandong University of Science and Technology, Qingdao , Shandong Province, P. R. China; 2. Department of Applied Mathematics, Dalian University of Technology, Dalian , Liaoning Province, P. R. China; 3. Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing , P. R. China) (Communicated by GUO Xing-ming) Abstract In this paper, we consider a new Monod type chemostat model with time delay and impulsive input concentration of the nutrient in a polluted environment. Using the discrete dynamical system determined by the stroboscopic map, we obtain a microorganism-extinction periodic solution. Further, we establish the sufficient conditions for the global attractivity of the microorganism-extinction periodic solution. Using new computational techniques for impulsive and delayed differential equation, we prove that the system is permanent under appropriate conditions. Our results show that time delay is profitless. Key words permanence, impulsive input, chemostat model, time delay for growth response, extinction Chinese Library Classification O Mathematics Subject Classification 34K45, 34C05, 92D25 Introduction The chemostat is a simple and well-adopted laboratory apparatus used to culture microorganisms. It can be used to investigate microbial growth and has the advantage that parameters are easily measurable. Chemostat with period inputs were studied 1 2, with periodic washout rate 3, and periodic input and washout 4. However, existing theories on chemostat model largely ignore effects of the environmental pollution. Environmental pollution caused by various industries and pesticide used in agriculture is one of the most important social and ecological problems at present. In order to use and regulate toxic substance wisely, we must assess the risk of the population exposed to toxicant. Therefore, it is important to study the effects of toxicant on populations and to find a theoretical threshold value, which determines permanence or extinction of a population community. Many mathematical chemostat models were formulated, and many good results were obtained Transient growth dynamics are of Received Sep. 5, 2007 / Revised Dec. 9, 2007 Project supported by the National Natural Science Foundation of China (Nos and ) and the Natural Science Foundation of Shandong University of Science and Technology (No. 05g016) Corresponding author MENG Xin-zhu, Associate Professor, Doctor, mxz721106@sdust.edu.cn

2 76 MENG Xin-zhu, ZHAO Qiu-lan and CHEN Lan-sun considerable importance in the study of how microorganisms respond to environmental changes, and are pertinent to understanding the control mechanisms for microbial growth 7. While the Monod model 8 has some success in describing steady state growth rates (see Refs. 5 and 9), it has been found inadequate to predict transients observed in chemostat experiments where the initial data is not at the globally attracting steady state. Lag phases occur in the growth response of microorganisms to changes in the environment 10. Many authors have directly incorporated time delays in the modelling equations and, as a result, the models take the form of delay differential equations We refer to Refs. 19,23, and references therein, for more detailed discussions on chemostat modeling approaches using delay differential equations. The more details can be seen in Refs. 5, In recent years, the microbial continuous culture has been investigated in Refs and some interesting results were obtained. Many scholars pointed out that it was necessary and important to consider models with periodic perturbations, since these models might be quite naturally exposed in many real world phenomena (for