Research Article | Open Access
Permanence and Positive Periodic Solutions of a Discrete Delay Competitive System
A discrete time non-autonomous two-species competitive system with delays is proposed, which involves the influence of many generations on the density of species population. Sufficient conditions for permanence of the system are given. When the system is periodic, by using the continuous theorem of coincidence degree theory and constructing a suitable Lyapunov discrete function, sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. As an application, examples and their numerical simulations are presented to illustrate the feasibility of our main results.
In recent years, the application of theories of functional differential equations in mathematical ecology has developed rapidly. Various delayed models have been proposed in the study of population dynamics, ecology, and epidemic. In fact, more realistic population dynamics should take into account the effect of delay. Also, delay differential equations may exhibit much more complicated dynamic behaviors than ordinary differential equations since a delay could cause a stable equilibrium to become unstable and cause the population to fluctuate (see ). One of the famous models for dynamics of population is the delay Lotka-Volterra competitive system. Owing to its theoretical and practical significance, various delay competitive systems have been studied extensively (see [2–8]). Although much progress has been seen for Lotka-Volterra competitive systems, such systems are not well studied in the sense that most results are continuous time versions related. Many authors [9–11] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have non-overlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. Therefore, the dynamic behaviors of population models governed by difference equations have been studied by many authors, see [12–18] and the references cited therein. Noting that some studies of the dynamics of natural populations indicate that the density-dependent population regulation probably takes place over many generations [19, 20], many authors have discussed the influence of many past generations on the density of species population and discussed the dynamic behaviors of competitive, predator-prey, and cooperative systems (see [21–24]).
Motivated by the above work [19–24], in this paper we will investigate the following discrete time non-autonomous two-species competitive system with delays: with the initial conditions where represents the density of population at the th generation, is the intrinsic growth rate of population at the th generation, measures the intraspecific influence of the th generation of population on the density of its own population, and stands for the interspecific influence of the th generation of population on population , and . The coefficients , , and are bounded nonnegative sequences. The exponential form of the equations in system (1.1) ensures that any forward trajectory of system (1.1) with initial conditions (1.2) remains positive for all . For the investigations of some continuous versions of (1.1) we refer to [8, 25, 26] and the references cited therein.
The principle aim of this paper is to study the dynamic behaviors of system (1.1), such as permanence, existence, and global attractivity of positive periodic solutions. To the best of our knowledge, no work has been done for the discrete non-autonomous difference system (1.1). The paper is organized as follows. In Section 2, we obtain sufficient conditions which guarantee the permanence of system (1.1). In Section 3, a good understanding of the existence and global attractivity of positive periodic solutions of system (1.1) is gained by using the method of coincidence degree theory and a Lyapunov discrete function. Some illustrative examples are given to demonstrate the feasibility of the obtained results in Section 4. To do this, we need to give the following notations and Definitions 1.1 and 1.2.
For the simplicity and convenience of exposition, throughout this paper we let , and denote the sets of all integers, nonnegative integers, nonnegative real numbers and two-dimensional Euclidian vector space, respectively. Meanwhile, we denote that , for any bounded sequence .
In this section, we will establish sufficient conditions for the permanence of system (1.1). To do this, we first give two lemmas which will be useful for establishing our main result in this section.
Lemma 2.1 (see [27, Lemma 1]). Assume that satisfies and for , where is a positive constant and . Then
Lemma 2.2 (see [27, Lemma 2]). Assume that satisfies and for , , and , where is a constant such that and . Then
Before stating Theorem 2.3, for the sake of convenience, we set where , , is a sufficiently small constant.
We are now in a position to state our main result of this section on the permanence of system (1.1).
Theorem 2.3. If the following assumptions: hold, then system (1.1) is permanent.
Proof. Clearly, any solution of system (1.1) satisfies , . The following two steps are considered.Step 1. According to Definition 1.1, we will prove that any positive solution of system (1.1) satisfies for .
It follows from the first equation of system (1.1) that
We will make a convention that if for any bounded sequence . For , , we can obtain that is, in other words,
Consequently, we have
By Lemma 2.1, we can derive that
Similar to the above argument, we can verify that Step 2. By a similar procedure to Step 1, we will prove that any positive solution of system (1.1) satisfies , where
For any sufficiently small , according to (2.5), there exists a positive integer such that for all . Thus, for , it follows from the first equation of system (1.1) that
Therefore, for all and , it follows that that is, where
Combining (2.5) and (2.17) with the first equation of system (1.1) leads to
And hence, by applying Lemma 2.2 and letting , it follows from (2.5)-(2.6) and (2.19) that
Analogously, from the second equation of system (1.1), we can verify that
The proof of Theorem 2.3 is completed by combining Steps 1 and 2.
3. Existence and Global Attractivity of Positive Periodic Solutions
In this section, we will give two main results. We first derive sufficient conditions for the existence of positive periodic solutions of system (1.1). We further assume that are positive -periodic for system (1.1), that is, for any , where , a fixed positive integer, denotes the prescribed common period of the parameters in system (1.1).
