PROPAGATING WAVES IN DISCRETE BISTABLE REACTION-DIFFUSION SYSTEMS

We consider a discrete bistable reaction-diffusion system modeled by N coupled Nagumo equations. We develop an asymptotic method to describe the phenomenon of propagation failure. The Nagumo model depends on two parameters: the coupling constant d and the bistability parameter a. We investigate the limit a --> 0 and d(a) --> 0 and construct traveling front solutions. We obtain the critical coupling constant d = d*(a) above which propagation is possible and determine the propagation speed c = c(d) if d > d*. We investigate two different cases for the initiation of a propagating front solution. Case 1 considers a uniform steady state distribution. A propagating front appears as the result of a fixed boundary condition. Case 2 also considers a uniform steady state distribution but a propagating front appears as the result of a localized perturbation.