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This thesis will consider the following assumptions which are based on a few insights about the artic climate: (1)the artic climate can be characterised by a growing season called summer and a dormat season called winter (2)in the summer season growing conditions are reasonably favourable and species are more likely to compete for plentiful resources (3)in the winter season there would be no further growth and the plant populations would instead by subjected to fierce weather events such as storms which is more likely to lead to the destruction of some or all of the biomass. Under these assumptions, is it possible to find those change in the environment that might cause mutualism (see section 1.9.2) from competition (see section 1.9.1) to change? The primary aim of this thesis to to provide a prototype simulation of growth of two plant species in the artic that: (1)take account of different models for summer and winter seasons (2)permits the effects of changing climate to be seen on each type of plant species interaction.

Since mammalian target of rapamycin (mTOR) and N-myristoltransferase (NMT) have been shown to be potentially related to breast cancer, mTOR-NMT signalling pathway is taken into specific consideration. In this thesis, mathematical models are developed to not only describe the mTORNMT signalling pathways, but also to analyze and predict the response to a treatment. Based on different biological hypotheses, candidate models are obtained by using an ordinary differential equation formalism. An optimization method called the Differential Evolution algorithm is applied to find the best parameter sets for our candidates. Doing so, will give the smallest distance between experimental data and simulated results. The experimental data are provided by Dr Shrivastav’s laboratory, Department of Biology, University of Winnipeg. Furthermore, the mathematical analysis for our candidate models has been found to show their asymptotic behaviours.To determine which candidate model is most likely to be the ”best” among the subgroup of models, model selection is used. Ultimately, the collaboration with Dr Shrivastav’s laboratory let us understand the simplified mTOR-NMT signalling pathway. / October 2016

The groups with which we associate influence our actions. This is often the case even when they are not deliberately organised but rather based on social categories, such as sex and skin colour, or cultural homogeneity, such as common language or customs. Group membership can cause widespread phenomena such as ingroup favouritism, polarisation of opinion and competition. Previous experiments have shown that these effects can be triggered by even completely arbitrary distinctions between groups. This thesis uses mathematical models to investigate under what circumstances these phenomena can arise. Using a game theoretical approach, the first three papers address the evolution of ingroup favouritism. Previous models have focused on the prisoners’ dilemma, interactions where the socially optimal behaviour is to co-operate, but where it is in the individual’s self-interest not to. The results presented here suggest that co-ordination problems may have been more important than those of co-operation in the evolution of an ingroup bias. In particular, this applies to common goals that require trust. It is also demonstrated in a behavioural experiment that such trust is most common within groups, but that it can emerge between groups through group reputation. The fourth paper focuses on a model on how cultural groups in contact can develop common norms, rather than polarise into different norm groups, by assuming a confirmation bias. The model is empirically tested on demographic and linguistic data from Mauritian Creole, a natural language developed from the mixing of parent languages. In the fifth paper, the group is defined by common preferences (e.g. for pop songs), which are transmitted in a random copying model. The competitive success of the groups, with respect to their size, is recorded on a toplist, the turnover rate of which is derived. In the final paper, people match up in pairs between groups according to their preferences, and all stable matchings are found under a specific assumption of bounded rationality, when people’s individual behaviour may be affected by the consequences for fellow group members.

The focus of this thesis is the inclusion of stochasticity into mathematical models of assembly with particular interest to the in vitro polymerisation of intermediate filaments, one of three components of the cytoskeleton. From the chemical master equation (CME), two additional models (the reaction rate equations or RREs and the two-moment approximation equations or 2MA equations) are derived. As analysis of the CME is generally intractable, we present the stochastic simulation algorithm (SSA) as a means of reproducing the most probable state of the CME at a given time. The results from the SSA are compared to simulations of both the RREs and the 2MA equations and we find that the three models are in good agreement. Further, the numerical results are compared to mean lengths and length distributions of experimental data which all models are shown to mimic. Mathematical analyses of the RREs demonstrate the conservation of mass in the system, and the unique positive equilibrium is proven to be globally asymptotically stable. Further, the 2MA equations are also shown to have conservation of mass and to possess an analogous equilibrium to the one found in the case of the RREs. In general, this study illustrates how randomness can be incorporated in polymerisation models and highlights the advantages and disadvantages of the different approaches.

The BACOLI package is a numerical software package for solving parabolic partial differential equations in one spatial dimension. It implements a B-spline collocation method for the spatial discretization of a system of partial differential equations. The resultant ordinary differential equations together with the boundary conditions form a system of differential-algebraic equations. The differential-algebraic equations are then solved using the DASSL solver. The BACOLI software package features adaptive error control in the temporal and spatial domains. The estimate of the temporal error is controlled through the DASSL solver. The estimate of the spatial error is controlled based on the difference between two solutions computed in the BACOLI software package. This difference gives an estimation of the error. If this error estimate does not meet the user-supplied tolerance, then the spatial mesh is changed. The BACOLI software package can only solve parabolic partial differential equations that depend on spatial derivatives. In this thesis, the BACOLI software package is modified to solve a broader spectrum of problems. In fact, after some modifications, the extended BACOLI software package can solve systems of parabolic partial differential equations and time-dependent equations that do not depend on spatial derivatives. We apply this extended software package to solve the monodomain model of cardiac electrophysiology. The monodomain model is a multi-scale mathematical model for the evolution of the electrical potential in cardiac tissue that couples the ionic currents at the cellular scale with their propagation at the tissue scale. Because of their local nature, the mathematical models of a single cell have no dependency on spatial derivatives whereas the models at the tissue level do. The heart models considered in our numerical experiments use various cardiac cell models. We find that solving the heart models through the extended BACOLI software package, in some cases, leads to a speed-up in comparison with the Chaste software package, which is a powerful, widely used, and well-respected software package for heart simulation.