instance, food supply, mating habits, harvesting). Systems with sudden perturbations are involving in impulsive differential equations, which have been studied intensively and systematically in Refs. 30,31. Authors 32 34, introduced some impulsive differential equations in population dynamics and obtained some interest results. The research on the chemostat model with impulsive perturbations is not too much yet (see Refs ,35 and references therein). However, this is an interest problem in mathematical biology. In view of our arguments above, it is very interestingtointroducedelayedgrowthresponse, impulsive input concentration of the nutrient and impulsive input concentration of the toxicant to chemostat model. While delay differential equations have been widely used in modelling population dynamics, some practical problems have to be overcome when applied to models of the chemostat. We remark that dynamics of impulsive and delayed differential equation are usually more difficult to study than those of the ordinary differential equation. As a result, fewer analytic tools are available for studying the dynamics of impulsive and delayed differential equation. So, chemostat model with impulse and delay have never been seen by now. In this paper, we consider the Monod type chemostat model with impulsive input concentration of the nutrient and delayed growth response in a polluted environment, and investigate how the impulsive perturbation of the substrate, time delay for growth response and impulsive input of the toxicant affect the dynamic behaviors of the chemostat system. 1 Model and preliminaries The following single-species Monod type chemostat model with impulsive input concentration of the nutrient was introduced by Chen 36 : S (t) =(S 0 S(t))D μ ms(t)x(t) δ(k m + S(t)), ( x (t) =x(t) μ m S(t) δ(k m + S(t)) D Recently, the following single-species chemostat model with impulsive perturbation on the nutrient concentration was introduced by Sun and Chen 29 : S (t) = DS(t) μ ms(t)x(t), t nt, n N, δ(k m + S(t)) ( ) x μm S(t) (t) =x(t) K m + S(t) D, t nt, n N, (2) S(t + )=S(t)+γS 0, t = nt, n N, x(t + )=x(t), t = nt, n N, S(0 + ) 0, x(0 + ) 0, ). (1)

3 Global qualitative analysis of new Monod type chemostat model 77 where S(t) denotes the concentration of the unconsumed nutrient in the growth vessel at time t and x(t) denotes the biomass of the population of microorganisms at time t. The function p(s) = μms(t) δ(k m+s(t)) represents the species specific per-capita nutrient uptake rate. It also models the rate of conversion of nutrient to viable biomass. S 0 and D are positive constants and denote, the concentration of the growth-limiting nutrient and the flow rate of the chemostat, respectively (see more details in Refs ,29). T = γ/d is the period of the pulsing, γs 0 is the amount of limiting substrate pulsed each T.WeaddDS 0 units of substrate in the chemostat, on average, per unit of time. In fact, the taken nutrient can not translate instantaneously into viable microorganisms, that is, there is a time delay in the growth response that describes the lag involved in the nutrient conversion process. At the same time impulsive input concentration of the toxicant may lead to the microorganisms species be extinct in the polluted chemostat environment. Therefore, the time delay and impulsive input concentration of the toxicant should be considered in the system (2). In this paper, we consider the polluted Monod type chemostat model with pulsed input and delayed growth response: S (t) = DS(t) μ ms(t)x(t), t nt, n N, δ(k m + S(t)) x (t) =e Dτ μ ms(t τ)x(t τ) (D + rc(t))x(t), t nt, n N, K m + S(t τ) c (t) = Dc(t), t