In this thesis we examine mathematical models which have been suggested as possibile mechanisms for forming certain biological patterns. We analyse them in detail attempting to produce the requisite patterns both analytically and numerically. A reaction diffusion system in two spatial dimensions with anisotropic diffusion is examined in detail and the results compared with certain snakeskin patterns. We examine two other variants to the standard reaction diffusion system: a system where the reaction kinetics and the diffusion coefficients depend upon the cell density suggested as a possible model for the segmentation sequence in Drosophila and a system where the model parameters have one dimensional spatial gradients. We also analyse a model derived from known cellular processes used to model the branching behaviour in bryozoans and show that, in one dimension, such a model can, in theory, give all the required solution behaviour. A genetic switch model for pattern elements on butterfly wings is also briefly examined to obtain expressions for the solution behaviour under coldshock.

No description available.

Mathematical models of isothermal chemical systems in reactor problems and Turing's theory of morphogenesis with an application in sea-shell patterning are studied. The reaction-diffusion systems describing these models are solved numerically. First- and second-order difference schemes are developed, which are economical and reliable in comparison to classical numerical methods. The linearization process decouples the reaction-diffusion equations thereby allowing the use of different time steps for each differential equation, which may be large due to the excellent stability properties of the methods. The methods avoid having to solve a non-linear algebraic system at each time step. The schemes are suitable for implementation on a parallel machine.

Thesis (Ph. D.)--University of Chicago, Dept. of Biology. / Includes bibliographical references. Also available on the Internet.

Structure, in its many forms, is a central theme in theoretical population ecology. At a mathematical level, it arises as nonuniformities in the topology of nonlinear dynamical systems. I investigate a mechanism wherein a chaotic time series can have episodes of nearly periodic dynamics interspersed with more 'typical' irregular dynamics. This phenomenon frequently appears in biological models, and may explain patterns of alternating biennial and irregular dynamics in measles epidemics. I investigate the interaction between spatial structure and density-dependent population regulation with a simple model of two logistic maps coupled by diffusive migration. I examine two different consequences of spatial structure: scale-dependent interactions ("nonlocal interactions") and spatial variation in resource quality ("environmental heterogeneity"). Nonlocal interactions allow three general dynamical regimes: in-phase, out-of-phase, and uncorrelated. With environmental heterogeneity, the dynamics of the total population size can be approximated by a logistic map with the mean growth parameter of the two patches; the dynamics within a single patch are often less regular. Adding environmental heterogeneity to non-local interactions has little qualitative effect on the dynamics when the differences between patches are small; when the differences are large, uncorrelated dynamics are most likely to be seen, and there are interesting consequences for the stability of source-sink systems. A third type of structure arises when individuals differ from one another. Accurate prediction of extinction risk in small populations requires that a distinction be made between demographic stochasticity (variation among individuals) and environmental stochasticity (variation among years or sites). I describe and evaluate two tests to determine whether all the variation in population survivorship can be explained by demographic stochasticity alone. Both tests have appropriate probabilities of type I error, unless the survival probability is very low or very high. Small amounts of environmental stochasticity are often not detected by the tests, but the hypothesis of demographic stochasticity alone is consistently rejected when environmental stochasticity is large. I also show how to factor out deterministic sources of variability, such as density-dependence. I illustrate these tests with data on a population of Acorn Woodpeckers.

