Permanence and global attractivity for facultative mutualism system with delay

Abstract

SUMMARYIn this paper, we consider a facultative mutualism system with di erent delays. Su cient criteria for permanence and global attractivity for the system are established. Ultimate uniform boundedness of the solutions ensures permanence. For the global attractivity of the system, magnitude of the delays plays a major role.

Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls

“…For example, Wang and Huang [1] analyzed permanence of a predator-prey model with harvesting predator. Mukherjee [2] addressed the permanence and global attractivity for facultative mutualism predator-prey model, Zhao and Jiang [3] considered the permanence and extinction for Lotka-Volterra model, Teng et al [4] established the permanence criteria for a delayed discrete species systems, Liu et al [5] studied the permanence and periodic solutions for reaction-diffusion food-chain system with impulsive effect. For more detailed research about this topic, one can see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Permanence criteria for Kolmogorov systems with delays

“…where A i is the ith row of the N × N matrix A, Jansen [11] (see also [7,Ch.13]) proved that (1) is permanent if there is a vector q ∈ intR N + such that the inequality q T (r + Ax) > 0 holds for every fixed pointx ∈ ∂R N + . Examples of permanence for special delayed Kolmogorov systems are given by Chen, Lu and Wang [5], Hou [8]- [10], Li and Teng [13], Liu and Chen [14], Lu, Lu and Enatsu [15], Mukherjee [16], Teng [19], and the references therein. In particular, for autonomous Lotka-Volterra differential systems with multiple delays, sufficient conditions for permanence, which are easily checkable inequalities involving the constant coefficients of the system, were obtained in [15].…”

Permanence in a Discrete Mutualism Model with Infinite Deviating Arguments and Feedback Controls

“…For example, Fan and Li [1] analyzed permanence of a delayed ratio-dependent predator-prey model with Holling type functional response. Mukherjee [2] addressed the permanence and global attractivity for facultative mutualism system with delay. Zhao and Jiang [3] focused on the permanence and extinction for nonautonomous Lotka-Volterra system.…”

“…where ( ) ( = 1, 2) is the density of mutualism species at the generation, { ( )}, { ( )}, { ( )}, { ( )} ( = 1, 2), 1 ( ), and 2 ( ) are bounded nonnegative sequences. Applying the comparison theorem of difference equation and some lemmas, they derived some sufficient conditions which guarantee the permanence of system (2). It is well known that ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates [35].…”