A front-fixing numerical method for a free-boundary nonlocal diffusion logistic model

Abstract

This paper introduces a numerical method, based on a front-fixing transformation together with a combination of the explicit finite difference schemes with the quadrature rules, to solve the Fisher-Kolmogorov, Petrovsky and Piskunov (KPP) population model that incorporates the combined complexities of nonlocal diffusion and free boundaries. The model utilizes nonlocal diffusion to capture intricate, potentially long-range, dispersal patterns of species. We propose two-stage front-fixing transformation that effectively maps the original integro-differential equation with two moving boundaries into a partial integro-differential equation on a fixed unit domain. The transformed system, which now includes an advection term and a spatially-scaled nonlocal integral, is then solved using a comparative analysis of several explicit finite difference schemes (explicit Euler scheme, upwind, and Lax?Wendroff) for the differential operator, coupled with Simpson's rule for numerical integration. Additionally, this work contributes to understanding accelerated spreading rates, particularly for fat-tailed kernels, by numerically validating theoretical predictions and providing new insights into how kernel properties influence population dynamics. The proposed method demonstrates considerable flexibility and accuracy across various kernel types and growth scenarios, confirming its robustness and computational efficiency, which is an important prerequisite for future extensions to more complex problems.