Propagation and its failure in a lattice delayed differential equation with global interaction
The authors study the following lattice delayed differential equation with global interaction \[ u_n^{\prime}=D[u_{n+1}+u_{n-1}-2u_n]-du_n+\sum\limits_{i=-\infty}^{\infty}J(i)b(u_{n-i}(t-r)), n=1,2,\dots \tag{1} \] Equation (1) describes the evolution of a single-species in one-dimensional patchy environment with infinite number of patches connected locally by diffusion. It is assumed in equation (1) that \(J(i)=J(-i)\geq 0, \sum_{i}J(i)=1\) and \(\sum_{i}| i| J(i)<+\infty,\) where \(\sum_{i}\) denotes the sum over \(i\in \mathbb{Z}\). It is also assumed that the birth function \(b\in C^1(\mathbb{R})\) and there exists a constant \(K>0\) such that \(b(0)=dK-b(K)=0.\) In the bistable case, under realistic assumptions on the birth function, it is proved that equation (1) admits a strictly monotone increasing travelling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotically stable with phase shift. Of particular interest is the phenomenon of ``propagation failure'' or ``pinning'' (i.e., wave speed \(c=0\)), some criteria for pinning are given in this paper, too.
scientific article; zbMATH DE number 2164826
scientific article