Dynamic Behaviors in a Droop Model for Phytoplankton Growth in a Chemostat with Nutrient Periodically Pulsed Input

Abstract

The dynamic behaviors in a droop model for phytoplankton growth in a chemostat with nutrient periodically pulsed input are studied. A series of new criteria on the boundedness, permanence, extinction, existence of positive periodic solution, and global attractivity for the model are established. Finally, an example is given to demonstrate the effectiveness of the results in this paper.

1. Introduction

The chemostat is a very important apparatus used to study the growth of microorganisms in a continuous cultured environment in a laboratory (see [1]). It may be viewed as a laboratory model of a simple lake with continuous stirring. Chemostat models have attracted widely the attention of the scientific community, since they have a wide range of applications, for example, waste-water treatment, production by genetically altered organisms (like production of insulin), and so forth. The growth in a chemostat is described by the systems of ordinary differential equations or functional differential equations. Many important and interesting results can be found in articles [2–10] and the references cited therein.

The droop model [11, 12] of phytoplankton growth is essential in theoretical phytoplankton ecology. The droop model takes into consideration that phytoplankton cells store nutrient and that the growth rate depends on the stored nutrient. Algae can uptake nutrient in excess of current needs and continue to grow during nutrient poor conditions. These nutrients are supplied from an external reservoir; in several earlier works, this concentration is assumed to be constant. The droop model of phytoplankton growth in a chemostat has been widely investigated in many literatures (see [13–19]). As in [16], the classical chemostat equations modeling phytoplankton population dynamics originally related the growth rate of the cells to the nutrient concentration in the medium. It is assumed that the nutrient uptake rate is proportional to the rate of reproduction. The constant of proportionality that converts units of nutrient to units of organisms is called the yield constant. Because of the assumed constant value of the yield, the classical Monod model is referred to as the constant-yield model in [15]. In [17], both uniform persistence and the existence of periodic coexistence state are established for a periodically forced droop model on two phytoplankton species competition in a chemostat under some appropriate conditions. In [18], the authors considered the nonautonomous droop model for phytoplankton growth in a chemostat in which the nutrient input varies nonperiodically. It is assumed that growth rate varies with the internal nutrient level of the cell and the uptake rate of phytoplankton depends on both the external and the internal nutrient concentrations. A series of new criteria on the positivity, boundedness, permanence, and extinction of the population is established.

This work was motivated by [19], where the droop model was presented with general functions for growth and uptake, which takes the following forms: 𝑑𝑆(𝑡)𝑆𝑑𝑡=𝐷⁰(𝑡)(𝑡)𝜌(),(𝑡)−𝑆−𝑁𝑆,𝑄𝑑𝑄()(𝑄)(𝑡)𝑑𝑡=𝜌𝑆,𝑄−𝜇𝑄,𝑑𝑁𝜇(𝑄).𝑑𝑡=𝑁−𝐷 (1.1) The author proved that the periodically forced droop model has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the subthreshold condition hold, the phytoplankton population is washed out of the chemostat. If the superthreshold condition holds, then there is a unique periodic solution to which all solutions approach asymptotically.

As 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 [20–30] 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.

However, we find that, up to now, few papers consider the droop model with nutrient periodically pulsed input. From the above discussion, this subject has broad biological significance. In pulse case, we will give some good extensions of the corresponding results given by Smith in [19]. We introduce the following droop model for phytoplankton growth in a chemostat with nutrient periodically pulsed input𝑑𝑆(𝑡)(𝑡)𝜌(),(𝑡)𝑑𝑡=−𝐷𝑆−𝑁𝑆,𝑄𝑑𝑄()(𝑄)(𝑡)𝑑𝑡=𝜌𝑆,𝑄−𝜇𝑄,𝑡≠𝑛𝑇,𝑑𝑁𝜇(𝑄),𝑆𝑡𝑑𝑡=𝑁−𝐷₊(𝑡)𝑄𝑡=𝑆+𝑝,₊(𝑡)𝑁𝑡=𝑄,𝑡=𝑛𝑇,𝑛∈𝑁,₊=𝑁(𝑡). (1.2) For model (1.2), we will investigate the permanence and extinction of the species, and the existence of positive 𝑇-periodic solution and global attractivity of the model. We also will give an example and numerical simulations to demonstrate the effectiveness of the results in this paper.

This paper is organized as follows. In the following section we will firstly introduce some basic assumption for model (1.2). Furthermore, we also will give several useful lemmas. In Section 3, we will state and prove an ultimate boundedness theorem of solutions for model (1.2). In Section 4, we will state and prove the sufficient condition on the permanence of species for model (1.2). In Section 5, the sufficient condition on the extinction of species for model (1.2) is given. In Section 6, the existence of positive 𝑇-periodic solution and the global attractivity of solutions for model (1.2) are established. Finally, in Section 7, we will discuss an example and give some numerical simulations.

2. Preliminaries

In model (1.2), 𝑁(𝑡) denotes the phytoplankton cells at biomass concentration, 𝑆(𝑡) denotes the nutrient at concentration in a growth medium. Each phytoplankton cell is assumed to possess an internal pool of stored nutrient, also called quota 𝑄(𝑡). 𝐷>0 is the input and output flow, and its referred to as the dilution rate; 𝑝>0 is the amount of the substrate concentration pulsed each 𝑛𝑇, where 𝑇>0 is a constant. Function 𝜇(𝑄) denotes the phytoplankton growth rate, function 𝜌(𝑆,𝑄) denotes the nutrient uptake rate. In this paper, for model (1.2), we always assume that the following condition holds.(H₁)Function 𝜇(𝑄) is continuously differentiable and 𝑑𝜇(𝑄)/𝑑𝑄>0 for all 𝑄≥𝑄_min, 𝜇(𝑄_min)=0, where 𝑄_min is some given positive constant.(H₂)Function 𝜌(𝑆,𝑄) is continuously differentiable and 𝜕𝜌(𝑆,𝑄)/𝜕𝑆>0, 𝜕𝜌(𝑆,𝑄)/𝜕𝑄≤0 for all 𝑆>0 and 𝑄>𝑄_min. Further, 𝜌(0,𝑄)≡0 for all 𝑄≥𝑄_min.

The initial condition in model (1.2) is given in the following form:𝑆0⁺=𝑆₀>0,𝑄0⁺=𝑄₀≥𝑄_min,𝑁0⁺=𝑁₀>0. (2.1)

Firstly, on the positivity of solutions for model (1.2), we have the following result.

Lemma 2.1. Any solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of 𝑚𝑜𝑑𝑒𝑙 (1.2) with initial condition (2.1) is positive, that is, 𝑆(𝑡)>0, 𝑄(𝑡)≥𝑄_min and 𝑁(𝑡)>0 for all 𝑡>0.

Proof. Obviously, for any 𝑡>0, we have 𝑁(𝑡)=𝑁0⁺exp^𝑡₀𝜇(𝑄(𝑠))−𝐷𝑑𝑠>0. (2.2)
We consider 𝑆(𝑡). If the conclusion is not true, then from 𝑆(0⁺)>0 and 𝑝>0 we can obtain that there is a 𝑡₁>0 such that 𝑆(𝑡₁)=0 and 𝑆(𝑡)>0 for all 𝑡∈(0,𝑡₁). From the mean-value theorem and assumption (𝐻₂), for any 𝑡∈(0,𝑡₁), there is a 𝜉(𝑡)∈(0,𝑆(𝑡)) such that 𝜌(𝑆(𝑡)(𝑡=(𝜉(𝑡)(𝑡,𝑄))𝜕𝜌,𝑄))𝑆(𝑡).𝜕𝑆 (2.3) Hence, from the first equation of model (1.2), we obtain 𝑑𝑆(𝑡)(𝑡)(𝜉(𝑡)(𝑡𝑑𝑡=−𝐷+𝑁𝜕𝜌,𝑄))𝑆(𝑡)𝜕𝑆∀𝑡∈0,𝑡₁,𝑡≠𝑛𝑇. (2.4) Choose an integer 𝑛₀≥0 such that 𝑡₁∈(𝑛₀𝑇,(𝑛₀+1)𝑇]. Obviously, 𝑆(𝑛₀𝑇⁺)>0 and integrating (2.4) from 𝑛₀𝑇 to 𝑡₁, 𝑆𝑡₁=𝑆𝑛₀𝑇⁺exp−^𝑡1_𝑛0𝑇(𝑢)(𝜉(𝑢)(𝑢𝐷+𝑁𝜕𝜌,𝑄))𝜕𝑆𝑑𝑢>0, (2.5) which leads to a contradiction.
Now, we consider 𝑄(𝑡). We have 𝑄(0⁺)≥𝑄_min and 𝜇(𝑄_min)=0. If 𝑄(0⁺)=𝑄_min, then since right derivative 𝑑𝑄(𝑡)𝑑𝑡_𝑡=0⁺=𝜌𝑆0⁺,𝑄_min>0, (2.6) there is a 𝑡₁>0 such that 𝑄(𝑡)>𝑄_min for all 𝑡∈(0,𝑡₁]. If there is a 𝑡₂>𝑡₁ such that 𝑄(𝑡₂)=𝑄_min and 𝑄(𝑡)>𝑄_min for all 𝑡∈(0,𝑡₂), then left derivative 𝑑𝑄(𝑡)/𝑑𝑡|_𝑡=𝑡⁻₂. But from assumption (𝐻₂), we have 𝑑𝑄(𝑡)𝑑𝑡_𝑡=𝑡⁻₂=𝜌𝑆𝑡₂,𝑄_min>0, (2.7) which leads to a contradiction. If 𝑄(0⁺)>𝑄_min, then similarly to the above, we also can obtain a contradiction. This completes the proof of Lemma 2.1.

