Wave propagation in a two-dimensional lattice dynamical system with global interaction

Spreading dynamics of a discrete Nicholson's blowflies equation with distributed delay

“…Besides, they are also natural outcomes of discretizing the corresponding spatial continuous model. In the literature, some discretizations of classical continuous models have been proposed, such as the discrete Fisher's equation [28], discrete Nagumo equation [27], discrete Allen-Cahn equation [3] and discrete Lotka-Volterra equation [7], etc., and there have been many studies focussed on the study of propagation dynamics of various types of discrete equations, see [4,9,10,14,[19][20][21][22][23][24][25][26] and the references therein. However, the approximation of continuous models by spatial discretization is a delicate issue, and it is well known that there could exist essential differences between a discrete model and its associated continuous version.…”

Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction

“…These can be seen in [9,14,29,37] and the references therein. Recently, there is a particular interest on studying the species population living in a patchy environment consisting of all integer nodes, see [7,8,34,35].…”

“…While The authors [7] studied the existence of the asymptotic speed of propagation, the existence of monotone traveling waves and the minimal wave and its relation with the asymptotic speed of propagation. Recently, Xu [35] further showed that for any given admissible speed, all the wave profiles propagating toward a fixed direction of (1.1) have the same asymptotic behavior when they approach the limiting states, which plays a very important role in the stability of traveling waves. Meanwhile, in the past decades, there are various surveys focusing on the existence, uniqueness, asymptotic behavior and stability of traveling waves for LDEs and its continuum RDEs( [1-4, 6-9, 11, 13, 16, 17, 19-22, 37-39]).…”

“…The existence of traveling wave fronts for (2.1) with the boundary condition (2.2) can be easily verified by using the monotone iteration technique combined with the sub-sup solutions, see [7]. Recently, Xu [35] derived the asymptotic behavior of traveling waves of (1.1), which is the key premise. for some m ∈ R, where l = 0 when c > c * and l = 1 when c = c * .…”

“…[35, Theorem 2.1] Assume (H1) − (H4) hold. For any c ≥ c * , let (c, ϕ) be a solution of (2.1) with boundary conditions (2.2), then lim x→−∞ ϕ(x) = (m − x) l e λ1(c)x…”