Turing theory plays an important role in real biological pattern formation problems, such as solid tumor growth and animal coat patterns. To understand how patterns form and develop over time due to growth, we consider spatiotemporal patterns, in particular Turing patterns, for reaction diffusion systems on growing surfaces with curvature. Of particular interest is isotropic growth of the sphere, where growth of the domain occurs in the same proportion in all directions. Applying a modified linear stability analysis and a separation of timescales argument, we derive the necessary and sufficient conditions for a diffusion driven instability of the steady state and for the emergence of spatial patterns. Finally, we explore these results using numerical simulations.

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This dissertation is in the area of complex dynamics, more specifically focused on the iteration of rational functions. Given a well-chosen family of rational functions, parameterized by a complex parameter, we are especially interested in regularity properties of the Hausdorff dimension of Julia sets of these polynomials considered as a function of the parameters. In this dissertation I deal with a family of polynomials of degree at least 3 depending in a holomorphic way on a parameter, focusing on the point where the dynamics and topology of the polynomials drastically change. In such a context proving continuity is quite challenging while real analyticity will most likely break. Our approach will, on the one hand, build on the existing methods of proving continuity of Hausdorff dimension, primarily based on proving continuity, in the weak* topology of measures on the Riemann sphere, of canonical conformal measures, but will also require methods which, up to my best knowledge, have not been implemented anywhere yet. Our main result gives a surprising example where the Hausdorff dimension of the Julia set is continuous in the parameter, but where the Julia set itself is not.

Master of Science / In this thesis, a recently developed physiologically-based model of the sleep-wake switch is analyzed and applied to a variety of clinically-relevant protocols. In contrast to phenomenological models, which have dominated sleep modeling in the past, the present work demonstrates the advantages of the physiologically-based approach. Dynamical and linear stability analyses of the Phillips-Robinson sleep model allow us to create a general framework for determining its response to arbitrary external stimuli. The effects of near-stable wake and sleep ghosts on the model’s dynamics are found to have implications for arousal during sleep, sleep deprivation, and sleep inertia. Impulsive sensory stimuli during sleep are modeled modeled according to their known physiological mechanism. The predicted arousal threshold variation matches experimental data from the literature. In simulating a sleep fragmentation protocol, the model simultaneously reproduces the body temperature and arousal threshold variation measured in another existing clinical study. In the second part of the thesis, we simulate sleep deprivation by introducing a wake-effort drive that is required to maintain wakefulness during normal sleeping periods. We interpret this drive both physiologically and psychologically, and demonstrate quantitative agreement between the model’s output and experimental subjective fatigue-related data. As well as subjective fatigue, the model is simultaneously able to reproduce adrenaline excretion and body temperature variations. In the final part of the thesis, the model is extended to include the orexinergic neurons of the lateral hypothalamic area. Due to the dynamics of the orexin group, the extended model exhibits sleep inertia, and an inhibitory circadian projection to the orexin group produces a postlunch dip in performance – both of which are well-known behavioral features. Including both homeostatic and circadian inputs to the orexin group, the model produces a waking arousal variation that quantitatively matches published clinical data.