nt, n N, (3) S(t + )=S(t)+γS 0, t = nt, n N, x(t + )=x(t), t = nt, n N, c(t + )=c(t)+γc 0, t = nt, n N, S(0 + ) 0, x(0 + ) 0, c(0 + ) 0, where c(t) is the concentration of the toxicant in the chemostat, r>0 is the rate of decrease of the intrinsic growth rate of the microorganisms. γc 0 is the amount of pulsed input concentration of the toxicant at each T. The constant τ 0 denotes the time delay involved in the conversion of nutrient to viable biomass. The positive constant, e Dτ, is required. Because it is assumed that the current change in biomass depends on the amount of nutrient consumed τ units of time in the past by the microorganisms that were in the growth vessel at that time and the amount of death of the microorganisms in τ units of time. S(nT + ) = lim n nt + S(t), and S(t) is left continuous at t = nt, i.e., S(nT ) = lim n nt S(t), x(t) is continuous for all t 0, c(nt + ) = lim n nt + c(t), and c(t) isleftcontinuousatt = nt, i.e., c(nt ) = lim n nt c(t), the details can be seen in books of Bainov and Simeonov 30 and Laksmikantham, et al. 31 Motivated by the application of the system (3) to population dynamics (refer to Ref. 37), we assume that solutions of the system (3) satisfy the initial conditions: (φ 1 (s),φ 2 (s),φ 3 (s)) C + = C( τ,0,r 3 +), φ i (0) > 0(i =1, 2, 3). (4) Lemma ,31 Consider the following impulse differential inequalities: w (t) ( )p(t)w(t)+q(t), t t k, w(t + k ) ( )d kw(t k )+b k, t = t k, k N, where p(t),q(t) C(R +, R),d k 0, and b k is the constant. Assume (A 0 ) the sequence {t k } satisfies 0 t 0 <t 1 <t 2 <, with lim t t k = ;

4 78 MENG Xin-zhu, ZHAO Qiu-lan and CHEN Lan-sun (A 1 ) w PC (R +, R) and w(t) is left-continuous at t k,k N. Then ( t ) w(t) ( )w(t 0 ) d k exp p(s)ds + ( ( t )) d j exp p(s)ds b k t 0<t k <t t 0 t 0<t k <t t k <t j<t t k t ( t ) + d k exp p(θ)dθ q(s)ds, t t 0. t 0 s<t k <t s Lemma Consider the following delay differential equation: dx(t) = r 1 x(t τ) r 2 x(t), dt where a, b and τ are all positive constants and x(t) > 0 for t τ,0. (i) If r 1 <r 2, then lim t x(t) =0. (ii) If r 1 >r 2, then lim t x(t) =+. For convenience, we give the basic properties of the following systems: S (t) = DS(t), t nt, n N, S(t + )=S(t)+γS 0, t = nt, n N, S(0 + )=S 10 0, (5) and c (t) = Dc(t), t nt, n N, c(t + )=c(t)+γc 0, t = nt, n N, c(0 + )=c (6) Lemma 1.3 The system (5) has a positive periodic solution S (t), and for any solution S(t) of (5) with initial value S 10 0, we get S(t) S (t) 0 as t, moreover, (i) if S 10 γs0, then S(t) S (t); (ii) if S 10 < γs0, then S(t) <S (t), where S (t) = γs0e D(t nt ),t (nt, (n +1)T, n N, S (0 + )= γs0. Proof It is clear that S (t) = γs 0e D(t nt ) 1 e is a positive periodic solution of (5), where t (nt, (n +1)T,n N,S (0 + )= γs0. The solution of (5) is S(t) = (S(0 + ) S (0 + ))e Dt + S (t),t (nt, (n +1)T,n N. Hence S(t) S (t) 0ast. And S(t) S (t) ifs 10 γs0 and S(t) < S (t) if S 10 < γs0. The proof is complete. Similarly, we have the following Lemmas. Lemma 1.4 System (6) has a positive periodic solution c (t) and for any solution c(t) of (6) with initial value c 10 0, we get c(t) c (t) 0 as t, moreover, (i) if c 10 γc0, then c(t) c (t); (ii) if c 10 < γc0, then c(t) <c (t), where c (t) = γc0e D(t nt ),t (nt, (n +1)T, n N, c (0 + )= γc0. Lemma 1.5 If (S(t),x(t),c(t)) is any solution of the system (3) with the initial condition (4), then there exist constants L>0, and ɛ>0 small enough, such that S(t) L, x(t) L and 0 <m 3 c(t) M 3 where m 3 = γc0e ɛ and M 3 = γc0 + ɛ for t large enough.

