Abstract

A discrete two-species competitive model is investigated. By using some preliminary lemmas and constructing a Lyapunov function, the existence and uniformly asymptotic stability of positive almost periodic solutions of the system are derived. In addition, an example and numerical simulations are presented to illustrate and substantiate the results of this paper.

1. Introduction

Gopalsamy has presented the following Lotka-Volterra competitive system with continuous time version in 1992 (see [1]): where represent the population densities of two competing species; are the intrinsic growth rates of species; stand for the rates of intraspecific competition of the first and second species, respectively; denote the competitive response function, respectively. All the coefficients above are continuous and bounded above and below by positive constants.

The discrete-time systems governed by difference equations recently have been won wide-spread attention and applied in studying population growth, the transmission of tuberculosis and HIV/AIDS and influenza prevention and control (see [2, 3]), just because discrete-time models conform better to the reality than the continuous ones, especially for the populations with a short life expectancy or non-overlapping generations. In addition, some works about the bifurcation, chaos, and complex dynamical behaviors of the discrete specie systems have been done (see [4, 5]). In practice, according to the discrete data measured, the discrete-time models commonly provide efficient computational models of continuous models for numerical simulations (see [2, 3, 6–11]). Therefore, we derive the discrete analogue of system (1) by using the same discretization method (see [11]): where (; ) denote the densities of species at the th generation, stand for the natural growth rates of species at the th generation, represent the self-inhibition rate, respectively, and and are the interspecific effects of the th generation of species on species .

From an evolutionary perspective, because of the selectivity of species evolution, the periodically varying environments are of vital importance for survival of the fittest. For instance, any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes (see [11–16]). Therefore, the coefficients of many systems constructed in ecology are usually considered as periodic functions (see [12, 13]). Not long ago, Wang (see [12]) studied a delayed predator-prey model with Hassell-Varley type functional responses and obtained the sufficient conditions for the existence of positive periodic solutions by applying the coincidence degree theorem. Many excellent results concerned with the discrete periodic systems are obtained (see [14–16]).

In nature, however, there hardly exists necessarily commensurate periods in the various environment components like seasonal weather change, food supplies, mating habits and harvesting, and so forth. Compared with the periodic systems, we can thus incorporate the assumption of almost periodicity of the coefficients of (1) to reflect the time-dependent variability of the environment (see [6, 8–10, 17]). Recently, Li et al. (see [18]) have proposed an almost periodic discrete predator-prey models with time delays and investigated permanence of the system and the existence of a unique uniformly asymptotically stable positive almost periodic sequence solution. Afterwards, by using Mawhins continuation theorem of the coincidence degree theory, reference [19] achieved some sufficient conditions for the existence of positive almost periodic solutions for a class of delay discrete models with Allee-effect.

Notice that the investigation of periodic solutions and almost periodic solutions is one of the most important topics in the qualitative theory of the difference equations. In this paper, based on the ideas mentioned above, for system (2), one carries out two main works.(i)Assume that all the coefficients , , , and are bounded nonnegative periodic sequences. We explore the existence and global stability of positive periodic solutions of system (2) with positive periodic coefficients.(ii)Furtherly, one discusses the almost periodic solutions of system (2) with positive almost periodic coefficients.

The organization of this paper is as follows. In Section 2, we present some notations and preliminary lemmas. In Section 3, we seek sufficient conditions which ensure the existence and global stability of positive periodic solutions of system (2). In Section 4, we further investigate the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solutions for system (2) above. In Section 5, we present an example and its numerical simulations are carried out to illustrate the feasibility of our main results. In Section 6, a conclusion is given to conclude this work.

2. Notations and Preliminaries Lemmas

Throughout this paper, the notations below will be used: where is a bounded sequence and .

Denote by , , , and the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. and are the cones of -dimensional and -dimensional real Euclidean spaces, respectively.

Definition 1 (see [10]). A sequence is called an almost periodic sequence provided that the following -translation set of is a relatively dense set in for all ; that is, for any given , there exists an integer such that each discrete interval of length contains a such that is referred to as the -translation number of .

