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Localized patterns in the Gray-Scott model : an asymptotic and numerical study of dynamics and stability Chen, Wan 2009

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Localized Patterns in the Gray-Scott Model An Asymptotic and Numerical Study of Dynamics and Stability by Wan Chen B.Sc., Wuhan University, China, 2003 Ph.D, The University of British Columbia, Canada, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July, 2009 c© Wan Chen 2009 Abstract Localized patterns have been observed in many reaction-diffusion systems. One well-known such system is the two-component Gray-Scott model, which has been shown numerically to exhibit a rich variety of localized spatio- temporal patterns including, standing spots, oscillating spots, self-replicating spots, etc. This thesis concentrates on analyzing the localized pattern for- mation in this model that occurs in the semi-strong interaction regime where the diffusivity ratio of the two solution components is asymptotically small. In a one-dimensional spatial domain, two distinct types of oscillatory instabilities of multi-spike solutions to the Gray-Scott model that occur in different parameter regimes are analyzed. These two instabilities relate to either an oscillatory instability in the amplitudes of the spikes, or an oscillatory instability in the spatial locations of the spikes. In the latter case a novel Stefan-type problem, with moving Dirac source terms, is shown to characterize the dynamics of a collection of spikes. From a numerical and analytical study of this problem, it is shown that an oscillatory motion in the spike locations can be initiated through a Hopf bifurcation. In a subregime of the parameters it is shown that this Stefan-type problem is quasi-steady, allowing for the derivation of an explicit set of ODE’s for the spike dynamics. In this subregime, a nonlocal eigenvalue problem analysis shows that spike amplitude oscillations can occur from another Hopf bifurcation. In a two-dimensional domain, the method of matched asymptotic ex- pansions is used to construct multi-spot solutions by effectively summing an infinite-order logarithmic expansion in terms of a small parameter. An asymptotic differential algebraic system of ODE’s for the spot locations is derived to characterize the slow dynamics of a collection of spots. Further- more, it is shown theoretically and from the numerical computation of cer- tain eigenvalue problems that there are three main types of fast instabilities for a multi-spot solution. These instabilities are spot self-replication, spot annihilation due to overcrowding, and an oscillatory instability in the spot amplitudes. These instability mechanisms are studied in detail and phase diagrams in parameter space where they occur are computed and illustrated for various spatial configurations of spots and several domain geometries. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Turing Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Experimental and Numerical Evidence of Spot Patterns . . . 3 1.3 A Brief History of the Gray-Scott Model . . . . . . . . . . . 4 1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.1 Turing Stability Analysis . . . . . . . . . . . . . . . . 7 1.4.2 Weakly Nonlinear Theory . . . . . . . . . . . . . . . . 8 1.4.3 Techniques for Localized Spike and Spot Patterns . . 9 1.5 Literature Review: Localized Spot Patterns for Reaction- Diffusion Systems . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5.1 Analysis in a One-Dimensional Domain . . . . . . . . 11 1.5.2 Analysis in a Two-Dimensional Domain . . . . . . . . 17 1.6 Objectives And Outline . . . . . . . . . . . . . . . . . . . . . 19 1.6.1 Thesis Outline: the GS Model in a 1-D Domain . . . 19 1.6.2 Thesis Outline: the GS Model in a 2-D Domain . . . 22 2 Dynamics and Oscillatory Instabilities of Spikes in the 1-D GS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 The Dynamics of k-Spike Quasi-Equilibria . . . . . . . . . . 29 2.1.1 Multi-Spike Quasi-Equilibria . . . . . . . . . . . . . . 32 iii Table of Contents 2.2 Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria . 35 2.2.1 Numerical Experiments . . . . . . . . . . . . . . . . . 40 2.3 Oscillatory Drift Instabilities of k-Spike Patterns . . . . . . . 43 2.3.1 Coupled ODE-PDE Stefan Problem . . . . . . . . . . 48 2.3.2 Oscillatory Drift Instabilities . . . . . . . . . . . . . . 52 2.3.3 Integral Equation Formulation . . . . . . . . . . . . . 55 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Dynamics and Spot-Replication: The 2-D GS Model . . . 59 3.1 k-spot Quasi-Equilibrium Solutions . . . . . . . . . . . . . . 59 3.2 The Spot-Splitting Instability . . . . . . . . . . . . . . . . . 67 3.3 The Slow Dynamics of Spots . . . . . . . . . . . . . . . . . . 70 3.4 The Direction of Splitting . . . . . . . . . . . . . . . . . . . . 74 3.5 Two-Spot Patterns in an Infinite Domain . . . . . . . . . . . 79 3.6 The Reduced-Wave and Neumann Green’s Functions . . . . 82 3.6.1 Green’s Function for a Unit Disk . . . . . . . . . . . . 82 3.6.2 Neumann Green’s Function for a Unit Disk . . . . . . 85 3.6.3 Green’s Function for a Rectangle . . . . . . . . . . . 86 3.6.4 Neumann Green’s Function for a Rectangle . . . . . . 87 3.7 Comparison of Theory with Numerical Experiments . . . . . 87 3.7.1 The Unit Square . . . . . . . . . . . . . . . . . . . . . 88 3.7.2 The Unit Disk . . . . . . . . . . . . . . . . . . . . . . 93 3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4 Competition and Oscillatory Profile Instabilities . . . . . . 104 4.1 Eigenvalue Problem for the Mode m = 0 . . . . . . . . . . . 104 4.1.1 Numerical Methods for the Eigenvalue Problem . . . 110 4.2 Nonlocal Eigenvalue Analysis . . . . . . . . . . . . . . . . . . 112 4.2.1 A One-spot Solution in an Infinite domain . . . . . . 112 4.2.2 Multi-Spot Patterns in a Finite Domain . . . . . . . . 114 4.2.3 Comparison of the Quasi-Equilibrium Solutions . . . 118 4.2.4 Comparison of the Eigenvalue Problem . . . . . . . . 120 4.3 Instabilities in an Unbounded Domain . . . . . . . . . . . . . 122 4.3.1 A One-Spot Solution . . . . . . . . . . . . . . . . . . 124 4.3.2 A Two-Spot Solution . . . . . . . . . . . . . . . . . . 127 4.4 A One-Spot Solution in a Finite Domain . . . . . . . . . . . 131 4.4.1 A One-Spot Solution in the Unit Disk . . . . . . . . . 131 4.4.2 A One-Spot Solution in a Square . . . . . . . . . . . 134 4.5 Symmetric k−Spot Patterns in a Finite Domain . . . . . . . 136 4.5.1 A Symmetric Circulant Matrix . . . . . . . . . . . . . 136 iv Table of Contents 4.5.2 A Ring of Spots in the Unit Disk . . . . . . . . . . . 138 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . 148 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2 Application to Other Systems . . . . . . . . . . . . . . . . . 151 5.2.1 Localized Patterns in Cardiovascular Calcification . . 151 5.2.2 The Brusselator Model with Superdiffusion . . . . . . 152 5.2.3 A Three-Component Reaction-Diffusion System . . . 152 5.2.4 A General Class of Reaction-Diffusion Models . . . . 153 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 v List of Tables 3.1 Correction terms and accuracy for the reduced-wave Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.2 Data for Experiment 3.3: . . . . . . . . . . . . . . . . . . . . 89 3.3 Data for Experiment 3.4 . . . . . . . . . . . . . . . . . . . . . 91 3.4 Data for Experiment 3.5 . . . . . . . . . . . . . . . . . . . . . 93 4.1 Equilibrium ring radius for a two-spot symmetric pattern in the unit disk . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 vi List of Figures 1.1 FIS experiment: Spot self-replication . . . . . . . . . . . . . . 6 1.2 Different types of instabilities of spike solutions . . . . . . . . 14 1.3 Slow dynamics of two-spike quasi-equilibrium solutions . . . . 20 1.4 Spot self-replication for the GS model . . . . . . . . . . . . . 26 2.1 Slow evolution of a two-spike pattern . . . . . . . . . . . . . . 42 2.2 Slow evolution of a three-spike pattern . . . . . . . . . . . . . 44 2.3 Drift instability threshold . . . . . . . . . . . . . . . . . . . . 48 2.4 Testing the stability threshold for a one-spike pattern . . . . 53 2.5 Large-scale oscillation of a one-spike pattern . . . . . . . . . . 54 2.6 Oscillatory drift instability and the breather mode . . . . . . 55 3.1 Numerical solution of the core problem (3.2) . . . . . . . . . . 60 3.2 One spot in a square: saddle node bifurcation . . . . . . . . . 63 3.3 Eigenvalue of (3.25) with modes m ≥ 2 . . . . . . . . . . . . . 69 3.4 Two spots in the infinite plane . . . . . . . . . . . . . . . . . 81 3.5 A three-spot pattern: slow drifting spots . . . . . . . . . . . . 89 3.6 Comparison of asymptotic and numerical results for the dy- namics of a three-spot pattern . . . . . . . . . . . . . . . . . . 90 3.7 A three-spot pattern: self-replication . . . . . . . . . . . . . . 92 3.8 Comparison of asymptotic and numerical results for the dy- namics of a four-spot pattern after a spot-splitting event . . . 