NUMERICAL ANALYSIS OF WEAK RANDOM DRIFT IN A CLINE

Mitchell Luskin ¹ and Thomas Nagylaki ²

¹ Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
² Department of Biophysics and Theoretical Biology, The University of Chicago, 920 East 58th Street, Chicago, Illinois 60637

The equilibrium state of a diffusion model for weak 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 ratio of the variance of either gene frequency to the product of the expected gene frequencies decreases monotonically to a non-zero 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. All the results are parameter free. Some asymptotic formulae are derived analytically.