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

<p style='text-indent:20px;'>This paper is concerned with the traveling wave fronts for a lattice dynamical system with global interaction, which arises in a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. We prove that all non-critical traveling wave fronts are globally exponentially stable in time, and the critical traveling wave fronts are globally algebraically stable by the weighted energy method combined with the comparison principle and the discrete Fourier transform.</p>

Stability of traveling wavefronts for a 2D lattice dynamical system arising in a diffusive population model

This paper is concerned with the traveling wavefronts of a 2D two-component lattice dynamical system. This problem arises in the modeling of a species with mobile and stationary subpopulations in an environment in which the habitat is two-dimensional and divided into countable niches. The existence and uniqueness of the traveling wavefronts of this system have been studied in (Zhao and Wu in Nonlinear Anal., Real World Appl. 12: 1178–1191, 2011). However, the stability of the traveling wavefronts remains unsolved. In this paper, we show that all noncritical traveling wavefronts with given direction of propagation and wave speed are exponentially stable in time. In particular, we obtain the exponential convergence rate.

Minimal Wave Speed in a Delayed Lattice Dynamical System with Competitive Nonlinearity

This paper deals with the minimal wave speed of delayed lattice dynamical systems without monotonicity in the sense of standard partial ordering in R2. By constructing upper and lower solutions appealing to the exponential ordering, we prove the existence of traveling wave solutions if the wave speed is not smaller than some threshold. The nonexistence of traveling wave solutions is obtained when the wave speed is smaller than the threshold. Therefore, we confirm the threshold is the minimal wave speed, which completes the known results.

Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models

In this paper, we study a two-component Lotka–Volterra competition systemon a one-dimensional spatial lattice. By the comparison principle, together with the weighted energy, we prove that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as j + ct → −∞, where j ∈ , t > 0, but the initial perturbation can be arbitrarily large on other locations. This partially answers an open problem by J.-S. Guo and C.-H.Wu.