Current biological knowledge supports the existence of a secondary group of cancer cells within the body of the tumour that exhibits stem cell-like properties. These cells are termed cancer stem cells, and as opposed to themore usual differentiated cancer cells, they exhibit highermotility, they are more resilient to therapy, and are able to metastasize to secondary locations within the organism and produce new tumours. The origin of the cancer stem cells is not completely clear; they seem to stem from the differentiated cancer cells via a transition process related to the epithelial-mesenchymal transition that can also be found in normal tissue. In the current work we model and numerically study the transition between these two types of cancer cells, and the resulting “ensemble” invasion of the extracellular matrix. This leads to the derivation and numerical simulation of two systems: an algebraic-elliptic system for the transition and an advection-reaction-diffusion system of Keller-Segel taxis type for the invasion.