{"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/non-vanishing-sharp-fronted-travelling-wave","title":"Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model","arxiv_id":"2107.05210","date":"2021-07-12","proceeding":null,"authors":["Maud El-Hachem","Scott W McCue","Matthew J Simpson"],"abstract":"The Fisher-KPP model, and generalisations thereof, is a simple reaction-diffusion models of biological invasion that assumes individuals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate $\\lambda$. Biologically-relevant initial conditions lead to long-time travelling wave solutions that move with speed $c=2\\sqrt{\\lambda D}$. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher-KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically-relevant initial conditions lead to travelling waves that move with speed $c=2\\sqrt{\\lambda D} > 0$. This means that, for biologically-relevant initial data, the Fisher-KPP model can not be used to study invasion with $c \\ne 2\\sqrt{\\lambda D}$, or retreating travelling waves with $c < 0$. Here, we reformulate the Fisher-KPP model as a moving boundary problem on $x < s(t)$, and we show that this reformulated model alleviates the key limitations of the Fisher-KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front, and can propagate with any wave speed, $-\\infty < c < \\infty$. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.","url_abs":"https://arxiv.org/abs/2107.05210v4","url_pdf":"https://arxiv.org/pdf/2107.05210v4.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":"non-vanishing-sharp-fronted-travelling-wave","repo_url":"https://github.com/ProfMJSimpson/TravellingWaves_ElHachem2021","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}