2021/07/12 by Maud El‐Hachem, Scott W. McCue, El-Hachem, Maud +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #92Bxx #Evolution and Genetic Dynamics #FOS: Biological sciences #FOS: Physical sciences #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Pattern Formation and Solitons (nlin.PS) #Populations and Evolution (q-bio.PE)
paper · pdf · doi:10.48550/arxiv.2107.05210
openalex publication_date 2021/07/12 · openalex created_date 2022/07/17 · openalex updated_date 2026/07/28
The Fisher-KPP model, and generalisations thereof, is a simple\nreaction-diffusion models of biological invasion that assumes individuals in\nthe population undergo linear diffusion with diffusivity D, and logistic\nproliferation with rate \λ. Biologically-relevant initial conditions\nlead to long-time travelling wave solutions that move with speed\nc=2\√(\λ D). Despite these attractive features, there are several\nbiological limitations of travelling wave solutions of the Fisher-KPP model.\nFirst, these travelling wave solutions do not predict a well-defined invasion\nfront. Second, biologically-relevant initial conditions lead to travelling\nwaves that move with speed c=2\√(\λ D) > 0. This means that, for\nbiologically-relevant initial data, the Fisher-KPP model can not be used to\nstudy invasion with c \≠ 2\√(\λ D), or retreating travelling waves\nwith c < 0. Here, we reformulate the Fisher-KPP model as a moving boundary\nproblem on x < s(t), and we show that this reformulated model alleviates the\nkey limitations of the Fisher-KPP model. Travelling wave solutions of the\nmoving boundary problem predict a well-defined front, and can propagate with\nany wave speed, -\∞ < c < \∞. Here, we establish these results using\na combination of high-accuracy numerical simulations of the time-dependent\npartial differential equation, phase plane analysis and perturbation methods.\nAll software required to replicate this work is available on GitHub.\n