Regulation of polarised cell growth is essential for many cellular processes including spatial coordination of cell morphology changes during the division cycle. We present a mathematical model of the core mechanism responsible for the regulation of polarised growth dynamics during the fission yeast cell cycle. The model is based on the competition of growth zones localised at the cell tips for a common substrate distributed uniformly in the cytosol. We analyse the bifurcations in this model as the cell length increases, and show that the growth activation dynamics provides an explanation for the new-end take-off (NETO) as a saddle-node bifurcation at which the cell sharply switches from monopolar to bipolar growth. We study the parameter sensitivity of the bifurcation diagram and relate qualitative changes of the growth pattern, e.g. delayed or absent NETO, to previously observed mutant phenotypes. We investigate the effects of imperfect asymmetric cell division, and show that this leads to distinct growth patterns that provide experimentally testable predictions for validating the presented competitive growth zone activation model. Finally we discuss extension of the model for describing mutant cells with more than two growth zones.