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Huxley and Fisher equations for gene propagation: An exact solution

Published online by Cambridge University Press:  17 February 2009

P. Broadbridge ,
B. H. Bradshaw ,
G. R. Fulford  and
G. K. Aldis
P. Broadbridge
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia; e-mail: [email protected].
B. H. Bradshaw
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia; e-mail: [email protected].
G. R. Fulford
Affiliation:
AgResearch Ltd., Wallaceville Animal Research Centre, PO Box 40063, Upper Hutt, New Zealand.
G. K. Aldis
Affiliation:
School of Mathematics and Statistics, Australian Defence Force Academy, Canberra ACT 2600, Australia.
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Abstract

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The derivation of gene-transport equations is re-examined. Fisher's assumptions for a sexually reproducing species lead to a Huxley reaction-diffusion equation, with cubic logistic source term for the gene frequency of a mutant advantageous recessive gene. Fisher's equation more accurately represents the spread of an advantaged mutant strain within an asexual species. When the total population density is not uniform, these reaction-diffusion equations take on an additional non-uniform convection term. Cubic source terms of the Huxley or Fitzhugh-Nagumo type allow special nonclassical symmetries. A new exact solution, not of the travelling wave type, and with zero gradient boundary condition, is constructed.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 11 - 20
Copyright
Copyright © Australian Mathematical Society 2002

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