The Fisher Equation

Gene Dispersion Within a Population

Sir Ronald Fisher

• 1890-1962

• Renown statistician and geneticist.

• Wrote mathematics for biologists, and biology for mathematicians.

Simplified Behavior

• We simplify the situation to only two variables and two parameters.

• Defining f(u) = s*u*(1-u)

• u’ = v

• v’ = -f(u) + c*v

• We get an interesting model.

• http://math.rice.edu/~dfield/dfpp.html

Slightly More Complicated View

• Assumptions:

– A population is distributed in a linear habitat.

– It is uniformly distributed.

– There are only two alleles present for the specified locus.

Variables and parameters

• p = frequency of the mutant gene.

• q = frequency of other allele.

• m = intensity of selection in favor of p.

• x = position along the habitat.

• t = time in generations.

• k = constant of diffusion.

• Assumption: p and m are independent.

Cases for c

• (a) c = 1

• (b) c is between 1 and sqrt(1/2)

• (c) c = sqrt(1/2)

• (d) c < sqrt(1/2)