Competition, mutation, expansion & extinction on a rough front
Reproductive competition (fitness) and mutation of two species, say A and B:
[For fixed population size the gA+gB=0 .]
Fractionf of A particles changes as
[More generally g(f) .]
Undirected invasion/expansion into neighboring spaces leads to a "diffusion term":
The more fit species (f=1) invades and replaces the less fit one (f=0), as
Stochasticity in reproduction (so-called demographic noise) broadens the above to
Extinction of the active phase (to the absorbing statef=0) belongs to the Directed Percolation universality class (described by the above field theory)
Layer by layer growth (each layer regarded as time slice) provides an example of this transition
How is this formulation modified for extinction at the rough front of range expansion?
In the same way that the local expansion velocity depends on local shape (through curvature and slope)
Competitive reproduction and mutation rates could in principle depend upon local front shape (its curvature and slope):
The expansion velocity, conversely, could depend upon local composition through f :
The extinction transition in range expansion with a rough front, can be studied systematically by
including the lowest order couplings in a gradient expansion between height and concentration fluctuations, leading to
"Bacterial range expansions on a growing front: Roughness, Fixation, and Directed Percolation,"
Reproductive competition (fitness) and mutation of two species, say A and B:
[For fixed population size the gA+gB=0 .]
Fraction f of A particles changes as
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