Michael Jeffrey Ward

In this thesis we investigate strongly localized solutions to systems of singularly perturbed reaction-diffusion equations arising in several new contexts. The first such context is that of bulk-membrane-coupled reaction diffusion systems in which reaction-diffusion systems posed on the boundary and interior of a domain are coupled. In particular we analyze the consequences of introducing bulk-membrane-coupling on the behaviour of strongly localized solutions to the singularly perturbed Gierer-Meinhardt model posed on the one-dimensional boundary of a flat disk and the singularly perturbed Brusselator model posed on the two-dimensional unit sphere. Using formal asymptotic methods we derive hybrid numerical-asymptotic equations governing the structure, linear stability, and slow dynamics of strongly localized solutions consisting of multiple spikes. By numerically calculating stability thresholds we illustrate that bulk-membrane coupling can lead to both the stabilization and the destabilization of strongly localized solutions based on intricate relationships between the bulk-membrane-coupling parameters. The remainder of the thesis focuses exclusively on the singularly perturbed Gierer-Meinhardt model in two new contexts. First, the introduction of an inhomogeneous activator boundary flux to the classically studied one-dimensional Gierer-Meinhardt model is considered. Using the method of matched asymptotic expansions we determine the emergence of \textit{shifted} boundary-bound spikes. By linearizing about such a shifted boundary-spike solution we derive a class of \textit{shifted} nonlocal eigenvalue problems parametrized by a shift parameter. We rigorously prove partial stability results and by considering explicit examples we illustrate novel phenomena introduced by the inhomogeneous boundary fluxed. In the second and final context we consider the Gierer-Meinhardt model in three-dimensions for which we use formal asymptotic methods to study the structure, stability, and dynamics of strongly localized solutions. Most importantly we determine two distinguished parameter regimes in which strongly localized solutions exist. This is in contrast to previous studies of strongly localized solutions in three-dimensions where such solutions are found to exist in only one parameter regime. We trace this distinction back to the far-field behaviour of certain core problems and formulate an appropriate conjecture whose resolution will be key in the rigorous study of strongly localized solutions in three-dimensional domains.

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Reaction-diffusion (RD) systems are indispensable models tounderstand pattern formation. To reproduce sophisticatedpattern-forming behaviors that resemble observed biologicalpatterns, we can increase the complexity of a RD model byintroducing spatial heterogeneity. In the first part of thisthesis, we consider two-component RD systems in thesingular limit of a large diffusivity ratio for which localized spotpatterns occur. We will use a hybrid asymptotic-numerical approachto analyze the existence, linear stability, and dynamics oflocalized spot patterns in the presence of certain spatialheterogeneities. More specifically, for the Schnakenberg model, we willinvestigate the effect on spot patterns that occur for distincttypes of localized spatial heterogeneity for which spots are eitherrepelled from or attracted to, respectively. For the Klausmeier model, we study the effect of water advection on the dynamics and steady-state behavior of localized patches or ``spots'' of vegetation. Moreover, we study the effect of a slowly varying rainfall rate on spot dynamics, and the induced bifurcation with delays. In many previous studies of spot instabilities, it has been observedfrom numerical PDE simulations that a linear shape-deforminginstability of a localized spot is the trigger of a nonlinearspot-replication event in the absence of heterogeneity. To provide a theoretical basis for theseobservations, we derive an amplitude equation from a weaklynonlinear analysis which confirms that a peanut-shaped instability ofa spot is subcritical. In the second part of this thesis, we study mean first passage time(MFPT) for a Brownian particle to be captured by small circulartraps in a 2-D confining domain. Our focus is to understand how thedeviations from a radially symmetric domain, which represents a domainheterogeneity in a general sense, alters the optimal spatialconfiguration of a collection of small circular traps that minimizesthe average MFPT. In this direction, we develop a numerical methodand perform asymptotic analysis to approximate the MFPT for general2-D domains. In particular, by deriving a new explicit analyticalformula for the Neumann Green's function, we demonstrate the fullpower of these tools for an elliptical domains of arbitrary aspectratio.

