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

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

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