Now, we consider the following linear impulsive differential equation𝑑𝜔(𝑡)(𝑡)𝜔𝑡𝑑𝑡=−𝐷𝜔−𝜂,𝑡≠𝑛𝑇,₊=𝜔(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁, (2.8) where 𝐷 and 𝑝 are defined in model (1.2) and 𝜂 is a constant. Clearly,𝜔^∗(𝑡)𝜂=−𝐷+𝑆^∗(𝑡),𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁 (2.9)

is the 𝑇-periodic solution of (2.8), where𝑆^∗(𝑡)=𝑝𝑒^{−𝐷(𝑡−𝑛𝑇)}1−𝑒^−𝐷𝑇,𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁. (2.10)

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 initial value 𝜔₀ with |𝜔₀|≤𝑀, one has𝜔𝑡,𝑡₀,𝜔₀−𝜔^∗(𝑡)<𝜀∀𝑡≥𝑡₀+𝑇(𝑀,𝜀), (2.11) where 𝜔(𝑡,𝑡₀,𝜔₀) is the solution of (2.8) with initial condition 𝜔(𝑡₀)=𝜔₀. We have following result.

Lemma 2.2 (see [27]). 𝑇-periodic solution 𝜔^∗(𝑡) of (2.8) is globally uniformly attractive.

For (2.8), when 𝜂=0, we obtain the subsystem of model (1.2) with 𝑁(𝑡)=0 as follows: 𝑑𝑆(𝑡)(𝑡)𝑆𝑡𝑑𝑡=−𝐷𝑆,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (2.12) Clearly, 𝑆^∗(𝑡) is the positive 𝑇-periodic solution of (2.12). From Lemma 2.2, we obtain that for any solution 𝑆(𝑡) of (2.12), one has𝑆(𝑡)−𝑆^∗(𝑡)⟶0as𝑡⟶∞. (2.13)

Putting 𝑆(𝑡)=𝑆^∗(𝑡) in the second equation of model (1.2), we obtain𝑑𝑄(𝑡)𝑆𝑑𝑡=𝜌^∗(𝑡),𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡). (2.14) From the expression of 𝑆^∗(𝑡), we can choose a constant 0<𝛼₀<𝑝𝑒^−𝐷𝑇/(1−𝑒^−𝐷𝑇) such thatinf_𝑡≥0𝑆^∗(𝑡)−𝛼₀≥𝑝𝑒^−𝐷𝑇1−𝑒^−𝐷𝑇−𝛼₀>0. (2.15)

For any 𝛼∈[−𝛼₀,𝛼₀], we consider the following equation:𝑑𝑄(𝑡)𝑆𝑑𝑡=𝜌^∗(𝑡)+𝛼,𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡). (2.16) For any initial points 𝑡₀≥0 and 𝑄₀≥𝑄_min, let 𝑄₀(𝑡) and 𝑄_𝛼(𝑡) be the solutions of (2.14) and (2.16) satisfying initial conditions 𝑄₀(𝑡₀)=𝑄₀ and 𝑄_𝛼(𝑡₀)=𝑄₀, respectively. We have the following result.

Lemma 2.3. (a)  There are constants 𝑀₁>1 and such that for any 𝑡₀≥0, 𝑄₀≥𝑄_min and 𝛼∈[−𝛼₀,𝛼₀]𝑄_min≤liminf_𝑡→∞𝑄_𝛼(𝑡)≤liminf_𝑡→∞𝑄_𝛼(𝑡)≤𝑀₁. (2.17)
(b) For each 𝛼∈[−𝛼₀,𝛼₀], (2.16) has a positive and globally uniformly attractive 𝑇-periodic solution 𝑄^∗_𝛼(𝑡).
(c) 𝑄^∗_𝛼(𝑡) converges to 𝑄^∗₀(𝑡) uniformly for 𝑡∈[0,𝑇] as 𝛼→0.

