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© Springer International Publishing Switzerland 2014. This paper reviews three mathematical modelling approaches that have recently been used to understand three different modes of collective cell motion in biology. Firstly, a cell-based model is presented for the study of cell motion in epithelial sheets, then a hybrid discrete cell-basedmodel is described for neural crest cell invasion and, finally, a traditional partial differential equationmodel is described for tumour cell invasion. It is shown that the behaviour of all of these models can, in limiting cases, be recapitulated by nonlinear diffusion equations where the particular nonlinearity of the diffusion coefficient captures, on the global scale, the inherent interactions on the local scale.

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