5 Global qualitative analysis of new Monod type chemostat model 79 Proof Let (S(t),x(t),c(t)) be any solution of the system (3) with the initial condition (4). Let W (t) =e Dτ S(t)+ 1 x(t + τ). δ The upper right derivative of W (t) along trajectories of (3) is Ẇ (t) = De Dτ D + rc(t + τ) S(t) x(t + τ) δ = DW(t) r c(t + τ)x(t + τ) δ DW(t). Consider the following impulse differential inequalities: by Lemma 1.1, we obtain W (t) DW(t), t nt, n N, W (t + )=W(t)+e Dτ γs 0, t = nt, n N, W (t) W (0 + )e Dt +e Dτ e D(t T ) e DT γs 0 1 e +e Dτ γs 0 e DT 1 e DT e Dτ γs 0 e DT 1, t. By the definition of W (t), we obtain that there exists a constant L>0 such that S(t) L and x(t) L for t large enough. From the system (6), we have that c (t) = γc 0e D(t nt ) 1 e is a globally asymptotically stable positive periodic solution of system (6), where t (nt, (n + 1)T,n N,c (0 + )= γc0. Hence, we have that By Lemma 1.4, we get that γc 0 e 1 e c (t) γc 0, t 0. 1 e 0 <m 3 = γc 0e 1 e ɛ c(t) M γc 0 3 = 1 e + ɛ for ɛ small enough and t large enough. The proof is complete. 2 Extinction In this section, we investigate the extinction of the microorganism species, that is, microorganisms are entirely absent from the chemostat permanently, i.e., x(t) =0, t 0. (7) This is motivated by the fact that x = 0 is an equilibrium solution for the variable x(t), as it leaves x (t) = 0. Under these conditions, we show below that the nutrient concentration oscillates with period T in synchronization with the periodic impulsive input concentration of the nutrient, and the concentration of the toxicant in the chemostat oscillates with period T in synchronization with the periodic impulsive input of the toxicant. By Lemma 1.5 and the second equation of the system (3), it follows that x (t) μ m e Dτ x(t τ) (D + rm 3 )x(t),

6 80 MENG Xin-zhu, ZHAO Qiu-lan and CHEN Lan-sun where m 3 = γc0e ɛ. Clearly, if μ m e Dτ <D+r γc0e, then μ m e Dτ <D+r γc0e ɛ for ɛ small enough. By Lemma 1.2, we have lim t x(t) =0ifμ m e Dτ <D+ r γc0e, which implies the microorganism species becomes ultimately extinct. This shows that the specific growth of the microorganism species can not supply the losing of the microorganism species flowing out and the death of the microorganism species which are killed by toxicant no matter how much input the nutrient. Therefore, we assume μ m e Dτ >D+ r γc0e in the rest of this paper. By Lemmas 1.3 and 1.4, we have the following lemma. Lemma 2.1 Systems (5) and (6) have a unique positive periodic solution S (t) and c (t), respectively, that is, the system (3) has a microorganism-extinction periodic solution (S (t), 0, c (t)) for t (nt, (n +1)T, n N. For any solution (S(t),x(t),c(t)) of (3) we have S(t) S (t) and c(t) c (t) as t. Theperiodicsolution(S (t), 0,c (t)) corresponds to washout of the single population of microorganism from the chemostat. We give conditions for extinction of the microorganisms species as below. Theorem 2.1 Periodic solution (S (t), 0,c (t)) of system (3) is globally attractive if or γc 0 > γs 0 < K ( ) m 1 e D ( e DT 1 ) + rγc 0 (μ m e Dτ D)(e DT (8) 1) rγc 0 ( e DT 1 ) ( γs 0 μm e Dτ D ) K m D ( ) 1 e r γs 0 + K m (1 e ) > 0, (9) where γ = TD. Proof Let (S(t),x(t),c(t)) be any solution of the system (3) with the initial condition (4). From (8) or (9), we have μ m e Dτ γs0 γc 0e <D+ r K m + 1 e. γs0 Since p(z) = μme Dτ z K m+z is strictly increasing for all z 0, we may choose a sufficiently small positive constant ε such that μ m e Dτ η K m + η <D+ rm 3, (10) where γs 0 η = 1 e + ε, m 3 = γc 0e ε. 1 e It follows from that the first equation of the system (3) that S (t) DS(t). So we consider the following impulse differential inequalities: S (t) DS(t), t nt, n N, S(t + )=S(t)+γS 0, t = nt, n N. By using Lemma 1.1, we have γs 0 lim sup S(t) t 1 e.

