Fractional forward Kolmogorov equations in population genetics
Fractional forward Kolmogorov equations in population genetics
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DOI：
10.1016/j.cnsns.2023.107432
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发表时间：
2023-07-22
影响因子：
3.9
通讯作者：
Mashayekhi，Somayeh
中科院分区：
文献类型：
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作者：
Mashayekhi，Somayeh
本文结合理论发展和数值方法来描述群体遗传学中异质性的影响。为了解决群体遗传学中的异质性问题，我们建立了群体遗传学中的第一个分数向前Kolmogorov方程，其中群体中给定一组基因座的等位基因频率分布是向前Kolmogorov方程的解。这一框架将得到实施，并将对模型进行计算研究。为了研究该模型，提出了一种求解分数阶偏微分方程的数值方法。该方法是基于通过勒让德多项式的最小二乘近似。利用勒让德多项式的Riemann-Liouville分数阶积分算子将分数阶偏微分方程的解化为一个代数方程组。给出了该方法的误差界和稳定性。包括说明性的例子来证明该技术的有效性和适用性。使用新的数值方法，我们推导出的等位基因频率分布的分数向前Kolmogorov方程的解决方案。我们研究了等位基因频率分布的行为，考虑到进化和人口力量的影响。这项研究表明，异质性改变了等位基因频率分布的行为。
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.