Global Stability of Positive Periodic Solutions and Almost Periodic Solutions for a Discrete Competitive System

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.

Dynamic Behaviors of a Competitive System with Beddington-DeAngelis Functional Response

“…Our main results Theorems 2 and 11 are in accordance with Lemma 2.2 and Theorem 3.1 in [2]; Theorems 3, 4, 8, and 9 are also consistent with Theorem 1-4 in [3], so our results generalize [2,3]. Moreover, when ( ) = 1 and ( ) = 0 ( = 1, 2), system (2) reduces to continuous analogue of system (7) investigated by Ma, Gao, and Xie [13]. Since we consider the permanence, global attractivity, and existence of unique almost periodic solution for the system which Ma et al [13] and Chen et al [14] have not studied, our results also supplement the main results of [13,14].…”

“…Hence, the goal of this paper is to obtain results on permanence, partial extinction, and the existence of a unique almost periodic solution of system (2) and (3). Our results supplement the main results of [13,14] and generalize [2,3]. Many important results concerned this direction; one could see [15][16][17][18] and so on.…”

“…Moreover, when ( ) = 1 and ( ) = 0 ( = 1, 2), system (2) reduces to continuous analogue of system (7) investigated by Ma, Gao, and Xie [13]. Since we consider the permanence, global attractivity, and existence of unique almost periodic solution for the system which Ma et al [13] and Chen et al [14] have not studied, our results also supplement the main results of [13,14]. Numeric simulations validate our analytical results.…”

“…We [11,12] investigated the effect of feedback control variables on permanence and extinction of (6). Motivated by Gopalsamy [1], Ma, Gao, and Xie [13] investigated the following discrete two-species competitive system:…”