The Permanence and Extinction of the Single Species with Contraception Control and Feedback Controls

Abstract

Population control has become a major problem in many wildlife species. Sterility control through contraception has been proposed as a method for reducing population size. In this paper, the single species with sterility control and feedback controls is considered. Sufficient conditions are obtained for the permanence and extinction of the system. The results show that the feedback controls do not influence the permanence of the species.

1. Introduction

Control of wildlife pest populations has usually relied on methods like chemical pesticides, biological pesticides, remote sensing and measure, computers, atomic energy, and so forth. Some brilliant achievements have been obtained. However, the warfare will never be over. Although a large variety of pesticide were used to control wildlife pest populations, the wildlife pests impairing crops are increasing especially because of resistance to the pesticide. So the pesticides are invalid. Moreover, wildlife pests will continue. On the other hand, the chemical pesticide kills not only wildlife pests but also their natural enemies. Therefore, wildlife pests are rampant. Now, sterile control to suppress wildlife pests is one of the most important measures in wildlife pest control. Sterile control [1–5] is especially for the purpose of suppressing the abundance of the pest in a new target region to a level at which it no longer causes economic damage. This can be achieved by releasing sterile insects into the environment in very large numbers in order to mate with the native insects that are present in the environment. A native female that mates with a sterile male will produce eggs, but the eggs will not hatch (the same effect will occur for the reciprocal cross). If there is a sufficiently high number of sterile insects than most of the crosses are sterile, and as time goes on, the number of native insects decreases and the ratio of sterile to normal insects increases, thus driving the native population to extinction. Sterile male techniques were first used successfully in 1958 in Florida to control Screwworm fly (Cochliomya omnivorax). A number of mathematical models have been done to assist the effectiveness of the SIT (see, e.g., [2–4]). Recently, Liu and Li [6] considered the following contraception control model: 𝑥^′₁(𝑡)=𝑥₁(𝑡)𝑏−𝑘𝑥₁(𝑡)+𝑥₂(𝑡),𝑥−𝜇_′2(𝑡)=𝜇𝑥₁(𝑡)−𝑑+𝑘𝑥₁(𝑡)+𝑥₂(𝑡)𝑥₂(𝑡), (1.1) where 𝑥₁(𝑡), 𝑥₂(𝑡) represent, respectively, the density of the fertile species and the sterile species at time 𝑡. The authors proved the equilibrium point of system (1.1) is stable under appropriate conditions. In view of the effects of a periodically changing environment, we consider the following nonautonomous contraception model: 𝑥^′₁(𝑡)=𝑥₁(𝑡)𝑏(𝑡)−𝑘(𝑡)𝑥₁(𝑡)+𝑥₂(𝑡)(𝑡),𝑥−𝜇_′2(𝑡)=𝜇(𝑡)𝑥₁(𝑡)−𝑑(𝑡)+𝑘(𝑡)𝑥₁(𝑡)+𝑥₂(𝑡)𝑥₂(𝑡). (1.2)

However, we note that ecosystem in the real world are continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecosystem is the question of whether or not an ecosystem can withstand those unpredictable forces which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. So it is necessary to study models with control variables which are so-called disturbance functions, and to find some suitable conditions to prevent a particular species from dying out. In 1993, Gopalsamy and Weng [7] introduced a feedback control variable into the delay logistic model and discussed the asymptotic behavior of solution in logistic models with feedback controls, in which the control variables satisfy certain differential equation.

In recent years, the population dynamical systems with feedback controls have been studied in many articles, for example, see [7–12] and references cited therein. However, to the best of the authors knowledge, to this day, still less scholars consider the nonautonomous single species with contraception control and feedback controls.

Motivated by the above works, we focus our attention on the permanence of species for the following single species nonautonomous systems with delays and feedback control: 𝑥^′₁(𝑡)=𝑥₁(𝑡)𝑏(𝑡)−𝑘(𝑡)𝑥₁(𝑡)+𝑥₂(𝑡)−𝜇(𝑡)−𝑐₁(𝑡)𝑢𝑡−𝜎₁(𝑡),𝑥^′₂(𝑡)=𝜇(𝑡)𝑥₁(𝑡)−𝑑(𝑡)+𝑘(𝑡)𝑥₁(𝑡)+𝑥₂(𝑡)+𝑐₂(𝑡)𝑢𝑡−𝜎₂(𝑡)𝑥₂(𝑡),𝑢^′(𝑡)=−𝑒(𝑡)𝑢(𝑡)+𝑓₁(𝑡)𝑥₁𝑡−𝜏₁(𝑡)+𝑓₂(𝑡)𝑥₂𝑡−𝜏₂(𝑡), (1.3) where 𝑥₁(𝑡), 𝑥₂(𝑡) represent, respectively, the density of the fertile population and the sterile population at time 𝑡. 𝑢(𝑡) is the control variable at time 𝑡. 𝑏(𝑡), 𝑘(𝑡) represent, respectively, the intrinsic growth rate and density-dependent rate of the species at time 𝑡, respectively. 𝜇(𝑡) is the migration rates from the fertile population to the sterile population. The function 𝑏(𝑡) is bounded continuous defined on 𝑅₊=[0,∞); functions 𝜇(𝑡), 𝑘(𝑡), 𝑑(𝑡), 𝑒(𝑡), 𝑐_𝑖(𝑡), 𝑓_𝑖(𝑡), 𝜎_𝑖(𝑡), and 𝜏_𝑖(𝑡)(𝑖=1,2) are continuous, bounded, and nonnegative defined on 𝑅₊. Let 𝜏=sup{𝜏₁(𝑡),𝜏₂(𝑡),𝜎₁(𝑡),𝜎₂(𝑡)∶𝑡≥0}.