Proof. Firstly, inequality 𝑄(𝑡)≥𝑄_min for all 𝑡∈[0,∞) can be proved by using the similar argument as in the proof of Lemma 2.1.
From assumptions (𝐻₁) and (𝐻₂), there are positive constants 𝐾 and 𝛿 such that for any 𝛼∈[−𝛼₀,𝛼₀]^𝑇₀𝜌𝑆^∗(𝑢)+𝛼,𝐾𝐾−𝜇(𝐾)𝑑𝑢<−𝛿. (2.18) If 𝑄_𝛼(𝑡)≥𝐾 for all 𝑡≥𝑡₀, then integrating (2.16), for any 𝑡≥𝑡₀, we have 𝑄_𝛼(𝑡)=𝑄_𝛼𝑡₀exp^𝑡_𝑡0𝜌𝑆^∗(𝑢)+𝛼,𝑄_𝛼(𝑢)𝑄_𝛼(𝑢)−𝜇𝑄_𝛼(𝑢)𝑑𝑢≤𝑄_𝛼𝑡₀exp^𝑡_𝑡0𝜌𝑆^∗(𝑢)+𝛼,𝐾𝐾−𝜇(𝐾)𝑑𝑢≤𝑄_𝛼𝑡₀exp−𝛿𝑡−𝑡₀. (2.19) Hence, 𝑄_𝛼(𝑡)→0 as 𝑡→∞, which leads to a contradiction. Therefore, there is a 𝑡₁>𝑡₀ such that 𝑄_𝛼(𝑡₁)<𝐾. Further, if there are 𝑡₂>𝑡₃>𝑡₁ such that 𝑄_𝛼(𝑡₂)>𝐾exp(𝑀^∗𝑇), 𝑄_𝛼(𝑡₃)=𝐾, and 𝑄_𝛼(𝑡)>𝐾 for all 𝑡∈(𝑡₃,𝑡₂], where 𝑀^∗=sup_𝑡∈𝑅+𝜌𝑆^∗(𝑡)+𝛼₀,𝐾𝐾+𝜇(𝐾), (2.20) then we can choose an integer 𝑞≥0 such that 𝑡₂∈(𝑡₃+𝑞𝑇,𝑡₃+(𝑞+1)𝑇] and integrating (2.16) to obtain 𝐾exp𝑀^∗𝑇<𝑄_𝛼𝑡₂=𝑄_𝛼𝑡₃exp^𝑡2_𝑡3𝜌𝑆^∗(𝑢)+𝛼,𝑄_𝛼(𝑢)𝑄_𝛼(𝑢)−𝜇𝑄_𝛼(𝑢)𝑑𝑢≤𝐾exp^𝑡2_𝑡^+𝑞𝑇₃+^𝑡2_{𝑡3+𝑞𝑇}𝜌𝑆^∗(𝑢)+𝛼,𝐾𝐾−𝜇(𝐾)𝑑𝑢≤𝐾exp^𝑡2_{𝑡3+𝑞𝑇}𝜌𝑆^∗(𝑢)+𝛼,𝐾𝐾(𝐾𝑀−𝜇)𝑑𝑢≤𝐾exp^∗𝑇, (2.21) which also leads to a contradiction. Therefore, we finally have 𝑄_𝛼(𝑡)≤𝐾exp𝑀^∗𝑇∀𝑡≥𝑡₁. (2.22) Choose constant 𝑀₁=𝐾exp(𝑀^∗𝑇), then we have limsup_𝑡→∞𝑄_𝛼(𝑡)≤𝑀₁ for all 𝛼∈[−𝛼₀,𝛼₀]. Thus, conclusion (𝑎) is proved.
From conclusion (a), directly using the main results given by Teng and Chen in [31], we can obtain that (2.16) for each 𝛼∈[−𝛼₀,𝛼₀] has a positive 𝑇-periodic solution 𝑄^∗_𝛼(𝑡).
For any constant 𝜂₀>1 and 𝑡₀≥0, let 𝑄_𝛼(𝑡) be a solution of (2.16) with initial value 𝑄_𝛼(𝑡₀)∈[𝜂₀⁻¹,𝜂₀]. By conclusion (𝑎), there is a constant 𝑀₁>1 such that 𝑀₁⁻¹≤𝑄^∗_𝛼(𝑡)≤𝑀₁,𝑀₁⁻¹≤𝑄_𝛼(𝑡)≤𝑀₁∀𝑡≥𝑡₀. (2.23) Consider Liapunov function 𝑉(𝑡)=ln𝑄_𝛼(𝑡)−ln𝑄^∗_𝛼(𝑡). (2.24) Calculating the Dini derivative 𝐷⁺𝑉(𝑡), from sign𝑄_𝛼(𝑡)−𝑄^∗_𝛼(𝑡)𝜌𝑆^∗(𝑡)+𝛼,𝑄_𝛼(𝑡)𝑄_𝛼(𝑡)−𝜌𝑆^∗(𝑡)+𝛼,𝑄^∗_𝛼(𝑡)𝑄^∗_𝛼(𝑡)≤0, (2.25) we can obtain for any 𝑡≥𝑡₀ and 𝑡≠𝑛𝑇𝐷⁺𝑉(𝑡)≤sign𝑄_𝛼(𝑡)−𝑄^∗_𝛼(𝑡)𝜇𝑄^∗_𝛼(𝑡)−𝜇𝑄_𝛼(𝑡). (2.26) Using the mean-value theorem, we further obtain 𝐷⁺𝑉(𝑡)(𝜉(𝑡≤−𝑑𝜇))𝑄𝑑𝑄^∗𝛼(𝑡)−𝑄_𝛼(𝑡), (2.27) where 𝜉(𝑡) is situated between 𝑄^∗_𝛼(𝑡) and 𝑄_𝛼(𝑡). When 𝑡=𝑛𝑇 and 𝑛∈𝑁, we obviously have 𝑉(𝑡⁺)=𝑉(𝑡). Hence 𝑉(𝑡)≤𝑉𝑡₀∀𝑡≥𝑡₀. (2.28) Consequently, by (2.23) we have ln𝑄_𝛼(𝑡)≤ln𝑄^∗_𝛼(𝑡)+𝑉𝑡₀≤ln𝜂₀𝑀²₁, (2.29) for all 𝑡≥𝑡₀. Hence 𝜂₀⁻¹𝑀₁⁻²≤𝑄_𝛼(𝑡)≤𝜂₀𝑀²₁forall𝑡≥𝑡₀. (2.30) Further, from (2.23) we obtain 𝜂₀⁻¹𝑀₁⁻²𝑉(𝑡)≤𝑄^∗_𝛼(𝑡)−𝑄_𝛼(𝑡)≤𝜂₀𝑀²₁𝑉(𝑡), (2.31) for all 𝑡≥𝑡₀. Consequently, from (2.27) it follows that 𝐷⁺𝑉(𝑡)≤−𝜂₀⁻¹𝑀₁⁻²𝑑𝜇(𝜉(𝑡))𝑉(𝑡)𝑑𝑄≤−𝑀₀𝑉(𝑡), (2.32) where 𝑀₀=𝜂₀⁻¹𝑀₁⁻²(𝜉(𝑡min𝑑𝜇))𝑑𝑄∶𝜂₀⁻¹𝑀₁⁻²≤𝑄≤𝜂₀𝑀²₁. (2.33) Hence, we further have 𝑉(𝑡)≤𝑉𝑡₀𝑒^{−𝑀0(𝑡−𝑡0)}≤ln𝜂₀𝑀₁𝑒^{−𝑀0(𝑡−𝑡0)},∀𝑡≥𝑡₀. (2.34) For any constant 𝜀>0, from (2.34), choose 𝑇𝜂₀1,𝜀=−𝑀₀𝜀ln𝜂₀𝑀²₁ln𝜂₀𝑀₁, (2.35) then for any 𝑡≥𝑡₀+𝑇(𝜂₀,𝜀) we can obtain 𝑉(𝑡)<𝜀𝜂₀𝑀²₁∀𝑡≥𝑡₀+𝑇𝜂₀,𝜀. (2.36) Therefore, 𝑄^∗_𝛼(𝑡)−𝑄_𝛼(𝑡)<𝜀∀𝑡≥𝑡₀+𝑇𝜂₀,𝜀. (2.37) This shows solution 𝑄^∗_𝛼(𝑡) is globally uniformly attractive. Thus, conclusion (𝑏) is proved.
Finally, we prove conclusion (𝑐). From assumptions (𝐻₁) and (𝐻₂), we can easily prove that the right hand of model (1.2) satisfies the uniform Lipschitz condition with respect to 𝛼∈[−𝛼₀,𝛼₀]. Hence, by the continuity of solutions with respect to parameter 𝛼, we can obtain that 𝑄^∗_𝛼(𝑡)→𝑄^∗₀(𝑡) uniformly for 𝑡∈[0,𝑇] as 𝛼→0. Thus, conclusion (𝑐) is proved. This completes the proof of Lemma 2.3.

When 𝛼=0, (2.16) degenerates into (2.14), from Lemma 2.3 we have the following result.

Corollary 2.4. Equation (2.14) has a unique positive 𝑇-periodic solution 𝑄^∗₀(𝑡) which is globally uniformly attractive.

The following lemma will be used in the proof of the result on the global attractivity of model (1.2).

Lemma 2.5. Let function 𝑓(𝑡) be continuous on [0,+∞) and differentiable for any 𝑡≠𝑛𝑇 ( 𝑛∈𝑁), where 𝑇>0 is a constant. If there exist a constant 𝑀>0, such that |𝑑𝑓(𝑡)/𝑑𝑡|<𝑀 for any 𝑡≠𝑛𝑇(𝑛∈𝑁), then 𝑓(𝑡) is uniformly continuous on [0,+∞).

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

3. Boundedness

On the ultimate boundedness of all positive solutions of model (1.2), we have the following result.

Theorem 3.1. Let (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) be any positive solution of 𝑚𝑜𝑑𝑒𝑙 (1.2), then we have that there exists a constant 𝑀₃>0, which is independent of any positive solution of 𝑚𝑜𝑑𝑒𝑙 (1.2), such that limsup_𝑡→∞𝑆(𝑡)<𝑀₃,limsup_𝑡→∞𝑄(𝑡)<𝑀₃,limsup_𝑡→∞𝑁(𝑡)<𝑀₃. (3.1)

