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Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion. (English) Zbl 1194.35219

Adhesion of cells to one another and their environment is an important regulator of many biological processes but has proved difficult to incorporate into continuum mathematical models. The authors improve further on the new modelling approach proposed by N. J. Armstrong et al. [A continuum approach to modelling cell-cell adhesion, J. Theor. Biol. 243, 98–113 (2006)]. The present models use an integro-partial differential equation for cell behaviour, in which the integral represents the sensing by cells of their local environment. This enables an effective representation of cell-cell adhesion, as well as random cell movement, and cell proliferation. This modelling approach is then used to investigate the ability of cell-cell adhesion to generate spatial patterns during cell aggregation. The model is also extended to give a new representation of cancer growth, whose solutions reflect the balance between cell-cell and cell-matrix adhesion in regulating cancer invasion. The non-local term in these models means that there is no standard theory from which one can deduce the boundedness required for biological realism: specifically, solutions for cell density must lie between zero and a positive density corresponding to close cell packing. Here the authors derive a number of conditions, each of which is sufficient for the required boundedness, and it is demonstrated numerically that cell density increases above the upper bound for some parameter sets not satisfying these conditions. Finally, the main mathematical challenges for future work on boundedness in models of this type is outlined.

MSC:

35K57 Reaction-diffusion equations
45K05 Integro-partial differential equations
92D25 Population dynamics (general)
92B05 General biology and biomathematics
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