Global Behaviors of a Chemostat Model with Delayed Nutrient Recycling and Periodically Pulsed Input

Abstract

The dynamic behaviors in a chemostat model with delayed nutrient recycling and periodically pulsed input are studied. By introducing new analysis technique, the sufficient and necessary conditions on the permanence and extinction of the microorganisms are obtained. Furthermore, by using the Liapunov function method, the sufficient condition on the global attractivity of the model is established. Finally, an example is given to demonstrate the effectiveness of the results in this paper.

1. Introduction

This paper is mainly concerned with single-species chemostat-type model with nutrient recycling. Usually, nutrient recycling is regarded as an instantaneous term by neglecting the time required to regenerate nutrient from dead biomass by bacterial decomposition. The motivation for such models is given by Beretta et al. in [1], where such systems are used to model the growth of planktonic communities in lakes, where the plankton feeds on a limiting nutrient supplied at a constant rate. The basic single-species chemostat model with delayed nutrient recycling is the following differential equation: 𝑑𝑆𝑑𝑡=𝐷(𝑆⁰−𝑆)−𝑚𝑈(𝑆)𝑁+𝑏𝐷₁^𝑡_−∞𝐹(𝑡−𝜏)𝑁(𝜏)𝑑𝜏,𝑑𝑁𝑑𝑡=𝑁[−(𝐷+𝐷₁)+𝑚₁𝑈(𝑆)]. (1.1) The chemostat models with nutrient recycling have been extensively investigated by many researchers. The studied main subjects are the persistence, permanence, and extinction of microorganisms, global stability and the existence of periodic oscillation of the systems, and so forth. Many important and interesting results can be found in [1–15] and the references cited therein. In [4], Freedman and Xu extended the single-species model proposed in [1] to two-species competition models with instantaneous and delayed nutrient recycling. They developed persistence and extinction criteria for the competing populations. In [10], by applying the method of Liapunov functionals they study the global asymptotic stability of the positive equilibria of the models in [4]. In [12], a chemostat model with distributed time delays both in material recycling and biotic species growth has been considered.

As it is well known, countless organisms live in seasonally or diurnally forced environment, in which the populations obtain food, so the effects of this forcing may be quite profound. Recently many papers studied chemostat model with variations in the supply of nutrients or the washout. The chemostat models with impulsive input perturbation have been studied in many articles see [16–22] and the references cited therein, where many important and interesting results on the persistence, permanence and extinction of microorganisms, global stability, the existence of periodic oscillation and dynamical complexity of the systems are discussed. Particular in [21], the following model of the lactic acid fermentation in membrane bioreactor with impulsive input is discussed:𝑑𝑆=𝑄𝑑𝑡𝑉(𝑆⁰−𝑆)−𝜇𝑆𝑥𝛿₁(𝐾_𝑠𝐾+𝑆)+(^𝐷𝛿₁−𝑚)𝑥,𝑑𝑥=𝑑𝑡𝜇𝑆𝑥𝐾_𝑠+𝑆−(𝐾_𝐷+𝑄₁𝑉)𝑥,𝑡≠𝑛𝑇,Δ𝑆=𝑄𝑆⁰𝑉,Δ𝑥=0.𝑡=𝑛𝑇,𝑛∈𝑁. (1.2) The model has both nutrient recycling and impulsive input substrate. Using Floquet's theory of impulsive periodic linear differential equations and small-amplitude perturbation, they obtain the biomass-free periodic solution is locally stable if some conditions are satisfied, see [21, Theorems 3.1]. In [18], the following Monod type chemostat model with nutrient recycling and impulsive input is studied: 𝑑𝑆(𝑡)𝜇𝑑𝑡=−𝐷𝑆(𝑡)−^𝑚𝑆(𝑡)𝑥(𝑡)𝛿(𝐾_𝑚+𝑆(𝑡))+𝑏𝛾𝑥(𝑡),𝑑𝑥(𝑡)=𝜇𝑑𝑡^𝑚𝑆(𝑡)𝑥(𝑡)𝐾_𝑚+𝑆(𝑡)−(𝐷+𝛾)𝑥(𝑡),𝑡≠𝑛𝑇,𝑆(𝑡⁺)=𝑆(𝑡)+𝐷𝑆⁰,𝑥(𝑡⁺)=𝑥(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁. (1.3) The sufficient and necessary conditions of the permanence and extinction of the microorganism species and the sufficient condition of the global asymptotic stability of the model are established.

However, we see that few authors consider the chemostat models with delayed nutrient recycling and periodically pulsed input. Based on the ideas given in [18], we develop model (1.1) into the following form by introducing impulsive input:𝑑𝑥(𝑡)𝑑𝑡=−𝑑𝑥−𝑚𝑈(𝑥)𝑦+𝑏𝑑₁∫⁰_−∞𝐹(𝑠)𝑦(𝑡+𝑠)𝑑𝑠,𝑑𝑦(𝑡)𝑑𝑡=𝑦[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥)],𝑡≠𝑛𝑇,𝑥(𝑡⁺)=𝑥(𝑡)+𝑝,𝑦(𝑡⁺)=𝑦(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁. (1.4) For system (1.4), we will investigate the permanence, extinction and the global attractivity. We will establish the sufficient and necessary conditions for the permanence, extinction.

The rest of this paper is organized as follows. In the following section we will firstly introduce the basic assumption for system (1.4). Next, we will give a equivalent form of system (1.4) by introducing a new variable. Further, we will give several useful lemmas. In Section 3 we will state and prove a boundedness result for system (1.4). In Section 4 we will state and prove an extinction result and a permanence result for system (1.4). In Section 5 we will state and prove a global attractivity result for system (1.4). Finally, in Section 6, we will discuss an example and give some numerical simulations.

2. Preliminaries

In system (1.4), 𝑥 is the concentration of a limiting nutrient and 𝑦 is a measure of the population of some organism; 𝑑>0 is the input and output flow, and is referred to as the wash-out rate; 𝑚>0 is the maximum uptake rate of nutrient, 𝑚₁>0 is the maximum specific growth rate of the organism; 𝑑₁>0 is the death rate; 𝑏∈(0,1) is the fraction of nutrient recycled after death of the species; 𝑝>0 is the amount of the substrate concentration pulsed each 𝑛𝑇, where 𝑇>0 is a constant; 𝑁 represents the set of all positive integers; 𝑈(𝑠) is the uptake function; Delay-kernel function 𝐹(𝑢) is a nonnegative bounded integrable function defined on 𝑅₊=[0,∞). In this paper, we always assume that 𝑑, 𝑚, 𝑑₁, 𝑚₁, 𝑏 and 𝑝 are constants, and 𝐹(𝑢)=𝑎𝑒^−𝑎𝑢, where 𝑎>0 is a constant.

