
The Geometry of Population Genetics
Ethan Akin(Author)
Springer (Publisher)
Published on 1. November 1979
Book
Paperback/Softback
IV, 208 pages
978-3-540-09711-2 (ISBN)
Description
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].
More details
Series
Edition
Softcover reprint of the original 1st ed. 1979
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Research
Illustrations
IV, 208 p.
Dimensions
Height: 244 mm
Width: 170 mm
Thickness: 12 mm
Weight
382 gr
ISBN-13
978-3-540-09711-2 (9783540097112)
DOI
10.1007/978-3-642-93128-4
Schweitzer Classification
Content
I. The Vectorfield Model of Population Genetics.- 1. The Equations of Selection, Recombination and Mutation.- 2. Multivariate Analysis and Types of Epistasis.- 3. Euclidean Vector Spaces and Riemannian Manifolds.- 4. The Shahshahani Metric.- 5. The Product Theorems and Epistasis.- 6. The Selection Field.- 7. The Recombination Field.- 8. The Mutation Field.- 9. The Combined Field.- II. The Geometry of Epistasis.- 1. Orthogonal Decompositions.- 2. The Product Model.- III. Selection, Recombination and Mutation.- 1. Selection and Epistasis.- 2. Recombination and Entropy.- 3. Recombination and Epistasis.- 4. Position Effects.- 5. Mutation.- IV. The Hopf Bifurcation.- 1. The Hessian.- 2. The Wright Conjecture.- 1. Proper Mappings.- 2. Partially Defined Distributions.- 3. Game Dynamics.