We will consider (1.3) together with initial conditions:𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃),𝑢(𝜃)=𝜓(𝜃), (1.4) where 𝜙_𝑖(𝜃),𝜓(𝜃)∈𝐵𝐶⁺[−𝜏,0] and 𝐵𝐶⁺[−𝜏,0]=𝜙∈𝐶([−𝜏,0],[0,+∞])∶𝜙(0)>0,𝜙(𝜃)isbounded. (1.5)

By the fundamental theory of functional differential equations [12], it is not difficult to see that the solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) of (1.3) is unique and positive if initial functions satisfy initial condition (1.4). So, in this paper, the solution of (1.3) satisfying initial conditions (1.4) is said to be positive.

The main purpose of this paper is to establish a new general criterion for the permanence and the extinction of (1.3), which is described by integral form and independent feedback controls. The paper is organized as follows. In the next section, we will give some assumptions and useful lemmas. In Section 3, some new sufficient conditions which guarantee the permanence of all positive solutions for (1.3) are obtained. In Section 4, we obtained some new sufficient conditions which guarantee the extinction of all positive solutions for (1.3).

2. Preliminaries

Throughout this paper, we will introduce the following assumptions:(H₁) there exist constants 𝜔_𝑖>0(𝑖=1,2) such that liminf_𝑡→∞^𝑡+𝜔1_𝑡𝑏(𝑠)−𝜇(𝑠)𝑑𝑠>0,liminf_𝑡→∞^𝑡+𝜔2_𝑡𝜇(𝑠)𝑑𝑠>0; (2.1)(H₂) there exist constants 𝜆_𝑖>0(𝑖=1,2), such that liminf_𝑡→∞^𝑡+𝜆1_𝑡𝑘(𝑠)𝑑𝑠>0,liminf_𝑡→∞^𝑡+𝜆2_𝑡𝑑(𝑠)𝑑𝑠>0; (2.2)(H₃) there exists a constant 𝛾>0 such that liminf_{𝑡→∞𝑡}^𝑡+𝛾𝑒(𝑠)𝑑𝑠>0. (2.3)

In addition, for a function 𝑔(𝑡) defined on set 𝐼⊂𝑅, we denote 𝑔^𝐿=inf_𝑡∈𝐼𝑔(𝑡),𝑔^𝑀=sup_𝑡∈𝐼𝑔(𝑡). (2.4)

Now, we state several lemmas which will be useful in the proving of main results in this paper.

First, we consider the following nonautonomous logistic equation: 𝑥^′(𝑡)=𝑥(𝑡)(𝑟(𝑡)−𝑎(𝑡)𝑥(𝑡)), (2.5) where functions 𝑎(𝑡), 𝑟(𝑡) are bounded and continuous on 𝑅₊. Furthermore, 𝑎(𝑡)≥0 for all 𝑡≥0. We have the following result which is given in [13] by Teng and Li.

Lemma 2.1. Suppose that there exist constants 𝜆, 𝜎 such that liminf_{𝑡→∞𝑡}^𝑡+𝜆𝑟(𝑠)𝑑𝑠>0,liminf_{𝑡→∞𝑡}^𝑡+𝜎𝑎(𝑠)𝑑𝑠>0. (2.6) Then,(a)there exist positive constants 𝑚 and 𝑀 such that 𝑚≤liminf_𝑡→∞𝑥(𝑡)≤limsup_𝑡→∞𝑥(𝑡)≤𝑀. (2.7) for any positive solution 𝑥(𝑡) of (2.5);(b)lim_𝑡→∞(𝑥⁽¹⁾(𝑡)−𝑥⁽²⁾(𝑡))=0 for any two positive solutions 𝑥⁽¹⁾(𝑡) and 𝑥⁽²⁾(𝑡) of (2.5).

