We investigate reaction-diffusion models for populations whose members undergo two specific processes: dispersal and settling. Systems of this type occur throughout biological science, in contexts ranging from ecology to cell biology.Here we consider three distinct applications, namely: / • animal translocation, / • the invasion of a domain by precursor and differentiated cells, and / • the development of tissue-engineered cartilage. / Mathematical modelling of these systems provides an understanding of the population-level patterns that emerge from the behaviour of individuals. / A multi-species reaction-diffusion model is developed and analysed for each of the three applications. We present numerical results, which are illuminated through analytical results derived for simplified or limiting cases. For these special cases, results are obtained using analytical techniques including perturbation analysis, travelling wave analysis and phase plane methods. These analytic results provide a more complete understanding of system behaviour than numerical results alone. Emphasis is placed on connecting modelling results with experimental observations. / The first application considered is animal translocations. Translocations are widely used to reintroduce threatened species to areas where they have disappeared. A variety of different dispersal and settling mechanisms are considered, and results compared. The model is applied to a case study of a double translocation of the Maud Island frog, Leiopelma pakeka. Results suggest that settling occurs at a constant rate, with repulsion playing a significantrole in dispersal. This research demonstrates that mathematical modelling of translocations is useful in suggesting design and monitoring strategies for future translocations, and as an aid in understanding observed behaviour. / The second application we investigate is the invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric (intestinal) nervous system, but is presented in general terms and is of broader applicability. Regions of the parameter space are characterised according to existence, shape and speed of travelling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, these results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells. / As a third application, we propose a model for the development of cartilage around a single chondrocyte. The limited ability of cartilage to repair when damaged has led to the investigation of tissue engineering as a method for reconstructing cartilage. As in healthy cartilage, the model predicts a balance between synthesis, transport, binding and decay of matrix components. Our observations could explain differences observed experimentally between various scaffold media. Modelling results are also used to predict the minimum chondrocyte seeding density required to produce functional cartilage. / In summary, we develop reaction-diffusion models for dispersing and settling populations for three biological applications. Numerical and analytical results provide an understanding of population-level behaviour. This thesis demonstrates that mathematical modelling of biological systems can further understanding of biological systems and help to answer questions posed by experimental research.

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The life cycle of Caulobacter crescentus is of interest because of the asymmetric nature of cell division that gives rise to progeny that have distinct morphology and function. One daughter called the stalked cell is sessile and capable of DNA replication, while the second daughter called the swarmer cell is motile but quiescent. Advances in microscopy combined with molecular biology techniques have revealed that macromolecules are localized in a non-homogeneous fashion in the cell cytoplasm, and that dynamic localization of proteins is critical for cell cycle progression and asymmetry. However, the molecular-level mechanisms that govern protein localization, and enable the cell to exploit subcellular localization towards orchestrating an asymmetric life cycle remain obscure. There are also instances of researchers using intuitive reasoning to develop very different verbal explanations of the same biological process. To provide a complementary view of the molecular mechanism controlling the asymmetric division cycle of Caulobacter, we have developed a mathematical model of the cell cycle regulatory network. Our reaction-diffusion models provide additional insight into specific mechanism regulating different aspects of the cell cycle. We describe a molecular mechanism by which the bifunctional histidine kinase PleC exhibits bistable transitions between phosphatase and kinase forms. We demonstrate that the kinase form of PleC is crucial for both swarmer-to-stalked cell morphogenesis, and for replicative asymmetry in the predivisional cell. We propose that localization of the scaffolding protein PopZ can be explained by a Turing-type mechanism. Finally, we discuss a preliminary model of ParA- dependent chromosome segregation. Our model simulations are in agreement with experimentally observed protein distributions in wild-type and mutant cells. In addition to predicting novel mutants that can be tested in the laboratory, we use our models to reconcile competing hypotheses and provide a unified view of the regulatory mechanisms that direct the Caulobacter cell cycle. / Ph. D.