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We consider a periodic reaction diffusion system which, because ofcompetition between $u$ and $v$, does not enjoy the comparison principle. Italso takes into account mutations, allowing $u$ to switch to $v$ and viceversa. Such a system serves as a model in evolutionary epidemiology where twotypes of pathogens compete in a heterogeneous environment while mutations canoccur, thus allowing coexistence.We first discuss the existence of nontrivialpositive steady states, using some bifurcation technics. Then, to sustain thepossibility of invasion when nontrivial steady states exist, we constructpulsating fronts. As far as we know, this is the first such construction in asituation where comparison arguments are not available.

Pulsating fronts for Fisher-KPP systems with mutations as models in evolutionary epidemiology