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The Geometry of Population Genetics

  • Book
  • © 1979

Overview

Authors:
  1. Ethan Akin

    1. Mathematics Department, The City College, New York City, USA

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Part of the book series: Lecture Notes in Biomathematics (LNBM, volume 31)

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Table of contents (5 chapters)

  1. Front Matter

    Pages I-IV
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  2. Introduction

    • Ethan Akin
    Pages 1-2

  3. The Vectorfield Model of Population Genetics

    • Ethan Akin
    Pages 3-79

  4. The Geometry of Epistasis

    • Ethan Akin
    Pages 80-118

  5. Selection, Recombination and Mutation

    • Ethan Akin
    Pages 119-172

  6. The Hopf Bifurcation

    • Ethan Akin
    Pages 173-190

  7. Back Matter

    Pages 191-208
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Keywords

About this book

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].

Authors and Affiliations

  • Mathematics Department, The City College, New York City, USA

    Ethan Akin

Bibliographic Information

  • Book Title: The Geometry of Population Genetics

  • Authors: Ethan Akin

  • Series Title: Lecture Notes in Biomathematics

  • DOI: https://doi.org/10.1007/978-3-642-93128-4

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1979

  • Softcover ISBN: 978-3-540-09711-2Published: 01 November 1979

  • eBook ISBN: 978-3-642-93128-4Published: 09 April 2013

  • Series ISSN: 0341-633X

  • Series E-ISSN: 2196-9981

  • Edition Number: 1

  • Number of Pages: IV, 208

  • Topics: Differential Geometry, Mathematical and Computational Biology

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