Research Article | Open Access
Periodic Solutions for -Species Lotka-Volterra Competitive Systems with Pure Delays
We study a class of periodic general -species competitive Lotka-Volterra systems with pure delays. Based on the continuation theorem of the coincidence degree theory and Lyapunov functional, some new sufficient conditions on the existence and global attractivity of positive periodic solutions for the -species competitive Lotka-Volterra systems are established. As an application, we also examine some special cases of the system, which have been studied extensively in the literature.
As we know well, in recent years the application of the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin to the existence of positive periodic solutions in population dynamical systems has been studied extensively [1–21]. For example, in [1–3, 12, 14–20], the authors studied existence of positive periodic solutions for population competition systems, in [4, 5, 8, 13, 21], the authors studied existence of positive periodic solutions for population cooperative systems, in , the authors studied existence of positive periodic solutions for population predator-prey system, and in [6, 11], the authors studied existence of positive periodic solutions for single species systems. The continuation theorem is a powerful tool to study the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with time-delayed problem, it is very convenient and the result is relatively simple [4, 5].
Frequently, the environments of most natural populations undergo temporal variation, causing changes in the growth characteristics of these populations. One of the methods of incorporating temporal nonuniformity of the environments in models is to assume that the parameters are periodic with the same period of the time variable .
However, in the real world, the growth rate of a natural species will not often respond immediately to changes in its own population or that of an interacting species but will rather do so after a time lag . Research [23, 24] has shown that time delays have a great destabilizing influence on the species population. Usually, time delays are of two types: discrete delay and distributed time delay. For a competition system, competitors have both an instantaneous competition and a memory competition in the past. Therefore, we should introduce distributed delay into model foundation, which will have more resemblance to the real ecosystem. In fact, during the last decades, most of the authors study dynamics of population with delays [1–7, 10–21, 25–31], which is useful for the control of the population of mankind, animals, and environment.
It is well known that the focus in theoretical models of population and community dynamics must be not only on how populations depend on their own population densities or the population densities of other organisms but also on how populations change in response to the physical environment . To consider periodic environmental factor, it is reasonable to study Lotka-Volterra systems with periodic coefficients. One of the celebrated population dynamical systems is Lotka-Volterra competition system. Since the Lotka-Volterra competition system has been established and was accepted by many scientists, now it has became the most important means to explain the ecological phenomenon. Recently, a great deal of Lotka-Volterra competition system with delays have been proposed to study the existence of periodic solutions [1–3, 12, 14–21, 25–31] and many good results were obtained by using of the continuation theorem [1–3, 12, 14–21].
In , the authors studied the following nonautonomous -species Lotka-Volterra competitive systems with continuous time delays: By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions of the existence and global attractivity of positive periodic solutions are established. In , the authors considered the following nonautonomous -species Lotka-Volterra competitive systems with distributed time delays, and the periodic state dependent delay Lotka-Volterra competition system, By using the Mawhin’s continuation theorem, the sufficient conditions on the existence of positive periodic solutions are established. In , the authors considered the following nonautonomous delay -species competitive systems with delays: By means of the Mawhin’s continuation theorem and Lyapunov function method, the sufficient conditions for the existence and global attractivity of positive periodic solutions are established. In , the authors studied the following nonautonomous -species Lotka-Volterra cooperative systems with continuous time delays and feedback controls: The sufficient conditions for the existence of positive periodic solutions are established, based on the Mawhin’s continuation theorem.
Motivated by the above works, in this paper, we investigate the following species periodic Lotka-Volterra type competitive systems with pure delays
By using the technique of coincidence degree developed by Gaines and Mawhin in , we will establish some new sufficient conditions, which guarantee that the system has at least one positive periodic solution. By means of the Lyapunov functionals we also will further establish the sufficient conditions on the global attractivity of the positive periodic solution.
In system (6), we have that represent the density of competitive species at time , respectively; represent the intrinsic growth rate of species at time , respectively; represent the intrapatch restriction density of species at time , respectively; , represent the competitive coefficients between species at time , respectively. In this paper, we always assume the following.
(H1) and , are continuous -periodic functions with and . , are continuous positive -periodic functions. are nonnegative integrable functions on satisfying .
From the viewpoint of mathematical biology, in this paper for system (6) we only consider the solution with the following initial conditions: where are nonnegative continuous functions defined on satisfying with .
In this paper, for any -periodic continuous function , we denote
In order to obtain the existence of positive -periodic solutions of system (6), we will use the continuation theorem developed by Gaines and Mawhin in . For the reader’s convenience, we will introduce the continuation theorem in the following.
Let and be two normed vector spaces. Let be a linear operator and a continuous operator. The operator is called a Fredholm operator of index zero, if and is a closed set in . If is a Fredholm operator of index zero, then there exist continuous projectors and such that and . It follows that is invertible and its inverse is denoted by and denote by an isomorphism of onto . Let be a bounded open subset of ; we say that the operator is -compact on , where denotes the closure of in , if is bounded and is compact. Such definitions can be found in [4, 5].
Lemma 1 (see ). Suppose with and , . Then the function has a unique inverse function satisfying , , .