Proof. Let (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) be any positive solution of model (1.2) with initial condition (2.1). Since 𝑑𝑆(𝑡)(𝑡)𝑆𝑡𝑑𝑡≤−𝐷𝑆,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁, (3.2) by Lemma 2.2 we directly have limsup_𝑡→∞𝑆(𝑡)≤limsup_𝑡→∞𝑆^∗(𝑡)≤𝑝1−𝑒^−𝐷𝑇. (3.3)
For any constant 𝜀>0, there is a 𝑇₀>0 such that 𝑆(𝑡)≤𝑝1−𝑒^−𝐷𝑇+𝜀∀𝑡≥𝑇₀. (3.4) From the second equation of model (1.2), we have 𝑑𝑄(𝑡)𝑝𝑑𝑡≤𝜌1−𝑒^−𝐷𝑇+𝜀,𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡) (3.5) for all 𝑡≥𝑇₀. Using the similar argument as in the proof of Lemma 2.3, we have that there is a positive constant 𝑀_𝑄>0, and 𝑀_𝑄 is independent of any solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of model (1.2), such that limsup_𝑡→∞𝑄(𝑡)≤𝑀_𝑄. (3.6)
Define a function 𝑈(𝑡) as follows: 𝑈(𝑡)=𝑆(𝑡)+𝑄(𝑡)𝑁(𝑡). (3.7) Calculating the derivative of 𝑈(𝑡) along solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of model (1.2), we have 𝑑𝑈(𝑡)(𝑡),𝑈𝑡𝑑𝑡=−𝐷𝑈𝑡≠𝑛𝑇,₊=𝑈(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (3.8) From Lemma 2.2, we obtain limsup_𝑡→∞𝑈(𝑡)≤limsup_𝑡→∞𝑆^∗(𝑡)≤𝑝1−𝑒^−𝐷𝑇. (3.9) From Lemma 2.1, we have 𝑄(𝑡)≥𝑄_min for all 𝑡≥0. Hence, by (3.9) we can obtain that 𝑁(𝑡) is ultimately bound. This completes the proof of Theorem 3.1.

Remark 3.2. Obviously, Theorem 3.1 is new and can serve as an extension of corresponding result given by Smith in [19].

4. Permanence

On the permanence of species for model (1.2), we have the following result.

Theorem 4.1. Suppose ^𝑇₀𝜇𝑄^∗₀(𝑡)−𝐷𝑑𝑡>0. (4.1) Then model (1.2) is permanent.

