A Fisher-KPP problem with nonlocal free boundary: Numerical simulations and machine learning classification for population dynamics

This paper investigates the population dynamics of a Fisher-KPP reaction-diffusion model with a nonlocal free boundary. Unlike classical Stefan-type problems, this model incorporates a nonlocal integral condition that accounts for long-range perception and memory effects in species movement. We develop an efficient numerical framework by first applying a front-fixing transformation to map the evolving habitat onto a fixed domain. Then, an Alternating Direction Explicit (ADE) scheme is constructed to solve the resulting nonlinear system. The long-term behavior of this model is characterized by a spreading-balancing-vanishing trichotomy. We propose a Physics-Informed Machine Learning (PIML) approach to predict the population’s asymptotic behavior. By training an ensemble classifier on data generated by our numerical scheme and embedding theoretical mathematical features, such as kernel mass and critical survival ratios, into the model, we achieve a 94% accuracy rate in classifying the trichotomy. This hybrid framework enables the instantaneous generation of high-resolution bifurcation maps, providing a powerful tool for predicting species survival and range dynamics without the need for numerical simulations.