In this work, we present our investigations into two different systems, both far from equilibrium. We first present the relaxation and ordering processes in magnetic skyrmion systems. This is followed by a study of the behavior of many species interacting on a spatially heterogeneous lattice. Magnetic skyrmions have been a subject of great interest in recent years. They have been proposed to be at the heart of next-generation computing and information storage devices. One interesting feature of magnetic skyrmions is the presence of the non-dissipative Magnus force. The Magnus force causes the skyrmions to be deflected from their direction of motion. In this work, we examine the effect the strength of this Magnus force has on the late-time ordering behavior of magnetic skyrmions. We show that the late-time ordering also shows enhanced relaxation with an increase in the Magnus force. We also studied the behavior of magnetic skyrmions when confined to a narrow channel. We show that, like before, the Magnus force helps the system order faster while experiencing a constant drive. Interestingly, when the drive was periodic, the Magnus force inhibited the relaxation in the system. Interacting populations have been a topic of scientific interest since the late eighteenth century. We studied the effect of spatial heterogeneity on a two-dimensional lattice. Using cyclic predator-prey interaction schemes, we numerically simulated the effect of asymmetric predation rates inside "habitats." We show that, due to the non-linearity of the system, the species that had a chance to escape predation did not see the largest benefit from this change. Instead, the predator of this prey saw the largest benefit from this change. The material on skyrmion systems is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award Number DE-SC0002308. The population dynamics research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF17-1-0156. / Doctor of Philosophy / In this work, we present our investigations into two different systems. Both of these systems are considered to be not in equilibrium. We first present is the behavior of magnetic skyrmions as the system settles into an arranged state. This is followed by a study of the behavior of multiple species interacting on a lattice where different parts of the lattice have different rules of interaction. Magnetic skyrmions are small defects that occur in specific types of magnetic materials. They have been proposed to be useful in next-generation computing devices. Similar to a curve-ball in baseball, but due to a different physical phenomenon, magnetic skyrmions follow curved paths when pushed. This effect, known as a Magnus force, causes the magnetic skyrmions to settle faster into a position relative to the other magnetic skyrmions in the system. We show that this effect also occurs when the magnetic skyrmions are being pushed through a narrow channel. If the push is periodically started and stopped, the Magnus force instead slows down the ability for magnetic skyrmions to settle into a position relative to the other magnetic skyrmions. Interacting populations have been a topic of scientific interest since the late eighteenth century. We studied the effect of changing the rules of species interaction based on where on a two-dimensional lattice the interaction occurred. Using cyclic predator-prey interaction schemes, we numerically simulated the effect of asymmetric predation rates inside "habitats." We showed that, due to the complex interaction scheme present in the system, the species that had a chance to escape predation did not see the largest benefit from this change. Instead, the predator of this prey saw the largest benefit from this change. The material on skyrmion systems is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award Number DE-SC0002308. The population dynamics research was sponsored by the Army Research Office and was accomplished under Grant No. W911NF17-1-0156.

By growing together in dense communities, microorganisms (microbes) have a huge impact on human life. Microbes also come in a wide variety of shapes, but we have yet to understand the importance of these shapes for community biology. How are multi- species communities, such as biofilms and colonies, affected by the morphologies of constituent cells? Which morphologies might these environments select for in turn? To address these questions, we use individual-based modelling to investigate the effects of cell shape on patterning and evolution within microbial communities. We develop a flexible simulation framework, coupling a continuum model of the biofilm chemical environment to a cellular-level description of biofilm growth mechanics. This modelling system allows competitions between different microbial cell shapes to be simulated and studied, in different community contexts. Our models predict that cell shape can strongly affect spatial structure and patterning within competitive communities. Rod cells perform better at colonising surfaces and the expanding edges of colonies, while round cells are better at dominating the upper surface of a community. Our predictions are supported by experiments using Escherichia coli and Pseudomonas aeruginosa bacteria, and demonstrate that particular shapes can confer a selective advantage in communities. In summary, the work presented in this thesis predicts and examines new mechanisms of self-organisation driven by cell shape, demonstrating a new significance for microbial morphology as a means for cells to succeed in a dense and competitive environment.

No description available.

The Turing mechanism for the production of a broken spatial symmetry in an initially homogeneous system of reacting and diffusing substances has attracted much interest as a potential model for certain aspects of morphogenesis such as pre-patterning in the embryo, and has also served as a model for self-organization in more generic systems. The two features necessary for the formation of Turing patterns are short-range autocatalysis and long-range inhibition which usually only occur when the diffusion rate of the inhibitor is significantly greater than that of the activator. This observation has sometimes been used to cast doubt on applicability of the Turing mechanism to cellular patterning since many messenger molecules that diffuse between cells do so at more-or-less similar rates. Here we show that stationary, symmetry-breaking Turing patterns can form in physiologically realistic systems even when the extracellular diffusion coefficients are equal; the kinetic properties of the 'receiver' and 'transmitter' proteins responsible for signal transduction will be primary factors governing this process.

We propose a dynamical systems approach to the study of weak turbulence(spatiotemporal chaos) based on the periodic orbit theory, emphasizing the role of recurrent patterns and coherent structures. After a brief review of the periodic orbit theory and its application to low-dimensional dynamics, we discuss its possible extension to study dynamics of spatially extended systems. The discussion is three-fold. First, we introduce a novel variational scheme for finding periodic orbits in high-dimensional systems. Second, we prove rigorously the existence of periodic structures (modulated amplitude waves) near the first instability of the complex Ginzburg-Landau equation, and check their role in pattern formation. Third, we present the extensive numerical exploration of the Kuramoto-Sivashinsky system in the chaotic regime: structure of the equilibrium solutions, our search for the shortest periodic orbits, description of the chaotic invariant set in terms of intrinsic coordinates and return maps on the Poincare section.