Further, consider the following nonautonomous linear equation: 𝑢^′(𝑡)=𝑟(𝑡)−𝑒(𝑡)𝑢(𝑡), (2.8) where functions 𝑟(𝑡) and 𝑒(𝑡) are bounded continuous defined on 𝑅₊, and 𝑟(𝑡)≥0 for all 𝑡≥0. One has the following result.

Lemma 2.2. Suppose that (H₃) holds. Then, (a) there exists a positive constant 𝑈 such that limsup_𝑡→∞𝑢(𝑡)≤𝑈 for any positive solution 𝑢(𝑡) of (2.8);(b)lim_𝑡→∞(𝑢⁽¹⁾(𝑡)−𝑢⁽²⁾(𝑡))=0 for any two positive solutions 𝑢⁽¹⁾(𝑡) and 𝑢⁽²⁾(𝑡) of (2.8).

The proof of Lemma 2.2 is very simple by making a transformation with 𝑢(𝑡)=1/𝑥(𝑡). This produces the calculations: 𝑢^′(𝑡)=−(1/𝑥²(𝑡))𝑥^′(𝑡) and 𝑥^′(𝑡)=𝑥(𝑡)(𝑒(𝑡)−𝑟(𝑡)𝑥(𝑡)). Then, according to the Lemma 2.1 one can obtain Lemma 2.2.

Lemma 2.3. Suppose that (H₃) holds. Then, for any constants 𝜖>0 and 𝑀>0, there exist constants 𝛿=𝛿(𝜖)>0 and 𝑇=𝑇(𝜖,𝑀)>0 such that for any 𝑡₀∈𝑅₊ and 𝑢₀∈𝑅 with |𝑢₀|≤𝑀, when |𝑟(𝑡)|<𝛿 for all 𝑡≥𝑡₀, one has 𝑢𝑡,𝑡₀,𝑢₀<𝜖∀𝑡≥𝑡₀+𝑇, (2.9) where 𝑢(𝑡,𝑡₀,𝑢₀) is the solution of (2.8) with initial condition 𝑢(𝑡₀)=𝑢₀.

The proof of Lemma 2.3 can be found as Lemma  2.4 in [11] by Wang et al.

3. Main Results

In this section, we study the permanence of species 𝑥₁(𝑡), 𝑥₂(𝑡) of (1.3). First, we have the theorem on the ultimate boundedness of all positive solutions of (1.3).

Theorem 3.1. Suppose that assumptions (H₁)–(H₃) hold. Then, any positive solution of (1.3) is ultimate bounded, in the sense that there exists a positive constant 𝑀>0 such that limsup_𝑡→∞𝑥₁(𝑡)<𝑀,limsup_𝑡→∞𝑥₂(𝑡)<𝑀,limsup_𝑡→∞𝑢(𝑡)<𝑀, (3.1) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) of (1.3).

