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We study the change in the speed of pushed and bistable fronts of thereaction diffusion equation in the presence of a small cut-off. We giveexplicit formulas for the shift in the speed for arbitrary reaction terms f(u).The dependence of the speed shift on the cut-off parameter is a function of thefront speed and profile in the absence of the cut-off. In order to determinethe speed shift we solve the leading order approximation to the front profileu(z) in the neighborhood of the leading edge and use a variational principlefor the speed. We apply the general formula to the Nagumo equation and recoverthe results which have been obtained recently by geometric analysis. Theformulas given are of general validity and we also apply them to a class ofreaction terms which have not been considered elsewhere.

Shift in the speed of reaction diffusion equation with a cut-off: pushed and bistable fronts