In order to obtain sufficient conditions for the existence of positive periodic solutions of system (1.1), we will use the method of coincidence degree. For convenience, we will summarize in the following a few concepts and results from  that will be useful in this section.
Let and be two Banach spaces. Consider an operator equation where is a linear operator and is a parameter. Let and denote two projectors such that
Denote that is an isomorphism of onto . Recall that a linear mapping with and will be called a Fredholm mapping if the following two conditions hold:(i) has a finite dimension;(ii) is closed and has a finite codimension.
Recall also that the codimension of is the dimension of , that is, the dimension of the cokernel coker of .
When is a Fredholm mapping, its index is the integer .
We will say that a mapping is -compact on if the mapping is continuous. is bounded and is compact, that is, it is continuous and is relatively compact, where is an inverse of the restriction of to , so that and .
Lemma 3.1 (see [28, Continuation Theorem]). Let and be two Banach spaces and let be a Fredholm mapping of index zero. Assume that is -compact on with open bounded in . Furthermore assume that(a)for each , , ,(b) for each ,(c).Then the equation has at least one solution in .
In what follows, we will use the following notations: where is an -periodic sequence of real numbers defined for .
Lemma 3.2 (see [29, Lemma 3.2]). Let be -periodic, that is, , then for any fixed and any , one has
Denote that For , define . Let denote the subspace of all -periodic sequences equipped with the usual supremum norm , that is, Then it follows that is a finite dimensional Banach space.
Let Then it follows that and are both closed linear subspaces of and
We are now in a position to state one of the main results of this section on the existence of positive periodic solutions of system (1.1).
Theorem 3.3. Assume that Then system (1.1) has at least one positive -periodic solution.
Proof. We first make the change of variables
By substituting (3.11) into system (1.1), we can get
It is easy to see that if system (3.12) has one -periodic solution, then system (1.1) has one positive -periodic solution. Therefore, to complete the proof, it is only to show that system (3.12) has at least one -periodic solution.
Set . Denote by the difference operator given by with and as follows: for any and . It is easy to see that is a bounded linear operator and then we get that is a Fredholm mapping of index zero.
Define It is not difficult to show that and are continuous projectors such that
Furthermore, the inverse (to ) exists and is given by
Then and are given by
In order to apply Lemma 3.1, we need to search for an appropriately open, bounded subset .
Corresponding to the operator equation , , we have Suppose that is a solution of (3.20) for a certain . Summing both sides of (3.20) from to with respect to , we can derive Since , there exist such that It follows from (3.21) that which implies where , besides, from (3.20) and (3.21) By (3.24), (3.25), and Lemma 3.2, we have
On the other hand, there also exist such that In view of (3.21), we can obtain Therefore, Then That is, By (3.25), (3.31), and Lemma 3.2, we have Inequalities (3.26) and (3.32) imply
Obviously, , and in (3.33) are independent of , respectively. Denote , where is taken sufficiently large such that any solution of the system of algebraic equations satisfies (If system (3.34) has at least one solution). Let , thus condition (a) in Lemma 3.1 holds. When , , is a constant vector in with . If system (3.34) has at least one solution, then If system (3.34) does not have one solution, then it is obvious that This implies that condition (b) in Lemma 3.1 is satisfied.
Now we prove that condition (c) in Lemma 3.1 holds. Define as follows: where is a parameter with . When , is a constant vector in with . We will show that , . If the conclusion is not true, then there is a constant vector with satisfying , that is, A similar argument to the above shows that , which is a contradiction. Using the property of topological degree and taking , , we have Obviously, the following equations: have the unique solution . Therefore, we have
Finally, we will show that is -compact on . For any , we have Hence, is bounded. Obviously, is continuous.
It is easy to see that For any , , without loss of generality, let , then we have
Thus, the set is equicontinuous and uniformly bounded. By using the Arzela-Ascoli theorem, we see that is compact. Consequently, is -compact.
By now, we know that verifies all the requirements in Lemma 3.1 and then system (3.12) has at least one -periodic solution. By the medium of (3.11), we derive that system (1.1) has at least one -periodic solution. This completes the proof of Theorem 3.3.
Next, by constructing a suitable Lyapunov-like discrete function, we further investigate the global attractivity of positive periodic solutions of system (1.1).
Theorem 3.4. In addition to (3.10), assume further that there exists a constant such that where (i=1,2) are defined in (2.5) and Then the positive periodic solution of system (1.1) is globally attractive.
Proof. Let be a positive periodic solution of system (1.1). To finish the proof of Theorem 3.4, we will consider the following two steps.Step 1. Let , then it follows from the first equation of system (1.1) that
By the mean value theorem, we have that is, where lies between and . Then we have And hence it follows from (3.47) and (3.50) that Step 2. Let
For the sake of convenience, we will make a convention that if for any bounded sequence . By a simple calculation, it derives that
Now, we are in a position to define by
Therefore, it follows from (3.51) and (3.53) that
By a similar argument, we can define by where Then it is easy to derive that where is between and .
Now we can define a Lyapunov-like discrete function by
It is easy to see that for all and . For the arbitrariness of and by (3.45), we can choose a small enough such that
Therefore, it follows from (3.55)–(3.60) that