发表机构
Mathematical Institute, University of Oxford; Department of Mathematical Sciences "G.L. Lagrange", Politecnico di Torino（牛津大学数学研究所; 都灵理工大学G.L. 拉格朗日数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过自由边界问题推导异质细胞群体分离波速的变分界，分析其稳定性，并发现圆形波稳定性不单由迁移率决定，可能引发指状不稳定性。
AI 中文摘要
我们考虑一个由具有不同迁移率的增殖细胞和非增殖细胞组成的异质细胞群体的最小压力模型。该模型被表述为一个反应-交叉-扩散方程组，描述细胞密度的时空动态。已知该模型允许具有严格分离分量的行波解：非增殖细胞占据前缘的有限区域，而增殖细胞保持在尾部。然而，这些波的速度、参数依赖性和稳定性仍然知之甚少。在这项工作中，我们通过将问题重新表述为广义多孔-费舍尔方程的自由边界问题，推导出波速的几乎显式的变分界。我们获得的估计适用于一般的压力定律和生长动力学，与数值模拟结果密切吻合，并在不可压缩极限中变得尖锐，在那里我们正式恢复波速的完全显式表征。然后我们分析波的稳定性，以表明仅当非增殖细胞比增殖细胞更具迁移性时，分离波才是稳定的。最后，受指状突起数值观察的启发，我们通过渐近形状扰动分析研究不可压缩分离圆形波的稳定性。这产生了压力、界面速度和角模式增长率的显式表达式，从而使可能导致指状不稳定性出现的失稳机制变得明显。有趣的是，我们发现，与一维情况相反，这种圆形波的稳定性不仅由迁移率系数的相对值决定，因此无论哪种细胞类型具有更大的迁移率，都可能出现不稳定性。
英文摘要
We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction--cross--diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous--Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.
