School of Mathematics and Statistics - Theses
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ItemMathematical modelling of layering in nature( 2012)This thesis examines convection-based models to simulate the formation of structured populations. Specifically, the thesis focuses on systems that exhibit characteristic layered patterns. Such structures are important throughout the natural world and range from the molecular layers formed on substrates to the shifting rock layers within the earth's crust. Our primary motivation here, however, stems from the formation of cortical layers within the mouse brain. The creation of layers within the cortex of the mammalian brain has been of interest to researchers in developmental biology. Experiments on mice have found a counter-intuitive layering sequence. The layers are formed in an inside-out arrangement, with the deepest layers formed first and the outermost last. It is believed that the glycoprotein reelin is linked to the mechanisms behind layering. In the reeler mutant, where this glycoprotein is absent, it is observed that the layer arrangement is reversed. This has led to a number of mechanisms being proposed for how reelin may affect the neurons that form the cortical layers. No conclusive theory of this phenomenon has been established so far. Mathematical modelling provides an elegant method to evaluate the feasibility of different layering mechanisms. In this context, the thesis provides models for the purpose of understanding layer formation in nature. The replication of both age-uniform and inverted structures is a key goal. We examine the effect that modelling perspective, population models versus the individual agent models, has on our results. Results from each perspective are utilised to strengthen and augment the work. Furthermore, the major issue of volume constraints is also addressed. Modelling is used to investigate the connections between the motion of invading agents and their conversion to settled states. Base models will be initially formulated from a quite general foundation. Published experimental results are utilised as inspiration for later extensions to these base models. This research culminates with a proposed theoretical framework for cortical layer formation.