Random perturbations of reaction–diffusion waves in biology

This paper considers the statistical properties of the traveling wave fronts of the scalar FitzHugh–Nagumo equation with random perturbations by two-parameter white noise ut=uxx+f(u)+εWxtut=uxx+f(u)+εWxt on the whole real line ℛℛ, where the traveling wave front connects two stable equilibria u=0u=0 and u=1u=1 of the reaction function f(u)f(u). As well as the method of Green’s function established by Tuckwell on a bounded domain, we get the asymptotic fluctuation behavior of two stable states which are two boundaries of the traveling wave front to the Nagumo equation by the fundamental solution. That is, the perturbations about the lower (upper) stable state reveal that the mean is increased (decreased) by zero mean white noise as t→+∞t→+∞.