Extinction on a rough front

red ballConsider growth of two species, say active and inactive, on a flat front

yellow ball The active particles have selective advantage  g=ga-gi  but mutate to inactive form at rate  μ;

In a well-mixed (mean-field) limit, the fraction   f   of active particles grows as

As  μ=>g the active fraction becomes extinct, giving way to an absorbing state at f =0.

Including spatial variations, diffusion, and stochasticity in reproduction events, leads to the field theory

Extinction transitions (to adsorbing states) belong to the Directed Percolation universality class, (near at f =0) described by

            

yellow ball How is this formulation modified for extinction at the rough front of range expansion?

                

Generic form of equation governing roughness of a growing front is

red ball Including lowest order couplings in a gradient expansion between height and concentration fluctuations at the front, leads to

     

    

yellow ball"Bacterial range expansions on a growing front: Roughness, Fixation, and Directed Percolation,"

J. Horowitz & M. Kardar PRE 99, 042134 (2019) (off-line)

Related equations were proposed and studied in connection with binary alloy ordering for a growing film:

"Interplay between phase ordering and roughening on growing films,"

B. Drossel & M. Kardar, Eur. J. Phys. B 36, 401 (2003)  (off-line)

yellow ballNon-linear terms are relevant below 4 dimensions, different criticality from standard directed percolation expected.

Renormalization group flows are to strong coupling, with no pertinent fixed point perturbatively accessible.