Proof. Let (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) be any positive solution of (1.3). We first prove that the component 𝑥₁(𝑡) of (1.3) is ultimately bounded. From the first equation of (1.3), we have 𝑑𝑥₁(𝑡)𝑑𝑡≤𝑥₁(𝑡)𝑏(𝑡)−𝜇(𝑡)−𝑘(𝑡)𝑥₁(𝑡). (3.2) We consider the following auxiliary equation: 𝑑𝑦(𝑡)(𝑡)𝑏(𝑡)(𝑡)(𝑡)𝑦(𝑡)𝑑𝑡=𝑦−𝜇−𝑘, (3.3) then by (H₁) and applying Lemma 2.1, there exists a constant 𝑀₁ such that limsup_𝑡→∞𝑦(𝑡)<𝑀₁ (3.4) for any positive solution 𝑦(𝑡) of (3.3). Let 𝑦^∗(𝑡) be the solution of (3.3) satisfying initial condition 𝑦^∗(𝑡₀)=𝑥₁(𝑡₀). Further, from comparison theorem, it follows that 𝑥₁(𝑡)<𝑦^∗(𝑡)∀𝑡>𝑡₀. (3.5) Thus, we finally obtain that limsup_𝑡→∞𝑥₁(𝑡)<𝑀₁. (3.6) From inequality (3.6), we obtain that there exists a positive constant 𝑇₁ such that 𝑥₁(𝑡)<𝑀₁∀𝑡≥𝑇₁. (3.7)
Hence, from the second equation of (1.3), one has 𝑑𝑥₂(𝑡)(𝑡)𝑀𝑑𝑡≤𝜇₁−𝑑(𝑡)𝑥₂(𝑡) (3.8) for all 𝑡≥𝑇₁. Further, consider the following auxiliary equation: 𝑑𝑣(𝑡)(𝑡)𝑀𝑑𝑡=𝜇₁−𝑑(𝑡)𝑣(𝑡), (3.9) from assumptions (H₁) and (H₂) and according to Lemma 2.2, there exists constant 𝑀₂>0 such that limsup_𝑡→∞𝑣(𝑡)<𝑀₂ (3.10) for the solution 𝑣(𝑡) of (3.9) with initial condition 𝑣(𝑇₁)=𝑥₂(𝑇₁). By the comparison theorem, we have 𝑥₂(𝑡)≤𝑣(𝑡)∀𝑡≥𝑇₁. (3.11) From this, we further obtain limsup_𝑡→∞𝑥₂(𝑡)<𝑀₂. (3.12) Then, we obtain that there exists constant 𝑇₂>𝑇₁ such that 𝑥₂(𝑡)<𝑀₂∀𝑡≥𝑇₂. (3.13)
From the third equation of (1.3), we have 𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)𝑑𝑡≤−𝑒+𝑀₁𝑓₁(𝑡)+𝑀₂𝑓₂(𝑡) (3.14) for all 𝑡≥𝑇₂+𝜏. Consider the following auxiliary equation: 𝑑𝑣(𝑡)(𝑡)𝑣(𝑡)𝑑𝑡=−𝑒+𝑀₁𝑓₁(𝑡)+𝑀₂𝑓₂(𝑡). (3.15) By assumption (H₃) and conclusions of Lemma 2.2, we can get that there exists a constant 𝑀₃>0 such that limsup_𝑡→∞𝑣(𝑡)<𝑀₃ (3.16) for the solution 𝑣(𝑡) of (3.15) with initial condition 𝑣(𝑇₂+𝜏)=𝑢(𝑇₂+𝜏). By the comparison theorem, we have 𝑢(𝑡)≤𝑣(𝑡)∀𝑡≥𝑇₂+𝜏. (3.17) Hence, we further obtain limsup_𝑡→∞𝑢(𝑡)<𝑀₃. (3.18) Choose the constant 𝑀=max{𝑀₁,𝑀₂,𝑀₃}, then we finally obtain limsup_𝑡→∞𝑥₁(𝑡)<𝑀,limsup_𝑡→∞𝑥₂(𝑡)<𝑀,limsup_𝑡→∞𝑢(𝑡)<𝑀. (3.19) This completes the proof.

Theorem 3.2. Suppose that assumptions (H₁)–(H₃) hold. Then, there exists a constant 𝜂>0, which is independent of any solution of (1.3), such that liminf_𝑡→∞𝑥₁(𝑡)>𝜂,liminf_𝑡→∞𝑥₂(𝑡)>𝜂, (3.20) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) of (1.3).