92 3.9 An asymmetric four-spot pattern: two self-replication events 94 3.10 Comparison of asymptotic and numerical results for the dy- namics of a four-spot pattern after two spot-splitting events . 94 3.11 Symmetric pattern: all k spots on a ring in a unit circle . . . 96 3.12 Four spots on a ring: spot-replication . . . . . . . . . . . . . 97 3.13 One center-spot and all other spots on a ring . . . . . . . . . 99 3.14 One center-spot and three spots on a ring: different splitting patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.15 One center-spot and three spots on a ring: different splitting patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 vii List of Figures 3.16 One center-spot and nine spots on a ring: dynamical splitting instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1 Principal eigenvalue of (4.2) with Nj bounded at infinity . . . 107 4.2 Eigenfunctions Φj and Nj of (4.2) with Nj bounded at infinity108 4.3 One-spot solution in the infinite plane: phase diagram and spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4 One-spot solution in the infinite plane: Comparison with re- sults from NLEP theory . . . . . . . . . . . . . . . . . . . . . 126 4.5 Two-spot solution in the infinite plane: Phase diagram . . . . 129 4.6 One-spot solution in the unit disk: dynamical profile instability133 4.7 One-spot solution in the unit disk: a test of the profile insta- bility threshold with A = 0.16, D = 4 . . . . . . . . . . . . . 133 4.8 One-spot solution in the unit disk: a test of the profile insta- bility threshold with A = 0.16, D = 6 . . . . . . . . . . . . . 134 4.9 One-spot solution in the unit square: Dynamical profile in- stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.10 One-spot solution in the unit square: a test of the dynamical profile instability . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.11 Two spots on a ring: a dynamical competition instability . . 139 4.12 Two spots on a ring: in-phase oscillation threshold . . . . . . 140 4.13 Two spots on a ring: a competition instability . . . . . . . . . 140 4.14 Two spots on a ring: a competition instability . . . . . . . . . 143 4.15 Two spots on a ring: the spot amplitudes . . . . . . . . . . . 144 4.16 Two spots on a ring: the phase diagram . . . . . . . . . . . . 145 4.17 k spots with k > 2 on a ring: the phase diagram . . . . . . . 146 viii Acknowledgements This thesis presents the work under the supervision of Dr. Michael Ward, I would like to thank him for his great guidance, support and encouragement. Also I would like to thank Dr. Brian Wetton for his useful suggestions and help. Finally, I would like to thank Prof. Juncheng Wei of the Chinese Uni- versity of Hong Kong for some helpful discussions on the theory of nonlocal eigenvalue problems. ix To my dear parents and my husband x Chapter 1 Introduction Various patterns have been observed in the physical world, such as spot or stripe patterns on animal skins, spiral waves in the Belousov-Zhabotinsky (BT) reaction, among many others. In 1952, the British mathematician Alan Turing [90] first proposed a simple reaction-diffusion system describ- ing chemical reaction and diffusion to account for morphogenesis, i.e. the development of patterns in biological systems. This study is the foundation of modern pattern formation. In [90], Turing employed linear analysis to determine the threshold for the instability of spatially homogeneous equilib- rium solutions of general two-component reaction-diffusion systems; a more specific summary of his work is discussed in §1.1. Turing’s original study has stimulated numerous theoretical and numerical studies of reaction-diffusion systems, which focus on pattern formation from a spatially uniform state that is near the transition from linear stability to linear instability. It was suggested in 1972 by Gierer et. al. [31] and Segel et.al. [83] that two key in- gredients for localization are positive feedback of the activator, which results in a self-production of activator substance, and a long-range inhibiting sub- stance, which suppresses the growth of activator. It is these two competing processes that give rise to different patterns. In 1993, Pearson [77] observed that for parameter values far from the Turing instability regime, the Gray-Scott model (cf. [35]) in a two-dimensional spatial domain can exhibit a rich variety of spatio-temporal patterns includ- ing, stationary spots, traveling spots, spot self-replication, spot-annihilation, growing stripes, labyrinthian patterns , stripe filaments, and spatial-temporal chaos, etc. The common feature in all of these patterns is that each con- sists of two distinct states of solutions: some localized regions where the chemical concentrations are very large, and a background ambient spatially homogeneous state. As time evolves, the localized regions of elevated chemi- cal concentrations can remain stable, or develop very complicated structures through drifting, splitting, breaking, etc., driven by intricate and unknown mechanisms that depend on the range of parameters in the reaction-diffusion model. The stability and dynamics of these localized patterns can not be analyzed by Turing’s approach based on a linearization around a spatially 1 1.1. Turing Patterns homogeneous equilibrium state. A detailed mathematical study of these localized structures could have significant applications in controlling chemical reactions for certain purposes, and in understanding and classifying patterns in biological systems. One ex- ample is the self-organized formation of either labyrinths, spots, or stripe patterns in the calcification and mineralization of cardiovascular stem cells, described by a prototype Gierer-Meinhardt model (cf. [105]). Another ex- ample is the stable or oscillatory spots arising in the BZ reaction in a closed system in which the reactants are mixed with octane and the surfactant aerosol OT (BZ-AOT) (cf. [41]). This reaction can be fabricated to design a re-writable memory device (cf. [41]). Over the past decade there has been a more systematic effort to under- take detailed numerical and theoretical studies of the dynamics and stabil- ity of localized patterns in reaction-diffusion systems. In this introductory chapter, in §1.1 we first review the pioneering work of Turing on station- ary, spatially periodic patterns, resulting from the interplay between pure diffusion and nonlinear reaction kinetics. Some experimental and numerical evidence of spot patterns and spot self-replication behavior are provided in §1.2. A brief history of the Gray-Scott model is given in §1.3. The various theoretical approaches for analyzing pattern formation are outlined in §1.4. Previous theoretical work for the existence, stability, and dynamics for one- and two-dimensional reaction-diffusion models are surveyed in §1.5. Finally in §1.6, we give an outline of this thesis and briefly highlight the analytical and numerical approaches that we have employed. 1.1 Turing Patterns A typical two-component reaction-diffusion (RD) system has the form τAAt = DA∆A+ F (A,H) , x ∈ Ω ; ∂nA = 0 , x ∈ ∂Ω , (1.1a) τHHt = DH∆H +G(A,H) , x ∈ Ω ; ∂nH = 0 , x ∈ ∂Ω . (1.1b) Such two-component, and even multi-component, systems arise in many physical applications including, chemical reaction theory with autocataly- sis, solid combustion theory, biological morphogenesis, population dynamics with spatial segregation, semiconductor gas-discharge systems, etc.. In many specific systems, A and H are the concentrations of activator and inhibitor, respectively; DA and DH are the corresponding diffusion co- efficients; τA and τH represent the corresponding reaction-time constants; F 2 1.2. Experimental and Numerical Evidence of Spot Patterns and G are the nonlinear reaction kinetics. This system has a spatially homo- geneous equilibrium solution AE ,HE when F (AE ,HE) = G(AE ,HE) = 0. If we ignore the diffusion terms, and only consider the temporal evolution of A and H, then the Jacobian matrix of the kinetics determines the linear stability of this spatially homogeneous solution AE and HE. However, in the presence of diffusion, the criteria for instability becomes more complicated. In 1952, Alan Turing (cf. [90]) studied (1.1) mathematically, and pro- posed that spatial concentration patterns could arise from an initial arbitrary configuration due to the interaction of reaction and diffusion, which are now known as Turing patterns. Then he hypothesized that such structures could play a role in the formation of patterns of leaf buds, skin markings and limbs, etc. The first experimental observation of Turing patterns occurred almost 40 years later. In [75] it was reported that a chlorite-iodide-malonic acid (CIMA) reaction can support either hexagonal, striped, or mixed, patterns that emerge spontaneously from an initially uniform background, and then remain stationary after propagating for a period of time. These results provided experimental evidence of Turing patterns that can be maintained indefinitely in a well-defined non-equilibrium state. In 1995, Kondo and Asai [58] identified that the skin pattern of a certain species of angelfish also developed according to Turing’s instability mechanism. 