A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 2015. / This dissertation analyses the reaction-di usion equations, in particular the modi ed Huxley model, arising in population dynamics. The focus is on determining the classical Lie point symmetries, and the construction of the conservation laws and group-invariant solutions for reaction-di usion equations. The invariance criterion for determination of classical Lie point symmetries results in a system of linear determining equations which can be solved analytically. Furthermore, the Lie point symmetries associated with the conservation laws are determined. Reductions by associated Lie point symmetries are carried out. Nonclassical symmetry techniques are also employed. Here the invariance criterion for symmetry determination results in a system of nonlinear determining equations which may be solved albeit di cult. Nonclassical symmetries results in exact solutions which may not be constructed by classical Lie point symmetries. The highlight in construction of exact solution using nonclassical symmetries is the introduction of the modi ed Hopf-Cole transformation. In this dissertation, the di usion term and the coe cient of the source term are given as quadratic functions of space variable in one case, and the coe cient as the generalised power law in the other. These equations admit a number of classical Lie point symmetries. The genuine nonclassical symmetries are admitted when the source term of the reaction-di usion equation is a cubic.

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.

We consider reaction-diffusion systems of two variables with Neumann boundary conditions on a finite interval with diffusion rates of different orders. Solutions of these systems can exhibit a variety of patterns and behaviours; one common type is called a mesa pattern; these are solutions that in the spatial domain exhibit highly localized interfaces connected by almost constant regions. The main focus of this thesis is to examine three different mechanisms by which the mesa patterns become unstable. These patterns can become unstable due to the effect of the heterogeneity of the domain, through an oscillatory instability, or through a coarsening effect from the exponentially small interaction with the boundary. We compute instability thresholds such that, as the larger diffusion coefficient is increased past this threshold, the mesa pattern transitions from stable to unstable. As well, the dynamics of the interfaces making up these mesa patterns are determined. This allows us to describe the mechanism leading up to the instabilities as well as what occurs past the instability threshold. For the oscillatory solutions, we determine the amplitude of the oscillations. For the coarsening behaviour, we determine the motion of the interfaces away from the steady state. These calculations are accomplished by using the methods of formal asymptotics and are verified by comparison with numerical computations. Excellent agreement between the asymptotic and the numerical results is found.

No description available.

No description available.

Title from PDF of title page (University of Missouri--Columbia, viewed on Feb 26, 2010). The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Dissertation advisor: Dr. Peter Pfeifer. Vita. Includes bibliographical references.

Predation and harvesting play critical roles in maintaining biodiversity in ecological communities. Too much harvesting may drive a species to extinction, while too little harvesting may allow a population to drive out competing species. The spatial features of a habitat can also significantly affect population dynamics within these communities. Here, we formulate and analyze three ordinary differential equation models for the population density of a single species. Each model differs in its assumptions about how the species is harvested. We then extend each of these models to analogous partial differential equation models that more explicitly describe the spatial habitat and the movement of individuals using reaction-diffusion equations. We study the existence and stability of non-zero equilibria of these models in terms of each model's parameters. Biological interpretations for these results are discussed.

No description available.

Reaction Diffusion Master Equation (RDME) framework, characterized by the discretization of the spatial domain, is one of the most widely used methods in the stochastic simulation of reaction-diffusion systems. Discretization sizes for RDME have to be appropriately chosen such that each discrete compartment is "well-stirred" and the computational cost is not too expensive. An efficient discretization size based on the reaction-diffusion dynamics of each species is derived in this dissertation. Usually, the species with larger diffusion rate yields a larger discretization size. Partitioning with an efficient discretization size for each species, a multiple grid discretization (MGD) method is proposed. MGD avoids unnecessary molecular jumping and achieves great simulation efficiency improvement. Moreover, reaction-diffusion systems with reaction dynamics modeled by highly nonlinear functions, show large simulation error when discretization sizes are too small in RDME systems. The switch-like Hill function reduces into a simple bimolecular mass reaction when the discretization size is smaller than a critical value in RDME framework. Convergent Hill function dynamics in RDME framework that maintains the switch behavior of Hill functions with fine discretization is proposed. Furthermore, the application of stochastic modeling and simulation techniques to the spatiotemporal regulatory network in Caulobacter crescentus is included. A stochastic model based on Turing pattern is exploited to demonstrate the bipolarization of a scaffold protein, PopZ, during Caulobacter cell cycle. In addition, the stochastic simulation of the spatiotemporal histidine kinase switch model captures the increased variability of cycle time in cells depleted of the divJ genes. / Ph. D.

Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 / In this study we present the numerical solution o fboundary value problems for the reaction-diffusion equations in 1-d and 2-d that model phenomena such as kinetics and population dynamics.These differential equations are solved nu- merically using the finite element method (FEM).The FEM was chosen due to several desirable properties it possesses and the many advantages it has over other numerical methods.Some of its advantages include its ability to handle complex geometries very well and that it is built on well established Mathemat- ical theory,and that this method solves a wider class of problems than most numerical methods.The Lax-Milgram lemma will be used to prove the existence and uniqueness of the finite element solutions.These solutions are compared with the exact solutions,whenever they exist,in order to examine the accuracy of this method.The adaptive finite element method will be used as a tool for validating the accuracy of theFEM.The convergence of the FEM will be proven only on the real line.