Proof. Let (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) be a solution of (1.3) satisfying initial condition (1.4). In view of Theorem 3.1, there exists a 𝑇₀ such that for all 𝑡>𝑇₀ we have 𝑥_𝑖(𝑡)<𝑀,𝑢(𝑡)<𝑀(𝑖=1,2). According to (H₁), we can choose constants 𝜀₀>0 and 𝑇₁>0 such that, for all 𝑡≥𝑇₁, we have ^𝑡+𝜔1_𝑡𝑏(𝑠)−𝜇(𝑠)−2𝑘(𝑠)𝜀₀−𝑐₁(𝑠)𝜀₀𝑑𝑠>𝜀₀. (3.21)
Next, we consider the following equation: 𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)𝑑𝑡=−𝑒+𝑓₁(𝑡)𝛼₀+𝑓₂(𝑡)𝛼₀, (3.22) where 𝛼₀ was given in the later. By (H₃), we have (3.22) satisfying all the conditions of Lemma 2.3. So, we can obtain that, for given constants 𝜀₀>0 and 𝑀>0 (𝑀 was given in Theorem 3.1), there exist constants 𝛿₀=𝛿₀(𝜀₀)>0 and 𝑇^∗=𝑇^∗(𝜀₀,𝑀)>0 such that for any 𝑡₀∈𝑅₊ and 0≤𝑢₀≤𝑀, when 𝛼₀𝑓₁(𝑡)+𝛼₀𝑓₂(𝑡)<𝛿₀ for all 𝑡≥𝑡₀, we have 𝑢𝑡,𝑡₀,𝑢₀<𝜀₀∀𝑡≥𝑡₀+𝑇^∗, (3.23) where 𝑢(𝑡,𝑡₀,𝑢₀) is the solution of (3.22) with initial condition 𝑢(𝑡₀)=𝑢₀.
Further, consider the following equation: 𝑑𝑧(𝑡)(𝑡)𝛼𝑑𝑡=𝜇₀−𝑑(𝑡)𝑧(𝑡), (3.24) where 𝛼₀ was given in (3.22). By (H₂), we have (3.24) satisfying all the conditions of Lemma 2.3, so by Lemma 2.3, for given constants 𝛼₀>0 and 𝑀>0, there exist constants 𝛿₁=𝛿₁(𝛼₀)>0 and 𝑇^∗∗=𝑇^∗∗(𝛼₀,𝑀)>0 such that for any 𝑡₀∈𝑅₊ and 0≤𝑧₀≤𝑀, when 𝜇(𝑡)𝛼₀<𝛿₁ for all 𝑡≥𝑡₀, we have 𝑧𝑡,𝑡₀,𝑧₀<𝛼₀∀𝑡≥𝑡₀+𝑇^∗∗, (3.25) where 𝑧(𝑡,𝑡₀,𝑧₀) is the solution of (3.24) with initial condition 𝑧(𝑡₀)=𝑧₀.
Let 𝛼₀≤min{𝜀₀,(𝛿₁/(𝜇^𝑀+1)),(𝛿₀/(𝑓^𝑀₁+𝑓^𝑀₂+1))} such that for all 𝑡≥𝑇₁, we have ^𝑡+𝜔1_𝑡𝑏(𝑠)−𝜇(𝑠)−2𝑘(𝑠)𝛼₀−𝑓₁(𝑠)𝜀₀𝑑𝑠>𝜀₀. (3.26)
We first prove that limsup_𝑡→∞𝑥₁(𝑡)≥𝛼₀. (3.27) In fact, if (3.27) is not true, then there exist a positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) of (1.3) and a constant 𝑇₂≥𝑇₁ such that 𝑥₁(𝑡)<𝛼₀, (3.28) for all 𝑡≥𝑇₂.
From the second equation of (1.3), we have 𝑑𝑥₂(𝑡)(𝑡)𝛼𝑑𝑡≤𝜇₀−𝑑(𝑡)𝑥₂(𝑡) (3.29) for all 𝑡≥𝑇₂. Let 𝑧(𝑡) be the solution of (3.24) with initial condition 𝑧(𝑇₂)=𝑥₂(𝑇₂), by the comparison theorem, we have 𝑥₂(𝑡)≤𝑧(𝑡) for all 𝑡≥𝑇₂. In (3.25), we choose 𝑡₀=𝑇₂ and 𝑧₀=𝑥₂(𝑇₂), since 𝜇(𝑡)𝛼₀<𝛿₁, we obtain 𝑧(𝑡,𝑡₀,𝑧₀)<𝛼₀ for all 𝑡≥𝑇₂+𝑇^∗∗. Hence, we further obtain 𝑥₂𝑡,𝑇₂,𝑥₂𝑇₂<𝛼₀∀𝑡≥𝑇₂+𝑇^∗∗. (3.30) By applying (3.28) and (3.30) to the third equation of (1.3), it follows that 𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)𝑑𝑡≤−𝑒+𝑓₁(𝑡)𝛼₀+𝑓₂(𝑡)𝛼₀, (3.31) for all 𝑡≥𝑇₂+𝑇^∗∗+𝜏. Let 𝑢(𝑡) be the solution of (3.22) with initial condition 