Lemma 2 (see ). Let be a Fredholm operator of index zero and let be -compact on . If(a)for each and , ,(b)for each , ,(c),then the operator equation has at least one solution lying in .
3. Main Results
Now, for convenience of statements we denote the functions
The following theorem is about the existence of positive periodic solutions of system (6).
Theorem 3. Suppose that assumption (H1) holds and there exists a constant , , such that where and the algebraic equation, has a unique positive solution. Then system (6) has at least one positive -periodic solution.
Proof. For system (6) we introduce new variables such that Then system (6) is rewritten in the following form: In order to apply Lemma 2 to system (14), we introduce the normed vector spaces and as follows. Let denote the space of all continuous function . We take with norm It is obvious that and are the Banach spaces. We define a linear operator and a continuous operator as follows: where Further, we define continuous projectors and as follows: We easily see and . It is obvious that is closed in and . Since for any there are unique and with such that , we have . Therefore, is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given in the following form: For convenience, we denote as follows: Thus, we have From formulas (23), we easily see that and are continuous operators. Furthermore, it can be verified that is compact for any open bounded set by using Arzela-Ascoli theorem and is bounded. Therefore, is -compact on for any open bounded subset .
Now, we reach the position to search for an appropriate open bounded subset for the application of the continuation theorem (Lemma 2) to system (14).
Corresponding to the operator equation with parameter , we have where are given in (22).
Assume that is a solution of system (24) for some parameter . By integrating system (24) over the interval , we obtain Consequently, Let ; then from Lemma 1 and (H1) we get that function has a unique periodic inverse function ; then for every , , we have One can see that are periodic functions. Then for every , , we have For each , and , we haveFrom ((26)–(30)), we obtainFrom the above equality we have From (32) we can obtain where Consequently, From the continuity of , there exist constants such that From (35) and (36), we further obtain From ((26)–(30)), we can obtain From (38) we can obtain From the assumptions of Theorem 3 and (35), (38), and (39) we further obtain where On the other hand, directly from system (14) we have where From (37), (40), and (42), we have for any Therefore, from (44) and (45), we have It can be seen that the constants are independent of parameter . For any , from (18) we can obtain We consider the following algebraic equation: From the assumption of Theorem 3, the equation has a unique positive solution . Hence, the equation has a unique solution .
Choosing constant large enough such that and , we define a bounded open set as follows: It is clear that satisfies conditions and of Lemma 2. On the other hand, by directly calculating we can obtain where From the assumption of Theorem 3, we have From this, we finally have This shows that satisfies condition of Lemma 2. Therefore, system (14) has a -periodic solution . Finally, we have system (6) which has a positive -periodic solution. This completes the proof.
Proof. From Theorem 3, we can obtain that system (6) has a positive periodic solution.
Let be a positive periodic solution of system (6). Choose positive constants , such that From the assumptions of Theorem 4, there exist constant such that for all we have Let be any solution of system (6); we define Lyapunov functional as follows: Calculating the upper right derivation of along system (6) for , we have Further, we define a Lyapunov function as follows:Calculating the upper right derivation of , from (58) we finally can obtain, for all , Integrating both sides of (60) from to and by (56), we derivethen By the definition of and (59) we have Therefore, for we have which, together with (55), lead to and hence . From the boundedness of and (64), it follows that are bounded for . It is obvious that both and satisfy the equations of system (9); then by system (9) and the boundedness of and we know that the derivatives and are bounded. Furthermore, we can obtain that and their derivatives remain bounded on . Therefore is uniformly continuous on . Thus from (62), we have Therefore, This completes the proof of Theorem 4.
Proof. From the global attractivity of bounded positive solutions, we can get that the permanence of system (6).
In this section, to illustrate the generality of our result, we apply Theorem 3 to some particular Lotka-Volterra type competition systems with pure delays. Consider the following periodic -species competition systems: For system ((68)–(70)), we assume the following:
(H2) , are continuous -periodic functions with and . are continuous, positive -periodic functions.
(H3) are -periodic continuous functions with ; , are positive -periodic continuous functions; are nonnegative integrable functions on satisfying .
(H4) are constants, are continuous -periodic functions with . are continuous, positive -periodic functions. Thus from Theorem 3 we have the following.
Corollary 6. Suppose that assumption (H2) holds and there exists a constant , , such that where and the algebraic equation, has a unique positive solution. Then system (68) has at least one positive -periodic solution.
Corollary 7. Suppose that assumption (H3) holds and there exists a constant , , such that where and the algebraic equation, has a unique positive solution. Then system (69) has at least one positive -periodic solution.
Corollary 8. Suppose that assumption (H4) holds and there exists a constant , , such that where and the algebraic equation, has a unique positive solution. Then system (70) has at least one positive -periodic solution.
5. One Example
In this section, we will give one example to illustrate the results obtained in this paper. From the example, we will see that if the conditions of Theorem 3 hold, then the system has a positive periodic solution. If the conditions of Theorem 4 hold, then the system has a positive periodic solution and which is globally attractive.
Example. We consider the following periodic three-species competition systems: