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The Fisher Equation

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.

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The Fisher Equation

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  1. The Fisher Equation Gene Dispersion Within a Population

  2. Sir Ronald Fisher • 1890-1962 • Renown statistician and geneticist. • Wrote mathematics for biologists, and biology for mathematicians.

  3. 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

  4. 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.

  5. 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.

  6. 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)

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