Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T04:04:38.506Z Has data issue: false hasContentIssue false
  • English
  • Français

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: pbroad@uow.edu.au.
B. H. Bradshaw
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia; e-mail: pbroad@uow.edu.au.
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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Ablowitz, M. J. and Zeppetella, A., “Explicit solutions of Fisher's equation for a special wave speed”, Bull. Math. Biol. 41 (1979) 835–840.CrossRefGoogle Scholar
[2]Aronson, D. G. and Weinberger, H. F., “Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation”, in Partial Differential Equations and Related Topics (ed. Goldstein, J. A.), Lect. Notes. in Math. 446, (Springer, Berlin, 1975) 5–49.CrossRefGoogle Scholar
[3]Arrigo, D. J., Hill, J. M. and Broadbridge, P., “Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source”, IMA J. Appl. Math. 52 (1994) 1–24.CrossRefGoogle Scholar
[4]Bluman, G. W. and Cole, J. D., “The general similarity solution of the heat equation”, J. Math. Mech. 18 (1969) 1025–42.Google Scholar
[5]Chen, Z. and Guo, B., “Analytic solutions of the Nagumo equation”, IMA J. Appl. Math. 48 (1992) 107–115.Google Scholar
[6]Clarkson, P. A. and Kruskal, M. D., “New similarity reductions of the Boussinesq equation”, J. Math. Phys. 30 (1989) 2201–13.CrossRefGoogle Scholar
[7]Clarkson, P. A. and Mansfield, E. L., “Symmetry reductions and exact solutions of a class of nonlinear heat equations”, Physica D 70 (1994) 250–288.CrossRefGoogle Scholar
[8]Fisher, R. A., “The wave of advance of advantageous genes”, Ann. Eugenics 7 (1937) 335–369.CrossRefGoogle Scholar
[9]Fulford, G., Forrester, P. and Jones, A., Modelling with Differential and Difference Equations (Cambridge University Press, New York, 1997).CrossRefGoogle Scholar
[10]Galaktionov, V. A., Dorodnitsyn, V. A., Elenin, G. G., Kurdyumov, S. P. and Samarskii, A. A., “A quasilinear heat equation with a source: peaking, localization, symmetry, exact solutions, asymptotics, structures”, J. Soviet. Math. 41 (1988) 1222–92.CrossRefGoogle Scholar
[11]Kudryashov, N. A., “Partial differential equations with solutions having movable first-order singularities”, Phys. Lett. A 169 (1992) 237–242.CrossRefGoogle Scholar
[12]Melgaard, D. K. and Sincovec, R. K., “General software for two-dimensional non-linear partial differential equations”, ACM Trans. Math. Software 7 (1981) 106–125.CrossRefGoogle Scholar
[13]Skellam, J. G., “The formulation and interpretation of mathematical models of diffusionary processes in population biology”, in The Mathematical Theory of the Dynamics of Biological Populations (eds. Bartlett, M. S. and Hiorns, R. W.), (Academic Press, London, 1973) 63–86.Google Scholar