The Influence of Spatial Inhomogeneities on Neutral Models of Geographical Variation III. Migration across a Geographical Barrier

The equilibrium structure of the infinite, one-dimensional stepping-stone model with a geographical barrier is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes symmetrically. Migration is reduced across the geographical barrier, but is otherwise uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. The two dimensionless parameters in the theory are β = 4ρ0[formula] and κ where ρ0, u, V0, and κ represent the population density, mutation rate, variance of gametic dispersion per generation, and penetrability of the barrier, respectively. The characteristic length is [formula]. Relative to a homogeneous infinite habitat, the barrier raises the probability of identity if the two points of observation are on the same side and lowers it if they are on opposite sides. The former effect is moderate or small, but the latter is large unless transmission is high (κ 1); genetic differentiation across the barrier is very strong for low transmission (κ 1). For points of observation on the same side, the influence of the barrier is significant only if the proximal point is within a few characteristic lengths. Upper and lower bounds on the probability of identity are established, and approximations are derived for four cases: (i) low expected homozygosity (β 1), (ii) high transmission, (iii) low transmission, and (iv) at least one point distant.