One of the major tasks that a cell faces during its lifecycle is how to spatially localize its components. Correct spatial organization of various proteins (cellular polarity) is fundamental not only for the correct cell shape but also to carry out essential cellular functions, such as the spatial coordination of cell division. We present a mathematical model of the core mechanism responsible for the regulation of polarized growth dynamics in the model organism, fission yeast. The model is based on the competition of growth zones of polarity protein Cdc42 localized at the cell tips for a common substrate (inactive Cdc42) that diffuses in the cytosol. To explore the underlying mechanism for oscillations and the effect of diffusion and noise, we consider three model frameworks including a 1D deterministic model, a 2D deterministic model, and a stochastic model. We simulate and analyze these models using numerical bifurcation tools, PDEs, and stochastic simulation algorithms.