1.2 Experimental and Numerical Evidence of Spot Patterns in Reaction-Diffusion Systems The motivation behind much theoretical work on localized pattern formation arises from the diverse laboratory and physical applications where localized patterns occur, and from the many supporting numerical simulations of various RD systems. A survey of experimental and theoretical studies of lo- calized spot patterns through reaction-diffusion modeling in various physical or chemical contexts is given in [92]. Since we are particularly interested in spot patterns (or spike patterns in a one-dimensional spatial domain) and the phenomena of self-replicating spots, the following examples in this sec- tion are carefully chosen to demonstrate the existence of spot patterns in various models. One example of localized spot patterns is for a certain semiconductor gas-discharge system (cf. [3] [4] [5]), which is modeled by a three-component reaction-diffusion system by Schenk et. al. (cf. [80]). In the laboratory ex- periment, when the current exceeds a certain critical value, the differential 3 1.3. A Brief History of the Gray-Scott Model resistance of the gas discharge domain becomes negative, and the homoge- neous state is then destabilized, which leads to spatio-temporal structures such as the birth, death, and scattering of localized regions of high current density. The experimental figures can be found in [4] and [5]. Our second example concerns in vitro experiments that show that vascular- derived mesenchymal stem cells can display self-organized calcified patterns such as labyrinths, stripes, and spots (cf. [30] [105]). In these experiments, the bone morphogenetic protein 2 (BMP-2) acts as an activator, and the ma- trix GLA protein (MGP) acts as the inhibitor, which is altered by external addition of MGP. Spot-replication patterns are also observed in numerical simulations of a wide range of RD systems. Muratov and Osipov [61] have performed an extensive numerical study of a prototype two-dimensional system with cubic nonlinearity. The activator nullcline has an ’N’-shape, which determines the properties of patterns and self-organization scenarios. When the ratio of diffusion coefficients and reaction-time constants satisfy DA/DH ≪ 1 and τA/τH ≪ 1, the self-replication event of a single spot is numerically studied in Fig. 10 of [61]. In addition, it has been shown numerically that above a critical temperature, a certain mathematical model of a diblock copolymer melt can exhibit localized patterns including spots, filaments, and spot self- replication phenomena (cf. [32]). 1.3 A Brief History of the Gray-Scott Model In this thesis, we concentrate on one specific RD model, the Gray-Scott model, which over the past ten years has been one of the most intensely studied RD system. It was first introduced to model an irreversible chemi- cal reaction u+ 2v → 3v, v → p, by Gray and Scott in [35]. It qualitatively models a chemical experiment that is set up in a thin transparent gel reac- tor, whose bottom surface is in contact with a well-stirred reservoir that is continuously fed by all reagents. In 1993, Pearson [77] discovered a large variety of spatio-temporal pat- terns by numerically computing solutions to the two-dimensional Gray-Scott (GS) model. This numerical study stimulated much interest in exploring new non-Turing type localized patterns in other RD systems. In [77] the following GS model with periodic boundary conditions was computed in a square of size 2.5 × 2.5: Ut = DU∆U − UV 2 + F (1− U) , (1.2a) Vt = DV∆V + UV 2 − (F + k)V . (1.2b) 4 1.3. A Brief History of the Gray-Scott Model Here k is the dimensionless rate of the second reaction and F is the dimen- sionless feed rate. The diffusion coefficients are Dv = 2Du = 2.0 × 10−5. For this ratio of diffusion coefficients used in [77] there are no stable Turing patterns. Clearly, this system has a trivial steady state at U = 1 and V = 0, which is linearly stable for any F and k. Initially the system was placed at this trivial state, then certain localized regions located symmetrically about the center of the grid were perturbed to U = 1/2 and V = 1/4. These initial conditions were further perturbed with ±1% random noise to break the symmetry. The system was then integrated for 200, 000 time steps and an image was saved. In this way, the numerical study of [77] revealed a variety of new and intricate spatial-temporal localized patterns in various regions of the two- dimensional (F, k) parameter space, which had not been seen previously in other RD systems. These patterns include stable spots, traveling spots, a mixture of spots and stripes, growing labyrinths, chaotic dynamics, and a continuous process of spot-birth through replication and spot-death through over-crowding etc. The corresponding phase diagram in the parameter space F vs. k for all patterns was plotted in [77]. Of all the new types of patterns observed, the spot self-replication process was considered the most qualita- tively interesting. Later in 1994, an autocatalytic ferrocyanide-iodate-sulphite (FIS) reac- tion experiment (cf. [59]) was performed to exhibit spot self-replication in the laboratory. In this experiment the control parameter was the concentration of ferrocyanide input in the reservoir. The repeated growth, self-replication, and annihilation, of spots was observed in this laboratory setting for a wide range of experimental parameters. The chemical reaction kinetics in this ex- perimental reaction, although much more involved, are qualitatively similar to that for a modified Gray-Scott model of the form Ut = DU∆U − UV 2 +A(U0 − U) , (1.3a) Vt = DV∆V + UV 2 +B(V0 − V ) . (1.3b) Here U and V are the concentrations of the two chemical species, the reagents in the reservoir are at fixed concentrations (U0, V0) with U0 = V0 = 1, while A and B represent the strengths of the coupling between the two chemical species. Fig. 1.1 shows the time evolution of spot patterns as copied from [59] (with permission). The top row is from the laboratory experiment, with blue (red) representing a state of high (low) pH level. The spot in the middle on the frame t = 0 develops a peanut shape at t = 4, and eventually 5 1.4. Methodology Figure 1.1: From [59] (with permission). The top row of figures are taken by video camera for the FIS reaction, with the blue (red) area representing high (low) pH levels. The bottom row are figures from the numerical simulation of the Gray-Scott model (1.3) (with permission). Similar spot self-replication processes are observed. splits. The same phenomena is observed for the spot in the middle of the last two frames. This behavior continues indefinitely, as long as the reactor conditions are maintained. The bottom row of Fig. 1.1 is from the numerical simulation of the dimensionless Gray-Scott model (1.3), with the parameters A = 0.02, B = 0.079 and DU = 2DV = 2.0× 10−5. The spots correspond to a region of high concentration V and low concentration U . It is obvious that the self-replication behaviors in the experiment and numerical computation are similar, which also suggests that replicating spots can also occur in other RD systems. In §1.5 we give a detailed summary of some more recent analytical and numerical studies of localized pattern formation in the GS model in various parameter regimes and from different mathematical viewpoints. 1.4 Methodology Theoretical approaches for the analysis of spatial-temporal pattern forma- tion have been developed over the past half century. Comprehensive surveys and examples of weakly nonlinear theory for various physical and chemical systems are given in [15], [14], [94], [66], and [95]. In this section, we describe some general analytic approaches that are applicable to many RD systems. 6 1.4. Methodology 1.4.1 Turing Stability Analysis For the finite domain problem, a Turing instability is determined by lin- earizing (1.1) around a spatially homogeneous steady state (AE ,HE), and then examining the behavior of discrete spatial Fourier modes as a function of a dimensionless bifurcation parameter (cf. [90]). For (1.1) posed in the one-dimensional interval 0 < x < 1 with ho- mogeneous Neumann boundary conditions at both endpoints, we perturb the spatially homogeneous state (AE ,HE) by introducing in (1.1) a si- nusoidal perturbation of the form A = AE + a cos(ωjx)e λt/τA and H = HE +h cos(ωjx)e λt/τH , where ωj = π j. By linearizing the resulting system, we obtain the eigenvalue problem λ ( a h ) = ( −ω2jDA +K1 K2 K3 −ω2jDH +K4 ) ( a h ) . Here K1 = FA(AE ,HE), K2 = FH(AE ,HE), K3 = GA(AE ,HE), and K4 = GH(AE ,HE). Assume that the spatially homogeneous equilibrium state is always stable, so that K1+K4 < 0 and K1K4−K2K3 > 0. In the presence of diffusion, the spatially homogeneous equilibrium state is stable when the following two conditions hold: −ω2j (DA +DH) +K1 +K4 < 0 , (K1 − ω2jDA)(K3 − ω2jDH)−K2K3 > 0 . When these conditions are not satisfied, the spatially homogeneous equi- librium state loses its stability to a spatially periodic Turing pattern of a certain wavelength. This type of diffusion-driven instability is well-known to occur when the ratio of inhibitor to activator diffusivity, i.e. DH/DA, is sufficiently large (cf. [90]). Many examples of this type of linear Turing stability theory are given in [66]. There are several common classifications of RD systems. If A is an activator it means that for some parameters the amplitudes of localized structures of A will grow when H is fixed. Mathematically this requires that ∂AF < 0 for certain values of A and H (cf. [42] [44] [43]). Moreover, if H is an inhibitor it means that H diffuses over a wide range to damp elevated regions of the activator A. These conditions are expressed mathematically in [42], [44], and [43], by ∂HG > 0 and ∂AG · ∂HF < 0 for all values of A and H. Then, according to the sign of ∂AG, the RD system (1.1) is classified into two categories: the activator-substrate model with a Jacobian matrix with sign structure ( + + − − ) , and the activator-inhibitor model with a Jacobian matrix with sign structure ( + − + − ) . 