Numerical solution and sensitivity analysis of a Fisher–KPP equation with nonlocal diffusion and free boundaries

Abstract

This paper considers a model describing population dynamics where diffusion is nonlocal and habitat boundaries expand based on population flux. The population distribution is governed by a nonlocal Fisher–KPP equation with free boundaries. We propose a Front-Tracking method coupled with a fourth-order Runge–Kutta scheme for time integration, which adaptively manages the computational grid. For comparison, we outline the Front-Fixing approach, which transforms the problem into a fixed domain. Numerical simulations are performed for different scenarios, comparing the accuracy and behavior of both proposed methods. Furthermore, a variance-based global sensitivity analysis using Sobol’ indices is conducted to identify the influence of key model parameters on the final habitat length. Results show that the initial habitat size and the boundary expansion rate are the most influential parameters affecting the final domain length, with their relative importance depending on the distributions of the chosen input parameters.