We consider a free boundary model of epithelial cell migration with logistic growth and nonlinear diffusion induced by mechanical interactions. Using numerical simulations, phase plane and perturbation analysis, we find and analyse travelling wave solutions with negative, zero, and positive wavespeeds. Unlike classical travelling wave solutions of reaction–diffusion equations, the travelling wave solutions that we explore have a well-defined front and are not associated with a heteroclinic orbit in the phase plane. We find leading order expressions for both the wavespeed and the density at the free boundary. Interestingly, whether the travelling wave solution invades or retreats depends only on whether the carrying capacity density corresponds to cells being in compression or extension.

我们考虑上皮细胞迁移的自由边界模型，其具有逻辑增长和机械相互作用引起的非线性扩散。使用数值模拟，相平面和扰动分析，我们找到并分析了具有负，零和正波速的行波解。与经典的反应扩散方程行波解决方案不同，我们探索的行波解决方案具有明确定义的前沿，并且与相平面中的非斜变轨道无关。我们发现自由边界处波速和密度的前导表达式。有趣的是，行波解是侵入还是后退仅取决于承载能力密度是否对应于处于压缩或伸展状态的细胞。