Influences of Allee effects in the spreading of malignant tumours

• We introduce one of the first mathematical models of tumour invasion with growth thresholds.
• Only biologically relevant travelling wave fronts exist, opposing earlier models.
• Experimental observations in tumour spread are uncovered in our model.
• We show the relevance of incorporating the Allee effect in tumour spread.

A recent study by Korolev et al. [Nat. Rev. Cancer, 14:371–379, 2014] evidences that the Allee effect—in its strong form, the requirement of a minimum density for cell growth—is important in the spreading of cancerous tumours. We present one of the first mathematical models of tumour invasion that incorporates the Allee effect. Based on analysis of the existence of travelling wave solutions to this model, we argue that it is an improvement on previous models of its kind. We show that, with the strong Allee effect, the model admits biologically relevant travelling wave solutions, with well-defined edges. Furthermore, we uncover an experimentally observed biphasic relationship between the invasion speed of the tumour and the background extracellular matrix density.