Numerical Solutions for Fractional Fixation Times in Evolutionary Models

The fixation time of alleles is a fundamental concept in population genetics, traditionally studied using the Wright–Fisher model and classical coalescent theory. However, these models often assume homogeneous environments and equal reproductive success among individuals, limiting their applicability to real-world populations where environmental heterogeneity plays a significant role. In this paper, we introduce a new forward-time model for estimating fixation time that incorporates environmental heterogeneity through the use of fractional calculus. By introducing a fractional parameter α, we capture the effects of heterogeneous environments on offspring production. To solve the resulting fractional differential equations, we develop a novel spectral method based on Eta-based functions, which are well-suited for approximating solutions to complex, high-variation systems. The proposed method reduces the problem to an optimization framework via the operational matrix of fractional derivatives. We demonstrate the effectiveness and accuracy of this approach through numerical examples and show that it consistently captures fixation dynamics across various scenarios. This work offers a robust and flexible framework for modeling evolutionary processes in heterogeneous environments.