  The propagation of unstable interfaces is at the origin of remarkable
patterns that are observed in various areas of science as chemical reactions,
phase transitions, growth of bacterial colonies. Since a scalar equation
generates usually stable waves, the simplest mathematical description relies on
two by two reaction-diffusion systems. Our interest is the extension of the
Fisher/KPP equation to a two species reaction which represents reactant
concentration and temperature when used for flame propagation, bacterial
population and nutrient concentration when used in biology. We study in which
circumstances instabilities can occur and in particular the effect of
dimension. It is observed numerically that spherical waves can be unstable
depending on the coefficients. A simpler mathematical framework is to study
transversal instability, that means a one dimensional wave propagating in two
space dimensions. Then, explicit analytical formulas give explicitely the range
of paramaters for instability.
