A robust cubic reaction-diffusion system for gene propagation

Continuum modelling of gene frequencies during spatial dispersion of a population arrives at a reaction-diffusion equation with cubic source term, rather than the quadratic equation that Fisher proposed in 1937. For the case of three possible alleles at one diploid locus, with general degrees of fitness for the six genotypes, we derive a new system of coupled cubic reaction-diffusion equations for two independent gene frequencies. When any number of preexisting alleles compete for a single locus, in the important case of partial dominance and shared disadvantage of preexisting alleles, the new mutant allele is described by a single equation if the total population is known. In the case of Mendelian inheritance considered by Fisher, this equation is the Huxley equation, a reaction-diffusion equation whose source term is degenerate cubic with two real roots. Some practical analytic solutions of the genetic dispersion equation are constructed by the method of nonclassical symmetry reduction. The obtained solutions satisfy specific boundary conditions and they are different from previously derived travelling wave solutions.