Optimizing Ecological Insights: A Comparative Study of Analytical and Numerical Approaches for Time-Fractional Fisher-KPP Equations

Abstract

Fractional reaction-diffusion equations provide a robust mathematical framework for modeling anomalous diffusion processes frequently encountered in ecological contexts. This paper presents a systematic comparative evaluation of four distinct solution methodologies for the time-fractional Fisher-Kolmogorov-Petrovsky-Piskunov equation: the Adomian Decomposition Method (ADM), the Homotopy Analysis Method (HAM), an L1-approximation finite-difference scheme (FD-L1), and a novel L1-discretization finite-volume scheme (FV-L1). The memory effects associated with sub-diffusive dispersal processes in fragmented habitats are effectively represented by the fractional derivative, computed using the Caputo formulation. Numerical experiments demonstrate that, while both ADM and HAM yield accurate analytical approximations in the short term, the FV-L1 scheme exhibits significantly enhanced long-term stability. It ensures mass preservation, a critical characteristic for isolated ecological systems. In contrast, the computationally efficient FD-L1 method experiences an artificial mass loss exceeding 2% over extended time intervals. Consequently, the findings advocate for the adoption of finite-volume discretization as the optimal approach for examining ecological conservation-related modeling, particularly in relation to invasion dynamics in heterogeneous landscapes. This study addresses the knowledge gap between the theoretical foundations of fractional calculus and practical ecological predictions, providing a framework for researchers seeking to explore memory-dependent biological phenomena.