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Propagating waves in discrete bistable reaction-diffusion systems. (English) Zbl 0787.92010
Summary: 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 \(a \to 0\) and \(d(a) \to 0\) 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.

MSC:
92C20 Neural biology
34B99 Boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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