In an attempt to describe how patterns emerge in biological systems, Alan Turing proposed a mathematical model encapsulating the properties of such processes. It details a partial differential equation governing the dynamics of two or more substances, called morphogens, reacting and diffusing in a specific manner, in turn generating what has now come to be denoted as Turing patterns. In recent years, evidence has accumulated to support Turing's claim and it has been proposed that it is responsible for the dynamical characteristics of phenomena such as skin pigmentation and branching of lungs in vertebrates. The aim of this paper is to study how the choice of model parameters and reaction kinetics influence the nature of patterns generated, as well as explore how boundary control can be employed to generate pre-defined patterns and the efficiency of this procedure. To simulate the patterns, the differential equation is solved in Python by means of a spectral method using discretized space and time domains. The model parameters were then studied to try to gain insight in their effects on the patterns yielded. The boundary control was implemented in MATLAB using a difference method. The metric used for efficiency was taken to be the energy expenditure of the boundary cells. The complex dynamics of the studied systems make it difficult to draw valuable conclusions on the influence of the parameters, but the results support the expected characteristics of the models used. The efficiency of the pattern generation is deemed to be closely related to the amount of boundary control utilized.

Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How the patchiness of landscapes affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. In this thesis, we focus on how landscape composition and the movement behaviour of individuals at an interface between patches of different quality affect the steady state of a single species. We consider a model of reaction-diffusion equations for the temporal evolution of the density of the population in space. Individual movement is described by a diffusion process, e.g., an uncorrelated random walk. Population net growth is encapsulated in the growth function that considers birth and death of individuals, including nonlinear effects that arise from competition and/or facilitation within the species. We consider the simplest case of two adjacent one-dimensional patches, e.g., two intervals on the real line that share one boundary point. Conditions are homogeneous within a patch but differ between patches. The movement behaviour of individuals between the two patches is incorporated into matching conditions of population flux and density at the interface between patches, i.e., the boundary point that the intervals share. These matching conditions turn out to be continuous in the flux but discontinuous in the density. Several authors have studied similar models recently. Most of these studies consider monostable dynamics on both patches, i.e., logistic growth. Under logistic growth, the net population growth rate is a strictly decreasing function of population density. Logistic population dynamics are very simple: the population extinction state is unstable and a positive steady state is globally asymptotically stable. In this work, we also include bistable dynamics, i.e., an Allee effect. Biologically, an Allee effect occurs when individuals cooperate at some level so that the net population growth rate is increasing with population density for at least some low or intermediate densities. Models with Allee growth typically exhibit bistability: there are two locally stable steady states, one at low density (possibly zero) and one at high density. This bistability makes mathematical analysis more challenging, but leads to more interesting results in return. Mathematically, most existing work on related models is based on linear stability analysis of the extinction state. We focus on the nonlinear models and specifically on positive steady states. We establish the existence, uniqueness and - in some cases - global asymptotic stability of a positive steady state. We classify the shape of these states depending on movement behaviour. We clarify the role of movement in this context. In particular, we investigate the following prior observation: a randomly diffusing population at steady state in a continuously varying habitat can exceed its carrying capacity. Our results clarify when and under which conditions this effect can arise in our two-patch landscape. The analysis of the model with an Allee effect on one of the two patches yields a rich and interesting structure of steady states. Under certain parameter conditions, some of these states are amenable to explicit stability calculations. These yield insights into the possible bifurcations that can occur in our system. Finally, we indicate how the model and analysis here can be extended to systems of reaction-diffusion equations on graphs that represent natural habitats with different geometries, for example watersheds.

Temperature isoclines are shifting with global warming. To survive, species with thermal niches must shift their geographical ranges to stay within the bounds of their suitable habitat, or acclimate to a new environment. Mathematical models that study range shifts are called moving-habitat models. The literature is rich and includes modelling with reaction-diffusion equations. Much of this literature represents space by the real line, with a handful studying 2-dimensional domains that are unbounded in at least one direction. The suitable habitat is represented by the set over which the demographics (reaction term) has a positive net growth rate. In some cases, this is a bounded set, in others, it is not. The unsuitable habitat is represented by the set over which the net growth rate is negative. The environmental shift is captured by an imposed shift of the suitable habitat. Individuals respond to their environment via their movement behaviour and many display habitat-dependent dispersal rates and a habitat bias. Such behaviour corresponds to a jump in density across the interface of suitable and unsuitable habitat. The questions motivating moving-habitat models are: when can a species track its shifting habitat and what is the impact of an environmental shift on a persisting species. Without closed form solutions, researchers rely on numerical methods to study the latter, and depending on the movement of the interface, the former may require numerical tools as well. We construct and analyse two numerical methods, a finite difference (FD) scheme and a finite element (FE) method in 1- and 2-dimensional space, respectively. The FD scheme can capture arbitrary movement of the boundary, and the FE method rather general shapes for the suitable habitat. The difficulty arises in capturing the jump across a shifting interface. We construct a reference frame in which the interfaces are fixed in time. We capture the jump in density with a clever placing of the nodes in the FD scheme, and through a Lagrange multiplier in the FE method. With biological applications, we demonstrate the power of our solvers in advancing research for moving-habitat models.

