Physical processes are often modelled using nonlinear dynamical systems. If such models are relevant then they should be capable of demonstrating behaviour observed in the physical process. In this thesis a new measure of model optimality is introduced: the distribution of ɩ-shadowing times defines the durations over which there exists a model trajectory consistent with the observations. By recognising the uncertainty present in every observation, including the initial condition, ɩ-shadowing distinguishes model sensitivity from model error; a perfect model will always be accepted as optimal. The traditional root mean square measure may confuse sensitivity and error, and rank an imperfect model over a perfect one. In a perfect model scenario a good variational assimilation technique will yield an ɩ-shadowing trajectory but this is not the case given an imperfect model; the inability of the model to ɩ-shadow provides information on model error, facilitating the definition of an alternative assimilation technique and enabling model improvement. While the ɩ-shadowing time of a model defines a limit of predictability, it does not validate the model as a predictor. Ensemble forecasting provides the preferred approach for evaluating the uncertainty in predictions, yet questions remain as to how best to construct ensembles. The formation of ensembles is contrasted in perfect and imperfect model scenarios in systems ranging from the analytically tractable to the Earth's atmosphere, thereby addressing the question of whether the apparent simplicity often observed in very high-dimensional weather models fails `even in or only in' low-dimensional chaotic systems. Simple tests of the consistency between constrained ensembles and their methods of formulation are proposed and illustrated. Specifically, the commonly held belief that initial uncertainties in the state of the atmosphere of realistic amplitude behave linearly for two days is tested in operational numerical weather prediction models and found wanting: nonlinear effects are often important on time scales of 24 hours. Through the kind consideration of the European Centre for Medium-range Weather Forecasting, the modifications suggested by this are tested in an operational model.

Frost heave describes the phenomenon whereby soil freezing causes upwards surface motion due to the action of capillary suction imbibing water from the unfrozen region below. The expansion of water on freezing is a small part of the overall surface heave and it is the flow of water towards the freezing front which is largely responsible for the uplift. In this thesis, we analyse a model of frost heave due to Miller (1972, 1978) which is referred to as `secondary frost heave'. Secondary frost heave is characterised by the existence of a `partially frozen zone', underlying the frozen soil, in which ice and water coexist in the pore space. In the first part of the thesis we follow earlier work of Fowler, Krantz and Noon where we show that the Miller model for incompressible soils can be dramatically simplified. The second part of the thesis then uses this simplification procedure to develop simplified models for saline and compressible soils. In the latter case, the development of the theory leads to the consideration of non-equilibrium soil consolidation theory and the formation of segregated massive ice within permafrost. The final part of the thesis extends the simplified Miller model to the analysis of differential frost heave and the formation of patterned ground (e.g. earth hummocks and stone circles). We show that an instability mechanism exists which provides a plausible theory for the formation of these types of patterned ground.

Features inside the living cell are complex and crowded; in such complex environments diffusion processes can be said to exhibit three distinct behaviours: pure or Fickian diffusion, superdiffusion and subdiffusion. Furthermore, the behaviour of biochemical processes taking place in these environments does not follow classical theory. Because of these factors, the task of modelling dynamical proceses in complex environments becomes very challenging and demanding and has received considerable attention from other researchers seeking to construct a coherent model. Here, we are interested to study the phenomenon of subdiffusion, which occurs when there is molecular crowding. The Reaction Diffusion Partial Differential Equations (RDPDEs) approach has been used traditionally to study diffusion. However, these equations have limitations due to their unsuitability for a subdiffusive setting. However, I provide models based on Fractional Reaction Diffusion Partial Differential Equations (FRDPDEs), which are able to portray intracellular diffusion in crowded environments. In particular, we will consider a class of continuous spatial models to describe concentrations of molecular species in crowded environments. In order to investigate the variability of the crowdedness, we have used the anomalous diffusion parameter $\alpha$ to mimic immobile obstacles or barriers. We particularly use the notation $D_t^{1-\alpha} f(t)$ to represent a differential operator of noninteger order. When the power exponent is $\alpha=1$, this corresponds to pure diffusion and to subdiffusion when $0<\alpha<1$. This thesis presents results from the application of fractional derivatives to the solution of systems biology problems. These results are presented in Chapters 4, 5 and 6. An introduction to each of the problems is given at the beginning of the relevant chapter. The introduction chapter discusses intracellular environments and the motivation for this study. The first main result, given in Chapter 4, focuses on formulating a variable stepsize method appropriate for the fractional derivative model, using an embedded technique~\cite{landman07,simpson07,simpson06}. We have also proved some aspects of two fractional numerical methods, namely the Fractional Euler and Fractional Trapezoidal methods. In particular, we apply a Taylor series expansion to obtain a convergence order for each method. Based on these results, the Fractional Trapezoidal has a better convergence order than the Fractional Euler. Comparisons between variable and fixed stepsizes are also tested on biological problems; the results behave as we expected. In Chapter 5, analyses are presented related to two fractional numerical methods, Explicit Fractional Trapezoidal and Implicit Fractional Trapezoidal methods. Two results, based on Fourier series, related to the stability and convergence orders for both methods have been found. The third main result of this thesis, in Chapter 6, concerns the travelling waves phenomenon modeled on crowded environments. Here, we used the FRDPDEs developed in the earlier chapters to simulate FRDPDEs coupled with cubic or quadratic reactions. The results exhibit some interesting features related to molecular mobility. Later in this chapter, we have applied our methods to a biological problem known as Hirschsprung's disease. This model was introduced by Landman~\cite{landman07}. However, that model ignores the effects of spatial crowdedness in the system. Applying our model for modelling Hirschsprung's disease allows us to establish an interesting result for the mobility of the cellular processes under crowded environmental conditions.

Thesis (M.S.)--University of Texas at El Paso, 2008. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [69]-72).

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 118-120).

Thesis (Ph. D.)--Rutgers University, 2008. / "Graduate Program in Mathematics." Includes bibliographical references (p. 140-143).

Thesis--Université de Paris.

Thesis (doctoral)--Stockholms universitet. / Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.