7 1.4. Methodology It follows readily that the GS system (1.2) is an activator-substrate model. The activator V increases the rate of a catalyzed reaction, while the chemical U is the substance acted upon by a catalyst, which is referred to as a substrate. The inhibitory effect results from the depletion of the substrate required to produce the activator. 1.4.2 Weakly Nonlinear Theory Perturbation Method For nonlinear differential equations, finding an analytical solution is impos- sible in general. However, when some parameters take extreme values, the original model can often be reduced to a simpler one for which the solu- tion can be calculated analytically. In the 1960’s and early 1970’s, many asymptotic and perturbation methods were developed and studied. The importance of these methods is that they can often greatly reduce the in- tricate mathematical models into a form that is more amenable to analyze (cf. [81]). Many formal asymptotic and perturbation methods are discussed and illustrated in [45]. One method, called the method of multiple scales, applies to wave-type problems where there are two time or space scales; a fast oscillation together with a slow modulation of the envelope of the fast oscillation. Usually it is difficult to represent all physical scales analytically, but this method allows for an accurate solution over asymptotically long time or space intervals. When the nonlinearities are weak and for the slow spatial and temporal solution behavior generic to parameter values near instability thresholds, the method of amplitude equations is often used to describe weakly nonlinear effects for the envelope function associated with the basic state (cf. [82] [67]). The derivation of these amplitude equations, which evolve over an asymp- totically long time interval, is based on a formal multiple-scale method. The analysis incorporates projections of the nonlinear terms on the discrete unstable Fourier modes in a perturbative way. For many infinite domain problems, the resulting amplitude equation characterizing a weakly non- linear instability near a bifurcation point is the complex Ginzburg-Landau model (CGL) partial differential equation ∂tA = µA− (1 + iα)|A|2A+ γ(1 + iβ)∂xxA . (1.4) Here i = √−1, while α, β, µ, γ are real constants with numerical values determined by the kinetics and diffusivities of each specific RD system. This CGL model provides a general and simple framework to address various 8 1.4. Methodology RD models in a unified manner, and thus it plays an important role in analyzing weakly nonlinear instabilities. However, the CGL model is only valid in a small region near instability thresholds of the base state, and may provide qualitatively misleading information if it is applied to explain pattern formation scenarios far from the instability threshold. Formal Method of Bifurcation Theory Another general method of analysis of RD systems uses qualitative bifurca- tion theory of differential equations to find general features of the solutions. Starting in the late 1960s there was an increased focus on the development of a bifurcation theory to describe the branching behavior of solutions to differ- ential equations as a function of dimensionless parameters, and to determine mathematical principles governing the exchange of stability of intersecting solution branches. This approach is geometrical and topological, and can be applied either at the level of fast oscillations, or to analyze the amplitude equations that are difficult to solve. An expansion scheme known as normal form theory (cf. [33]) is often considered to preserve the essential features of the original model and it represents a universal description of dynamics near bifurcation points. This theoretical framework is very useful in complicated situations with degener- ate or symmetric bifurcations, but it is usually restricted to ideal, spatially periodic solutions. A related approach is to derive a reduced dynamical description near a bifurcation point by projecting the dynamics onto a lower dimensional space (the center manifold). For finite domain problems, center manifold theory (cf. [9]) allows for the rigorous analysis of ODE amplitude equations. 1.4.3 Techniques for Localized Spike and Spot Patterns In contrast to the extensive development and successful use of weakly nonlin- ear theory to characterize small-amplitude pattern formation near bifurca- tion points in the 1970’s and 1980’s, there are few general theoretical results addressing the dynamics and stability of spatially localized patterns for RD models, such as those described in §1.2. The role of analytical theories to explain the dynamics and stability of localized patterns, including localized spikes and spots, stripes, spiral waves and interfacial patterns etc., for which the singular perturbation method is essential, has gained increasing interest since the mid-1980s. With regards to the rich collection of localized patterns computed for 9 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems the GS model by him in [77], Pearson comments: ”Most work in this field has focused on pattern formation from a spatially uniform state that is near the transition from linear stability to linear instability. With this restric- tion, standard bifurcation-theoretic tools such as amplitude equations have been used with considerable success. It is unclear whether the patterns pre- sented here will yield to these standard technologies.” Further emphasizing this point, Knobloch in his survey [47] remarks that ”The question of the stability of finite amplitude structures, be they periodic or localized, and their bifurcation properties is a major topic that requires new insights.” In particular, to characterize localized spike and spot patterns the method of matched asymptotic expansions must be used to incorporate the two dis- tinct spatial scales, and to resolve localized regions or boundary layers where solution behavior changes rapidly. Similar singular perturbation techniques often allow for the reduction of an RD system to a finite-dimensional dynam- ical system for certain collective coordinates that evolve slowly in time. The evolution of these coordinates often characterize the dynamics of certain fea- tures of quasi-equilibrium localized patterns, such as slowly drifting spots. In order to analyze the stability of spatially localized quasi-equilibrium so- lutions, one typically must analyze the spectrum of certain singularly per- turbed eigenvalue problems. These eigenvalue problems, with two distinct spatial scales, are often very challenging to investigate analytically. In cer- tain limiting cases, a class of nonlocal eigenvalue problems determines the threshold for the instability of localized patterns. This general theoretical framework is the one used in this thesis to investigate some specific aspects of localized pattern formation in the GS model. 