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Almost Periodic Solution of a Discrete Schoener’s Competition Model with DelaysDOI: 10.1155/2014/256094 Abstract: We consider an almost periodic discrete Schoener’s competition model with delays. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result. 1. Introduction In 2009, Wu et al. [1] had studied a discrete Schoener’s competition mode with delays: where , and are real positive bounded sequences and are positive integers, . Sufficient conditions which guarantee the permanence and the global attractivity of positive solutions for system (1) are obtained. By the biological meaning, the system (1) is considered together with the following initial condition: where . Let be any solution of system (1) with the initial condition (2). One could easily see that , for all . Schoener’s competition system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated, and many excellent results have been derived (see [1–6] and the references cited therein). Recently, a few papers investigate the global stability of the pure-delay model (see [7–13]). Notice that the investigation of almost periodic solutions for difference equations is one of most important topics in the qualitative theory of difference equations due to the applications in biology, ecology, neural network, and so forth (see [14–21] and the references cited therein). But to the best of the author’s knowledge, to this day, still no scholars have studied the almost periodic version which is corresponding to system (1). Therefore, with stimulation from the works of [12, 19], the main purpose of this paper is to derive a set of sufficient conditions ensuring the existence of a unique strictly positive almost periodic solution of system (1) which is globally attractive. Denote by and the set of integers and the set of nonnegative integers, respectively. For any bounded sequence defined on , define . Throughout this paper, we assume the following.(H1) , and are bounded positive almost periodic sequences such that The remaining part of this paper is organized as follows. In Section 2, we will introduce some definitions and several useful lemmas. In Section 3, by applying the theory of difference inequality, we present the permanence results for system (1). In Section 4, we establish the sufficient conditions for the existence of a unique
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