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Global Positive Periodic Solutions for Periodic Two-Species Competitive Systems with Multiple Delays and Impulses
A set of easily verifiable sufficient conditions are derived to guarantee the existence and the global stability of positive periodic solutions for two-species competitive systems with multiple delays and impulses, by applying some new analysis techniques. This improves and extends a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.
Throughout this paper, we make the following notation and assumptions:
let be a constant and , with the norm being defined by ; , with the norm being defined by ; , if exists, ; ; , with the norm being defined by ; , with the norm being defined by ;
then those spaces are all Banach spaces. We also denote that
In this paper, we investigate the existence, uniqueness, and global stability of the positive periodic solution for two corresponding periodic Lotka-Volterra competitive systems involving multiple delays and impulses: with initial conditions where , , , , , and are all in . Also , , , and are all in with , , , , , , , , , , . Furthermore, the intrinsic growth rates , are with , . For the ecological justification of (2) and (3) and similar types refer to [1–10].
In , Freedman and Wu proposed the following periodic single-species population growth models with periodic delay:
They had assumed that the net birth , the self-inhibition rate , and the delay are continuously differentiable -periodic functions, and , , , and for . The positive feedback term in the average growth rate of species has a positive time delay (the sign of the time delay term is positive), which is a delay due to gestation (see [1, 2]). They had established sufficient conditions which guarantee that system (5) has a positive periodic solution which is globally asymptotically stable.
In , Fan and Wang investigated the following periodic single-species population growth models with periodic delay:
They had assumed that the net birth , the self-inhibition rate , and the delay are continuously differentiable -periodic functions, and , , , and for . The negative feedback term in the average growth rate of species has a negative time delay (the sign of the time delay term is negative), which can be regarded as the deleterious effect of time delay on a species growth rate (see [4–6]). They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (6). But the discussion of global attractivity is only confined to the special case when the periodic delay is constant.
They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (7) by using differential inequalities and topological degree, respectively. In fact, in many practical situations the time delay occurs so often. A more realistic model should include some of the past states of the system. Therefore, in , Liu et al. considered two corresponding periodic Lotka-Volterra competitive systems involving multiple delays: where , , , , , , , , , and are -periodic functions. Here, the intrinsic growth rates are -periodic functions with . They had derived the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the above two competitive systems by using Gaines and Mawhin's continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional.
However, the ecological system is often deeply perturbed by human exploitation activities such as planting, harvesting, and so on, which makes it unsuitable to be considered continually. For having a more accurate description of such a system, we need to consider the impulsive differential equations. The theory of impulsive differential equations not only is richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modeling of many real world phenomena [11–13]. Recently, some impulsive equations have been recently introduced in population dynamics in relation to population ecology [14–26] and chemotherapeutic treatment [27, 28]. However, to the best of the authors' knowledge, to this day, few scholars have done works on the existence, uniqueness, and global stability of positive periodic solution of (2) and (4). One could easily see that systems (5)–(9) are all special cases of systems (2) and (3). Therefore, we propose and study the systems (2) and (3) in this paper.
For the sake of generality and convenience, we always make the following fundamental assumptions. , , , , , , , and are all in ; , , , and are all in with , , , , , , , , and . satisfies , , are constants, and there exists a positive integer such that . Without loss of generality, we can assume that and , and then . is a real sequence such that , , is an -periodic function.
Definition 1. A function , is said to be a positive solution of (2) and (3), if the following conditions are satisfied:(a) is absolutely continuous on each ;(b)for each , and exist, and ;(c) satisfies the first equation of (2) and (3) for almost everywhere (for short a.e.) in and satisfies for , .
Under the above hypotheses ()–(), we consider the following nonimpulsive delay differential equation: with the initial conditions where
Proof. It is easy to see that is absolutely continuous on every interval , , , and
On the other hand, for any , ,
which implies that is a solution of (2); similarly, we can prove that is also a solution of (3). Therefore, , are solutions of (2)–(4) on . Similarly, if is a solution of (10)–(12) on , we can prove that are solutions of (2)–(4) on .
