{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/pattern-formation-and-front-stability-for-a","title":"Pattern formation and front stability for a moving-boundary model of biological invasion and recession","arxiv_id":"2207.03053","date":"2022-07-07","proceeding":null,"authors":["Alexander K. Y. Tam","Matthew J. Simpson"],"abstract":"We investigate pattern formation in a two-dimensional (2D) Fisher--Stefan model, which involves solving the Fisher--KPP equation on a compactly-supported region with a moving boundary. By combining the Fisher--KPP and classical Stefan theory, the Fisher--Stefan model alleviates two limitations of the Fisher--KPP equation for biological populations. In this work, we investigate whether the 2D Fisher--Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher--KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts ($c > 0$) of the Fisher--Stefan model are linearly stable to perturbations of all wave numbers. However, our analysis demonstrates that receding planar fronts ($c < 0$) of the Fisher--Stefan model are linearly unstable for all wave numbers. This is analogous to unstable solutions for planar solidification in the classical Stefan problem. Introducing a surface tension regularisation stabilises receding fronts for short-wavelength perturbations, giving rise to a range of unstable modes and a most unstable wave number. We supplement linear stability analysis with level-set numerical solutions that corroborate theoretical results. Overall, front instability in the Fisher--Stefan model suggests a new mechanism for pattern formation in receding biological populations.","url_abs":"https://arxiv.org/abs/2207.03053v1","url_pdf":"https://arxiv.org/pdf/2207.03053v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"pattern-formation-and-front-stability-for-a","repo_url":"https://github.com/alex-tam/2d_fisher-stefan_level-set_stability","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}