For system (1.4), we always assume that uptake function 𝑈(𝑠) satisfies the following assumption: (𝐇)𝑈(0)=0, 𝑈(𝑠) is continuously differentiable for all 𝑠≥0 and 𝑑𝑈(𝑠)/𝑑𝑡>0 for all 𝑠∈[0,𝑀₀], where 𝑀₀=𝑝/(1−𝑒^−𝑑𝑇).

The initial conditions in system (1.4) are given in the following form: 𝑥(0)=𝑥₀>0,𝑦(𝜃)=𝜑(𝜃)≥0,𝜃∈𝑅₋,𝜑(0)>0,sup_{𝜃∈𝑅−}𝜑(𝜃)<∞, (2.1) where 𝑅₋=(−∞,0]. It is easy to prove that solution (𝑥(𝑡),𝑦(𝑡)) of system (1.4) with initial condition (2.1) is positive, that is, 𝑥(𝑡)>0 and 𝑦(𝑡)>0 in the interval of the existence.

Now, for system (1.4) we introduce a new variable 𝑧(𝑡) as follows: 𝑧(𝑡)=^𝑡_−∞𝐹(𝑡−𝜏)𝑦(𝜏)𝑑𝜏. (2.2) Then, system (1.4) is equivalent to the following system: 𝑑𝑥(𝑡)𝑑𝑡=−𝑑𝑥(𝑡)−𝑚𝑈(𝑥(𝑡))𝑦(𝑡)+𝑏𝑑₁𝑧(𝑡),𝑑𝑦(𝑡)𝑑𝑡=𝑦(𝑡)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥(𝑡))],𝑑𝑧(𝑡)𝑑𝑡=−𝑎𝑧(𝑡)+𝑎𝑦(𝑡),𝑡≠𝑛𝑇,𝑥(𝑡⁺)=𝑥(𝑡)+𝑝,𝑦(𝑡⁺)=𝑦(𝑡),𝑧(𝑡⁺)=𝑧(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁, (2.3) where variable 𝑧(𝑡) can be interpreted as an intermediate component. Besides, initial condition (2.1) is changed into the following form 𝑥(0)=𝑥₀>0,𝑦(0)=𝑦₀>0,𝑧(0)=𝑧₀>0. (2.4) Therefore, in the rest of this paper we will mainly discuss system (2.3).

Firstly, on the positivity of solutions for system (2.3), we have the following result.

Lemma 2.1. The solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (2.3) with initial condition (2.4) is positive, that is, 𝑥(𝑡)>0, 𝑦(𝑡)>0 and 𝑧(𝑡)>0 for any 𝑡≥0.

The proof of Lemma 2.1 is simple, we hence omit it here.

We consider the following linear impulsive differential equation: 𝑑𝜔(𝑡)𝑑𝑡=−𝑑𝜔(𝑡)−𝜂,𝑡≠𝑛𝑇,𝜔(𝑡⁺)=𝜔(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁,𝜔(0⁺)=𝜔₀, (2.5) where 𝜂 is a constant. Clearly, 𝜔^∗𝜂(𝑡)=−𝑑+𝑥^∗(𝑡),𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁 (2.6) is the 𝑇-periodic solution of system (2.5), where 𝑥^∗(𝑡)=𝑝𝑒^{−𝑑(𝑡−𝑛𝑇)}1−𝑒^−𝑑𝑇,𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁. (2.7) We say that 𝜔^∗(𝑡) is globally uniformly attractive, if for any constants 𝑀>0 and 𝜀>0 there is a constant 𝑇(𝑀,𝜀)>0 such that for any initial time 𝑡₀≥0 and any solution 𝜔(𝑡) of system (2.5) with |𝑤(𝑡₀)|≤𝑀, one has |𝑤(𝑡)−𝑤^∗(𝑡)|<𝜀∀𝑡≥𝑡₀+𝑇(𝑀,𝜀). (2.8) We have the following result.

Lemma 2.2. 𝑇-periodic solution 𝜔^∗(𝑡) of system (2.5) is globally uniformly attractive.

Proof. Let 𝜔(𝑡) be any solution of system (2.5) with initial value 𝜔(𝑡₀). Define 𝑢(𝑡)=𝜔(𝑡)−𝜔^∗(𝑡), then we have 𝑑𝑢(𝑡)𝑑𝑡=−𝑑𝑢(𝑡),𝑡≠𝑛𝑇,𝑢(𝑡⁺)=𝑢(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁,𝑢(𝑡₀)=𝜔(𝑡₀)−𝜔^∗(𝑡₀). (2.9) The solution of system (2.9) is 𝑢(𝑡)=𝑢(𝑡₀)𝑒^{−𝑑(𝑡−𝑡0)}, 𝑡≥𝑡₀. For any constant 𝑀>0 and 𝜀>0, when |𝜔(𝑡₀)|≤𝑀, we have |𝑢(𝑡)|=|𝑢(𝑡₀)|𝑒^{−𝑑(𝑡−𝑡0)}≤(𝑀+𝑀^∗)𝑒^{−𝑑(𝑡−𝑡0)},𝑡≥𝑡₀, (2.10) where 𝑀^∗=𝜂/𝑑+𝑝/(1−𝑒^−𝑑𝑇). Choose 𝑇(𝑀,𝜀)=−(1/𝑑)ln(𝜀/(𝑀+𝑀^∗)); then for any 𝑡₀≥0 we can obtain |𝑢(𝑡)|≤(𝑀+𝑀^∗)𝑒^{−𝑑(𝑡−𝑡0)}≤0 (2.11) for all 𝑡≥𝑡₀+𝑇(𝑀,𝜀). This shows that solution 𝜔^∗(𝑡) is globally uniformly attractive. This completes the proof.

In system (2.5), when 𝜂=0, then we obtain the subsystem of system (2.3) with 𝑦(𝑡)=0 and 𝑧(𝑡)=0 as follows: 𝑑𝑥(𝑡)𝑑𝑡=−𝑑𝑥(𝑡),𝑡≠𝑛𝑇,𝑥(𝑡⁺)=𝑥(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (2.12) Clearly, 𝑥^∗(𝑡) is the positive 𝑇-periodic solution of system (2.12). Therefore, system (2.3) has the semitrivial 𝑇-periodic solution (𝑥^∗(𝑡),0,0). The solution of system (2.12) with initial condition 𝑥(0⁺)=𝑥₀ is 𝑥(𝑡)=(𝑥₀−𝑝1−𝑒^−𝑑𝑇)𝑒^−𝑑𝑡+𝑥^∗(𝑡),𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁. (2.13) From Lemma 2.2, we obtain that for any solution 𝑥(𝑡) of system (2.12), one has |𝑥(𝑡)−𝑥^∗(𝑡)|→0as𝑡→∞. (2.14)

3. Boundedness

On the ultimate boundedness of all positive solutions of system (2.3) we have the following result.