7 Global qualitative analysis of new Monod type chemostat model 81 Hence, there exist a positive integer n 1 and a arbitrarily small positive constant ε such that for all t n 1 T, γs 0 S(t) 1 e + ε =: η, c(t) γc 0e 1 e ε = m 3. (11) From (11) and the second equation of (3), we get that, for t>n 1 T + τ, x (t) μ mηe Dτ K m + η x(t τ) (D + rm 3)x(t). Consider the following comparison equation: dz(t) dt By Lemma 1.2 and (10), we obtain that = μ mηe Dτ K m + η z(t τ) (D + rm 3)z(t). lim z(t) =0. t Since x(s) =z(s) =φ 2 (s) > 0 for all s τ,0, by the comparison theorem in differential equation and the nonnegativity of solution (with x(t) 0),wehavethatx(t) 0ast. Without loss of generality, we may assume that 0 <x(t) <εfor all t 0, by the first equation of the system (3), we have S (t) ( D + μ ) mε S(t). δk m Consider the following impulse system: ( z 1 (t) = D + μ ) mε z 1 (t), δk m t nt, n N, z 1 (t + )=z 1 (t)+γs 0, t = nt, n N, z 1 (0 + )=S(0 + ). Then we have that, for nt < t (n +1)T, z 1 (t) = γs 0e (D+ μmε δkm )(t nt ) μmε (D+ 1 e δkm )T is a unique globally asymptotically stable positive periodic solution of the system (12). Hence, z 1 (t) S(t) and z 1 (t) S (t) asε 0. By using comparison theorem of impulsive equation (see Theorem in Ref. 24), for any ε 1 > 0thereexistssuchaT 1 > 0that,fort>T 1, On the other hand, from the first equation of (3), it follows that (12) S(t) > z 1 (t) ε 1. (13) S (t) DS(t). Consider the following comparison system: z 2(t) = Dz 2 (t), t nt, n N, z 2 (t + )=z 2 (t)+γs 0, t = nt, n N, z 2 (0 + )=S(0 + ). (14)

8 82 MENG Xin-zhu, ZHAO Qiu-lan and CHEN Lan-sun Then we have S(t) < z 2 (t)+ε 1, (15) as t and z 2 (t) =S (t), where z 2 (t) is a unique positive periodic solution of (14). Let ε 0, then it follows from (13) and (15) that S (t) ε 1 <S(t) <S (t)+ε 1, for t large enough, which implies S(t) S (t) ast. Since the variables S and x do not appear in the third and sixth equations of the system (3), then we only need to consider the subsystem of (3) as follows: c (t) = Dc(t), t nt, n N, c(t + )=c(t)+γc 0, t = nt, n N, c(0 + )=c According to Lemma 1.4, we get that c(t) c (t) ast. This completes the proof. Corollary 2.1 Periodic solution (S (t), 0,c (t)) of system (3) is globally attractive if τ> 1 ( D ln μ m γs 0 e DT 1 ) K m (1 e )+γs 0 D(e DT 1) + rγc 0, where γ = TD. 3 Permanence Definition 3.1 The system (3) is said to be permanent if there exists a compact region D intd such that every solution of the system (3) with the initial condition (4) will eventually enter and remain in region D, whered = {(S(t),x(t),c(t)) S(t) 0,x(t) 0,c(t) 0}. Theorem 3.1 The system (3) is permanent if γs 0 > K ( m e DT 1 ) D ( 1 e ) + rγc 0 (μ m e Dτ D)(1 e > 0 (16) ) rγc 0 or ( ) ( 1 e γs 0 μm e Dτ D ) K m D ( e DT 1 ) γc 0 < r γs 0 + K m (e DT. (17) 1) Proof Suppose that (S(t),x(t),c(t)) is any positive solution of the system (3) with the initial condition (4). The second equation of the system (3) may be rewritten as x (t) = μ m e Dτ S(t) (D + rc(t)) x(t) μ m e Dτ d t S(θ)x(θ) dθ. (18) K m + S(t) dt t τ K m + S(θ) Define t V (t) =x(t)+μ m e Dτ S(θ)x(θ) t τ K m + S(θ) dθ. Calculating the derivative of V (t) along the solution of (3), it follows from (18) that dv (t) = μ m e Dτ S(t) (D + rc(t)) x(t). dt K m + S(t)