Proof. From Lemma 2.1 we directly have liminf_𝑡→∞𝑄(𝑡)≥𝑄_min>0. Hence, 𝑄(𝑡) is obviously permanent. We now prove the permanence of 𝑁(𝑡). From (4.1), we can choose a constant 𝜀₀>0 small enough such that ^𝑇₀𝜇𝑄^∗₀(𝑡)−𝜀₀−𝐷𝑑𝑡>𝜀₀. (4.2) Let (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) be any solution of model (1.2) with initial condition (2.1). From Theorem 3.1, for any constant 𝜀>0 there is a 𝑇₀>0 such that 𝑆(𝑡)≤𝑀₃+𝜀,𝑄(𝑡)≤𝑀₃+𝜀,𝑁(𝑡)≤𝑀₃+𝜀 (4.3) for all 𝑡≥𝑇₀. For any 𝛼∈[−𝛼₀,𝛼₀], we consider the following assistant equation: 𝑑𝑄(𝑡)𝑆𝑑𝑡=𝜌^∗(𝑡)+𝛼,𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡). (4.4) By Lemma 2.3, we obtain that (4.4) has a unique positive 𝑇-periodic solution 𝑄^∗_𝛼(𝑡), which is globally uniformly attractive and 𝑄^∗_𝛼(𝑡) converges to 𝑄^∗₀(𝑡) uniformly for 𝑡∈𝑅₊ as 𝛼→0. Hence, there is an 𝛼^∗>0 and 𝛼^∗<𝜀₀ such that 𝑄^∗_𝛼(𝑡)>𝑄^∗₀(𝑡)−𝜀₀2∀𝑡∈𝑅₊,|𝛼|<𝛼^∗. (4.5)
Choose positive constants 𝜀₁ and 𝜀₂ such that inf_𝑡∈𝑅+𝑆^∗(𝑡)−𝜀₂−𝜀₁𝜌𝑀₃+𝜀,𝑄_min𝐷𝜀>0,₂+𝜀₁𝜌𝑀₃+𝜀,𝑄_min𝐷<𝛼^∗. (4.6)(4.7) We first proof that lim_𝑡→∞sup𝑁(𝑡)>𝜀₁ (4.8) for any solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of model (1.2) with initial condition (2.1).
In fact, if (4.8) is not true, then there is a solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of model (1.2) such that limsup_𝑡→∞𝑁(𝑡)≤𝜀₁. Hence, there is a 𝑇₁>𝑇₀ such that 𝑁(𝑡)<𝜀₁, for all 𝑡≥𝑇₁. From the fist equation of model (1.2) and (4.3), we have 𝑑𝑆(𝑡)(𝑡)𝑑𝑡≥−𝐷𝑆−𝜀₁𝜌𝑀₃+𝜀,𝑄_min,𝑡≥𝑇₁𝑆𝑡,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (4.9) From the comparison theorem of impulsive differential equations and Lemma 2.2, for above 𝜀₂>0, there is an 𝑛₁∈𝑁 and 𝑛₁𝑇>𝑇₁ such that 𝑆(𝑡)𝜀≥−¹𝜌𝑀₃+𝜀,𝑄_min𝐷+𝑆^∗(𝑡)−𝜀₂∀𝑡>𝑛₁𝑇. (4.10) From the second equation of model (1.2), we have 𝑑𝑄(𝑡)𝑆𝑑𝑡≥𝜌^∗(𝑡)−𝜀₁𝜌𝑀₃+𝜀,𝑄_min𝐷−𝜀₂,𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡) (4.11) for all 𝑡>𝑛₁𝑇. Let 𝛼₁=−𝜀₁𝜌(𝑀₃+𝜀,𝑄_min)/𝐷−𝜀₂, from Lemma 2.3 and the comparison theorem, there exists an 𝑛₂>𝑛₁ such that 𝑄(𝑡)≥𝑄^∗_𝛼1(𝑡)−𝜀₀2≥𝑄^∗₀(𝑡)−𝜀₀∀𝑡>𝑛₂𝑇. (4.12) Then, from the third equation of model (1.2), we have 𝑑𝑁(𝑡)(𝑡)𝜇𝑄𝑑𝑡≥𝑁^∗₀(𝑡)−𝜀₀−𝐷∀𝑡>𝑛₂𝑇. (4.13) Integrating (4.13) from 𝑛₂𝑇 to 𝑡>𝑛₂𝑇, we obtain 𝑁(𝑡)≥𝑁𝑛₂𝑇exp^𝑡_𝑛2𝑇𝜇𝑄^∗₀(𝑠)−𝜀₀−𝐷𝑑𝑠. (4.14) Obviously, from (4.2) and (4.14), we obtain 𝑁(𝑡)→∞ as 𝑡→∞, which leads to a contradiction. Therefore, (4.8) is true.
Now, we prove that there is a constant 𝑚₁>0 such that liminf_𝑡→∞𝑁(𝑡)≥𝑚₁ (4.15) for any solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of model (1.2) with initial condition (2.1).
Assume that (4.15) is not true, then there exists a sequence of initial values 𝑧_𝑘=(𝑆_𝑘,𝑄_𝑘,𝑁_𝑘), which satisfy 𝑆_𝑘>0, 𝑄_𝑘≥𝑄_min, and 𝑁_𝑘>0 such that for solution (𝑆(𝑡,𝑧_𝑘),𝑄(𝑡,𝑧_𝑘),𝑁(𝑡,𝑧_𝑘)) of model (1.2) with initial condition 𝑆(0⁺,𝑧_𝑘), 𝑄(0⁺,𝑧_𝑘), and 𝑁(0⁺,𝑧_𝑘)liminf_𝑡→∞𝑁𝑡,𝑧_𝑘<𝜀₁𝑘²∀𝑘=1,2,…. (4.16) From (4.8) and (4.16), we obtain that there exist two sequences {𝑣_𝑞^(𝑘)} and {𝑡_𝑞^(𝑘)} such that for each 𝑘=1,2,…0<𝑣₁^(𝑘)<𝑡₁^(𝑘)<𝑣₂^(𝑘)<𝑡₂^(𝑘)<⋯<𝑣_𝑞^(𝑘)<𝑡_𝑞^(𝑘)𝑣<⋯,_𝑞(𝑘)⟶∞,𝑣_𝑞^(𝑘)𝑁𝑡⟶∞as𝑞⟶∞,_𝑞(𝑘),𝑧_𝑘=𝜀₁𝑘²,𝑁𝑣_𝑞^(𝑘),𝑧_𝑘=𝜀₁𝑘𝜂,𝑘²<𝑁𝑡,𝑧_𝑘<𝜀₁𝑘∀𝑡∈𝑣_𝑞^(𝑘),𝑡_𝑞^(𝑘). (4.17)(4.18)(4.19) From Theorem 3.1, there is a 𝑇^(𝑘)≥𝑇₀ such that 𝑆𝑡,𝑧_𝑘≤𝑀₃+𝜀,𝑄𝑡,𝑧_𝑘≤𝑀₃+𝜀,𝑁𝑡,𝑧_𝑘≤𝑀₃+𝜀 (4.20) for all 𝑡≥𝑇^(𝑘). Further for each 𝑘, from (4.17) there is an integer 𝐾^(𝑘)>0 such that 𝑣_𝑞^(𝑘)>𝑇^(𝑘) for all 𝑞>𝐾^(𝑘). Hence for any 𝑡∈[𝑣_𝑞^(𝑘),𝑡_𝑞^(𝑘)] and 𝑞>𝐾^(𝑘), we have 𝑑𝑁𝑡,𝑧_𝑘𝑑𝑡=𝑁𝑡,𝑧_𝑘𝜇𝑄𝑡,𝑧_𝑘−𝐷≥−𝛾𝑁𝑡,𝑧_𝑘, (4.21) where 𝛾=𝐷+𝜇(𝑀₃+𝜀). Therefore, for any 𝑞≥𝐾^(𝑘) and 𝑘=1,2,…, integrating (4.21) on [𝑣_𝑞^(𝑘),𝑡_𝑞^(𝑘)], we obtain from (4.18) 𝜀₁𝑘²=𝑁𝑡_𝑞^(𝑘),𝑧_𝑘≥𝑁𝑣_𝑞^(𝑘),𝑧_𝑘exp−𝛾𝑡_𝑞^(𝑘)−𝑣_𝑞^(𝑘)=𝜀₁𝑘exp−𝛾𝑡_𝑞^(𝑘)−𝑣_𝑞^(𝑘). (4.22) Consequently, 𝑡_𝑞^(𝑘)−𝑣_𝑞^(𝑘)≥ln𝑘𝛾∀𝑞≥𝐾^(𝑘),𝑘=1,2,…. (4.23)
Consider the following equation: 𝑑𝑢(𝑡)(𝑡)𝑑𝑡=−𝐷𝑢−𝜀₁𝜌𝑀₃+𝜀₀,𝑄_min𝑢𝑡,𝑡≠𝑛𝑇,₊=𝑢(𝑡)+𝑝,𝑡=𝑛𝑇. (4.24) From Lemma 2.2, there is a constant 𝑇^∗>0 such that for any 𝑡₀≥0 and 0≤𝑢₀≤𝑀₃+𝜀𝑢(𝑡)≥𝑆^∗(𝑡)−𝜀₁𝜌𝑀₃+𝜀,𝑄_min𝐷−𝜀₂∀𝑡≥𝑡₀+𝑇^∗, (4.25) where 𝑢(𝑡) is the solution of (4.24) with initial value 𝑢(𝑡₀)=𝑢₀.
Further consider the following equation: 𝑑𝑣(𝑡)𝑆𝑑𝑡=𝜌^∗(𝑡)+𝛼₁,𝑣(𝑡)−𝜇(𝑣(𝑡))𝑣(𝑡). (4.26) From Lemma 2.3, there is a constant 𝑇^∗∗>0 such that for any 𝑡₀≥0 and 0≤𝑣₀≤𝑀₃+𝜀𝑣(𝑡)≥𝑄^∗_𝛼1(𝑡)−𝜀₀2∀𝑡≥𝑡₀+𝑇^∗∗, (4.27) where 𝑣(𝑡) is the solution of (4.26) with initial value 𝑣(𝑡₀)=𝑣₀.
From (4.23), we can choose an integer 𝑁₀>0 such that 𝑡_𝑞^(𝑘)−𝑣_𝑞^(𝑘)>𝑇^∗+𝑇^∗∗+𝑇∀𝑘>𝑁₀,𝑞>𝐾^(𝑘). (4.28) Since for any integer 𝑘, 𝑞≥𝐾^(𝑘), 𝑡∈[𝑣_𝑞^(𝑘),𝑡_𝑞^(𝑘)] and 𝑞≥𝐾^(𝑘) we have 𝑑𝑆𝑡,𝑧_𝑘𝑑𝑡=−𝐷𝑆𝑡,𝑧_𝑘−𝑁𝑡,𝑧_𝑘𝜌𝑆𝑡,𝑧_𝑘,𝑄𝑡,𝑧_𝑘≥−𝐷𝑆𝑡,𝑧_𝑘−𝜀₁𝜌𝑀₃+𝜀,𝑄_min𝑆𝑡,𝑡≠𝑛𝑇,₊,𝑧_𝑘=𝑆𝑡,𝑧_𝑘+𝑝,𝑡=𝑛𝑇, (4.29) from the comparison theorem and inequality (4.25), we have 𝑆𝑡,𝑧_𝑘(𝑡)𝜀≥𝑢≥−¹𝜌𝑀₃+𝜀,𝑄_min𝐷+𝑆^∗(𝑡)−𝜀₂=𝑆^∗(𝑡)+𝛼₁ (4.30) for all 𝑡∈[𝑣_𝑞^(𝑘)+𝑇^∗,𝑡_𝑞^(𝑘)] and 𝑞≥𝐾^(𝑘). Further, since 𝑑𝑄𝑡,𝑧_𝑘𝑆𝑑𝑡≥𝜌^∗(𝑡)+𝛼₁,𝑄𝑡,𝑧_𝑘−𝜇𝑄𝑡,𝑧_𝑘𝑄𝑡,𝑧_𝑘 (4.31) for all 𝑡∈[𝑣_𝑞^(𝑘)+𝑇^∗,𝑡_𝑞^(𝑘)] and 𝑞≥𝐾^(𝑘). From the comparison theorem and inequality (4.27), 𝑄𝑡,𝑧_𝑘≥𝑄^∗_𝛼1(𝑡)−𝜀₀2≥𝑄^∗₀(𝑡)−𝜀₀, (4.32) for all 𝑡∈[𝑣_𝑞^(𝑘)+𝑇^∗+𝑇^∗∗,𝑡_𝑞^(𝑘)] and 𝑞≥𝐾^(𝑘). Hence, for any 𝑞≥𝐾^(𝑘), 𝑘≥𝑁₀ and 𝑡∈[𝑡_𝑞^(𝑘)−𝑇,𝑡_𝑞^(𝑘)], we have 𝑑𝑁𝑡,𝑧_𝑘𝑑𝑡=𝑁𝑡,𝑧_𝑘𝜇𝑄𝑡,𝑧_𝑘−𝐷≥𝑁𝑡,𝑧_𝑘𝜇𝑄^∗₀(𝑡)−𝜀₀−𝐷. (4.33) Integrating (4.33) from 𝑡_𝑞^(𝑘)−𝑇 to 𝑡_𝑞^(𝑘), by (4.19) we obtain 𝜀₁𝑘²=𝑁𝑡_𝑞^(𝑘),𝑧_𝑘≥𝑁𝑡_𝑞^(𝑘)−𝑇,𝑧_𝑘exp^{𝑡𝑞(𝑘)}_{𝑡𝑞(𝑘)−𝑇}𝜇𝑄^∗₀(𝑡)−𝜀₀≥𝜀−𝐷𝑑𝑡₁𝑘²exp𝜀₀>𝜀₁𝑘², (4.34) which is contradictory. This contradiction shows that (4.15) is true for any positive solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of model (1.2).
Next, we prove that 𝑆(𝑡) in model (1.2) is permanent. Let (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) be any solution of model (1.2) with initial condition (2.1). From the assumption of 𝜌(𝑆,𝑄) and the mean-value theorem, for any 𝑡≥0, there exists a 𝜉(𝑡)∈(0,𝑆(𝑡)) such that 𝜌(𝑆(𝑡),𝑄(𝑡))𝑆(𝑡)=𝜌(𝑆(𝑡),𝑄(𝑡))−𝜌(0,𝑄(𝑡))𝑆(𝑡)=𝜕𝜌(𝜉(𝑡),𝑄(𝑡)).𝜕𝑆 (4.35) Further, from assumption (𝐻₂) and the boundedness of solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)), there exists a constant 𝑀_𝑠>0 such that 𝜌(𝑆(𝑡),𝑄(𝑡))𝑆(𝑡)=𝜕𝜌(𝜉(𝑡),𝑄(𝑡))𝜕𝑆<𝑀_𝑠∀𝑡≥0. (4.36) From the first equation of model (1.2), we obtain 𝑑𝑆(𝑡)(𝑡)(𝑡)𝜌(𝑆(𝑡)(𝑡𝑀𝑑𝑡=−𝐷𝑆−𝑁,𝑄))≥−𝐷+₃+𝜀𝑀_𝑠𝑆(𝑡), (4.37) for all 𝑡≥𝑇₀ and 𝑡≠𝑛𝑇, and 𝑆𝑡⁺=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (4.38) Using the comparison theorem and Lemma 2.2, we can obtain liminf_𝑡→∞𝑆(𝑡)≥𝑝exp−𝐷+𝑀₃+𝜀𝑀_𝑠𝑇−𝐷+𝑀₃+𝜀𝑀_𝑠𝑇. (4.39) This shows that 𝑆(𝑡) is permanent. This completes the proof of Theorem 4.1.

