Advance of advantageous genes for a multiple-allele population genetics model

Abstract This paper extends the classical result of Fisher (1937) from the case of two alleles to the case of multiple alleles. Consider a population living in a homogeneous one-dimensional infinite habitat. Individuals in this population carry a gene that occurs in k forms, called alleles. Under the joint action of migration and selection and some additional conditions, the frequencies of the alleles, p i , i = 1 , … , k , satisfy a system of differential equations of the form (1.2) . In this paper, we first show that under the conditions A 1 A 1 is the most fit among the homozygotes, (1.2) is cooperative, the state that only allele A 1 is present in the population is stable, and the state that allele A 1 is absent and all other alleles are present in the population is unstable, then there exists a positive constant, c ⁎ , such that allele A 1 propagates asymptotically with speed c ⁎ in the population as t → ∞ . We then show that traveling wave solutions connecting these two states exist for | c | ≥ c ⁎ . Finally, we show that under certain additional conditions, there exists an explicit formula for c ⁎ . These results allow us to estimate how fast an advantageous gene propagates in a population under selection and migration forces as t → ∞ . Selection is one of the major evolutionary forces and understanding how it works will help predict the genetic makeup of a population in the long run.