𝑢(𝑇₂+𝑇^∗∗+𝜏)=𝑢(𝑇₂+𝑇^∗∗+𝜏), by the comparison theorem, we have 𝑢(𝑡)≤𝑢(𝑡)∀𝑡≥𝑇₂+𝑇^∗∗+𝜏. (3.32) In (3.23), we choose 𝑡₀=𝑇₂+𝑇^∗∗+𝜏 and 𝑢₀=𝑢(𝑇₂+𝑇^∗∗+𝜏), since 𝛼₀𝑓₁(𝑡)+𝛼₀𝑓₂(𝑡)<𝛿₀ for all 𝑡≥𝑇₂+𝑇^∗∗+𝜏, we obtain 𝑢𝑡,𝑇₂+𝑇^∗∗+𝜏,𝑢𝑇₂+𝑇^∗∗+𝜏<𝜀₀∀𝑡≥𝑇₂+𝑇^∗∗+𝜏+𝑇^∗. (3.33) Hence, from (3.32), we further obtain 𝑢𝑡,𝑇₂+𝑇^∗∗+𝜏,𝑢𝑇₂+𝑇^∗∗+𝜏<𝜀₀∀𝑡≥𝑇₂+𝑇^∗∗+𝜏+𝑇^∗. (3.34) Hence, by (3.30) and (3.34) it follows that 𝑑𝑥₁(𝑡)𝑑𝑡≥𝑥₁(𝑡)𝑏(𝑡)−𝜇(𝑡)−2𝑘(𝑡)𝛼₀−𝑐₁(𝑡)𝜀₀, (3.35) for any 𝑡>𝑇₂+𝑇^∗∗+𝑇^∗+2𝜏. Integrating (3.35) from 𝑇₃=𝑇₂+𝑇^∗∗+𝑇^∗+2𝜏 to 𝑡 we have 𝑥₁(𝑡)≥𝑥₁𝑇₃exp^𝑡_𝑇3𝑏(𝑠)−𝜇(𝑠)−2𝑘(𝑠)𝛼₀−𝑐₁(𝑠)𝜀₀𝑑𝑠. (3.36) Thus, from (3.26), we have 𝑥₁(𝑡)→∞ as 𝑡→∞, which leads to a contradiction. So, (3.27) holds.
Now, we prove the conclusion of Theorem 3.2. In fact, if it is not true, then there exists an initial functions sequence {𝑍^(𝑚)}={(𝜙₁^(𝑚)(𝜃),𝜙₂^(𝑚)(𝜃),𝜓^(𝑚)(𝜃))},𝜃∈[−𝜏,0] such that liminf_𝑡→∞𝑥₁𝑡,𝑍^(𝑚)<𝛼₀(𝑚+1)²∀𝑚=1,2,…, (3.37) for the solution (𝑥₁(𝑡,𝑍^(𝑚)),𝑥₂(𝑡,𝑍^(𝑚)),𝑢(𝑡,𝑍^(𝑚))) of (1.3). From (3.27) and (3.37), for every 𝑚, there are two time sequences {𝑠_𝑞^(𝑚)} and {𝑡_𝑞^(𝑚)}, satisfying 0<𝑠₁^(𝑚)<𝑡₁^(𝑚)<𝑠₂^(𝑚)<𝑡₂^(𝑚)<⋯<𝑠_𝑞^(𝑚)<𝑡_𝑞^(𝑚)<⋯ and lim_𝑞→∞𝑠_𝑞^(𝑚)=∞, such that 𝑥₁𝑠_𝑞^(𝑚),𝑍^(𝑚)=𝛼₀𝑚+1,𝑥₁𝑡_𝑞^(𝑚),𝑍^(𝑚)=𝛼₀(𝑚+1)², (3.38)𝛼₀(𝑚+1)²≤𝑥₁𝑡,𝑍^(𝑚)≤𝛼₀𝑠𝑚+1∀𝑡∈_𝑞^(𝑚),𝑡_𝑞^(𝑚). (3.39)
From Theorem 3.1, we can choose a positive constant 𝑇^(𝑚) such that 𝑥₁(𝑡,𝑍^(𝑚))<𝑀 and 𝑥₂(𝑡,𝑍^(𝑚))<𝑀 and 𝑢(𝑡,𝑍^(𝑚))<𝑀, for all 𝑡>𝑇^(𝑚). Further, there is an integer 𝐾₁^(𝑚)>0 such that 𝑠_𝑞^(𝑚)>𝑇^(𝑚)+𝜏 for all 𝑞≥𝐾₁^(𝑚). Let 𝑞≥𝐾₁^(𝑚), for any 𝑡∈(𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚)), we have 𝑑𝑥₁𝑡,𝑍^(𝑚)𝑑𝑡≥𝑥₁𝑡,𝑍^(𝑚)𝑏(𝑡)−𝜇(𝑡)−𝑘(𝑡)𝑀−𝑘(𝑡)𝑀−𝑐₁(𝑡)𝑀≥−𝛾₁𝑥₁𝑡,𝑍^(𝑚), (3.40) where 𝛾₁=sup_𝑡∈𝑅+{|𝑏(𝑡)−𝜇(𝑡)−2𝑘(𝑡)𝑀−𝑐₁(𝑡)𝑀|}. Integrating the above inequality from 𝑠_𝑞^(𝑚) to 𝑡_𝑞^(𝑚), we further have 𝛼₀(𝑚+1)²=𝑥₁𝑡_𝑞^(𝑚),𝑍^(𝑚)≥𝑥₁𝑠_𝑞^(𝑚),𝑍^(𝑚)exp−𝛾₁𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)=𝛼₀𝑚+1exp−𝛾₁𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚). (3.41) Consequently, 𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)≥ln(𝑚+1)𝛾₁∀𝑞≥𝐾₁^(𝑚),𝑚=1,2,…, (3.42) we can choose a large enough 𝑁₀ such that 𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)≥𝑇^∗∗+𝑇^∗+2𝜏+𝜔₁∀𝑚≥𝑁₀,𝑞≥𝐾₁^(𝑚). (3.43) For any 𝑚≥𝑁₀,𝑞≥𝐾₁^(𝑚), and 𝑡∈[𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚)], from (3.39), we can obtain 𝑑𝑥₂𝑡,𝑍^(𝑚)(𝑡)𝛼𝑑𝑡≤𝜇⁰(𝑡)𝑥𝑚+1−𝑑₂𝑡,𝑍^(𝑚)≤𝜇^𝑀𝛼₀−𝑑(𝑡)𝑥₂𝑡,𝑍^(𝑚). (3.44) Let 