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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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