Permanence for the discrete mutualism model with time delays

Permanence of a nonlinear competition model with delay

“…For example, Chen and Shi [27] investigated the following nonlinear model: 6) where i = 1, 2, ..., n, j = 1, 2, ..., m, x i (t) denotes the density of prey species X i at time t, y j (t)…”

Almost periodic sequence solution of a discrete Hassell–Varley predator-prey system with feedback control

“…From Lemma 2.2 in [5], we can get the following lemma directly: Proof. By (H 1 ) and the first two equations of (1.3), we see that x i (k) > 0, i = 1, 2.…”

“…Discrete time models can also provide efficient computational models of continuous models for numerical simulations. Indeed, much progress in the discrete time models have been made by many scholars, see, for examples, [3][4][5]7,[9][10][11] and references cited therein. Recently, Wu and Li [4] studied the dynamic of the discrete predator-prey system with Hassell-Varley type functional response as follows:…”

Global attractivity and almost periodic solution of a discrete mutualism model with delays

“…Furthermore, the limit sequence is also an almost periodic sequence. Lemma () Assume that { x ( n )} satisfies x ( n ) > 0 and for n ∈ N , where a ( n ) and b ( n ) are non‐negative sequences bounded above and below by positive constants. Then, Lemma () Assume that { x ( n )} satisfies and x ( N 0 ) > 0, where a ( n ) and b ( n ) are non‐negative sequences bounded above and below by positive constants and N 0 ∈ N . Then, …”