𝑧(𝑡) be solution of (3.24) with initial condition 𝑧(𝑠_𝑞^(𝑚))=𝑥₂(𝑠_𝑞^(𝑚)), by the comparison theorem, we have 𝑥₂(𝑡)≤𝑧(𝑡) for all 𝑡≥𝑠_𝑞^(𝑚). In (3.25), we choose 𝑡₀=𝑠_𝑞^(𝑚) and 𝑧₀=𝑧(𝑠_𝑞^(𝑚)), since 𝜇^𝑀𝛼₀<𝛿₁ and 𝑥₂(𝑠_𝑞^(𝑚)<𝑀, we obtain 𝑧(𝑡,𝑠_𝑞^(𝑚),𝑧(𝑠_𝑞^(𝑚)))<𝛼₀ for all 𝑡≥𝑠_𝑞^(𝑚)+𝑇^∗∗. By Lemma 2.3, we can obtain that 𝑇^∗∗=𝑇^∗∗(𝛼₀,𝑀) is independent of 𝑚. Hence, we further obtain 𝑥₂𝑡,𝑠_𝑞^(𝑚),𝑥₂𝑠_𝑞^(𝑚)<𝛼₀, (3.45) for all 𝑡>𝑠_𝑞^(𝑚)+𝑇^∗∗. From the third equation of (1.3), we obtain 𝑑𝑢𝑡,𝑍^(𝑚)(𝑡)𝑢𝑑𝑡=−𝑒𝑡,𝑍^(𝑚)+𝑓₁(𝑡)𝑥₁𝑡−𝜏₁(𝑡),𝑍^(𝑚)+𝑓₂(𝑡)𝑥₂𝑡−𝜏₂(𝑡),𝑍^(𝑚)≤−𝑒(𝑡)𝑢𝑡,𝑍^(𝑚)+𝑓₁(𝑡)𝛼₀+𝑓₂(𝑡)𝛼₀, (3.46) for all 𝑡≥𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏. Assume that 𝑢(𝑡) is the solution of (3.22) with initial condition 𝑢(𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏)=𝑢(𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏), then we have 𝑢𝑡,𝑍^(𝑚)≤𝑢(𝑡)∀𝑡∈𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏,𝑡_𝑞^(𝑚),𝑚≥𝑁₀,𝑞≥𝐾₁^(𝑚). (3.47) In (3.23), we choose 𝑡₀=𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏 and 𝑢₀=𝑢(𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏). Obviously, 𝛼₀(𝑓₁(𝑡)+𝑓₂(𝑡))<𝛿₀ for all 𝑡≥𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏. So, we have 𝑢(𝑡)=𝑢𝑡,𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏,𝑢𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏<𝜀₀ (3.48) for all 𝑡∈[𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏+𝑇^∗,𝑡_𝑞^(𝑚)]. Using the comparison theorem, it follows that 𝑢𝑡,𝑍^(𝑚)<𝜀₀ (3.49) for all 𝑡∈[𝑠_𝑞^(𝑚)+𝑇^∗∗+𝜏+𝑇^∗,𝑡_𝑞^(𝑚)], 𝑞≥𝐾₁^(𝑚), and 𝑚≥𝑁₀.
So, for any 𝑚≥𝑁₀, 𝑞≥𝐾₁^(𝑚), and 𝑡∈[𝑠_𝑞^(𝑚)+𝑇^∗∗+𝑇^∗+2𝜏,𝑡_𝑞^(𝑚)], from (3.26), (3.45), and (3.49), it follows 𝑑𝑥₁𝑡,𝑍^(𝑚)𝑑𝑡=𝑥₁𝑡,𝑍^(𝑚)𝑏(𝑡)−𝜇(𝑡)−𝑘(𝑡)𝑥₁𝑡,𝑍^(𝑚)+𝑥₂𝑡,𝑍^(𝑚)−𝑐₁(𝑡)𝑢𝑡−𝜎₁(𝑡),𝑍^(𝑚)≥𝑥₁𝑡,𝑍^(𝑚)𝑏(𝑡)−𝜇(𝑡)−2𝑘(𝑡)𝛼₀−𝑐₁(𝑡)𝜀₀. (3.50) Integrating the above inequality from 𝑡_𝑞^(𝑚)−𝜔₁ to 𝑡_𝑞^(𝑚), then from (3.26), we obtain 𝛼₀(𝑚+1)²=𝑥₁𝑡_𝑞^(𝑚),𝑍^(𝑚)≥𝑥₁𝑡_𝑞^(𝑚)−𝜔₁,𝑍^(𝑚)exp^{𝑡𝑞(𝑚)}_{𝑡𝑞(𝑚)−𝜔1}𝑏(𝑡)−𝜇(𝑡)−2𝑘(𝑡)𝛼₀−𝑐₁(𝑡)𝜀₀≥𝛼𝑑𝑡₀(𝑚+1)²exp𝛼₀>𝛼₀(𝑚+1)², (3.51) which leads to a contradiction. Therefore, this contradiction shows that there exists constant 𝜂₁>0 such that liminf_𝑡→∞𝑥₁(𝑡)>𝜂₁, (3.52) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) of (1.3). Therefore, there exists constant 𝜂₂>0 such that liminf_𝑡→∞𝑥₂(𝑡)>𝜂₂, (3.53) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) of (1.3). This completes the proof.