Theorem 3.1. Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any positive solution of system (2.3) if 𝑏0<<𝑚𝑎−𝑑𝑎𝑚₁, (3.1) then limsup_𝑡→∞𝑥(𝑡)<𝑀₀,limsup_𝑡→∞𝑚𝑦(𝑡)<¹𝑚𝑀₀,limsup_𝑡→∞𝑧(𝑡)<(𝑎−𝑑)𝑀₀𝑏𝑑₁, (3.2) where 𝑀₀=𝑝/(1−𝑒^−𝑑𝑇).

Proof. From condition (3.1), there is a constant 𝑐>0 such that 𝑏𝑑₁+𝑐𝑑−𝑎𝑐<0,𝑎𝑐−𝑚𝑑₁𝑚₁<0. (3.3) Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any solution of system (2.3) define Liapunov function 𝑉(𝑡) as follows: 𝑚𝑉(𝑡)=𝑥(𝑡)+𝑚₁𝑦(𝑡)+𝑐𝑧(𝑡). (3.4) Calculating the derivative of 𝑉(𝑡) along solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (2.3), we have 𝑑𝑉(𝑡)=𝑑𝑡𝑑𝑥(𝑡)+𝑚𝑑𝑡𝑚₁𝑑𝑦(𝑡)𝑑𝑡+𝑐𝑑𝑧(𝑡)𝑑𝑡=−𝑑𝑉(𝑡)+(𝑎𝑐−𝑚𝑑₁𝑚₁)𝑦+(𝑏𝑑₁+𝑐𝑑−𝑎𝑐)𝑧≤−𝑑𝑉(𝑡) (3.5) for all 𝑡≥0 and 𝑡≠𝑛𝑇, and 𝑉(𝑡⁺)=𝑉(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (3.6) From the comparison theorem of impulse differential equations, we have 𝑉(𝑡)≤𝑢(𝑡),∀𝑡≥0, (3.7) where 𝑢(𝑡) is the solution of system (2.12) with initial value 𝑢(0⁺)=𝑉(0⁺). From Lemma 2.2, we have 𝑢(𝑡)→𝑥^∗(𝑡) as 𝑡→∞. Hence, we further obtain limsup_𝑡→∞𝑝𝑉(𝑡)≤1−𝑒^−𝑑𝑇=𝑀₀. (3.8) From this, we finally obtain limsup_𝑡→∞𝑥(𝑡)<𝑀₀,limsup_𝑡→∞𝑚𝑦(𝑡)<¹𝑚𝑀₀,limsup_𝑡→∞𝑧(𝑡)<(𝑎−𝑑)𝑀₀𝑏𝑑₁. (3.9) This completes the proof of Theorem 3.1.

Return to the original system (1.4), we have the following corollary as a consequence of Theorem 3.1.

Corollary 3.2. Let (𝑥(𝑡),𝑦(𝑡)) be any positive solution of system (1.4) if inequality (3.1) holds, then limsup_𝑡→∞𝑥(𝑡)<𝑀₀,limsup_𝑡→∞𝑚𝑦(𝑡)<¹𝑚𝑀₀, (3.10) where 𝑀₀=𝑝/(1−𝑒^−𝑑𝑇).

Remark 3.3. Compare Theorem 3.1 with Lemma 2.3 given in [18] that we can see the ultimate boundedness is quite different between model with delayed nutrient recycling and the model with instantaneous nutrient recycling.

4. Extinction and Permanence

On the extinction of the microorganism species of system (2.3), we have the following result.

Theorem 4.1. Suppose that inequality (3.1) holds and ^𝑇₀[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗(𝑡))]𝑑𝑡≤0. (4.1) Then periodic solution (𝑥^∗(𝑡),0,0) of system (2.3) is globally attractive.

