  We consider a periodic reaction diffusion system which, because of
competition between $u$ and $v$, does not enjoy the comparison principle. It
also takes into account mutations, allowing $u$ to switch to $v$ and vice
versa. Such a system serves as a model in evolutionary epidemiology where two
types of pathogens compete in a heterogeneous environment while mutations can
occur, thus allowing coexistence.We first discuss the existence of nontrivial
positive steady states, using some bifurcation technics. Then, to sustain the
possibility of invasion when nontrivial steady states exist, we construct
pulsating fronts. As far as we know, this is the first such construction in a
situation where comparison arguments are not available.
