Huxley and Fisher equations for gene propagation: an exact solution. (English) Zbl 1043.35041

Broadbridge, P.; Bradshaw, B. H.; Fulford, G. R.; Aldis, G. K.

ANZIAM J. 44, No. 1, 11-20 (2002).

Summary: 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.

MSC:

35C05	Solutions to PDEs in closed form
92D25	Population dynamics (general)
35K57	Reaction-diffusion equations

Keywords:

gene-transport equations; reaction-diffusion equations; non-uniform convection term; Cubic source terms; zero gradient boundary condition

Software:

PDETWO

References:

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