9 Global qualitative analysis of new Monod type chemostat model 83 AccordingtoLemma1.5,foranyε>0 small enough, there exists a positive constant T 1 such that for t T 1, γc 0 c(t) 1 e + ɛ = M 3. Hence dv (t) μ m e Dτ S(t) dt K m + S(t) (D + rm 3) x(t) μm e Dτ S(t) =(D + rm 3 ) D + rm 3 K m + S(t) 1 x(t), for t T 1. (19) Let ( ( m 2 = 1 δk m 1 2 μ m T ln 1+ γs ( 0 μm e Dτ D )( 1 e ) ) ) rγc 0 K m D (1 e D. )+rγc 0 From the inequality (17) or (18), it is clear that m 2 > 0. For this m 2, we can choose a positive constant ε 1 small enough such that μ m e Dτ ϱ > 1, (20) (D + rm 3 )(K m + ϱ) where «ϱ = γs 0e D+ μmm 2 T δkm γc 0 «ε 1 > 0, M 3 = 1 e D+ μmm 2 T 1 e + ɛ 1. δkm For any positive constant t 0, we claim that the inequality x(t) <m 2 cannot hold for all t t 0. Otherwise, there is a positive constant t 0, such that x(t) <m 2 for all t t 0. From the first and fourth equations of the system (3), we have ( S (t) D + μ mm ) 2 S(t), t nt, δk m S(t + )=S(t)+γS 0, t = nt. By Lemma 1.1, there exists such T 2 t 0 + τ, fort T 2 that S(t) > γs 0e 1 e From (19) and (21), we have dv (t) μm e Dτ > (D + rm 3 ) dt D + rm 3 Let x l = D+ μmm 2 δkm D+ μmm 2 δkm «T «T ε 1 =: ϱ. (21) ϱ K m + ϱ 1 x(t), for t T =max{t 1,T 2 }. (22) min x(t). t T,T +τ We show that x(t) x l for all t T. Otherwise, there exists a nonnegative constant T 3 such that x(t) x l for t T,T + τ + T 3,x(T + τ + T 3 )=x l and x (T + τ + T 3 ) 0. Thus from the second equation of (3) and (20), we easily see that x (T + τ + T 3 ) > μ m e Dτ ϱ K m + ϱ (D + rm 3) x l μm e Dτ =(D + rm 3 ) D + rm 3 ϱ K m + ϱ 1 x l > 0,

10 84 MENG Xin-zhu, ZHAO Qiu-lan and CHEN Lan-sun which is a contradiction. Hence we get that x(t) x l > 0 for all t T. From (22), we have dv (t) μm e Dτ ϱ > (D + rm 3 ) dt D + rm 3 K m + ϱ 1 x l > 0, which implies V (t) + as t +. This is a contradiction to V (t) (1 + μ m τe Dτ )L. Therefore, for any positive constant t 0, the inequality x(t) <m 2 cannot hold for all t t 0. On the one hand, if x(t) m 2 holds true for all t large enough, then our aim is obtained. On the other hand, x(t) is oscillatory about m 2. Let { } m m 2 =min 2 2,m 2e (D+rM3)τ. We claim x(t) m 2. We shall prove that x(t) m 2 below. There exist two positive constants t and ω such that x( t) =x( t + ω) =m 2 and x(t) <m 2, for t <t< t + ω. When t is large enough, inequalities S(t) >ϱand c(t) M 3 hold true for t <t< t + ω. Since x(t) is continuous and bounded and is not effected by impulses, we conclude that x(t) is uniformly continuous. Hence there exists a constant T 4 (with0<t 4 <τand T 4 is independent of the choice of t) such that x(t) > m 2 2 for all t t t + T 4. If ω T 4, our aim is obtained. If T 4 <ω τ, from the second equation of (3) we have that x (t) (D + rm 3 )x(t) for t <t t + ω. Then we have x(t) m 2 e (D+rM3)τ for t <t t + ω t + τ since x( t) =m 2. It is clear that x(t) m 2 for t <t t + ω. If ω τ, then we have that x(t) m 2 for t < t t + τ. Thus, proceeding exactly as the proof for the above claim, we can obtain x(t) m 2 for t + τ t t + ω. Since the interval t, t + ω is arbitrarily chosen (we only need t to be large), we get that x(t) m 2 for t large enough. In view of our arguments above, the choice of m 2 is independent of the positive solution of (3) which satisfies that x(t) m 2 for sufficiently large t. By Lemma 2.5, we have x(t) L for t large enough. Hence, from the first equation of (3), we have that, ( S (t) D + μ ) ml S(t). δk m Then we have S(t) z 3 (t), where z 3 (t) is unique a globally asymptotically stable positive periodic solution of ( z 3(t) = D + μ ) ml z 3 (t), t nt, n N, δk m z 3 (t + )=z 3 (t)+γs 0, t = nt, n N, z 3 (0 + )=S(0 + ) > 0. There exists an ε>0 small enough such that for sufficiently large t, S(t) z 3 (t) ε γs 0e (D+ μml δkm )T ε =: m μml (D+ 1 e δkm )T 1 > 0. By Lemma 1.5, we have that 0 <m 3 c(t) M 3 where m 3 = γc0e ɛ and M 3 = γc0 for sufficiently large t and sufficient small ɛ. Set D = {(S, x, c) R+ m 3 1 S(t) L, m 2 x(t) L, m 3 c(t) M 3 }. +ɛ

