Fractional forward Kolmogorov equations in population genetics

This paper combines theoretical development and numerical methods to characterize the effect of heterogeneity in population genetics. To address heterogeneity in population genetics, we build the first fractional forward Kolmogorov equations in population genetics, where the distribution of the allele frequencies of a given set of loci in a population is a solution of forward Kolmogorov equations. This framework will be implemented, and the model will be studied computationally. To study the model, a new numerical method for solving the fractional partial differential equations is presented. The method is based upon the least squares approximation via Legendre polynomials. The Riemann–Liouville fractional integral operator for Legendre polynomials is utilized to reduce the solution of the fractional partial differential equations to a system of algebraic equations. The error bound and the stability of the method are presented. Illustrative examples are included to demonstrate the validity and applicability of the technique. Using the new numerical method, we derive the allele frequency distribution as a solution of the fractional forward Kolmogorov equations. We study the behavior of allele frequency distribution by considering the effect of evolutionary and demographic forces. This study shows that heterogeneity changes the behavior of the distribution of the allele frequencies. •We build the first fractional forward Kolmogorov equations in population genetics.•The distribution of the allele frequencies is a solution of Kolmogorov equations.•A method for solving the fractional partial differential equations is presented.•The method is based upon the least squares approximation via Legendre polynomials.•We study the behavior of allele frequency using the new numerical method.