Proof. From condition (3.1), there is a constant 𝑐>0 such that 𝑏𝑑₁+𝑐𝑑−𝑎𝑐<0,𝑎𝑐−𝑚𝑑₁𝑚₁<0. (4.2) Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any solution of system (2.3), define Liapunov function 𝑉(𝑡) as follows 𝑚𝑉(𝑡)=𝑥(𝑡)+𝑚₁𝑦(𝑡)+𝑐𝑧(𝑡). (4.3) Then similar to the proof of Theorem 3.1. We obtain 𝑉(𝑡)≤𝑢(𝑡) for all 𝑡≥0, where 𝑢(𝑡) is the solution of system (2.12) with initial value 𝑢(0⁺)=𝑉(0⁺) and 𝑢(𝑡)→𝑥^∗(𝑡) as 𝑡→∞. Hence, there exists a function 𝛼(𝑡)∶𝑅₊→𝑅 satisfying 𝛼(𝑡)→0 as 𝑡→∞ such that 𝑉(𝑡)≤𝑢(𝑡)=𝑥^∗(𝑡)+𝛼(𝑡) (4.4) for all 𝑡≥0. From the definition of 𝑉(𝑡), we further have 𝑥(𝑡)≤𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝑦(𝑡)−𝑐𝑧(𝑡). (4.5) From the second equation of system (2.3), we obtain 𝑑𝑦(𝑡)𝑑𝑡≤𝑦(𝑡)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝑦(𝑡)−𝑐𝑧(𝑡))]. (4.6) From condition (4.1), we obtain for any 𝜀₀>0^𝑇₀[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)−𝑚₁𝜀₀)]𝑑𝑡<0. (4.7) Since lim_𝑡→∞𝛼(𝑡)=0, we can obtain limsup_{𝑡→∞𝑡}^𝑡+𝑇[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀−𝑐𝑧(𝑡))]𝑑𝑡≤lim_{𝑡→∞𝑡}^𝑡+𝑇[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀=)]𝑑𝑡_𝑇0[−(𝑑+𝑑₁)+𝑚𝑝(𝑠^∗1(𝑡)−𝑚𝜀₀)]𝑑𝑡<0. (4.8) Hence, there exist constants 𝜂>0 and 𝑇₀>0 such that when 𝑡≥𝑇₀, _𝑡^𝑡+𝑇[−(𝑑+𝑑₁)+𝑚𝑝(𝑠^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀−𝑐𝑧(𝑡))]≤−𝜂. (4.9) and |𝛼(𝑡)|<1.
If 𝑦(𝑡)≥𝜀₀ for all 𝑡≥𝑇₀, then from (4.6) we obtain 𝑑𝑦(𝑡)𝑑𝑡≤𝑦(𝑡)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀−𝑐𝑧(𝑡))]. (4.10) For any 𝑡≥𝑇₀, we choose an integer 𝑝≥0 such that 𝑡∈(𝑇₀+𝑝𝑇,𝑇₀+(𝑝+1)𝑇]; then integrating (4.10) from 𝑇₀ to 𝑡, from (4.9), we can obtain 𝑦(𝑡)≤𝑦(𝑇₀)exp{^𝑡_𝑇0[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀−𝑐𝑧(𝑡))]𝑑𝑡}≤𝑦(𝑇₀)exp(−𝜂𝑝)exp{^𝑡_{𝑇0+𝑝𝑇}[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀−𝑐𝑧(𝑡))]𝑑𝑡}≤𝑦(𝑇₀)exp(−𝜂𝑝)exp([𝑈(𝑀₀𝑚+1−𝑚₁𝜀₀)−(𝑑+𝑑₁)]𝑇), (4.11) where constant 𝑀₀ is given in Theorem 3.1. Since 𝑝→∞ as 𝑡→∞, from (4.11), we obtain 𝑦(𝑡)→0 as 𝑡→∞ which leads to a contradiction. Hence, there is a 𝑡^∗≥𝑇₀ such that 𝑦(𝑡^∗)<𝜀₀.
Now, we claim that there exists a constant 𝑀₁>1 such that 𝑦(𝑡)≤𝜀₀𝑀₁ for all 𝑡≥𝑡^∗. In fact, if there exists a 𝑡₁>𝑡^∗ such that 𝑦(𝑡₁)>𝜀₀𝑀₁, then there exists a 𝑡₂∈(𝑡^∗,𝑡₁) such that 𝑦(𝑡₂)=𝜀₀ and 𝑦(𝑡)>𝜀₀ for 𝑡∈(𝑡₂,𝑡₁). Choose an integer 𝑝≥0 such that 𝑡₁∈[𝑡₂+𝑝𝑇,𝑡₂+(𝑝+1)𝑇). Since for any 𝑡∈(𝑡₂,𝑡₁)𝑑𝑦(𝑡)𝑑𝑡≤𝑦(𝑡)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀−𝑐𝑧(𝑡))], (4.12) integrating this inequality from 𝑡₂ to 𝑡₁, from (4.9) we can obtain 𝑦(𝑡₁)≤𝑦(𝑡₂)exp{^𝑡1_𝑡2[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀−𝑐𝑧(𝑡))]𝑑𝑡}≤𝑦(𝑡₂)exp(−𝜂𝑝)exp{^𝑡1_{𝑡2+𝑝𝑇}[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑚(𝑡)+𝛼(𝑡)−𝑚₁𝜀₀)]𝑑𝑡}≤𝜀₀exp([𝑈(𝑀₀𝑚+1−𝑚₁𝜀₀)−(𝑑+𝑑₁)]𝑇). (4.13) Obviously, choose constant 𝑀₁=exp([𝑈(𝑀₀𝑚+1−𝑚₁𝜀₀)−(𝑑+𝑑₁)]𝑇); (4.14) then from (4.13) we obtain a contradiction. Hence, we have 𝑦(𝑡)≤𝜀₀𝑀₁ for all 𝑡≥𝑡^∗. Since 𝜀₀ is arbitrary, we finally have lim_𝑡→∞𝑦(𝑡)=0. Obviously, we can obtain from the third equation of system (2.3) lim_𝑡→∞𝑧(𝑡)=0 and further from the first equation of system (2.3) we can obtain easily that lim_𝑡→∞(𝑥(𝑡)−𝑥^∗(𝑡))=0. This completes the proof of Theorem 4.1.

On the permanence of the nutrient and the microorganism for system (2.3), we have the following result.

Theorem 4.2. Suppose that inequality (3.1) holds and ^𝑇₀[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗(𝑡))]𝑑𝑡>0, (4.15) where 𝑥^∗(𝑡) is the unique positive 𝑇-periodic solution of system (2.12). Then system (2.3) is permanent.