Since is absolutely continuous on every interval , , , and in view of (15), it follows that for any , which implies that is continuous on . It is easy to prove that is absolutely continuous on . Similarly, we can prove that is absolutely continuous on . Similar to the proof of , we can check that are solutions of (10)–(12) on . If is a solution of (2)–(4) on by the same method, we can prove that are solutions of (10)–(12) on . The proof of Lemma 2 is completed.
From Lemma 2, if we want to discuss the existence and global asymptotic stability of positive periodic solutions of systems (2)–(4), we only discuss the existence of the existence and global asymptotic stability of positive periodic solutions of systems (10)–(12).
The organization of this paper is as follows. In Section 2, we introduce several useful definitions and lemmas. In Section 3, first, we study the existence of at least one periodic solution of systems (2)–(4) by using continuation theorem proposed by Gaines and Mawhin (see ). Second, we investigate the global asymptotic stability of positive periodic solutions of the above systems by using the method of Lyapunov functional. As applications in Section 4, we study some particular cases of systems (2)–(4) which have been investigated extensively in the references mentioned previously.
In this section, we will introduce some concepts and some important lemmas which are useful for the next section.
Let , be two real Banach spaces, let be a linear mapping, and let be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that , , it follows that is invertible; we denote the inverse of that map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exist isomorphisms . Let denote the space of -periodic functions which are continuous for , are continuous from the left for , and have discontinuities of the first kind at point . We also denote that .
Definition 3 (see ). The set is said to be quasiequicontinuous in if for any there exists such that if , , , , and , then .
Proof. Since , , and is continuous on , it follows that has a unique inverse function on . Hence, it suffices to show that , . For any , by the condition , one can find that exists as a unique solution and exists as a unique solution ; that is, and ; that is, and .
As it follows that . Since , we have and . We can easily obtain that if , , , then , , where is the unique inverse function of , which together with implies that . The proof of Lemma 6 is completed.
Lemma 7 (see ). Let and be two Banach spaces, and let be a Fredholm operator with index zero. is an open bounded set, and let be L-compact on . Suppose that (a) for each and ;(b) for each ;(c). Then, equation has at least one solution lying in .
Lemma 8 (see ). The set is relatively compact if only if(1) is bounded, that is, , for each , and some ;(2) is quasiequicontinuous in .
Lemma 9 (see ). Assume that , are continuous nonnegative functions defined on the interval ; then there exists such that .
Lemma 11 (see Barbalat's Lemma ). Let be a nonnegative function defined on such that is integrable and uniformly continuous on ; then .
In the following section, we only discuss the existence and global asymptotic stability of positive periodic solutions of systems (10)–(12).
3. Existence and Global Asymptotic Stability
Since , , , , , by Lemma 6, we see that all have their inverse functions. Throughout the following part, we set , , , and to represent the inverse function of , , , and , respectively. Obviously, , , , . We also denote that
Proof. Since the solutions of systems (11) and (12) remain positive for , we carry out the change of variable ; then (11) can be transformed to
It is easy to see that if system (23) has one -periodic solution , then is a positive -periodic solution of system (10); that is to say, is a positive -periodic solution of system (2). Therefore, it suffices to prove that system (23) has a -periodic solution. In order to use Lemma 7 to (23), we take
Then and are Banach spaces when they are endowed with the norm . Let withand define
It is not difficult to show that
and . So, is closed in , and is a Fredholm mapping of index zero. It is trivial to show that , are continuous projectors such that
Furthermore, the generalized inverse (to L) exists and is given by
Clearly, and are continuous. By applying Ascoli-Arzela theorem, one can easily show that , are relatively compact for any open bounded set . Moreover, is obviously bounded. Thus, is -compact on for any open bounded set . Now, we reach the position to search for an appropriate open bounded set for the application of Lemma 7. Corresponding to the operating equation , , we have Since is a -periodic function, we need only to prove the result in the interval . Integrating (33) over the interval leads to the following: Hence, we have Note that , and then there exists , such that Since , we can let , that is, ; then According to Lemma 7, we know that . Thus, By (37) and (38), we have Similarly, we obtain