5. Extinction

On the extinction of species of model (1.2), we have the following result.

Theorem 5.1. Suppose ^𝑇₀𝜇𝑄^∗₀(𝑡)−𝐷𝑑𝑡<0. (5.1) Then for any positive solution (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) of model (1.2) lim_𝑡→∞𝑆(𝑡)=𝑆^∗(𝑡),lim_𝑡→∞𝑄(𝑡)=𝑄^∗₀(𝑡),lim_𝑡→∞𝑁(𝑡)=0. (5.2)

Proof. From (5.1), there exists a constant 𝜀₀>0 such that ^𝑇₀𝜇𝑄^∗₀(𝑡)+𝜀₀−𝐷𝑑𝑡<0. (5.3) For any 𝛼∈[−𝛼₀,𝛼₀], we consider the following equation: 𝑑𝑄(𝑡)𝑆𝑑𝑡=𝜌^∗(𝑡)+𝛼,𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡). (5.4) By Lemma 2.3, we obtain that positive 𝑇-periodic solution 𝑄^∗_𝛼(𝑡) of (5.4) is globally uniformly attractive and 𝑄^∗_𝛼(𝑡) converges to 𝑄^∗₀(𝑡) uniformly for 𝑡∈𝑅₊ as 𝛼→0. Hence, there is an 𝛼^∗>0 and 𝛼^∗<𝜀₀ such that 𝑄^∗_𝛼(𝑡)<𝑄^∗₀(𝑡)+𝜀₀∀𝑡∈𝑅₊,|𝛼|<𝛼^∗. (5.5) Let (𝑆(𝑡),𝑄(𝑡),𝑁(𝑡)) be any solution of model (1.2) with initial condition (2.1). Since 𝑑𝑆(𝑡)(𝑡)𝑆𝑡𝑑𝑡≤−𝐷𝑆,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇, (5.6) for any constant 0<𝜀₁<𝛼^∗, from Lemma 2.2, there is a 𝑇₀>0 such that 𝑆(𝑡)≤𝑆^∗(𝑡)+𝜀₁,𝑡≥𝑇₀. (5.7) From the second equation of model (1.2), we have 𝑑𝑄(𝑡)𝑆𝑑𝑡≤𝜌^∗(𝑡)+𝜀₁,𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡)∀𝑡≥𝑇₀. (5.8) From Lemma 2.3 and the comparison theorem, there exists a function 𝛽(𝑡)∶𝑅₊→𝑅 satisfying 𝛽(𝑡)→0 as t→∞ such that 𝑄(𝑡)≤𝑄^∗_𝜀1(𝑡)+𝛽(𝑡)≤𝑄^∗₀(𝑡)+𝛽(𝑡)+𝜀₀∀𝑡≥𝑇₀. (5.9) Since lim_𝑡→∞𝛽(𝑡)=0, we can obtain lim_{𝑡→∞𝑡}^𝑡+𝑇𝜇𝑄^∗₀(𝑡)+𝛽(𝑡)+𝜀₀−𝐷𝑑𝑡=^𝑇₀𝜇𝑄^∗₀(𝑡)+𝜀₀−𝐷𝑑𝑡<0. (5.10) Hence, there exist constants 𝜂>0 and 𝑇₁>𝑇₀ such that when 𝑡≥𝑇_1𝑡^𝑡+𝑇𝜇𝑄^∗₀(𝑡)+𝛽(𝑡)+𝜀₀−𝐷𝑑𝑡≤−𝜂 (5.11) and |𝛽(𝑡)|<1.
For any 𝑡≥𝑇₁, from the third equation of model (1.2), we have 𝑑𝑁(𝑡)(𝑡𝜇𝑄𝑑𝑡≤𝑁)^∗₀(𝑡)+𝛽(𝑡)+𝜀₀−𝐷∀𝑡≥𝑇₁. (5.12) We choose an integer 𝑟_𝑡≥0 such that 𝑡∈(𝑇₁+𝑟_𝑡𝑇,𝑇₁+(𝑟_𝑡+1)𝑇]. Then, integrating (5.12) from 𝑇₁ to 𝑡, we obtain 𝑁(𝑡)≤𝑁𝑇₁exp^𝑡_𝑇1𝜇𝑄^∗₀(𝑡)+𝛽(𝑡)+𝜀₀𝑇−𝐷𝑑𝑡≤𝑁₁exp^{𝑇1+𝑟𝑡T}_𝑇1+^𝑡_{𝑇1+𝑟𝑡𝑇}𝜇𝑄^∗₀(𝑡)+𝛽(𝑡)+𝜀₀𝑇−𝐷𝑑𝑡≤𝑁₁exp−𝜂𝑟_𝑡exp𝜇𝑀₁+1+𝜀₀−𝐷𝑇. (5.13) Therefore, by (5.11) we have 𝑁(𝑡)→0 as 𝑡→∞.
For any constant 𝜀>0, there is a 𝑇^∗>0 such that 𝑆(𝑡)≤𝑀₃+𝜀 and 𝑁(𝑡)≤𝜀 for all 𝑡>𝑇^∗. When 𝑡>𝑇^∗, from model (1.2) we obtain 𝑑𝑆(𝑡)(𝑡)𝑆𝑡𝑑𝑡≤−𝐷𝑆,𝑡≠𝑛𝑇,₊(𝑡)(𝑡)=𝑆+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁,𝑑𝑆(𝑡)𝑀𝑑𝑡≥−𝐷𝑆−𝜀𝜌₃+𝜀,𝑄_min𝑆𝑡,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝑝,𝑡=𝑛𝑇,𝑛∈𝑁. (5.14) Using the comparison theorem and Lemma 2.2, we can obtain that there is a 𝑇>𝑇^∗ such that −𝜀𝜌𝑀₃+𝜀,𝑄_min𝐷+𝑆^∗(𝑡)−𝜀≤𝑆(𝑡)≤𝑆^∗(𝑡)+𝜀 (5.15) for all 𝑡≥𝑇. From the arbitrariness of 𝜀, we finally obtain 𝑆(𝑡)⟶𝑆^∗(𝑡)as𝑡⟶∞. (5.16) Finally, from Lemma 2.3, we can easily obtain 𝑄(𝑡)⟶𝑄^∗₀(𝑡)as𝑡⟶∞. (5.17) This completes the proof of Theorem 5.1.

Remark 5.2. Obviously, system (1.2) has a semitrivial 𝑇-periodic solution (𝑆^∗(𝑡),𝑄^∗₀(𝑡),0) at which microorganism culture fails. Theorem 5.1 shows that 𝑇-periodic solution (𝑆^∗(𝑡),𝑄^∗₀(𝑡),0) of system (1.2) is global attractivity.

Remark 5.3. The biological meaning of Theorems 3.1–5.1 is very significant, which can be seen in [18] (see Remarks 3 and  5 in [18]).

Remark 5.4. We notice that it is quite difficult to obtain the sufficient and necessary conditions on the extinction and permanence of species 𝑁. So, we take this problem in the future.

