Almost Periodic Solution of a Discrete Commensalism System

Abstract

A nonautonomous discrete two-species Lotka-Volterra commensalism system with delays is considered in this paper. Based on the discrete comparison theorem, the permanence of the system is obtained. Then, by constructing a new discrete Lyapunov functional, a set of sufficient conditions which guarantee the system global attractivity are obtained. If the coefficients are almost periodic, there exists an almost periodic solution and the almost periodic solution is globally attractive.

Permanence, partial survival, extinction, and global attractivity of a nonautonomous harvesting Lotka–Volterra commensalism model incorporating partial closure for the populations

“…Recently, many scholars [12][13][14][15][16][17][18][19][20][21] studied the dynamic behavior of the commensalism model; however, none of them consider the influence of harvesting. Stimulated by the recent works of Chakraborty, Das, and Kar [24], we propose a nonautonomous nonselective commensalism model incorporating partial closure to the population.…”

“…Only recently scholars paid attention to such a kind of relationship; see [12][13][14][15][16][17][18][19][20] and the references therein. Topics such as the existence of the positive periodic solution [17], the existence of a positive almost periodic solution [14], the existence and stability of the positive equilibrium [16], the influence of the impulsive [15] were investigated, and many excellent results were obtained. However, as was pointed out by Georgescu and Maxin [20], "One would think that the stability of the coexisting equilibria for two-species models of commensalism would follow immediately from the corresponding results for models of mutualism, when these results are available, .…”

Allee effect increasing the final density of the species subject to the Allee effect in a Lotka–Volterra commensal symbiosis model

“…During the last decades, many scholars investigated the dynamic behaviors of the mutualism model or commensalism model . Such topics as the stability of the positive equilibrium [1-3, 8-12, 14-17, 19, 20, 24], the persistence of the system [4,6,7,13], the existence of the positive periodic solution [18,21,22,25], the extinction of the species [5,23] etc. have been extensively investigated.…”

Dynamic behaviors of a stage structure amensalism system with a cover for the first species

“…During the last decades, many scholars investigated the dynamic behavior of the commensalism or amensalism model [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Such topics as the local stability of the equilibrium [2-4, 7, 8, 10-16, 18, 19], the existence of the positive periodic solution [5,17] the existence and stability of the almost periodic solution [6], extinction of the species [8,11,14], and the influence of the cover [14,16,18] have been studied, and many excellent results are obtained. Recently, Xiong, Wang, and Zhang [13] proposed the following amensalism model:…”