1.5 Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems The experimental and numerical evidence highlighted above for the occur- rence of spot patterns and various spatio-temporal structures have, over the past decade, been the motivation for many theoretical investigations concerning the existence, stability, and dynamics of localized patterns in reaction-diffusion systems. For the study of localized pattern formation, there are two main asymp- totic regimes of the diffusion coefficients in (1.1) that can be distinguished: the weak interaction regime with DA ≪ 1, DA/DH = O(1) where the origi- nal numerical simulation of the Gray-Scott model (cf. [77]) was performed, and the semi-strong interaction regime DA/DH ≪ 1, where many analytical 10 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems studies, such as those described below, have been focused. In the semi-strong interaction regime, a prototypical reaction-diffusion system of activator-substrate type is the Gray-Scott (GS) model (1.2) where the chemical u is a fast-diffusing substrate, and is consumed by a slowly- diffusing activator v. This mechanism drives sharply localized spatial spikes (or spots) of activator coupled with nearby shallow dips of substrate. In contrast, the Gierer-Meinhardt (GM) model was suggested in [31] as a typi- cal activator-inhibitor reaction-diffusion system. For this model the reaction terms in (1.1) have the form F (A,H) = −A+ A p Hq , G(A,H) = −H+ A r Hs , 1 < qr (p− 1)(s + 1) , 1 < p . In the semi-strong interaction regime, the time-dependent solutions to the GM model are characterized by sharply localized spatial spikes (or spots) of activator coupled with nearby shallow peaks of inhibitor. The inhibitor has a long range interaction and mediates the creation of additional spikes (or spots) of activator concentration. Over the past decade there have been many theoretical studies relating to the existence, stability, and dynamics, of localized structures for (1.1) in the semi-strong interaction regime, for various choices of the kinetics. Most of these previous studies have been focused on pattern formation in a one-dimensional domain. In contrast, owing to the significantly increased mathematical complexity of higher dimensional analysis, there have been relatively few analytical studies of localized pattern formation in more re- alistic two-dimensional domains. In §1.3.1 we give a literature survey of some analytical studies of spike-type pattern formation in one-dimensional domains, while in §1.3.2 we highlight some analytical work for spot-type pattern formation in two-dimensional domains. 1.5.1 Analysis in a One-Dimensional Domain Since early 1990s, there have been many analytical studies for spike patterns of the Gray-Scott (GS) model (1.2). A convenient alternative dimensionless form of the GS model was put forth in [64], by introducing v = V/ √ F , x = −1 + 2X/L and t = (F + k)T , where L is the length of the spatial interval, i.e. X ∈ [0, L]. Then, on |x| < 1, (1.2) becomes vt = ε 2 vxx − v +Auv2, vx(±1, t) = 0 , (1.5a) τut = Duxx + (1− u)− uv2, ux(±1, t) = 0 , (1.5b) 11 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems where the positive parameters A, τ , D, and 0 < ε≪ 1, are defined in terms of the positive parameters F , k, DU , DV , L of (1.2) as D ≡ 4DU FL2 , ε2 ≡ 4DV L2(F + k) , τ ≡ F + k k , A ≡ √ F F + k . The new system (1.5) is particularly convenient in that it shows that the asymptotic construction of equilibrium solutions in the semi-strong limit ε → 0 with D = O(1) depends only on the so-called feed-rate parameter A and the diffusion coefficient D. Alternatively, the reaction-time constant τ only influences the stability of the solutions. The effect of the finite domain and the strength of inter-spike interactions is mediated by the diffusivity D. The problem (1.5) for a one-spike solution with D ≪ 1 and the spike interior to (−1, 1) is essentially equivalent to the problem for a one-spike solution on the infinite line x ∈ (−∞,∞). For k−spike patterns, the effect of the finite domain and inter-spike interactions are significant only when k √ D = O(1). When k √ D ≪ 1, an equilibrium k−spike solution is closely approximated by a solution consisting of k identical copies of a one-spike equilibrium for the infinite line problem. In the weak interaction regime where D/ε2 = O(1), Nishiura et. al [72] have studied self-replicating patterns in (1.2) for the specific diffusivity val- ues DU = 2×10−5 and DV = 10−5. Starting from a localized initial pattern of one spike, self-replication was viewed as a complicated transient process leading to a stable stationary or oscillating Turing pattern. The mechanism underlying spike self-replication suggested in [72] was based on a hierarchi- cal saddle-node bifurcation structure for the global bifurcation branches of multi-spike solutions to (1.2). The saddle-node bifurcation values for multi- spike solution branches were found to nearly coincide. When the parameters in the GS model were chosen to be near this saddle-node bifurcation value, a single initial spike was found to undergo a self-replication process, whereby the two spikes at the extremities of the spike pattern underwent repeated replication. This edge-splitting self-replication process terminated with a stable spatially periodic Turing pattern. Some necessary conditions for the occurrence of self-replicating spike patterns were formulated in [72]. See [72] for further details. The related investigations of [68] and [73] for the GS model in the weak interaction regime gave a qualitative explanation for the occurrence of spatio-temporal chaos of spikes based on the inter-relationship of global bifurcation branches of ordered patterns with respect to supply (feed-rate parameter F ) and removal rates (reaction time constant k). As a result, 12 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems the chaotic dynamics in the GS model were suggested to be driven by at least two mechanisms. One is the heteroclinic cycle built up by three dif- ferent processes; long wave instability (the unstable homogeneous state to background state (1, 0) with small perturbation), self-replication, and self- destruction (from stationary spike patterns to the homogeneous state). The other possible mechanism was the combination of replication and annihila- tion (two spikes collide and generate a new one). In the semi-strong interaction regime D ≫ O(ε2), the substrate u in the GS model varies globally over the domain. In contrast, the activator v is localized and consists of a a sequence of spikes, each of which is lo- calized within a narrow layer of order O(ε) around some point interior to the interval. In this semi-strong regime, there are many results from for- mal asymptotic analysis for spike solutions for the GS model, including, the construction of single spike solutions and pulse-splitting instability [78] [19] [55], the slow dynamics of traveling spikes [22] [16] [17] [64] [86] [53], and the oscillatory instability of the spike amplitudes [54], etc. We now describe some of this work in terms of the different parameter regimes for the dimensionless system (1.5). In terms of feed-rate parameter A, three distinct regimes for the GS model (1.5) in a one-dimensional domain, each with very different solution behavior, have been identified in [63]. They are the high feed-rate regime A = O(1) (also called the pulse-splitting regime), the low feed-rate regime A = O(ε1/2), and the intermediate regime O(ε1/2) ≪ A ≪ 1, which sepa- rates the former two regimes in A. In the high feed-rate regime A = O(1), the central phenomena is pulse- splitting, as shown in Fig. 1.2(a). Referring to Fig. 1.2(a), there is initially one spike as depicted by the blue curve, whose peak gradually deforms and splits into two spikes as shown by the green curve. The resulting spikes then slowly drift towards their equilibrium location (shown at some later instant by the red curve) as time evolves. Another possible instability that can occur in the high feed-rate regime is the oscillatory drift instability. This instability relates to the oscillatory motion of spot locations, as seen in Fig. 1.2(b), where the blue spike keeps its basic shape, but oscillates repeatedly about its equilibrium location at x = 0. The realization of either type of instability in a numerical simulation depends on the actual parameter values of A = O(1), D, and τ . In the high feed-rate regime, the asymptotic study of [55] constructed a certain core problem for k-spike equilibria in a finite domain. The far-field conditions for this core problem were derived from the asymptotic matching of outer and inner expansions for the global variable u. An alternative 13 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems −1 −0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (a) Splitting −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ← → (b) Drift −1 −0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 (c) Competition −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ↓ ↑ (d) Profile Figure 1.2: An illustration of different instabilities of spike solutions v vs. x. As time evolves, each spike pattern is plotted first by the blue curve, then the green curve and finally the red curve. (a) Splitting instability: one initial spike (in blue) splits into two and the resulting two spikes slowly drift toward their equilibrium locations; (b) Oscillatory drift instability: the location of the spike oscillates about the equilibrium location at x = 0; (c) Competition instability: the amplitude of the center spike decays as a result of the overcrowding effect; (d) Oscillatory profile instability: the amplitude of a spike oscillates repeatedly. 