Proof. Let (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) be any solution of system (2.3) with initial value (2.4). Since inequality (3.1) holds, from Theorem 3.1, for any 𝜀>0 there is a 𝑇₁>0 such that 𝑥(𝑡)≤𝑀₀𝑚+𝜀,𝑦(𝑡)≤¹𝑚(𝑀₀+𝜀),∀𝑡≥𝑇₁. (4.16) From assumption (H) and the theorem of mean value, for all 𝑡≥0, there exists a 𝜉(𝑡)∈(0,𝑥(𝑡)) such that 𝑈(𝑥(𝑡))=(𝑑𝑈(𝜉(𝑡))/𝑑𝑠)𝑥(𝑡). Since 𝑑𝑈(𝑠)/𝑑𝑠 is continuous for 𝑠≥0, there exists a constant 𝑀₁>0 such that 𝑈(𝑥(𝑡))=𝑥(𝑡)𝑑𝑈(𝜉(𝑡))𝑑𝑠≤𝑀₁,∀𝑡≥0. (4.17) From the first equation of system (2.3), we obtain 𝑑𝑥(𝑡)𝑑𝑡≥−𝑑𝑥(𝑡)−𝑚𝑈(𝑥(𝑡))𝑦(𝑡)≥−𝑑𝑥(𝑡)−𝑚₁(𝑀₀+𝜀)𝑑𝑈(𝜉(𝑡))𝑑𝑠𝑥(𝑡)≥−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁)𝑥(𝑡) (4.18) for all 𝑡≥𝑇₁ and 𝑡≠𝑛𝑇, and 𝑥(𝑡⁺)=𝑥(𝑡)+𝑝, 𝑡=𝑛𝑇, 𝑛∈𝑁. Using the comparison theorem of impulsive differential equation, we obtain 𝑥(𝑡)≥𝑣(𝑡) for all 𝑡≥𝑇₁, where 𝑣(𝑡) is the solution of the following impulsive equation: 𝑑𝑣(𝑡)𝑑𝑡=−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁)𝑣(𝑡),𝑡≠𝑛𝑇,𝑣(𝑡⁺)=𝑣(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁, (4.19) with initial condition 𝑣(𝑇⁺₁)=𝑥(𝑇⁺₁). Further from Lemma 2.2, we have lim_𝑡→∞(𝑣(𝑡)−𝑣^∗(𝑡))=0, (4.20) where 𝑣^∗(𝑡) is the unique 𝑇-periodic solution of (4.19) and 𝑣^∗(𝑡)=𝑝exp{−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁)(𝑡−𝑛𝑇)}1−exp{−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁)𝑇} (4.21) for all 𝑡∈(𝑛𝑇,(𝑛+1)𝑇] and 𝑛∈𝑁. Therefore, we further obtain liminf_𝑡→∞𝑥(𝑡)≥liminf_𝑡→∞𝑣(𝑡)≥𝑝exp(−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁)𝑇)1−exp(−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁.)𝑇) (4.22) This shows that 𝑥(𝑡) in system (2.3) is permanent.
Next, we prove that there exists a constant 𝑚₂>0 such that liminf_𝑡→∞𝑦(𝑡)≥𝑚₂ for any solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (2.3) with initial value (2.4). From assumption (H), we can choose a constant 𝜀₀ such that 𝑑𝑈(𝑠)/𝑑𝑠>0 for all 𝑠∈[0,𝑀₀+𝜀₀]. According to (4.6), we can choose positive constants 𝜀, 𝜀₁ and 𝜀₂, and 𝜀≤𝜀₀, such that ^𝑇₀[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗(𝑡)−𝑚𝑈(𝑀₀+𝜀)𝑑𝜀₁−𝜀₂)]𝑑𝑡>𝜀₂. (4.23) We first prove that there is a constant 𝜂>0 and 𝜂<𝜀₁ such that limsup_𝑡→∞𝑦(𝑡)≥𝜂 (4.24) for any solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (2.3) with initial value (2.4).
In fact, if (4.24) is not true, then there is a solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (2.3) such that limsup_𝑡→∞𝑦(𝑡)≤𝜀₁. Hence, there is a 𝑇₀>0 such that 𝑦(𝑡)<𝜀₁, for all 𝑡≥𝑇₀. Further, from Theorem 3.1, there exists a constant 𝑇₁≥𝑇₀ such that for all 𝑡≥𝑇₁𝑥(𝑡)<𝑀₀𝑚+𝜀,𝑦(𝑡)<¹𝑚(𝑀₀+𝜀). (4.25) From the first equation of system (2.3), we have 𝑑𝑥(𝑡)𝑑𝑡≥−𝑑𝑥(𝑡)−𝑚𝑈(𝑥(𝑡))𝑦(𝑡)≥−𝑑𝑥(𝑡)−𝜀₁𝑚𝑈(𝑀₀+𝜀),𝑡≥𝑇₁,𝑡≠𝑛𝑇,𝑥(𝑡⁺)=𝑥(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (4.26) From the comparison theorem of impulse differential equations and Lemma 2.2, for above 𝜀₂>0, there is a 𝑛₁∈𝑁 and 𝑛₁𝑇>𝑇₁ such that 𝑥(𝑡)≥−𝑚𝑈(𝑀₀+𝜀)𝑑𝜀₁+𝑥^∗(𝑡)−𝜀₂ (4.27) for all 𝑡≥𝑛₁𝑇. Then from the second equation of system (2.3), we have that 𝑑𝑦(𝑡)𝑑𝑡≥𝑦(𝑡)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗(𝑡)−𝑚𝑈(𝑀₀+𝜀)𝑑𝜀₁−𝜀₂)] (4.28) for all 𝑡>𝑛₁𝑇. Integrating (4.29) from 𝑛₁𝑇 to 𝑡≥𝑛₁𝑇, we obtain 𝑑𝑦(𝑡)𝑑𝑡≥𝑦(𝑛₁𝑇)exp^𝑡_𝑛1𝑇[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗(𝑡)−𝑚𝑈(𝑀₀+𝜀)𝑑𝜀₁−𝜀₂)]𝑑𝑡. (4.29) Obviously, from (4.23) and (4.29), we obtain 𝑦(𝑡)→∞ as 𝑡→∞, which leads to a contradiction. Therefore, (4.24) is true.