Remark 3.3. In Theorem 3.2, we note that (H₁)–(H₃) are decided by (1.3), which is independent of the feedback controls. So, the feedback controls have no influence on the permanence of (1.3).

4. Extinction

In this section, we discuss the extinction of the component 𝑥₁(𝑡),𝑥₂(𝑡) of (1.3).

Theorem 4.1. Suppose that there exist constants 𝜎₁,𝜎₂, such that limsup_𝑛→∞^𝑡+𝜎1_𝑡𝑏(𝑠)−𝜇(𝑠)𝑑𝑠<0;liminf_𝑡→∞^𝑡+𝜎2_𝑡𝑘(𝑠)𝑑𝑠>0 (4.1) hold. Then, lim_𝑡→∞𝑥₁(𝑡)=0,lim_𝑡→∞𝑥₂(𝑡)=0, (4.2) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑢(𝑡)) of (1.3).

Proof. By the condition, for every given positive constant 𝜀, there exist constants 𝜀₁ and 𝑇₁ such that ^𝑡+𝜎1_𝑡𝑏(𝑠)−𝜇(𝑠)−𝑘(𝑠)𝜀𝑑𝑠<−𝜀₁, (4.3) for all 𝑡>𝑇₁. First, we show that there exists a 𝑇₂>𝑇₁, such that 𝑥₁(𝑇₂)<𝜀. Otherwise, we have 𝑥₁(𝑡)≥𝜀,∀𝑡>𝑇₁. (4.4) Hence, for all 𝑡≥𝑇₁, one has 𝑥^′₁(𝑡)<𝑥₁(𝑡)𝑏(𝑡)−𝑘(𝑡)𝜀−𝜇(𝑡). (4.5) Thus, as 𝑡→+∞, we have 𝜀≤𝑥₁(𝑡)<𝑥₁𝑇₁exp^𝑡_𝑇1𝑏(𝑠)−𝜇(𝑠)−𝑘(𝑠)𝜀𝑑𝑠<𝑥₁𝑇₁exp−𝜀₁𝑡−𝑇₁⟶0. (4.6) So, 𝜀<0, which leads to a contradiction. Therefore, there exists a 𝑇₂>𝑇₁, such that 𝑥₁(𝑇₂)<𝜀.
Second, we show that 𝑥₁(𝑡)<𝜀exp𝜇₁𝜎₁∀𝑡>𝑇₂, (4.7) where 𝜇₁=max_{𝑡∈[0,+∞]}{𝑏(𝑡)+𝜇(𝑡)+𝑘(𝑡)𝜀}. Otherwise, there exists a 𝑇₃>𝑇₂, such that 𝑥₁(𝑇₃)>𝜀exp{𝜇₁𝜎₁}. By the continuity of 𝑥₁(𝑡), there must exist 𝑇₄∈(𝑇₂,𝑇₃) such that 𝑥₁(𝑇₄)=𝜀 and 𝑥₁(𝑡)>𝜀 for 𝑡∈(𝑇₄,𝑇₃). Let 𝑃₁ be the nonnegative integer such that 𝑇₃∈(𝑇₄+𝑃₁𝜎₁,𝑇₄+(𝑃₁+1)𝜎₁). Further, we obtain that 𝜀exp𝜇₁𝜎₁<𝑥₁𝑇₃<𝑥₁𝑇₄exp^𝑇3_𝑇4𝑏(𝑠)−𝜇(𝑠)−𝑘(𝑠)𝜀𝑑𝑠=𝑥₁𝑇₄exp^{𝑇4+𝑃1𝜎1}_𝑇4𝑏(𝑠)(𝑠)(𝑠)𝜀+−𝜇−𝑘𝑑𝑠_{𝑇3𝑇4+𝑃1𝜎1}𝑏(𝑠)(𝑠)(𝑠)𝜀𝜇−𝜇−𝑘𝑑𝑠<𝜀exp₁𝜎₁, (4.8) which leads to contradiction. This shows that (4.7) holds. By the arbitrariness of 𝜀, it immediately follows that 𝑥₁(𝑡)→0 as 𝑡→+∞. Further, we can obtain that 𝑥₂(𝑡)→0 as 𝑡→+∞. This completes the proof of Theorem 4.1.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (11071283), Research Project supported by Shanxi Scholarship Council of China (2011-093) and the Sciences Foundation of Shanxi (2009011005-3).