Huxley and Fisher equations for gene propagation: An exact solution

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. © Australian Mathematical Society 2002.