14 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems analysis, based on geometric singular perturbation theory, was given in [19]. It is shown in [55] that there are no k-spike equilibria to (1.5) when A > Apk with Apk = 1.347 coth(1/(k √ D)), and that a pulse-splitting process will be initiated in an O(1) time when A > Apk. A conjecture based on these critical values Apk was used in [55] to predict the the final number of equilibrium spikes. In addition, in [55] the stability of k-spike equilibria with respect to the small eigenvalues of O(ε) associated with the oscillatory drift instability was studied. The instability leads to the oscillation of spike locations as shown in Fig. 1.2(b). The scaling law for the Hopf bifurcation value associated with the oscillatory drift instability was found to be τTW = O(ε−1A−2), with the exact value for τTW depending on A,D, ε, k. In the low feed-rate regime A = O(ε1/2), there are two possible mecha- nisms for instabilities of spike patterns. One is the competition instability, whereby the interaction between spikes may annihilate some of them due to an overcrowding effect. One example of this is given in Fig. 1.2(c), where for an initial pattern of three spikes (in blue), the center spike gradually decays (in green), and eventually is annihilated (in red). Alternatively, the amplitude of a spike may oscillate repeatedly, which is referred to as an oscillatory profile instability. This instability is depicted in Fig. 1.2(d). For equilibrium spike solutions, these instabilities were analyzed in [54]. In the low feed-rate regime, with new parameter A = ε−1/2A = O(1) and variable ν = ε1/2v, we reformulate the GS model (1.5) on |x| < 1 as νt = ε 2νxx − v +Auν2 , vx(±1, t) = 0 , (1.6a) τut = Duxx + (1− u)− ε−1uν2 , ux(±1, t) = 0 . (1.6b) It was shown in [54] (see also [22]), that there exists a saddle-node bifur- cation structure of k-spike equilibria. In addition, a nonlocal eigenvalue problem (NLEP) for the large eigenvalues with λ = O(1) was analyzed in [54], which showed that these large eigenvalues can enter the unstable right half-plane either along the real axis or through a Hopf bifurcation leading, respectively, to either a competition instability (also called an overcrowding or annihilation instability) as shown in Fig. 1.2(c)), or to an oscillatory pro- file instability respectively (as shown in Fig. 1.2(d)). The type of instability obtained depends on the values of A, τ , and D. Other related work for a one-spike solution for the infinite line problem in the low feed-rate regime was given in [63]. An alternative analysis, based on geometric singular per- turbation theory, for a periodic spike pattern on the infinite line was given in [93]. In [86], the dynamics and instabilities of quasi-equilibrium two-spike so- lutions were studied in the low feed-rate regime on a finite domain. The 15 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems spikes were found to slowly drift towards there equilibrium locations on a finite interval with speed O(ε2). It was also shown the parameter thresholds for competition and oscillatory profile instabilities for the two-spike pattern depend on the parameters A, D, and τ , as well as the instantaneous loca- tions of the two spikes. This dependence of the stability thresholds on the spike locations suggests that, for a two-spike pattern that is initially stable, the pattern can be de-stabilized at some later time if, as a result of the spike motion, the stability boundary is crossed before the spikes reach their equi- librium locations. Such an intrinsic dynamically triggered instability was studied in [86]. The companion study [53] analyzed the stability problem of k-spike equilibria with respect to the small eigenvalues λ = O(ε2), which are associated with slow oscillatory drift instabilities of the equilibrium spike locations. An oscillatory drift instability refers to an oscillatory behavior of the location of a spike around its equilibrium value on a finite domain. In the intermediate regime, O(1) ≪ A ≪ O(ε−1/2), it was shown in [54] that no competition instabilities can occur for a k-spike equilibrium solution when the spikes are separated by O(1) distances. In addition, a single universal NLEP problem independent of D and k was derived to provide a scaling law for the stability of a symmetric k-spike equilibrium solution with respect to oscillatory profile instabilities. The critical value of τ where such profile instabilities occur has the scaling law τH = O(A4)≫ 1. In contrast, in [53] it was shown that the Hopf bifurcation value associated with the small eigenvalues λ = O(ε2), governing oscillatory drift instabilities, has the scaling law τTW = O(ε −2A−2). By comparing these two scaling laws, we conclude that there are the following two subregimes in the intermediate regime for A where different solution behavior occur: O(1)≪ A≪ O(ε−1/3), τH ≪ τTW , profile instability dominates; O(ε−1/3)≪ A≪ O(ε−1/2), τTW ≪ τH , drift instability dominates. In the first subregime, an oscillatory profile instability occurs before the onset of an oscillatory drift instability as τ is increased. In contrast, the drift instability dominates in the second subregime as τ is increased. For the infinite-line problem, this result was also observed in [65]. In this thesis, our contribution to the study of spike solutions for the one-dimensional problem focuses on analyzing these two distinct oscillatory instabilities in the intermediate regime for the evolution of multi-spike patterns on a finite interval. In a more general context, there have been many formal asymptotic studies of spike motion for other two-component reaction-diffusion singularly 16 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems perturbed reaction-diffusion systems including, k−spike dynamics for an elliptic-parabolic limit of the Gierer-Meinhardt (GM) model (cf. [39]), two- spike dynamics for a class of problems including a regularized GM model on the infinite line (cf. [20]), and two-spike dynamics for the GM and GS models (cf. [86]). In addition to the largely formal asymptotic studies outlined above, there are only a few rigorous theories for spike solutions to two-component reaction-diffusion systems. The existence and stability of single-spike and multi-spike stationary state on the infinite line was studied in [22] and [18]. Recently, in [21] a renormalization method was used to rigorously analyze two-spike dynamics for a regularized GM model on the infinite line. 1.5.2 Analysis in a Two-Dimensional Domain For a two-dimensional spatial domain, there are only a few analytical results characterizing spot dynamics of reaction-diffusion systems, such as [13], [52] and [88] for a one-spot solution of the GM model, and [26] [27] and [28] for exponentially weakly interacting spots in various contexts. With regards to the stability of equilibrium multi-spot patterns for sin- gularly perturbed two-component reaction diffusion systems, an analytical theory based on the rigorous derivation and analysis of certain nonlocal eigenvalue problems (NLEP) has been developed in [103] [102] [99] [101] [100] for the GM and GS models. In [99], the existence and stability of a one-spot solution was analyzed in the infinite domain Ω = R2 for the following GS model: vt = ε 2∆v − v +Auv2, x ∈ Ω, (1.7a) τut = D∆u+ (1− u)− uv2, ∂nu = ∂nv = 0, x ∈ ∂Ω. (1.7b) In [101] the one-spot analysis was extended to treat the case of a k−spot pat- terns on a bounded domain. In [101] it was shown that there is a saddle-node bifurcation for equilibrium solution branches, similar to that for multi-spike patterns for the GS model in one space dimension. This saddle-node bifur- cation was found to occur in the low feed-rate regime, characterized by the scaling A = O(ε(− ln ε)1/2). In this regime for A, in [101] a leading order asymptotic theory in powers of ν ≡ −1/ ln ε was developed to characterize the stability of a k-spot pattern for a fixed τ independent of ε. This lead- ing order theory predicts that the stability threshold for the spot pattern is independent of the spot locations and that there are no oscillatory instabil- ities in the spot amplitudes for fixed τ independent of ε. Rigorous results of 17 1.5. Literature Review: Localized Spot Patterns for Reaction-Diffusion Systems the existence and stability of asymmetric multiple spot patterns for the GS model in R2 were given in [100]. Finally, a survey of this theory, together with a further application of it to the Schnakenburg model, is described in [104]. However, these previous theoretical results based on the leading order theory in powers of ν have been found not to agree rather closely with sta- bility thresholds computed from full numerical simulations of the GS model. This discrepancy between the previous theoretical results and full numerical results occurs since ν = −1/ ln ε is not very small unless ε is extremely small, for which numerical computations are extremely stiff. Therefore, a stability theory for multiple spot solutions that accounts for all terms in powers of ν is required. In addition, since the scaling regime A = O(ε(− ln ε)) where a spot-replication instability can occur (cf. [63]) is logarithmically close to the low feed-rate regime A = O(ε(− ln ε)1/2) where competition instabili- ties can occur, it is highly desirable to develop an asymptotic theory that incorporates these two slightly different scaling regimes into a single param- eter regime where both types of instabilities be studied