11 Global qualitative analysis of new Monod type chemostat model 85 Then D is a bounded compact region which has positive distance from coordinate axes. In view of our arguments above, one obtains that every solution of the system (3) with the initial condition (4) eventually enters and remains in the region D. The proof is complete. Corollary 3.1 The system (3) is permanent if τ< 1 ( D ln μ m γs ) 0 1 e K m (e DT 1) + γs 0 D(1 e )+rγc 0, where γ = TD. 4 Discussions In Section 2, we give the conditions for the population of microorganisms which will eventually be washed out of the chemostat. Theorem 2.1 shows if the impulsive periodic input nutrient concentration γs 0 is under certain value or the impulsive periodic input concentration of the toxicant γc 0 is over certain value, then the population of microorganisms will be eventually extinct. In Section 3, we give the conditions for permanence of the microorganisms species. Theorem 3.1 shows if the impulsive periodic input concentration of the nutrient γs 0 is over certain value or the impulsive periodic input concentration of the toxicant γc 0 is under certain value, then the microorganisms species is permanent. In this case, the microorganism is obtained. Obviously, if both the continuous culture and the impulsive culture can obtain the microorganism, the latter is better than the former since the impulsive culture can save the substrate. Whether the microorganism is extinct or not is determined completely by the input amount of the substrate γs 0 and concentration of the toxicant γc 0 for the fixed period nt.the environment with no pollution is in favor of living of the microorganism species. Otherwise, the polluted environment can lead to the microorganism species be extinct. This shows that the input concentration of the toxicant greatly affects the dynamics behaviors of the model. From Corollaries 2.1 and 3.1, we can see the extinction and permanence of the microorganism are dependent on the time delay for growth response of the microorganism. Ultimately, when thetimedelayforgrowthresponseistoolong, the permanence of system disappears and the consumer population of the microorganism dies out, then we call it profitless time delay. This shows the sensitivity of the model dynamics on time delay (growth response). The ultimate scenario makes intuitive biological sense: if it takes too long to grow then the highest possible recruitment rate to the microorganism species (μ m e Dτ ) will drop below the losing rate of flowing out, D, and the death of the microorganism population which are killed by the toxicant leading to the extinction of x. This implies that time delay has the significant influence to the dynamics behaviors of the model. References 1 Hsu S B. A competition model for a seasonally fluctuating nutrientj. Journal of Mathematical Biology, 1980, 9(2): Smith H L. Competitive coexistence in an oscillating chemostatj. SIAM Journal on Applied Mathematics, 1981, 40(3): Butler G J, Hsu S B, Waltman P. A mathematical model of the chemostat with periodic washout ratej. SIAM Journal on Applied Mathematics, 1985, 45(3): Pilyugin S S, Waltman P. Competition in the unstirred chemostat with periodic input and washoutj. SIAM Journal on Applied Mathematics, 1999, 59(4): Simth H L, Waltman P. The theory of the chemostatm. Cambridge: Cambridge University Press, Hsu S B, Hubbell S P, Waltman P. A mathematical theory for single nutrient competition in continuous cultures of micro-organismsj. SIAM Journal on Applied Mathematics, 1977, 32(2):

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