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Motivated by the spatial segregation of intracellular proteins between the cytoplasm and the cellular membrane, we investigate the spatio-temporal dynamics of coupled bulk-surface reaction-diffusion models. For such models, a passive diffusion process occurring inside a bounded bulk domain is coupled to a nonlinear reaction-diffusion process restricted to the domain boundary. We first consider a model with 2-D circular bulk geometry, for which we perform a systematic weakly nonlinear analysis of spatio-temporal patterns. Amplitude equations near Hopf, pitchfork and codimension-two pitchfork-Hopf bifurcations are derived from a multi-scale expansion. For both Brusselator and Schnakenberg surface kinetics, we detect a variety of sub- and supercritical bifurcations leading to the formation of stationary Turing patterns and radially symmetric oscillations. Unstable mixed-mode oscillations near codimension-two pitchfork-Hopf bifurcations are also identified. We then consider the synchronized oscillatory dynamics of two identical pointlike boundaries, modeled by nonlinear ODEs, that are spatially segregated and coupled via a 1-D PDE bulk diffusion field. For such coupled PDE--ODE systems, we perform a weakly nonlinear analysis near Hopf bifurcations associated with even and odd perturbations. This allows us to detect stable in-phase and anti-phase oscillations when assuming Sel’kov kinetics in each compartment. We also consider the PDE diffusive coupling of two chaotic Lorenz oscillators. Through a computation of Lyapunov exponents, we establish that complete synchronization of the two chaotic trajectories is possible whenever the coupling strength and the bulk diffusivity are large enough. Lastly, we study a mass-conserved membrane-bulk model for a specific intracellular protein pattern-forming system, the Cdc42-GEF system. For both 1-D and 2-D bulk geometries, we investigate the formation of spatio-temporal patterns as diffusion levels and membrane-bulk coupling rates are varied. As a result of mass conservation, only anti-phase oscillations are observed on a 1-D interval, with no instabilities leading to in-phase oscillations detected. For the 2-D circular bulk case, no instabilities leading to radially symmetric oscillations are found. Instead, a variety of stationary Turing patterns and traveling waves, that interact near Bogdanov-Takens bifurcations, are observed.

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In this thesis, we develop novel numerical and analytical techniques for calculating the MFPT for a Brownian particle to be captured by either small stationary or mobile traps inside a bounded 2-D domain. Of particular interest is identifying the optimal arrangements of small traps that minimize the average MFPT. Although the MFPT and the associated optimal trap arrangement problem havebeen well-studied for disk-shaped domains, there are very few analytical or numerical results available for general star-shaped domains or thin domains with large aspect ratio. We develop an embedded numerical method for both stationary and periodic mobile trap problems, based on the Closest Point Method (CPM), to perform MFPT simulations on various confining 2-D domains. Optimal trap arrangements are identified numerically through either a refined discrete samplingapproach or a particle-swarm optimization procedure. To confirm some of the numerical findings, novel perturbation approaches are developed to approximate the average MFPT and identify optimal trap configurations for a class of near-disk confining domains or an arbitrary thin domain of large aspect ratio.We also analyze cell-bulk coupled ODE-PDE models for describing the communication between localized spatially segregated dynamically active signaling compartments or cells, coupled through a passive extracellular bulk diffusion field. In a 2-D bounded domain, where the cells are small disks of a common radius ε
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In this thesis, the phenomenon of localized crime hotspots in models capturing the features of repeat and near-repeat victimization of urban crime was studied. Stability, insertion, slow movement of crime hotspots and the effect of police patrol modelled by an extra equation derived from biased random walk were studied by means of matched asymptotic expansions, nonlocal eigenvalue problem (NLEP) stability analysis, and numerical computations. In the absence of police, we confirmed the linear stability of the far-from equilibrium steady-states with crime hotspots in the original parameters regimes as observed in [47]. The results hold for both the supercritical and subcritical regimes distinguished by a Turing bifurcation (cf. [48, 49]). Moreover, the phenomenon of peak insertion was characterized by a simple nonlinear equation computable by quadratures and a normal form equation identical to that of the self-replication of Mesa patterns [28] was derived. Slow dynamics of unevenly-spaced configurations of hotspots were described by a system of differential-algebraic systems (DAEs), which was derived from resolving an intricate triple-deck structure of boundary layers formed between the hotspots and their neighbouring regions. In the presence of police, which was modelled by a simple interaction with criminals, single and multiple hotspots patterns were constructed in a near-shadow limit of criminal diffusivity. While a single hotspot was found to be unconditionally stable, the linear stability behaviour of multiple-hotspot patterns was found to depend on two thresholds, between which we also observe a novel Hopf bifurcation phenomenon leading to asynchronous oscillations. For one particular, but representative, parameter value in the model, the determination of the spectrum of the NLEP was found to reduce to the study of a quadratic equation for the eigenvalue. For more general parameter values, where this reduction does not apply, a winding number analysis on the NLEP was used to determine detailed stability properties associated with multiple hotspot steady-state solutions.

