The differential equations which model the action of selection and
recombination are nonlinear equations which are impossible to It is
even difficult to describe in general the solve explicitly.
Recently, Shahshahani began using qualitative behavior of
solutions. differential geometry to study these equations [28].
with this mono graph I hope to show that his ideas illuminate many
aspects of pop ulation genetics. Among these are his proof and
clarification of Fisher's Fundamental Theorem of Natural Selection
and Kimura's Maximum Principle and also the effect of recombination
on entropy. We also discover the relationship between two classic
measures of 2 genetic distance: the x measure and the arc-cosine
measure. There are two large applications. The first is a precise
definition of the biological concept of degree of epistasis which
applies to general (i.e. frequency dependent) forms of selection.
The second is the unexpected appearance of cycling. We show that
cycles can occur in the two-locus-two-allele model of selection
plus recombination even when the fitness numbers are constant (i.e.
no frequency dependence). This work is addressed to two different
kinds of readers which accounts for its mode of organization. For
the biologist, Chapter I contains a description of the entire work
with brief indications of a proof for the harder results. I imagine
a reader with some familiarity with linear algebra and systems of
differential equations. Ideal background is Hirsch and Smale's text
[15].
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