simultaneously. The leading order theory in [101] is not sufficiently accurate to treat both types of instabilities in a single parameter regime. To our knowledge, the first attempt to asymptotically analyze the mech- anism of self-replicating spot patterns in a two-dimensional domain is [56] for the Schnakenburg model. In the semi-strong diffusion limit of this model, a differential algebraic (DAE) system of ODE’s was derived to describe the dy- namical behavior of multi-spot patterns. This asymptotic analysis is based on constructing a quasi-equilibrium solution that has the effect, in the outer region, of representing the spots as logarithmic singularities of certain un- known source strengths at unknown spot locations. Asymptotic matching, based on summing all of the logarithmic terms in the asymptotic expan- sion, is then used to derive a DAE system for the spot locations and source strengths. Related methods to account for all logarithmic terms in singularly perturbed elliptic problems have been formulated previously for eigenvalue problems in [96] and for other related problems in [85]. With regards to spot self-replication, in [56] the numerical computation of an eigenvalue problem, which was derived by matched asymptotic analy- sis, was performed to determine the source strength value for the onset of a spot-replication event. This critical value of the source strength was found to depend on the the parameters in the Schnakenburg model, together with the domain geometry and the spot locations through a certain Neumann Green’s function. 18 1.6. Objectives And Outline 1.6 Objectives And Outline The overall general objective of this thesis is to characterize quantitatively the dynamics and instability mechanisms of spatio-temporal localized pat- terns for the GS model in various parameter regimes. The main analytical method that we use here is a formal matched asymptotic analysis, which is a very powerful tool for problems involving disparate spatial scales, such as those for localized patterns in reaction-diffusion systems. This method provides a way to reduce intricate mathematical models into a form more amenable to analysis. The central role of this asymptotic method in applied mathematics was emphasized by Segel in [81], and in his later paper [84] on quasi-steady state analysis. More specifically, this thesis consists of certain analytical and numerical results for the GS model in one-dimensional and two-dimensional domains. For a one-dimensional domain, we study the open problem of analyzing the two distinct types of oscillatory instabilities discussed previously to- gether with the dynamics of multi-spike patterns in the intermediate feed- rate regime O(1) ≪ A ≪ O(ε−1/2) for the semi-strong interaction limit D = O(1) with ε → 0. In a two-dimensional domain, we analyze the dy- namics, self-replication behavior, and competition and oscillatory profile in- stabilities of spot patterns in the semi-strong interaction regime D = O(1) and ε → 0 for the GS model. Phase diagrams in parameter space high- lighting where these diverse instabilities can occur are constructed through a combination of analytical and numerical methods from the spectrum of certain eigenvalue problems. The objectives and structure of the thesis are given below in greater detail. 1.6.1 Thesis Outline: the GS Model in a 1-D Domain In Chapter §2 we analyze the dynamics and oscillatory instabilities of multi- spike solutions to the GS model (1.6) in the intermediate regime O(1) ≪ A ≪ O(ε−1/2) of the feed-rate parameter A. The novelty and significance of this intermediate parameter regime is that, in terms of the reaction-time parameter τ , there are two distinct subregimes in A where qualitatively different types of spike dynamics and instabilities occur. These instabilities are either a breathing instability in the spike amplitude, or a breathing instability in the spike location. In addition, in the intermediate parameter regime a formal singular perturbation analysis, as summarized in Principal Result 2.1, reveals that an ODE-PDE coupled Stefan-type problem with moving Dirac source terms determines the time-dependent locations of the 19 1.6. Objectives And Outline spike trajectories. The derivation and study of this Stefan problem is a new result for spike dynamics in the GS model. Finally, in contrast to previous studies, our study is not limited to the special case of two-spike dynamics. In our analysis we can readily treat an arbitrary number of spikes for (1.6) in the regime O(1)≪ A≪ O(ε−1/2). 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.000.750.500.250.00−0.25−0.50−0.75−1.00 v, u x (a) v and u vs. x 1.00 0.75 0.50 0.25 0.00 −0.25 −0.50 −0.75 −1.00 20151050 xj σ × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × (b) xj vs. σ Figure 1.3: Slow dynamics, for ε = 0.01, A = 8, and D = 0.2, of a two-spike quasi-equilibrium solution with x1(0) = −0.2 and x2(0) = 0.3. Left figure: plot of v (solid curves) and u (dotted curves) vs. x at σ = 0, σ = 2.5, and σ = 30. Right figure: plot of xj vs. σ = ε 2A2t. As σ increases, the spikes drift to their steady state limits at x = ±1/2. The solid curves are the asymptotic results and the crosses indicate full numerical results computed from (1.6) using the software of [8]. In the subregime O(1)≪ A ≪ O(ε−1/3) with τ ≪ O(ε−2A−2), the Ste- fan problem is quasi-steady and we derive an explicit differential-algebraic (DAE) system for the spike trajectories. The result is given in Principal Result 2.2. For the case of a two-spike evolution, in Fig. 1.3 we illustrate our result by plotting the quasi-equilibrium solution for u and v together with the spike trajectories for a particular set of the parameter values. In this figure the asymptotic DAE system results for the spike trajectories are shown to compare very favorably with full numerical results computed from (1.6) using the method of [8]. In this subregime, where the speed of the spikes is O(ε2A2) ≪ 1, we show from the analysis of a nonlocal eigenvalue problem (NLEP) that the instantaneous quasi-equilibrium spike solution first loses its stability to a Hopf bifurcation in the spike amplitudes when τ = τH = O(A4). This bifurcation leads to oscillations on an O(1) time scale in the amplitudes of the spikes. Our stability results are given in Principal 20 1.6. Objectives And Outline Results 2.3 and 2.4. These NLEP stability results are the first such results for multi-spike quasi-equilibrium patterns for the GS model on a finite do- main with an arbitrary number of spikes. An important remark is that our NLEP stability analysis in the intermediate regime for a multi-spike pat- tern with O(1) inter-spike distances can be reduced to the study of a single NLEP. This feature, which greatly simplifies the stability analysis, is in dis- tinct contrast to the stability analysis of [86] and [54] for the GS model in the low feed-rate regime A = O(1), and for the corresponding Gierer-Meinhardt model (see [39], [21], [97]), where k distinct NLEP’s govern the stability of k-spike patterns. Next, we study spike dynamics in the subregime O(ε−1/3) ≪ A ≪ O(ε−1/2) with τ = τ0ε−2A−2, for some O(1) bifurcation parameter τ0. In this regime, the time-dependent spike locations are determined from the full numerical solution of a Stefan-type problem with moving Dirac source terms concentrated at the unknown spike locations. Similar moving bound- ary problems arise in the study of the immersed boundary method (see [6] and [89]). The numerical method that we use for our Stefan problem relies on the approach of [89] involving a high-order spatial discretization of singular Dirac source terms. Such high spatial accuracy is needed in our problem in order to accurately calculate the average flux for u at each source point, which determines the speed of each spike. An explicit time integration scheme is then used to advance the spike trajectories each time step. With this numerical approach we compute large-scale time-dependent oscillatory motion in the spike locations when the reaction-time parameter τ0 exceeds some critical bifurcation value. Although the overall scheme has a high spatial order of accuracy, the explicit time integration step renders our numerical scheme not particularly suitable for studying large-scale drift instabilities over very long time intervals. By linearizing the Stefan problem around an equilibrium spike solution, we analytically calculate a critical value of τ0 at which the equilibrium so- lution becomes unstable to small-scale oscillations in the equilibrium spike location. This bifurcation value of τ0, given in Proposition 2.5 below, sets the threshold value for an oscillatory drift instability. The result for τ0, based on a linearization of the Stefan problem, agrees with the result de- rived in [53] using the singular limit eigenvalue problem (SLEP) method of [69]. The SLEP method has been extensively used to study similar oscillatory drift instabilities that lead to the destabilization of equilibrium transition layer solutions for Fitzhugh-Nagumo type systems (cf. [70], [38]). With