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We formulate and investigate a relatively new modeling paradigm by which spatially segregated dynamically active units communicate with each other through a signaling molecule that diffuses in the bulk medium between active units. The modeling studies start with a simplified setting in a one-dimensional space, where two dynamically active compartments are located at boundaries of the domain and coupled through the feedback term to the local dynamics together with flux boundary conditions at the two ends. For the symmetric steady state solution, in-phase and anti-phase synchronizations are found and Hopf bifurcation boundaries are studied using a winding number approach as well as parameter continuation methods of bifurcation theory in the case of linear coupling. Numerical studies show the existence of double Hopf points in the parameter space where center manifold and normal form theory are used to reduce the dynamics into a system of amplitude equations, which predicts the configurations of the Hopf bifurcation and stability of the two modes near the double Hopf point. The system with a periodic chain of cells is studied using Floquet theory. For the case of a single active membrane bound component, rigorous spectral results for the onset of oscillatory dynamics are obtained and in the finite domain case, a weakly nonlinear theory is developed to predict the local branching behavior near the Hopf bifurcation point. A previously developed model by Gomez et al.\cite{Gomez-Marin2007} is analyzed in detail, where the phase diagrams and the Hopf frequencies at onset are provided analytically with slow-fast type of local kinetics. A coupled cell-bulk system, with small signaling compartments, is also studied in the case of a two-dimensional bounded domain using the method of asymptotic expansions. In the very large diffusion limit we reduce the PDE cell-bulk system to a finite dimensional dynamical system, which is studied both analytically and numerically. When the diffusion rate is not very large, we show the effect of spatial distribution of cells and find the dependence of the quorum sensing threshold on influx rate.

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In this thesis we present an analysis of the Gierer-Meinhardt model with saturation (GMS) on various curve geometries in ℝ². We derive a boundary fitted coordinate framework which translates an asymptotic two-component differential equation into a single component reaction diffusion equation with singular interface conditions. We create a numerical method that generalizes the solution of such a system to arbitrary two-dimensional curves and show how it extends to other models with singularity properties that are related to the Laplace operator. This numerical method is based on integrating logarithmic singularities which we handle by the method of product integration where logarithmic singularities are handled analytically with numerically interpolated densities. In parallel with the generalized numerical method, we present some analytical solutions to the GMS model on a circular and slightly perturbed circular curve geometry. We see that for the regular circle, saturation leads to a hysteresis effect for two dynamically stable branches of equilibrium radii. For the near circle we show that there are two distinct perturbations, one resulting from the introduction of a angular dependent radius, and one caused by Fourier mode interactions which causes a vertical shift to the solution. We perform a linear stability analysis to the true circle solution and show that there are two classes of eigenvalues leading to breakup or zigzag instabilities. For the breakup instabilities we show that the saturation parameter can completely stabilize perturbations that we show are always unstable without saturation and for the zigzag instabilities we show that the eigenvalues are given by the near circle curve normal velocity. The breakup analysis is based on the reduction of an implicit non-local eigenvalue problem (NLEP) to a root finding problem. We derive conditions for which this eigenvalue problem can be made explicit and use it to analyze a stripe and ring geometry. This formulation allows us to classify certain technical properties of NLEPs such as instability bands and a Hopf bifurcation condition analytically.