6. Global Attractivity

Now, we discuss the global attractivity of all positive solutions and the existence of positive 𝑇-periodic solution of model (1.2); we have the following result.

Theorem 6.1. Suppose inequality (4.1) holds. Then 𝑚𝑜𝑑𝑒𝑙 (1.2) has a unique positive 𝑇-periodic solution, which is globally attractive.

Proof. Define Liapunov function 𝑊(𝑡) as follows: 𝑊(𝑡)=𝑆(𝑡)+𝑄(𝑡)𝑁(𝑡). (6.1) From the proof of Theorem 3.1, we have 𝑊(𝑡)→𝑆^∗(𝑡) as 𝑡→∞. Hence, 𝑊(𝑡)=𝑆^∗(𝑡)+𝛼(𝑡), where 𝛼(𝑡) is defined on 𝑅⁺ with lim_𝑡→∞𝛼(𝑡)=0. Substitute 𝑆(𝑡)=𝑆^∗(𝑡)+𝛼(𝑡)−𝑄(𝑡)𝑁(𝑡) into model (1.2), we obtain 𝑑𝑄(𝑡)𝑆𝑑𝑡=𝜌^∗(𝑡)(𝑡)(𝑡)𝑁(𝑡)(𝑡)(𝑄(𝑡𝑄(𝑡),(𝑡)+𝛼−𝑄,𝑄−𝜇))𝑑𝑁(𝑡𝜇(𝑄(𝑡.𝑑𝑡=𝑁)))−𝐷 (6.2) Further, let 𝑢(𝑡)=𝑄(𝑡)𝑁(𝑡); then system (6.2) changes into the following form: 𝑑𝑢(𝑡)(𝑡𝜌𝑆𝑑𝑡=𝑢)^∗(𝑡)+𝛼(𝑡)−𝑢(𝑡),𝑄(𝑡)𝑄(𝑡),(𝑡)−𝐷𝑑𝑄𝑆𝑑𝑡=𝜌^∗(𝑡)+𝛼(𝑡)−𝑢(𝑡),𝑄(𝑡)−𝜇(𝑄(𝑡))𝑄(𝑡). (6.3) Let (𝑢₁(𝑡),𝑄₁(𝑡)) and (𝑢₂(𝑡),𝑄₂(𝑡)) be any two positive solutions of system (6.3), for any constant 𝜖₀>0 with 𝑚−𝜖₀>0, from Theorems 4.1 and 5.1, there exits a constant 𝑇₀>0 such that 𝑚−𝜖₀≤𝑄_𝑖(𝑡)≤𝑀₃+𝜖₀,𝑚−𝜖₀²≤𝑢_𝑖(𝑡)≤𝑀₃+𝜖₀²,𝑖=1,2, (6.4) for all 𝑡≥𝑇₀. Define the Liapunov function as follows: 𝑉(𝑡)=ln𝑢₁(𝑡)−ln𝑢₂(𝑡)+ln𝑄₁(𝑡)−ln𝑄₂(𝑡). (6.5) Obviously, 𝑉(𝑡⁺)=𝑉(𝑡) for all 𝑡=𝑛𝑇 and 𝑛∈𝑁. Calculating the Dini derivative of 𝑉(𝑡), for any 𝑡≥𝑇₀ and 𝑡≠𝑛𝑇, we have 𝐷⁺𝑉(𝑡)=sign𝑢₁(𝑡)−𝑢₂(𝑡×𝜌𝑆_∗)(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)+sign𝑄₁(𝑡)−𝑄₂(𝑡)×𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)+𝜇𝑄₂(𝑡)−𝜇𝑄₁(𝑡)=sign𝑢₁(𝑡)−𝑢₂(𝑡)+sign𝑄₁(𝑡)−𝑄₂(𝑡)×𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)−𝑑𝜇(𝜉(𝑡))𝑄𝑑𝑄¹(𝑡)−𝑄₂(𝑡), (6.6) where 𝜉(𝑡) is situated between 𝑄₁(𝑡) and 𝑄₂(𝑡). We claim that sign𝑢₁(𝑡)−𝑢₂(𝑡)+sign𝑄₁(𝑡)−𝑄₂(𝑡)×𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)≤0. (6.7) In fact, there are the following several possible cases:(1)𝑢₁(𝑡)<𝑢₂(𝑡),𝑄₁(𝑡)<𝑄₂(𝑡),(2)𝑢₁(𝑡)<𝑢₂(𝑡),𝑄₁(𝑡)=𝑄₂(𝑡),(3)𝑢₁(𝑡)<𝑢₂(𝑡),𝑄₁(𝑡)>𝑄₂(𝑡),(4)𝑢₁(𝑡)=𝑢₂(𝑡),𝑄₁(𝑡)<𝑄₂(𝑡),(5)𝑢₁(𝑡)=𝑢₂(𝑡),𝑄₁(𝑡)=𝑄₂(𝑡),(6)𝑢₁(𝑡)=𝑢₂(𝑡),𝑄₁(𝑡)>𝑄₂(𝑡),(7)𝑢₁(𝑡)>𝑢₂(𝑡),𝑄₁(𝑡)<𝑄₂(𝑡),(8)𝑢₁(𝑡)>𝑢₂(𝑡),𝑄₁(𝑡)=𝑄₂(𝑡),(9)𝑢₁(𝑡)>𝑢₂(𝑡),𝑄₁(𝑡)>𝑄₂(𝑡).We only need to prove case (1).  Cases (2)–(8) can be easily proved and case (9) can be proved similarly to case (1). For case (1), we have sign𝑢₁(𝑡)−𝑢₂(𝑡)+sign𝑄₁(𝑡)−𝑄₂(𝑡)=−2. (6.8) From assumption (𝐻₂), we have 𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)=𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)+𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)≥𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)≥0. (6.9) Hence, (6.7) is true. Therefore, 𝐷⁺𝑉(𝑡)(𝜉(𝑡≤−𝑑𝜇))𝑄𝑑𝑄¹−𝑄₂≤−𝐴𝑄₁−𝑄₂, (6.10) for all 𝑡≥𝑇₀ and 𝑡≠𝑛𝑇, where (𝑄(𝑡𝐴=inf𝑑𝜇))𝑑𝑄∶𝑚−𝜖₀≤𝑄(𝑡)≤𝑀₃+𝜖₀. (6.11) For any 𝑡≥𝑇₀, integrating from 𝑇₀ to 𝑡, we have 𝑉(𝑡)+^𝑡_𝑇0𝐴𝑄₁(𝑠)−𝑄₂(𝑠)𝑑𝑠≤𝑉𝑇₀. (6.12) Hence, ^∞_𝑇0𝑄₁(𝑠)−𝑄₂(𝑠)𝑑𝑠<∞. (6.13) From Theorem 3.1 and system (6.3), we easily see that 𝑑𝑄_𝑖(𝑡)/𝑑𝑡(𝑖=1,2) are bounded for 𝑡≥𝑇₀ and 𝑡≠𝑛𝑇. Since |𝑄₁(𝑡)−𝑄₂(𝑡)| is continuous for all 𝑡≥𝑇₀, by Lemma 2.5, |𝑄₁(𝑡)−𝑄₂(𝑡)| is uniformly continuous on [𝑇₀,∞). Therefore, from Barbalat lemma, we obtain lim_𝑡→∞𝑄₁(𝑡)−𝑄₂(𝑡)=0. (6.14) Let 𝐺()=𝜌()𝑆,𝑄𝑆,𝑄𝑄,()𝛼=inf𝜕𝜌𝑆,𝑄𝑚𝜕𝑆∶𝑆∈−𝜖₀,𝑀₃+𝜖₀,𝑄∈𝑚−𝜖₀,𝑀₃+𝜖₀,()𝛽=sup𝜕𝐺𝑆,𝑄𝑚𝜕𝑄∶𝑆∈−𝜖₀,𝑀₃+𝜖₀,𝑄∈𝑚−𝜖₀,𝑀₃+𝜖₀. (6.15) From assumption (𝐻₂), we have 𝛼>0 and 𝛽>0. Further, we consider the following Liapunov function: 𝑉₁(𝑡)=ln𝑢₁(𝑡)−ln𝑢₂(𝑡). (6.16) Obviously, 𝑉₁(𝑡⁺)=𝑉₁(𝑡) for all 𝑡=𝑛𝑇 and 𝑛∈𝑁. Calculating the Dini derivative of 𝑉₁(𝑡), for any 𝑡≥𝑇₀ and 𝑡≠𝑛𝑇, from the mean-value theorem, we have 𝐷⁺𝑉₁(𝑡)=sign𝑢₁(𝑡)−𝑢₂(𝑡×𝜌𝑆_∗)(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)=sign𝑢₁(𝑡)−𝑢₂(𝑡)×𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)+𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)−𝜌𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝑄₂(𝑡)𝑄₂(𝑡)𝜉≤−𝜕𝜌₁(𝑡),𝑄₁(𝑡)𝑄₁(𝑡)𝑢𝜕𝑆¹(𝑡)−𝑢₂(𝑡+𝑆)𝜕𝐺^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡),𝜉₂(𝑡)𝑄𝜕𝑄¹(𝑡)−𝑄₂(𝑡𝑚)≤−𝛼−𝜖₀²ln𝑢₁(𝑡)−ln𝑢₂(𝑡(𝑡)𝑚)+𝛿=−𝛼−𝜖₀²𝑉₁(𝑡)+𝛿(𝑡), (6.17) where 𝜉₁(𝑡) is situated between 𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₁(𝑡) and 𝑆^∗(𝑡)+𝛼(𝑡)−𝑢₂(𝑡), 𝜉₂(𝑡) is situated between 𝑄₁(𝑡) and 𝑄₂(𝑡) and 𝛿(𝑡)=𝛽|𝑄₁(𝑡)−𝑄₂(𝑡)|. By the comparison theorem and the formula of variational constant for one-order linear differential equations, we can obtain 𝑉₁(𝑡)≤𝑉𝑇₀𝑒^{−𝛼(𝑚−𝜀0)2(𝑡−𝑇0)}+^𝑡_𝑇0𝛿(𝑠)𝑒^{𝛼(𝑚−𝜀0)2(𝑠−𝑡)}𝑑𝑠, (6.18) for all 𝑡≥𝑇₀. From lim_𝑡→∞𝛿(𝑡)=0, we easily obtain 𝑉₁(𝑡)→0 as 𝑡→∞. Consequently, lim_𝑡→∞𝑢₁(𝑡)−𝑢₂(𝑡)=0. (6.19) For any two positive solutions (𝑆_𝑖(𝑡),𝑄_𝑖(𝑡),𝑁_𝑖(𝑡))(𝑖=1,2), from 𝑁_𝑖(𝑡)=𝑢_𝑖(𝑡)/𝑄_𝑖(𝑡) and 𝑆_𝑖(𝑡)=𝑆^∗(𝑡)+𝛼(𝑡)−𝑄_𝑖(𝑡)𝑁_𝑖(𝑡)(𝑖=1,2), we can further obtain lim_𝑡→∞𝑆₁(𝑡)−𝑆₂(𝑡)=0,lim_𝑡→∞𝑁₁(𝑡)−𝑁₂(𝑡)=0. (6.20) This shows that all solutions of model (1.2) is globally attractive.
Let 𝑋(𝑡,𝑧)=(𝑆(𝑡,𝑧),𝑄(𝑡,𝑧),𝑁(𝑡,𝑧)) be some positive solution of model (1.2) with initial condition (𝑆(0),𝑄(0),𝑁(0))=𝑧, where 𝑧=(𝑧₁,𝑧₂,𝑧₃). From Theorems 3.1 and 5.1, there exist two constants 𝑀>𝑚>0 such that 𝑚≤liminf_𝑡→∞𝑆(𝑡,𝑧)≤limsup_𝑡→∞𝑆(𝑡,𝑧)≤𝑀,𝑚≤liminf_𝑡→∞𝑄(𝑡,𝑧)≤limsup_𝑡→∞𝑄(𝑡,𝑧)≤𝑀,𝑚≤liminf_𝑡→∞𝑁(𝑡,𝑧)≤limsup_𝑡→∞𝑁(𝑡,𝑧)≤𝑀. (6.21) Hence, there exists an integer 𝑛₀>0 such that sequence of vectors {𝑋(𝑛𝑇,𝑧)∈𝑅³∶𝑛≥𝑛₀} is bounded. Hence, there exists a subsequence {𝑛_𝑘} and a 𝑧^∗∈𝑅³ such that lim_𝑘→∞𝑋𝑛_𝑘𝑇,𝑧=𝑧^∗. (6.22) Let 𝑋(𝑡,𝑧^∗)=(𝑆(𝑡,𝑧^∗),𝑄(𝑡,𝑧^∗),𝑁(𝑡,𝑧^∗)) be the solution of model (1.2) with initial condition (𝑆(0),𝑄(0),𝑁(0))=𝑧^∗. By the continuity of solutions of system (2.1) with respect to initial values, we obtain lim_𝑘→∞𝑋𝑛_𝑘+1𝑇,𝑧=lim_𝑘→∞𝑋𝑇,𝑋𝑛_𝑘𝑇,𝑧=𝑋𝑇,𝑧^∗. (6.23) On the other hand, we easily prove that 𝑋(𝑡+𝑇,𝑧) also is the solution of model (1.2). Hence, from the global attractivity of system (1.2) we can obtain lim_𝑡→∞(𝑋(𝑡,𝑧)−𝑋(𝑡+𝑇,𝑧))=0. (6.24) Consequently, lim_𝑘→∞𝑋𝑛_𝑘𝑇,𝑧−𝑋𝑛_𝑘+1𝑇,𝑧=0. (6.25) Further from (6.22) and (6.23), we obtain 𝑋(𝑇,𝑧^∗)=𝑧^∗. This shows that 𝑋(𝑡,𝑧^∗) is a positive 𝑇-periodic solution of model (1.2).
Finally, from above discussion we obtain that model (1.2) has a unique positive 𝑇-periodic solution, which is globally attractive. This completes the proof of Theorem 6.1.

