Thesis (Ph. D.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 101-109). Also available in print.

Using a partial differential equation model derived from the ideas of the Meyerowitz and Traas groups on the role of the growth hormone auxin and those of Green and his group on the role compressive stresses can play in plants, we demonstrate how all features of spiral phyllotaxis can be recovered by the passage of a pushed pattern forming front. The front is generated primarily by a PIN1 mediated instability of a uniform auxin concentration and leaves in its wake an auxin fluctuation field at whose maxima new primordia are assumed to be initiated. Because it propagates through a slowly changing metric, the patterns have to make transitions between spirals enumerated by decreasing parastichy numbers. The point configurations of maxima coincide almost exactly with those configurations generated by the use of discrete algorithms based on optimal packing ideas which suggests that pushed pattern forming fronts may be a general mechanism by which natural organisms can follow optimal strategies. We also describe in detail a numerical method that is used to efficiently and accurately integrate the model equations while preserving the variational structure from which they are derived.

This dissertation is devoted to better understanding the role that fluids play in the selection of the shapes and functions of objects and creatures in nature. Toward that end, three specific examples are considered: stalactites, icicles, and species of colonial green algae known as Volvox. In the cases of stalactites and icicles, the object's growth is considered as a free-boundary problem. For stalactites, the coupling of thin-film fluid dynamics with calcium carbonate chemistry leads to a local, geometric growth law that is proportional to the thickness of the water layer covering the surface at any point. Application of this law to a uniformly translating shape allows a universal stalactite form to be derived; the comparison of this shape to images of actual stalactites supports the theory. In the case of icicles, the transport of the latent heat of fusion is coupled with the dynamics of both the thin-film of water encompassing the icicle and a thermally buoyant boundary layer in the immediately surrounding air. The uniformly translating shape solution is found to be parameter-free, and is, in fact, the same shape exhibited by stalactites. A comparison between this shape and icicle images validates the theory. The final example considers how advection of nutrients due to the stirring of water by the flagella of a Volvox colony leads to a metabolite uptake rate that is much greater than would occur by diffusion alone. Moreover, nutrient acquisition by pure diffusion would limit the size of Volvox species to a certain bottleneck radius at the point where diffusional uptake just meets metabolic demands, whereas advection increases the uptake in such a way as to avoid this problem entirely, thus enabling the evolution of the larger Volvox species.

Mes activités de recherche actuelles se situent dans le cadre de la morphogenèse optique et plus généralement de la dynamique non-linéaire. Les systèmes étudiés sont les milieux Kerr (cristaux liquides et fibres optiques) en cavité ou avec feedback optique. J'y étudie plus particulièrement les phénomènes d'instabilités temporelles et spatio-temporelles tels que : - la formation de structures transverses et les instabilités modulationnelles - les solitons dissipatifs et les structures localisées - les systèmes convectifs et leurs instabilités convectives et absolues - les effets du bruit sur ces instabilités, tels que les structures entretenues par le bruit.

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.