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In the first part of this thesis, we study the existence and stability of multi-spot patterns on the surface of a sphere for a singularly perturbed Brusselator and Schnakenburg reaction-diffusion model. The method of matched asymptotic expansions, tailored to problems with logarithmic gauge functions, is used to construct both symmetric and asymmetric spot patterns. There are three distinct types of instabilities of these patterns that are analyzed: self-replication instabilities, amplitude oscillations of the spots, and competition instabilities. By using a combination of spectral theory for nonlocal eigenvalue problems together with numerical computations, parameter thresholds for these three different classes of instabilities are obtained. For the Brusselator model, our results point towards the existence of cycles of creation and destruction of spots, and possibly to chaotic dynamics. For the Schnakenburg model, a differential-algebraic ODE system for the motion of the spots on the surface of the sphere is derived.In the second part of the thesis, we study the existence and stability of mesa solutions in one spatial dimension and the corresponding planar mesa stripe patterns in two spatial dimensions. An asymptotic analysis is used in the limit of a large diffusivity ratio to construct mesa patterns in one spatial dimension for a general class of two-component reaction-diffusion systems that includes the well-known Gierer Meinhardt activator-inhibitor model with saturation (GMS model), and a predator-prey model. For such one-dimensional patterns, we study oscillatory instabilities of the pattern by way of a Hopf bifurcation and from a reduction to a limiting ODE-PDE system. In addition, explicit thresholds are derived characterizing transverse instabilities of planar mesa-stripe patterns in two spatial dimensions. The results of our asymptotic theory as applied to the GMS and predator-prey systems are confirmed with full numerical results.

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In Applied Mathematics, linear and nonlinear eigenvalue problems arise frequently when characterizing the equilibria of various physical systems. In this thesis, two specific problems are studied, the first of which has its roots in micro engineering and concerns Micro-Electro Mechanical Systems (MEMS). A MEMS device consists of an elastic beam deflecting in the presence of an electric field. Modelling such devices leads to nonlinear eigenvalue problems of second and fourth order whose solution properties are investigated by a variety of asymptotic and numerical techniques.The second problem studied in this thesis considers the optimal strategy for distributing a fixed quantity of resources in a bounded two dimensional domain so as to minimize the probability of extinction of some species evolving in the domain. Mathematically, this involves the study of an indefinite weight eigenvalue problem on an arbitrary two dimensional domain with homogeneous Neumann boundary conditions, and the optimization of the principal eigenvalue of this problem. Under the assumption that resources are placed on small patches whose area relative to that of the entire domain is small, the underlying eigenvalue problem is solved explicitly using the method of matched asymptotic expansions and several important qualitative results are established.

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Localized patterns have been observed in many reaction-diffusionsystems. One well-known such system is the two-component Gray-Scottmodel, which has been shown numerically to exhibit a rich variety oflocalized spatio-temporal patterns including, standing spots,oscillating spots, self-replicating spots, etc. This thesisconcentrates on analyzing the localized pattern formation in this model thatoccurs in the semi-strong interaction regime where the diffusivity ratio ofthe two solution components is asymptotically small. In a one-dimensional spatial domain, two distinct types of oscillatoryinstabilities of multi-spike solutions to the Gray-Scott model thatoccur in different parameter regimes are analyzed. These twoinstabilities relate to either an oscillatory instability in theamplitudes of the spikes, or an oscillatory instability in the spatiallocations of the spikes. In the latter case a novel Stefan-typeproblem, with moving Dirac source terms, is shown to characterize thedynamics of a collection of spikes. From a numerical and analyticalstudy of this problem, it is shown that an oscillatory motion in thespike locations can be initiated through a Hopf bifurcation. In asubregime of the parameters it is shown that this Stefan-type problemis quasi-steady, allowing for the derivation of an explicit set ofODE's for the spike dynamics. In this subregime, a nonlocal eigenvalueproblem analysis shows that spike amplitude oscillations can occurfrom another Hopf bifurcation.In a two-dimensional domain, the method of matched asymptoticexpansions is used to construct multi-spot solutions by effectivelysumming an infinite-order logarithmic expansion in terms of a smallparameter. An asymptotic differential algebraic system of ODE's for thespot locations is derived to characterize the slow dynamics of a collection of spots. Furthermore, it is showntheoretically and from the numerical computation of certain eigenvalueproblems that there are three main types of fast instabilities for amulti-spot solution. These instabilities are spotself-replication, spot annihilation due to overcrowding, and anoscillatory instability in the spot amplitudes. These instabilitymechanisms are studied in detail and phase diagrams in parameter space where they occur are computed and illustrated for various spatialconfigurations of spots and several domain geometries.

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