  We study a one-dimensional reaction-diffusion system which describes an
isothermal autocatalytic chemical reaction involving both a quadratic (A + B ->
2B) and a cubic (A + 2B -> 3B) autocatalysis. The parameters of this system are
the ratio D = D_B/D_A of the diffusion constants of the reactant A and the
autocatalyst B, and the relative activity k of the cubic reaction. First, for
all values of D > 0 and k >= 0, we prove the existence of a family of
propagating fronts (or travelling waves) describing the advance of the
reaction. In particular, in the quadratic case k=0, we recover the results of
Billingham and Needham [BN]. Then, if D is close to 1 and k is sufficiently
small, we prove using energy functionals that these propagating fronts are
stable against small perturbations in exponentially weighted Sobolev spaces.
This extends to our system part of the stability results which are known for
the scalar Fisher equation.
