We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition. Unlike many reaction–diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion. Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed $c$, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative $x$-directions. Unlike the well-studied Fisher–Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed $c\ge c^{\ast }>0$, the moving boundary model leads to travelling wave solutions with $\left|c\right|<c^{\ast \ast }$. We interpret these travelling wave solutions in the phase plane and show that they are associated with several features of the classical Fisher–Kolmogorov phase plane that are often disregarded as being nonphysical. We show, numerically, that the phase plane analysis compares well with long time solutions from the full partial differential equation model as well as providing accurate perturbation approximations for the shape of the travelling waves.

我们分析了一种新型的恶性侵袭数学模型，该模型以两阶段移动边界问题的形式描述了恶性细胞群体向背景组织（例如皮肤）群体的侵袭。两种种群中的细胞均发生扩散迁移和逻辑增殖。两个种群之间的界面根据两阶段Stefan条件移动。与许多恶性浸润的反应扩散模型不同，移动边界模型明确描述了癌症与周围组织之间的锋利前沿的运动，而无需引入简并的非线性扩散。数值模拟表明，该模型引起了非常有趣的行波解，并随着速度而移动。$C$，并且该模型支持恶性入侵和恶性撤退，其中行波可以正向或负向移动 $X$-方向。与经过深入研究的Fisher–Kolmogorov和Porous-Fisher模型不同，行波以最小波速移动$C\ge C^{\ast }>0$，运动边界模型导致行波解 $\left|C\right|<C^{\ast \ast }$。我们在相平面中解释了这些行波解，并表明它们与经典的Fisher-Kolmogorov相平面的一些特征相关联，这些特征通常被视为非物理的。我们从数字上表明，相平面分析与完整的偏微分方程模型的长期解比较好，并为行波的形状提供了精确的扰动近似。