Now, we prove 𝑦(𝑡) in system (2.3) is permanent. Assume that it is not true, then there exists a sequence of initial values {𝜑_𝑘}={(𝑥₀^(𝑘),𝑦₀^(𝑘),𝑧₀^(𝑘))} which satisfies initial condition (2.4) such that for solution (𝑥(𝑡,𝜑_𝑘),𝑦(𝑡,𝜑_𝑘),𝑧(𝑡,𝜑_𝑘)), of system (2.3),liminf_𝑡→∞𝑦(𝑡,𝜑_𝑘𝜂)<𝑘²,∀𝑘=1,2,…. (4.30) From (4.24) and (4.30) we obtain that there exist two time sequences {𝑠_𝑞^(𝑘)} and {𝑡_𝑞^(𝑘)} such that for each 𝑘=1,2,…,0<𝑠₁^(𝑘)<𝑡₁^(𝑘)<𝑠₂^(𝑘)<𝑡₂^(𝑘)<⋯<𝑠_𝑞^(𝑘)<𝑡_𝑞^(𝑘)𝑠<⋯,_𝑞(𝑘)→∞,𝑡_𝑞^(𝑘)→∞as𝑞→∞,𝑦(𝑡_𝑞^(𝑘),𝜑_𝑘𝜂)=𝑘²,𝑦(𝑠_𝑞^(𝑘),𝜑_𝑘𝜂)=𝑘𝜂,𝑘²<𝑦(𝑡,𝜑_𝑘𝜂)<𝑘∀𝑡∈(𝑠_𝑞^(𝑘),𝑡_𝑞^(𝑘)). (4.31)(4.32)(4.33) From Theorem 3.1, there is a 𝑇^(𝑘)≥𝑇₀ such that 𝑦(𝑡,𝜑_𝑘𝑚)≤¹𝑚(𝑀₀+𝜀),𝑥(𝑡,𝜑_𝑘)≤𝑀₀+𝜀∀𝑡≥𝑇^(𝑘). (4.34) Further, from (4.31) for every 𝑘 there is an integer 𝐾^(𝑘)>0 such that 𝑠_𝑞^(𝑘)>𝑇^(𝑘) for all 𝑞≥𝐾^(𝑘). Hence, for any 𝑡∈[𝑠_𝑞^(𝑘),𝑡_𝑞^(𝑘)] and 𝑞≥𝐾^(𝑘), we have 𝑑𝑦(𝑡,𝜑_𝑘)𝑑𝑡=𝑦(𝑡,𝜑_𝑘)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥(𝑡,𝜑_𝑘))]≥−𝛾𝑦(𝑡,𝜑_𝑘), (4.35) where 𝛾=𝑑+𝑑₁+𝑚₁𝑈(𝑀₀+𝜀). Therefore for any 𝑞≥𝐾^(𝑘) and 𝑘=1,2,… , integrating (4.35) on [𝑠_𝑞^(𝑘),𝑡_𝑞^(𝑘)], we obtain from (4.32) 𝜂𝑘²=𝑦(𝑡_𝑞^(𝑘),𝜑_𝑘)≥𝑦(𝑠_𝑞^(𝑘),𝜑_𝑘)exp[−𝛾(𝑡_𝑞^(𝑘)−𝑠_𝑞^(𝑘)=𝜂)]𝑘exp[−𝛾(𝑡_𝑞^(𝑘)−𝑠_𝑞^(𝑘))]. (4.36) Consequently, 𝑡_𝑞^(𝑘)−𝑠_𝑞^(𝑘)≥ln𝑘𝛾∀𝑞≥𝐾^(𝑘),𝑘=1,2,…. (4.37) Since for any integer 𝑘, 𝑞≥𝐾^(𝑘), 𝑡∈[𝑠_𝑞^(𝑘),𝑡_𝑞^(𝑘)] and 𝑞≥𝐾^(𝑘) we have 𝑑𝑥(𝑡,𝜑_𝑘)𝑑𝑡=−𝑑𝑥(𝑡,𝜑_𝑘)−𝑚𝑦(𝑡,𝜑_𝑘)𝑈(𝑠(𝑡,𝜑_𝑘))+𝑏𝑑₁𝑧(𝑡,𝜑_𝑘)≥−𝑑𝑥(𝑡,𝜑_𝑘)−𝑚𝑦(𝑡,𝜑_𝑘)𝑈(𝑥(𝑡,𝜑_𝑘))≥−𝑑𝑥(𝑡,𝜑_𝑘)−𝑚𝜀₁𝑈(𝑀₀+𝜀). (4.38) By the comparison theorem it follows that 𝑥(𝑡,𝜑_𝑘)≥𝑥(𝑡), forall 𝑡∈[𝑠_𝑞^(𝑘),𝑡_𝑞^(𝑘)], where 𝑥(𝑡) is the solution of the following system with initial condition 𝑥(𝑠_𝑞^(𝑘))=𝑥(𝑠_𝑞^(𝑘),𝜑_𝑘): 𝑑𝜔(𝑡)𝑑𝑡=−𝑑𝜔(𝑡)−𝑚𝜀₁𝑈(𝑀₀+𝜀),𝑡≠𝑛𝑇,𝜔(𝑡⁺)=𝜔(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (4.39) By Lemma 2.2, system (4.39) has a unique positive 𝑇-periodic solution 𝜔^∗(𝑡)=𝑥^∗(𝑡)−𝑚𝑈(𝑀₀+𝜀)𝑑𝜀₁, (4.40) and 𝜔^∗(𝑡) is globally uniformly attractive. Hence, for above 𝜀₂, there is a constant 𝑇^∗>0, and 𝑇^∗ is independent of any 𝑛 and 𝑞≥𝐾^(𝑘), such that 𝑥(𝑡)≥𝑥^∗(𝑡)−𝑚𝑈(𝑀₀+𝜀)𝑑𝜀₁−𝜀₂∀𝑡≥𝑠_𝑞^(𝑘)+𝑇^∗. (4.41) Choose an integer 𝑁₀>0, such that when 𝑘≥𝑁₀ and 𝑞≥𝐾^(𝑘) we have 𝑡_𝑞^(𝑘)−𝑠_𝑞^(𝑘)≥𝑇^∗+𝑇. Hence, for any 𝑞≥𝐾^(𝑘), 𝑘≥𝑁₀ and 𝑡∈[𝑡_𝑞^(𝑘)−𝑇,𝑡_𝑞^(𝑘)], we have 𝑑𝑦(𝑡,𝜑_𝑘)𝑑𝑡=𝑦(𝑡,𝜑_𝑘)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥(𝑡,𝜑_𝑘))]≥𝑦(𝑡,𝜑_𝑘)[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗(𝑡)−𝑚𝑈(𝑀₀+𝜀)𝑑𝜀₁−𝜀₂)]. (4.42) Integrating (4.42) from 𝑡_𝑞^(𝑘)−𝑇 to 𝑡_𝑞^(𝑘), by (4.33) we obtain 𝜂𝑘²=𝑦(𝑡_𝑞^(𝑘),𝜑_𝑘)≥𝑦(𝑡_𝑞^(𝑘)−𝑇,𝜑_𝑘)exp^{𝑡𝑞(𝑘)}_{𝑡𝑞(𝑘)−𝑇}[−(𝑑+𝑑₁)+𝑚𝑝(𝑥^∗(𝑡)−𝑝(𝑀₀+𝜀)𝑑𝜀₁−𝜀₂≥𝜂)]𝑑𝑡𝑘²exp(𝜀₂)>𝜂𝑘², (4.43) which is contradictory. This contradiction shows that there exists a constant 𝑚₂>0 such that liminf_𝑡→∞𝑦(𝑡)≥𝑚₂ (4.44) for any positive solution (𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (2.3). Choose 𝜀₃>0 such that 𝑚₂−𝜀₃>0, then there exists a 𝑇₃>0 such that 𝑦(𝑡)≥𝑚₂−𝜀₃ for all 𝑡>𝑇₃. From the third equation of system (2.3), we have 𝑧(̇𝑡)≥−𝑎𝑧(𝑡)+(𝑚₂−𝜀₃),∀𝑡>𝑇₃. (4.45) By the comparison theorem it follows that 𝑧(𝑡)≥𝑧(𝑡), for all 𝑡>𝑇₃, where 𝑧(𝑡) is the solution of the following system with initial condition 𝑧(𝑇₃)=𝑧(𝑇₃)>0: 𝑢̇(𝑡)=−𝑎𝑢(𝑡)+(𝑚₂−𝜀₃). (4.46) By simple calculation we have 𝑚𝑧(𝑡)≥²−𝜀₃𝑎,∀𝑡>𝑇₃. (4.47) Hence, we further obtain liminf_𝑡→∞𝑚𝑧(𝑡)≥²−𝜀₃𝑎. (4.48) Let 𝑚=min{𝑝exp(−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁)𝑇)1−exp(−(𝑑+𝑚₁(𝑀₀+𝜀)𝑀₁)𝑇),𝑚₂,𝑚₂−𝜀₃𝑎}; (4.49) then we have liminf_𝑡→∞𝑥(𝑡)≥𝑚,liminf_𝑡→∞𝑦(𝑡)≥𝑚,liminf_𝑡→∞𝑧(𝑡)≥𝑚. (4.50) This completes the proof of Theorem 4.2.