Definition 2 (see [10]). Suppose , where is an open set in . is said to be almost periodic in uniformly for or uniformly almost periodic for short, if for any and any compact set in there exists a positive integer such that any interval of length contains an integer for which is called the -translation number of .

Lemma 3 (see [10]). is an almost periodic sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as . Thus, the limit sequence is also an almost periodic sequence.

Furthermore, we consider the following almost periodic difference system: where , , and is almost periodic in uniformly for and is continuous in .

The product system of (7) is in the following form: and [20] obtained the following lemma, where is a solution of (8).

Lemma 4 (see [20]). Suppose there exists a Lyapunov function defined for , , and satisfying the following conditions:(1), where with and is continuous, increasing in ;(2), where is a constant;(3), where is a constant and .Moreover, suppose that there exists a solution of system (7) such that for all ; then there exists a unique uniformly asymptotically stable almost periodic solution of system (7) which satisfies . In particular, if is periodic of period , then system (7) has a unique uniformly asymptotically stable periodic solution with period .

Lemma 5 (see [21]). Any positive solution of system (2) satisfies

Lemma 6 (see [21]). Let system (2) satisfy the following assumptions: Then, any positive solution of system (2) satisfies

3. Existence and Stability of Positive Periodic Solutions

Apparently, the permanence of system (2) can be obtained according to Lemmas 5 and 6. In the following, we will show the existence and stability of positive periodic solutions of system (2). To this end, let us assume that all the coefficients of system (2) are -periodic; namely,

Lemma 7 (see [16]). If the assumption (10) holds, then system (2) has at least one strictly positive -periodic solution and is denoted by .

Definition 8. A positive periodic solution of system (2) is globally stable if each other solution with positive initial value defined for all satisfies Now, we present the main results.

Theorem 9. Let the following assumption and (10) hold; then the positive periodic solution of system (2) is globally stable.

Proof. Let be a positive periodic solution of system (2).
Denote and ; then we have which, according to the mean value, yields where all the constants . Obviously, together with (14) we can find a sufficiently small such that It follows from Lemmas 5 and 6 that there exists an such that ; we have Then one obtains the fact that both and are between and . Similarly, both and are between and . From the first equation of (2), one has Similar to the arguments as above, we must have We denote ; then . Therefore, provided that , Consequently, , where . By Definition 8, it follows that the positive periodic solution of system (2) is globally stable. This completes the proof.

4. Existence and Stability of Positive Almost Periodic Solutions

In this section, we discuss the existence of positive almost periodic solutions of system (2).

Lemma 10. If assumption (10) is true, then .

Proof. According to an inductive argument, system (2) is actually described as follows: Combining Lemmas 5 and 6, for any solution of system (2) and an arbitrarily small constant , there must be which is sufficiently large such that Assuming that is any positive integer sequence such that as , we can prove that there is a subsequence of still denoted by , such that , uniformly in on any finite subset of as , where , , and is a finite number.
In fact, for any finite subset , , , when is large enough. Therefore, , ; that is, are uniformly bounded when is sufficiently large.
Next, for , we choose a subsequence of such that and uniformly converge on for sufficient large.
Similar to the arguments of , for , one can select a subsequence of such that and uniformly converge on for large enough .
Repeating above-mentioned process, for , one obtains a subsequence of such that and uniformly converge on for sufficiently large .
Based on the above, one selects the sequence which is a subsequence of still denoted by ; then, for , one gets , uniformly in as . So the conclusion holds truely due to the arbitrariness of .
Different from Section 3, we suppose that all the coefficients , , , and , , are bounded nonnegative almost periodic sequences; for the above sequence , as , there exists a subsequence denoted by such that as uniformly on .
For any , assume that when is large enough. By an inductive argument of system (2) from to , where , one obtains Hence, (25) yields Let ; one has It is easy to see that is a solution of system (2) on for arbitrary , and Then we get (29) due to which is an arbitrarily small positive constant: This completes the proof.