Remark 6.2. According to Theorem 6.1, we can find that model (1.2) has a unique positive 𝑇-periodic solution that is globally attractive as long as it is only permanent.

Remark 6.3. We easily find that Theorem 6.1 is a very good extension in pulse case of the corresponding results given by Smith in [19].

7. Numerical Examples

In this section, we will give an example to demonstrate the effectiveness of our main results. We consider model (1.2) with the following parameters:𝜇(𝑄)=21−0.8𝑄()𝑆,𝜌𝑆,𝑄=21+𝑆,𝑝=2,𝐷=0.6,𝑇=2. (7.1)

Obviously, assumptions (𝐻₁) and (𝐻₂) hold. We consider the following subsystem:𝑑𝑆(𝑡)(𝑡),(𝑡)𝑑𝑡=−𝐷𝑆𝑑𝑄(𝑆(𝑡)(𝑡(𝑄(𝑡𝑄(𝑡),𝑆𝑡𝑑𝑡=𝜌,𝑄))−𝜇))𝑡≠𝑛𝑇,₊(𝑡)𝑄𝑡=𝑆+𝑝,₊=𝑄(𝑡),𝑡=𝑛𝑇,𝑛∈𝑁. (7.2) From Lemmas 2.2 and 2.3, system (7.2) has a periodic solution (𝑆^∗(𝑡),𝑄^∗₀(𝑡)). Further, let 𝐼=^𝑇₀𝜇𝑄^∗₀(𝑡)−𝐷𝑑𝑡. (7.3)

Since the precise expression of 𝑄^∗₀(𝑡) is quite complex, it is hard to determine the value of 𝐼. Therefore, here we will use the method of numerical simulation. Applying MatLab software, we can obtain the numerical simulation of 𝑄^∗₀(𝑡), see Figure 1.

Figure 1 

Time series of 𝑄∗0(𝑡).

Furthermore, by numerical calculation, we can obtain^𝑇₀𝜇𝑄^∗₀(𝑡)−𝐷𝑑𝑡≈0.93. (7.4)

Therefore, we have 𝐼>0. Thus, by Theorem 6.1 we finally obtain that model (1.2) has a unique positive 𝑇-periodic solution, which is globally attractive, see Figure 2.

Figure 2 

Dynamical behavior of system (1.2).

Acknowledgments

Supported by the National Natural Science Foundation of China (10961022) and Doctoral Scientific Research Fund of Xinjiang Medical University.