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    Cell-population growth modelling and nonlocal differential equations (2007)

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    Type of Content
    Theses / Dissertations
    UC Permalink
    http://hdl.handle.net/10092/1165
    http://dx.doi.org/10.26021/1971
    Thesis Discipline
    Mathematics
    Degree Name
    Doctor of Philosophy
    Publisher
    University of Canterbury. Mathematics and Statistics
    Collections
    • Engineering: Theses and Dissertations [2462]
    Authors
    Begg, Ronald Evanshow all
    Abstract

    Aspects of the asymptotic behaviour of cell-growth models described by partial differential equations, and systems of partial differential equations, are considered. The models considered describe the evolution of the size-distribution or age-distribution of a population of cells undergoing growth and division. First, the relationship between the behaviour, with and without dispersion, of a single-compartment size-distribution model of cell-growth with fixed-size cell division (where cells can only divide at a single, critical size) is considered. In this model dispersion accounts for stochastic variation in the growth process of each individual cell. Existence, uniqueness and the asymptotic stability of the solution is shown for a size-distribution model of cell-growth with dispersion and fixed-size cell division. The conditions for the analysis to hold for a more general class of division behaviours are also discussed. A class of nonlocal ordinary differential equations is studied, which contains as a subset the nonlocal ordinary differential equations describing the steady size-distributions of a single-compartment model of cell-growth. Existence of solutions to these equations is found to be implied by the existence of 'upper' and 'lower' solutions, which also provide bounds for the solution. A multi-compartment, age-distribution model of cell-growth is studied, which describes the evolution of the age-distribution of cells in different phases of cell-growth. The stability of the model when periodic solutions exist is examined. Sufficient conditions are given for the existence of stable steady age-distributions, as well as for stable periodic solutions. Finally, a multi-compartment age-size distribution model of cell-growth is studied, which describes the evolution of the age-size distribution of cells in different phases of cell-growth. Sufficient conditions are given for the existence of steady age-size distributions. An outline of the analysis required to prove stability of the steady age-size distributions of the model is also given. The analysis is based on ideas introduced in the previous chapters.

    Keywords
    differential equations; nonlocal; stability; cell-growth; partial differential equations; pde
    Rights
    Copyright Ronald Evan Begg
    https://canterbury.libguides.com/rights/theses
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