The realm of the nano-world begins when things start getting smaller in size than one thousandth of the thickness of the human hair. Surface patterning at the nanoscale has started to find applications in information storage, self-cleaning of surfaces due to the "lotus effect", biocompatible materials based on surface roughness and many more. Several methods such as particle-beam writing, optical lithography, stamping and various kinds of self-assembly are widely used to serve the purpose of patterning smaller surface structures. However, globally much research is going into developing more efficient, reproducible and simple methods of patterning surfaces and in better controlling the order of these nanostructures. Researchers have always looked upon Nature to get inspiration and to mimic its model in engineering novel architectures. One of the methods used by this greatest artist (Nature) to make beautiful patterns around is through reaction diffusion based non-linear processes. Non-linear systems driven away from equilibrium sustain pattern only during the continuous dissipation of a regular flow of energy and are different from equilibrium processes that are converging towards a minimum in free energy (a. k. a. self-assembly). Dissipative pattern formation from micrometer to kilometers scale has been known but ordered patterns at nanoscale have never been achieved. In the process of thoroughly characterizing suitable substrates for nanoelectronics applications, we came across a remarkable process leading to the formation of highly ordered arrays of dimples on tantalum. The pattern formation happens in a narrow electrochemical windows which are functions of many parameters such as concentration, external applied voltage, temperature etc. After investigating the formation of dimples by performing spatio-temporal studies, we found that the underlying principles behind this unique way of engineering nano-structures have their roots in nonlinear interaction/reaction electro-hydrodynamics. We then have demonstrated the generality of this process by extending it to titanium, tungsten and zirconium surfaces. The pattern similar to Rayleigh-Bernard convection cells originates inside the electrochemical solution due to coupling among electrolyte ions during their migration across the electrochemical double layer (Helmholtz layer) and simultaneously imprints on the surface due to dissolution of metal oxide via etching. Based on these results we further postulate that, given appropriate electropolishing chemistry; these patterns can be formed on virtually any metal or semiconductor surface. The application of these nanostructures as nanobeakers for placing metal nanoparticles is also elucidated Highly porous materials such as mesoporous oxides are of technological interest for catalytic, sensing, optical and filtration applications: the mesoporous materials (with pores of size 2-50 nm) in the form of thin films can be used as membranes due large surface area. In the second part of this thesis, a new technique of making detachable ultrathin membranes of transition metal oxides is presented. The underlying concepts behind the detachment of membranes from the underlying substrate surface are discussed. The control on the size of the pores by modulating the voltage and concentration is also elucidated. The method is generalized by showing the similar detachment behavior on other metal oxide membranes.Thus, the results of this work introduces new techniques of engineering nanostructures on surfaces based on reaction-diffusion adaptive systems and contribute to the better understanding of electrochemical self-organization phenomena due to migration coupling induced electro-hydrodynamics. / Thesis / Doctor of Philosophy (PhD)

This thesis represents a small contribution to our understanding of metastable patterns in various stochastic models from physics and biology. By a \emph{metastable pattern}, we mean a pattern that appears to persist in a regular fashion on some timescale, but disappears or undergoes an irregular change on a longer timescale. Metastable patterns frequently result from stochastic perturbations of patterns that are stable without perturbation. In this thesis, we study stochastic perturbations of stable spatiotemporal patterns in several classes of PDE and integral equations. In particular, we address two major questions: \begin{enumerate}[Q1.] \item When perturbed by noise, for how long does a pattern that is stable without noise persist? \item How does the stochastic perturbation affect the average behaviour of a pattern on the timescale where it appears to persist? \end{enumerate} To address these questions, we pursue two lines of inquiry: the first based on the theory of \emph{quasi-ergodic measures}, and the second based on \emph{phase decomposition techniques}. In our first line of inquiry we present novel, rigorous connections between metastability of general infinite dimensional stochastic evolution systems and the spectral properties of their sub-Markov generators using the theory of quasi-ergodic measures. To do so, we develop a novel $L^p$-approach to the study of quasi-ergodic measures. We are then able to draw conclusions about the metastability of travelling waves and other patterns in a class of stochastic reaction-diffusion equations. For instance, we obtain a rigorous definition of the \emph{quasi-asymptotic speed}~of a travelling wave in a stochastic PDE. We moreover find that stochastic perturbations of amplitude $\sigma>0$ cause the quasi-asymptotic speed of certain travelling waves to deviate from the deterministic wave speed by a constant that is approximately proportional to $\sigma^2$. In our second line of inquiry, the dynamics of our (infinite dimensional) stochastic evolution system are projected onto a finite dimensional manifold that captures some property of a metastable pattern. While most previous studies using phase reduction techniques have used the \emph{variational phase}, we take an approach based on the \emph{isochronal phase}, inspired by classical work on finite dimensional oscillatory systems. When the pattern in question is a travelling wave, the isochronal phase captures the position of the wave at a given point in time. By exploiting the regularity properties of the isochronal phase, we are able to prove several novel results about the metastable behaviour of the reduced dynamics in the small noise regime in a very large class of stochastic evolution systems. These results allow us to moreover compute the noise-induced changes in the speed of stochastically perturbed travelling waves and other patterns. The results we obtain using this approach are numerically precise, and may be applied to a very general class of stochastic evolution systems.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23675号 / 理博第4765号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 泉 正己, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM