  Directed Percolation (DP) is a classic model for nonequilibrium phase
transitions into a single absorbing state (fixation). It has been extensively
studied by analytical and numerical techniques in diverse contexts. Recently,
DP has appeared as a generic model for the evolutionary/ecological dynamics of
competing bacterial populations. Range expansion -- the stochastic reproduction
of bacteria competing for space to be occupied by their progeny -- leads to a
fluctuating and rough growth front, which is known from experiment and
simulation to affect the underlying critical behavior of the DP transition. In
this work, we employ symmetry arguments to construct a pair of non-linear
stochastic partial differential equations describing the co-evolution of
surface roughness with the composition field of DP. Macroscopic manifestations
(phenomenology) of these equations on growth patterns and genealogical tracks
of range expansion are discussed; followed by a renormalization group analysis
of possible scaling behaviors at the DP transition.
