In order to combat the spread of cancer in the human body it is important to understand the properties of this disease. The interaction between the newly produced cancerous cells and the surrounding healthy tissue is crucial in the avascular phase of tumour development. The mathematical model we propose investigates this invasion of cancer into the surrounding tissue in presence and absence of acidity changes in the tumour surrounding.The model describes the interaction between the growing tumour and the surrounding normal tissue in the immmediate vicinity of the tumour-host interface. We investigate variations to the key parameters responsible for change of the tumour from stable to invasive and also the effect of functional forms for the diffusion coefficient of the cancer cells, which reflect different assumptions on the movement rules of the cells. We show that the model exhibits travelling fronts with speeds which can be determined by analysis similar to the classical Fisher's travelling wave case and then confirm this numerically. We comment on the biological applicability of the model.