Returning to the original system (1.4), as a consequence of Theorems 4.1 and 4.2 we have the following corollary.

Corollary 4.3. Suppose that inequality (3.1) holds. Then,(a)solution (𝑥^∗(𝑡),0) of system (1.4) is globally attractive if and only if ^𝑇₀[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗(𝑡))]𝑑𝑡≤0; (4.51)(b)system (1.4) is permanent if and only if ^𝑇₀[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑠^∗(𝑡))]𝑑𝑡>0. (4.52)

Remark 4.4. From [18, Theorem 3.2], we find that the results on permanence is similar between model with delayed nutrient recycling and the model with instantaneous nutrient recycling.

Remark 4.5. According to Corollary 4.3, the sufficient and necessary conditions on the permanence and extinction of the microorganisms are obtained for a chemostat models with delayed nutrient recycling and periodically pulsed input. Obviously, it is an very good extension of the corresponding results given in [18].

5. Global Attractivity

Now, we discuss the global attractivity of all positive solutions of system (2.3), we have the following result.

Theorem 5.1. Suppose that inequalities (3.1) and (4.15) hold. If 𝛼𝑑₁−𝛽𝑑<0, (5.1) where 𝛼=sup_{𝜃∈[0,𝑀0]}𝑈^′(𝜃) and 𝛽=inf_{𝜃∈[0,𝑀0]}𝑈^′(𝜃), then for any two positive solutions (𝑥₁(𝑡),𝑦₁(𝑡),𝑧₁(𝑡)) and (𝑥₂(𝑡),𝑦₂(𝑡),𝑧₂(𝑡)) of system (2.3), lim_𝑡→∞(𝑥₁(𝑡)−𝑥₂(𝑡))=0,lim_𝑡→∞(𝑦₁(𝑡)−𝑦₂(𝑡))=0,lim_𝑡→∞(𝑧₁(𝑡)−𝑧₂(𝑡))=0. (5.2)

Proof. From inequality (3.1) and (5.1), there exist two constants 𝑐>0 and 𝑐₁>0 such that 𝑏𝑑₁+𝑐𝑑−𝑎𝑐<0,𝑎𝑐−𝑚𝑑₁𝑚₁<0,𝑑−𝑐₁𝑚₁𝛼>0,𝑐₁𝑚𝛽−𝑚𝑑₁𝑚₁>0. (5.3) Further, we can choose a constant 𝜀₀>0 such that 𝑑−𝑐₁𝑚₁𝛼_𝜀0𝑚>0,−𝜀₀𝑐₁(𝑐₁𝑚𝛽_𝜀0−𝑚𝑑₁𝑚₁)>0, (5.4) where 𝛼_𝜀0=sup_{𝜃∈[0,𝑀0+𝜀0]}𝑈^′(𝜃) and 𝛽_𝜀0=inf_{𝜃∈[0,𝑀0+𝜀0]}𝑈^′(𝜃).
Let 𝑚𝑉(𝑡)=𝑥(𝑡)+𝑚₁𝑦(𝑡)+𝑐𝑧(𝑡), (5.5) then system (2.3) is equivalent to the following system: 𝑑𝑉(𝑡)𝑑𝑡=−𝑑𝑉(𝑡)+(𝑎𝑐−𝑚𝑑₁𝑚₁)𝑦(𝑡)+(𝑏𝑑₁+𝑑𝑐−𝑎𝑐)𝑧(𝑡),𝑑𝑦(𝑡)𝑑𝑡=𝑦(𝑡)[−(𝑑+𝑑₁)+𝑚₁𝑚𝑈(𝑉(𝑡)−𝑚₁𝑦(𝑡)−𝑐𝑧(𝑡))],𝑡≠𝑛𝑇,𝑑𝑧(𝑡)𝑑𝑡=𝑎𝑦(𝑡)−𝑎𝑧(𝑡),𝑉(𝑡⁺)=𝑉(𝑡)+𝑝,𝑦(𝑡⁺)=𝑦(𝑡),𝑧(𝑡⁺)=𝑧(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁. (5.6) Let (𝑥₁(𝑡),𝑦₁(𝑡),𝑧₁(𝑡)) and (𝑥₂(𝑡),𝑦₂(𝑡),𝑧₂(𝑡)) be any two positive solutions of system (2.3); from Theorems 3.1 and 4.1 we have 𝑚≤liminf_𝑡→∞𝑥_𝑖(𝑡)≤limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀₀,𝑚≤liminf_𝑡→∞𝑦_𝑖(𝑡)≤limsup_𝑡→∞𝑦_𝑖𝑚(𝑡)≤¹𝑚𝑀₀,𝑚≤liminf_𝑡→∞𝑧_𝑖(𝑡)≤limsup_𝑡→∞𝑧_𝑖(𝑡)≤(𝑎−𝑑)𝑀₀𝑏𝑑₁ (5.7) for 𝑖=1,2. Hence, there exists a 𝑇>0 such that 𝑚−𝜀₀≤𝑥_𝑖(𝑡)≤𝑀₀+𝜀₀,𝑚−𝜀₀≤𝑦_𝑖𝑚(𝑡)≤¹𝑚𝑀₀+𝜀₀,𝑚−𝜀₀≤𝑧_𝑖(𝑡)≤(𝑎−𝑑)𝑀₀𝑏𝑑₁+𝜀₀ (5.8) for all 𝑡≥𝑇. From above inequalities and the theorem of mean value, we can obtain |𝑦₁(𝑡)−𝑦₂(𝑡)|≥(𝑚−𝜀₀)|ln𝑦₁(𝑡)−ln𝑦₂(𝑡)|,𝑈(𝑥₁(𝑡))−𝑈(𝑥₂(𝑡))=𝑑𝑈(𝜉(𝑡))𝑑𝑠(𝑥₁(𝑡)−𝑥₂(𝑡)), (5.9) for all 𝑡≥𝑇, where 𝜉(𝑡) is situated between 𝑥₁(𝑡) and 𝑥₂(𝑡).
Let𝑉_𝑖(𝑡)=𝑥_𝑖𝑚(𝑡)+𝑚₁𝑦_𝑖(𝑡)+𝑐𝑧_𝑖(𝑡),𝑖=1,2, (5.10) then (𝑉_𝑖(𝑡),𝑦_𝑖(𝑡),𝑧_𝑖(𝑡)) is the solution of system (5.6). Define the Liapunov function as follows: 𝑊(𝑡)=|𝑉₁(𝑡)−𝑉₂(𝑡)|+𝑐₁|ln𝑦₁(𝑡)−ln𝑦₂(𝑡)|+𝑐|𝑧₁(𝑡)−𝑧₂(𝑡)|. (5.11) Calculating the derivative of 𝑊(𝑡), from (5.9) we have 𝑑𝑊(𝑡)𝑑𝑡=sign(𝑉₁(𝑡)−𝑉₂(𝑡))×[−𝑑(𝑉₁(𝑡)−𝑉₂(𝑡))+(𝑎𝑐−𝑚𝑑₁𝑚₁)(𝑦₁(𝑡)−𝑦₂(𝑡))+(𝑏𝑑₁+𝑑𝑐−𝑎𝑐)(𝑧₁(𝑡)−𝑧₂(𝑡))]+𝑐₁𝑚₁sign(𝑦₁(𝑡)−𝑦₂(𝑡))×[𝑈(𝑉₁𝑚(𝑡)−𝑚₁𝑦₁(𝑡)−𝑐𝑧₁(𝑡))−𝑈(𝑉₂𝑚(𝑡)−𝑚₁𝑦₂(𝑡)−𝑐𝑧₂(𝑡))]+sign(𝑧₁(𝑡)−𝑧₂(𝑡))[𝑎𝑐(𝑦₁(𝑡)−𝑦₂(𝑡))−𝑎𝑐(𝑧₁(𝑡)−𝑧₂(𝑡))]≤−𝑑|𝑉₁(𝑡)−𝑉₂(𝑡)|+(𝑚𝑑₁𝑚₁−𝑎𝑐)|𝑦₁(𝑡)−𝑦₂(𝑡)|+(𝑎𝑐−𝑏𝑑₁−𝑑𝑐)|𝑧₁(𝑡)−𝑧₂(𝑡)|+𝑐₁𝑚₁sign(𝑦₁(𝑡)−𝑦₂(𝑡))𝑑𝑈(𝜉(𝑡))𝑑𝑠×[(𝑉₁(𝑡)−𝑉₂𝑚(𝑡))−𝑚₁(𝑦₁(𝑡)−𝑦₂(𝑡))−𝑐(𝑧₁(𝑡)−𝑧₂(𝑡))]+𝑎𝑐|𝑦₁(𝑡)−𝑦₂(𝑡)|−𝑎𝑐|𝑧₁(𝑡)−𝑧₂(𝑡)|≤−(𝑑−𝑐₁𝑚₁𝛼_𝜀0)|𝑉₁(𝑡)−𝑉₂−𝑚(𝑡)|−𝜀₀𝑐₁(𝑐₁𝑚𝛽_𝜀0−𝑚𝑑₁𝑚₁)𝑐₁|ln𝑦₁(𝑡)−ln𝑦₂(𝑡)|−(𝑑−𝑐₁𝑚₁𝛼_𝜀0+𝑏𝑑₁𝑐)𝑐|𝑧₁(𝑡)−𝑧₂(𝑡)|≤−𝜌𝑊(𝑡), (5.12) where 𝜌=min{𝑑−𝑐₁𝑚₁𝛼_𝜀0,𝑚−𝜀₀𝑐₁(𝑐₁𝑚𝛽_𝜀0−𝑚𝑑₁𝑚₁)}. (5.13) From (5.4), we obtain 𝜌>0. On the other hand, we directly obtain 𝑊(𝑡⁺)=𝑊(𝑡),∀𝑡=𝑛𝑇,𝑛∈𝑁. (5.14) Hence, for any 𝑡>𝑇 we have 𝑊(𝑡)≤𝑊(𝑇)exp(−𝜌𝑡). (5.15) Consequently, lim_𝑡→∞𝑊(𝑡)=0. From this, we finally obtain lim_𝑡→∞(𝑥₁(𝑡)−𝑥₂(𝑡))=0,lim_𝑡→∞(𝑦₁(𝑡)−𝑦₂(𝑡))=0,lim_𝑡→∞(𝑧₁(𝑡)−𝑧₂(𝑡))=0. (5.16) This completes the proof of Theorem 5.1.

