This thesis consists of two parts. In the first part, we develop an ODE cell differentiation population model to study the dynamics of hair follicles (HFs). In particular, we wish to understand the mechanisms underlying the cyclical regenerations that this mammalian organ undergoes throughout an organism's lifetime. We'd also like to study the effect of ionizing radiation (IR) on the regeneration processes. In brief, the cycle of a HF consists of three consecutive phases: anagen—the active proliferation phase, catagen—the degeneration phase, and telogen—the resting phase, and while HFs undergo irreversible degeneration during the catagen phase, recent experimental research on mice shows that when anagen HFs are subject to IR, they undergo a transient degeneration, followed by a nearly full regeneration that makes the HFs return to homeostatic state. In our model we propose various feedback mechanisms and study their role in determining the degenerative and regenerative behavior of HF cells. The model is built based on current theoretical knowledge in biology and model parameters are calibrated to IR experimental data. We perform bifurcation and sensitivity analyses to determine the effect of IR exposure on the stability of the HF homeostatic steady state and compare with the dynamics of the irreversible degeneration during catagen. The second part is motivated by the biological process of cell polarization. Cell polarity refers to the asymmetrical spatial distribution of molecules and substances within a cell or cell membrane, which occurs as a response to internal or external stimuli. In this work, we study autonomous reaction-diffusion models to find mechanisms that can lead to polarization at a single cell-membrane. In particular, we investigate the role that positive and negative feedbacks play in the early polarization process. We first perform and document ample tests for various reaction-diffusion models using three numerical methods; two of them are based on fast Fourier transform (FFT) differentiation, while the third model is of finite-difference type. The performance of the numerical methods is compared while simultaneously selecting those reaction-diffusion models that are likely to attain polarization. We then present a more detailed investigation of two reaction-diffusion models which include diffusion-inhibiting negative feedback and a positive production feedback; these models are studied using one of the FFT-based numerical methods. A similar analysis for the remaining numerical methods is pending and will be detailed in the discussion section of this part of the thesis.