Cells adhere to each other and to the extracellular matrix (ECM) through protein molecules on the surface of the cells. The breaking and forming of adhesive bonds, a process critical in cancer invasion and metastasis, can be influenced by the mutation of cancer cells. In this paper, we develop a nonlocal mathematical model describing cancer cell invasion and movement as a result of integrin-controlled cell-cell adhesion and cell-matrix adhesion, for two cancer cell populations with different levels of mutation. The partial differential equations for cell dynamics are coupled with ordinary differential equations describing the ECM degradation and the production and decay of integrins. We use this model to investigate the role of cancer mutation on the possibility of cancer clonal competition with alternating dominance, or even competitive exclusion (phenomena observed experimentally). We discuss different possible cell aggregation patterns, as well as travelling wave patterns. In regard to the travelling waves, we investigate the effect of cancer mutation rate on the speed of cancer invasion.

中文翻译：

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癌细胞生长和入侵的两人口模型中的聚集和行波动力学。

细胞通过细胞表面上的蛋白质分子彼此粘附并粘附至细胞外基质（ECM）。粘附键的断裂和形成是癌细胞入侵和转移的关键过程，可能会受到癌细胞突变的影响。在本文中，我们针对两个突变水平不同的癌细胞群体，开发了一个因整合素控制的细胞-细胞粘附和细胞-基质粘附而导致癌细胞入侵和运动的非局部数学模型。细胞动力学的偏微分方程与描述ECM降解以及整联蛋白产生和衰变的普通微分方程耦合。我们使用此模型来调查癌症突变在具有交替优势的癌症克隆竞争中的作用，甚至竞争性排斥（实验观察到的现象）。我们讨论了不同的可能的小区聚合模式以及行波模式。关于行波，我们研究了癌症突变率对癌症侵袭速度的影响。