生物入侵的Fisher-KPP型模型：开源计算工具、关键概念与分析
Fisher-KPP-type models of biological invasion: Open source computational tools, key concepts and analysis
AI总结：
本综述提供开源Julia计算工具，结合相平面与摄动方法分析Fisher-KPP、Porous-Fisher和Fisher-Stefan三种生物入侵反应扩散模型的行波解，并揭示其前沿特性与消退机制。
摘要：
本综述提供了开源计算工具，支持一系列数学方法来分析用于研究生物入侵的三个相关标量反应扩散模型。从经典的Fisher-Kolmogorov（Fisher-KPP）模型开始，我们说明了如何使用计算方法来探索随时间变化的偏微分方程（PDE）解，并结合相平面和正则摄动技术来探索以无量纲速度$c \ge 2$移动的入侵行波解。为了克服Fisher-KPP模型解中缺乏明确锐前沿的问题，我们还回顾了两种替代的建模方法。第一种是Porous-Fisher模型，其中线性扩散项被退化的非线性扩散项取代。使用相平面和正则摄动方法，我们探索了以无量纲速度$c \ge 1/\sqrt{2}$移动的锐前沿和光滑前沿入侵行波之间的区别。第二种替代方法是将Fisher-KPP模型重新表述为在$0 < x < L(t)$上的移动边界问题，从而得到Fisher-Stefan模型，该模型具有源自带线性扩散项的PDE模型的锐前沿行波解。随时间变化的PDE解和相平面方法表明，Fisher-Stefan模型的行波解可以同时描述生物入侵（$c > 0$）和生物消退（$c < 0$）。用于复现本综述中所有计算结果的开源Julia代码可在GitHub上获取；我们鼓励研究人员直接使用此代码，或根据更复杂模型的需要改编代码。
英文摘要：
This review provides open-access computational tools that support a range of mathematical approaches to analyse three related scalar reaction-diffusion models used to study biological invasion. Starting with the classic Fisher-Kolmogorov (Fisher-KPP) model, we illustrate how computational methods can be used to explore time-dependent partial differential equation (PDE) solutions in parallel with phase plane and regular perturbation techniques to explore invading travelling wave solutions moving with dimensionless speed $c \ge 2$. To overcome the lack of a well-defined sharp front in solutions of the Fisher-KPP model, we also review two alternative modeling approaches. The first is the Porous-Fisher model where the linear diffusion term is replaced with a degenerate nonlinear diffusion term. Using phase plane and regular perturbation methods, we explore the distinction between sharp- and smooth-fronted invading travelling waves that move with dimensionless speed $c \ge 1/\sqrt{2}$. The second alternative approach is to reformulate the Fisher-KPP model as a moving boundary problem on $0 < x < L(t)$, leading to the Fisher-Stefan model with sharp-fronted travelling wave solutions arising from a PDE model with a linear diffusion term. Time-dependent PDE solutions and phase plane methods show that travelling wave solutions of the Fisher-Stefan model can describe both biological invasion $(c > 0)$ and biological recession $(c < 0)$. Open source Julia code to replicate all computational results in this review is available on GitHub; we encourage researchers to use this code directly or to adapt the code as required for more complicated models.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
