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.
| Identifer | oai:union.ndltd.org:ADTP/269992 |
| Date | January 2008 |
| Creators | Trewenack, Abbey Jane |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
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