ANZIAM J.
44 (2002), 11-20
|
Huxley and Fisher equations for gene propagation: An exact solution
|
P. Broadbridge
School of Mathematics and
Applied Statistics
University of Wollongong
Wollongong NSW 2522
Australia
pbroad@uow.edu.au
|
|
B. H. Bradshaw
School of Mathematics and
Applied Statistics
University of Wollongong
Wollongong NSW 2522
Australia
|
|
G. R. Fulford
AgResearch Ltd.
Wallaceville Animal Research Centre
PO Box 40063, Upper Hutt
New Zealand
|
|
|
|
|
G. K. Aldis
School of Mathematics and Statistics
Australian Defence Force Academy
Canberra ACT 2600
Australia
|
|
|
Abstract
|
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.
|
Download the article in PDF format (size 68 Kb)
|
|
|
|