As a direct consequence of Theorem 5.1, we have the following corollary on global attractivity of all positive solutions for the original system (1.4).

Corollary 5.2. Suppose that all the conditions of Theorem 5.1 hold. Then for any two positive solutions (𝑥₁(𝑡),𝑦₁(𝑡)) and (𝑥₂(𝑡),𝑦₂(𝑡)) of system (1.4), lim_𝑡→∞(𝑥₁(𝑡)−𝑥₂(𝑡))=0,lim_𝑡→∞(𝑦₁(𝑡)−𝑦₂(𝑡))=0. (5.17)

6. Numerical Examples

In this section, we will give an example to demonstrate the effectiveness of our main results.

Consider system (1.4) with the following parameters: 𝑈(𝑥)=12𝑥exp(−0.2𝑥)0.5+𝑥,𝐹(𝑡)=2𝑒^−2𝑡,𝑑=0.5,𝑑₁=0.2,𝑚=0.9,𝑚₁=0.8,𝑏=0.4,𝑝=0.5,𝑇=1,𝑎=2. (6.1) By calculating, we obtain 𝑏=4𝑎−𝑑,𝑚15𝑎𝑚₁=98₁₀,[−(𝑑+𝑑₁)+𝑚₁𝑈(𝑥^∗𝑥(𝑡))]𝑑𝑡=0.1351>0,_∗(𝑡)=0.5𝑒^{−0.5(𝑡−𝑛)}1−𝑒^−0.5,𝑡∈(𝑛,𝑛+1],𝑛∈𝑁,𝛼𝑑₁−𝛽𝑑=2.4947>0. (6.2) Therefore, inequalities (4.15) hold, but (5.1) does not hold. From Theorem 4.1, we obtain that system (1.4) is permanent. But, from the numerical simulation (see Figures 1 and 2) we see that there exists a unique positive 𝑇-periodic solution (𝑥^∗(𝑡),𝑦^∗(𝑡)) of system (1.4) such that any solution (𝑥(𝑡),𝑦(𝑡)) of system (1.4) with initial value (𝑥₀,𝜑(𝜃)) tends to (𝑥^∗(𝑡),𝑦^∗(𝑡)) as 𝑡→∞. Therefore, we can guess that if only inequality (4.15) holds then system (1.4) has a unique positive 𝑇-periodic solution which is globally attractive.

Figure 1 

Time series of 𝑥(𝑡).

Figure 2 

Time series of 𝑦(𝑡).

Open Problem 6. From the above numerical simulation, it prompts us to be able to put forward the following important and interesting open problem, that is, when inequality (3.1) holds then system (1.4) is also globally attractive as long as it is permanent.

Acknowledgment

This paper was supported by The National Natural Science Foundation of China [10961022,10901130,11071283], The Scientific Research Programmes of Colleges in Xinjiang [XJEDU2007G01, XJEDU2006I05, XJEDU2008S10, XJEDU2009S21], and the Sciences Foundation of Shanxi [2009011005-3].