NUMERICAL ANALYSIS OF RANDOM DRIFT IN A CLINE

Thomas Nagylaki ¹ and Bradley Lucier ²

¹ Dept. of Biophysics and Theoretical Biology, The University of Chicago, Chicago, Illinois 60637
² Dept. of Mathematics, The University of Chicago, Chicago, Illinois 60637

The equilibrium state of a diffusion model for random genetic drift in a cline is analyzed numerically. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneous, symmetric and independent of genotype. A single diallelic locus with a step environment is investigated in the absence of dominance and mutation. The flattening of the expected cline due to random drift is very slight in natural populations. The ratio of the variance of either gene frequency to the product of the expected gene frequencies decreases monotonically to a nonzero constant. The correlation between the gene frequencies at two points decreases monotonically to zero as the separation is increased with the average position fixed; the decrease is asymptotically exponential. The correlation decreases monotonically to a positive constant depending on the separation as the average position increasingly deviates from the center of the cline with the separation fixed. The correlation also decreases monotonically to zero if one of the points is fixed and the other is moved outward in the habitat, the ultimate decrease again being exponential. Some asymptotic formulae are derived analytically.——The loss of an allele favored in an environmental pocket is investigated by simulating a chain of demes exchanging migrants, the other assumptions being the same as above. For most natural populations, provided the allele would be maintained in the population deterministically, this process is too slow to have evolutionary importance.