Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher–KPP equation. While the Fisher–KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher–KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher–Stefan model, which is a generalisation of the Fisher–KPP model obtained by reformulating the Fisher–KPP model as a moving boundary problem. The nondimensional Fisher–Stefan model involves just one parameter, \(\kappa \), which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher–Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and \(\kappa \) so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter \(\kappa \). Interestingly, when we reinterpret the Fisher–KPP model as a moving boundary problem, many overlooked features of the classical Fisher–KPP phase plane take on a new interpretation since travelling waves solutions with \(c < 2\) are normally disregarded. This means that our analysis of the Fisher–Stefan model has both practical value and an inherent mathematical value.

生物入侵，即运动和增殖个体的种群导致入侵空置区域的移动前沿，通常使用基于经典 Fisher-KPP 方程的偏微分方程模型进行研究。尽管 Fisher-KPP 模型和扩展已成功用于模拟一系列入侵现象，包括生态和细胞入侵，但 Fisher-KPP 模型经常被忽视的局限性是它不能用于模拟生物衰退，其中空间人口的范围随着时间的推移而减少。在这项工作中，我们研究了Fisher-Stefan模型，它是通过将 Fisher-KPP 模型重新表述为移动边界问题而获得的 Fisher-KPP 模型的推广。无量纲 Fisher–Stefan 模型只涉及一个参数，\(\kappa \)，它将移动边界处的密度前沿的形状与相关行波的速度c相关联。使用数值模拟、相平面和微扰分析，我们为缓慢侵入和后退行波以及快速后退行波构建了 Fisher-Stefan 模型的近似解。这些近似值使我们能够确定c和\(\kappa \)之间的关系，以便通常报告的c 的实验估计值可用于提供未知参数\(\kappa \) 的估计值. 有趣的是，当我们将 Fisher-KPP 模型重新解释为移动边界问题时，经典 Fisher-KPP 相平面的许多被忽视的特征有了新的解释，因为具有\(c < 2\) 的行波解通常被忽略。这意味着我们对 Fisher-Stefan 模型的分析既有实用价值，也有内在的数学价值。