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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 589202, 14 pages
The Permanence and Extinction of the Single Species with Contraception Control and Feedback Controls
1Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, China
2Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830051, China
Received 8 February 2012; Accepted 3 April 2012
Academic Editor: Elena Braverman
Copyright © 2012 Qiuying Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  considered the following contraception control model: where , represent, respectively, the density of the fertile species and the sterile species at time . 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:
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  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: where , represent, respectively, the density of the fertile population and the sterile population at time . is the control variable at time . , represent, respectively, the intrinsic growth rate and density-dependent rate of the species at time , respectively. is the migration rates from the fertile population to the sterile population. The function is bounded continuous defined on ; functions , , , , , , , and are continuous, bounded, and nonnegative defined on . Let .
We will consider (1.3) together with initial conditions: where and
By the fundamental theory of functional differential equations , it is not difficult to see that the solution 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).
Throughout this paper, we will introduce the following assumptions:(H1) there exist constants such that (H2) there exist constants , such that (H3) there exists a constant such that
In addition, for a function defined on set , we denote
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: where functions , are bounded and continuous on . Furthermore, for all . We have the following result which is given in  by Teng and Li.
Further, consider the following nonautonomous linear equation: where functions and are bounded continuous defined on , and for all . One has the following result.
Lemma 2.3. Suppose that holds. Then, for any constants and , there exist constants and such that for any and with , when for all , one has where is the solution of (2.8) with initial condition .
3. Main Results
Theorem 3.1. Suppose that assumptions – hold. Then, any positive solution of (1.3) is ultimate bounded, in the sense that there exists a positive constant such that for any positive solution of (1.3).
Proof. Let be any positive solution of (1.3). We first prove that the component of (1.3) is ultimately bounded. From the first equation of (1.3), we have
We consider the following auxiliary equation:
then by and applying Lemma 2.1, there exists a constant such that
for any positive solution of (3.3). Let be the solution of (3.3) satisfying initial condition . Further, from comparison theorem, it follows that
Thus, we finally obtain that
From inequality (3.6), we obtain that there exists a positive constant such that
Hence, from the second equation of (1.3), one has for all . Further, consider the following auxiliary equation: from assumptions and and according to Lemma 2.2, there exists constant such that for the solution of (3.9) with initial condition . By the comparison theorem, we have From this, we further obtain Then, we obtain that there exists constant such that
From the third equation of (1.3), we have for all . Consider the following auxiliary equation: By assumption and conclusions of Lemma 2.2, we can get that there exists a constant such that for the solution of (3.15) with initial condition . By the comparison theorem, we have Hence, we further obtain Choose the constant , then we finally obtain This completes the proof.
Proof. Let be a solution of (1.3) satisfying initial condition (1.4). In view of Theorem 3.1, there exists a such that for all we have . According to , we can choose constants and such that, for all , we have
Next, we consider the following equation: where was given in the later. By , we have (3.22) satisfying all the conditions of Lemma 2.3. So, we can obtain that, for given constants and ( was given in Theorem 3.1), there exist constants and such that for any and , when for all , we have where is the solution of (3.22) with initial condition .
Further, consider the following equation: where was given in (3.22). By , we have (3.24) satisfying all the conditions of Lemma 2.3, so by Lemma 2.3, for given constants and , there exist constants and such that for any and , when for all , we have where is the solution of (3.24) with initial condition .
Let such that for all , we have
We first prove that In fact, if (3.27) is not true, then there exist a positive solution of (1.3) and a constant such that for all .
From the second equation of (1.3), we have for all . Let be the solution of (3.24) with initial condition , by the comparison theorem, we have for all . In (3.25), we choose and , since , we obtain for all . Hence, we further obtain By applying (3.28) and (3.30) to the third equation of (1.3), it follows that for all . Let be the solution of (3.22) with initial condition , by the comparison theorem, we have In (3.23), we choose and , since for all , we obtain Hence, from (3.32), we further obtain Hence, by (3.30) and (3.34) it follows that for any . Integrating (3.35) from to we have Thus, from (3.26), we have as , 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 such that for the solution of (1.3). From (3.27) and (3.37), for every , there are two time sequences and , satisfying and , such that
From Theorem 3.1, we can choose a positive constant such that and and , for all . Further, there is an integer such that for all . Let , for any , we have where . Integrating the above inequality from to , we further have Consequently, we can choose a large enough such that For any , and , from (3.39), we can obtain Let be solution of (3.24) with initial condition , by the comparison theorem, we have for all . In (3.25), we choose and , since and , we obtain for all . By Lemma 2.3, we can obtain that is independent of . Hence, we further obtain for all . From the third equation of (1.3), we obtain for all . Assume that is the solution of (3.22) with initial condition , then we have In (3.23), we choose and . Obviously, for all . So, we have for all . Using the comparison theorem, it follows that for all , , and .
So, for any , , and , from (3.26), (3.45), and (3.49), it follows Integrating the above inequality from to , then from (3.26), we obtain which leads to a contradiction. Therefore, this contradiction shows that there exists constant such that for any positive solution of (1.3). Therefore, there exists constant such that for any positive solution of (1.3). This completes the proof.
In this section, we discuss the extinction of the component of (1.3).
Theorem 4.1. Suppose that there exist constants , such that hold. Then, for any positive solution of (1.3).
Proof. By the condition, for every given positive constant , there exist constants and such that
for all . First, we show that there exists a , such that . Otherwise, we have
Hence, for all , one has
Thus, as , we have
So, , which leads to a contradiction. Therefore, there exists a , such that .
Second, we show that where . Otherwise, there exists a , such that . By the continuity of , there must exist such that and for . Let be the nonnegative integer such that . Further, we obtain that which leads to contradiction. This shows that (4.7) holds. By the arbitrariness of , it immediately follows that as . Further, we can obtain that as . This completes the proof of Theorem 4.1.
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).
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