this method, Hopf bifurcation values for the onset of the instability can be cal- 21 1.6. Objectives And Outline culated and the dominant translation instability, either zigzag or breather, can be identified (cf. [70]). For spatially extended systems on the infinite line, it is then often possible to perform a center manifold reduction, valid near the Hopf bifurcation point, to develop a weakly nonlinear normal form theory for large-scale oscillatory drift instabilities (see [25], [24] and the ref- erences therein). In contrast to this normal form theory, we emphasize that our Stefan problem with moving sources provides a description of large-scale oscillatory drift instabilities for values of τ0 not necessarily close to the Hopf bifurcation point. Similar Stefan problems with moving Dirac source terms have appeared in a few other contexts. In particular, such a problem determines a flame- front interface in the thin reaction zone limit of a certain PDE model of solid fuel combustion on the infinite line (cf. [76]). By using the heat ker- nel, this Stefan problem was reformulated in [76] into a nonlinear integro- differential equation for the moving flame-front interface. By solving this integro-differential equation numerically, a periodic doubling cascade and highly irregular relaxation oscillations of the flame-front interface were com- puted in [76]. For a related Stefan problem arising from solid combustion theory, a three-term Galerkin type-truncation was used in [29] to qualita- tively approximate the Stefan problem by a more tractable finite dimen- sional dynamical system, which can then be readily analyzed. Finally, we remark that in [46] a time-dependent moving source with prescribed speed was shown to prevent blowup behavior for a certain class of nonlinear heat equation. An outline of Chapter §2 is as follows. In §2.1 we derive the Stefan problem governing spike dynamics in the intermediate regime O(1)≪ A≪ O(ε−1/2). In §2.1.1 we analyze the quasi-steady limit of this problem. In §2.2 we analyze the stability of the quasi-equilibrium spike patterns of §2.1.1 in the subregime O(1) ≪ A ≪ O(ε−1/3). In §2.3 we compute numerical solu- tions to the Stefan problem showing large-scale oscillatory drift instabilities in the subregime O(ε−1/3) ≪ A ≪ O(ε−1/2). In addition, a critical value of τ0 for the onset of this instability is determined analytically. Concluding remarks are made in §2.4. 1.6.2 Thesis Outline: the GS Model in a 2-D Domain In Chapter §3 we construct quasi-equilibrium spot patterns to the GS model in a two-dimensional domain. In addition, we derive a DAE system for the evolution of a k-spot pattern and we study self-replicating instabilities for this pattern. 22 1.6. Objectives And Outline In §3.1, the method of matched asymptotic expansions is used to con- struct a k−spot quasi-equilibrium solution to the GS model in an arbi- trary bounded 2-D spatial domain. Let the jth spot be centered at xj for j = 1, · · · , k. The local solution in the vicinity of each spot is, to leading order, radially symmetric. Upon introducing a new local spatial variable, and by rescaling u and v, we derive the following radially symmetric core problem that holds in the inner, or local, region near the jth spot: ∆ρVj − Vj + UjV 2j = 0, ∆ρUj − UjV 2j = 0, 0 < ρ <∞, (1.8a) Vj → 0, Uj → Sj ln ρ+ χ(Sj) +O(ρ−1) , as ρ→∞ . (1.8b) Here Uj, Vj are inner variables corresponding to u, v, and ∆ρ is the radi- ally symmetric part of the Laplacian. In (1.8), the local solution Uj has a logarithmic growth in the far-field, where the unknown source strength Sj > 0 is introduced to measure the logarithmic growth of the j th spot. This far-field logarithmic behavior for Uj must asymptotically match with a logarithmic singularity of the Green’s function for the reduced-wave equa- tion in the 2-D spatial domain. Recall that this Green’s function in 2-D can be decomposed as G(x;xj) ∼ − 12pi ln |x− xj | + Rjj, where Rjj is the reg- ular, or self-interaction, part of G. The core problem, without the explicit far-field condition in (1.8b), was first identified in [64] in the context of the GS model in R2, and later in the study [56] of the Schnakenburg model. The boundary value problem (1.8) is then solved by using the collocation software COLSYS (cf. [2]), for a range of values of the source strength Sj. In the outer region, the nonlinear terms in (1.7b) can be represented as the sum of singular Dirac source terms, which results from the localization of the activator v near each xj. Therefore, in the quasi-steady limit, the outer variable u can be explicitly expressed in terms of the Green’s function of the reduced-wave equation. By asymptotically matching the inner and outer solutions for u, a system of algebraic equations for the source strengths Sj for j = 1, · · · , k is obtained. This system depends on the parameters A, D, and ε together with the spot locations and domain geometry through the Green’s function. Analytically, we show that this nonlinear algebraic system has a saddle node bifurcation in terms of the parameter A. After calculating the source strengths, the quasi-equilibrium solution can then be explicitly represented in terms of A,D, ε and the core solution Uj, Vj , Sj , j = 1, · · · , k valid near each spot. In §3.3 we asymptotically match higher order terms in the asymptotic expansion of the inner and outer solutions to derive an ODE system gov- erning the dynamics of a collection of spots. This ODE system is coupled 23 1.6. Objectives And Outline to the nonlinear algebraic system for the source strengths. The resulting DAE system for the spot locations shows that the speed of each spot is slow on the order O(ε2), and that the motion of each spot is proportional to a linear combination of the gradients of a certain Green’s function at the other spot locations. This DAE system is asymptotically valid for any quasi-equilibrium spot pattern for parameter values away from the instabil- ity thresholds for spot-splitting, spot-oscillation, and spot-annihilation. In §3.2 we study the stability of the core solution near the jth spot to a local perturbation with angular dependence eimθ, wherem is a non-negative integer. Assuming that τλ ≪ O(ε−2), we then derive a radially symmetric eigenvalue problem in which m ≥ 0 appears as a parameter. The angular modem = 1 corresponds to translation, which trivially has a zero eigenvalue with multiplicity two for any source strength Sj. In contrast, spot-splitting instabilities are associated with modes m ≥ 2, whereas instabilities in the spot amplitudes correspond to m = 0. In §3, we only consider the eigenvalue problem with m ≥ 2, which corresponds to spot self-replication behavior. For this mode, the local eigenvalue problems near each spot are coupled together only through the nonlinear algebraic system that determine the source strengths Sj for j = 1, . . . , k. For a given source strength Sj the solution to the core problem (1.8) for the inner variables Uj and Vj is computed. We then compute the spectrum of the local eigenvalue problem near the jth spot in terms of the param- eters m ≥ 0 and Sj . The quasi-equilibrium solution is unstable to spot- replication with mode m when the principal eigenvalue λ0 of this problem satisfies Re(λ0) > 0. From a numerical computation of the spectrum of this eigenvalue problem, obtained by using the linear algebra package LAPACK (cf. [1]), we determine the splitting threshold by seeking the critical value Σm of the source strength Sj for mode m for which the real part of the first eigenvalue λ0 enters the positive half-plane (i.e. Re(λ0) = 0). We compute that Σ2 < Σ3 < Σ4, where Σ2 ≈ 4.31. Therefore, we predict that when Sj satisfies Σ2 < Sj < Σ3, a peanut-splitting instability with mode m = 2 for the jth spot is initiated through a linear instability. This initial instability is found numerically to be the precursor to a nonlinear spot self-replication event. This self-replication instability occurs in the high feed-rate regime, characterized by A = O ( −ε ln ε/√D ) . In §3.4 we determine the direction of spot-splitting relative to the direc- tion of spot motion. In §3.5 we analyze the motion and spot self-replication instability for a two-spot pattern in R2. In order to obtain an explicit analytical theory for certain bounded domains, in §3.6 we give analytical 24 1.6. Objectives And Outline expressions for the required Green’s function for the unit circle and for a rectangular domain, which is valid for the two relevant asymptotic lim- its D = O(1) and D ≫ 1. For these specific domains where the Green’s function is readily available, we illustrate our asymptotic theory by taking different initial spot patterns, and we compare the asymptotic results for the spot trajectories with corresponding full numerical results computed from (1.7). For instance in Fig. 1.6.2, we show numerical solutions to (1.7) in a square domain [0, 1] × [0, 1] for the parameter values A = 2.35, D = 1, and ε = 0.02, starting from an initial 4-spot pattern with spots equally spaced on a ring of radius 0.2 centered at (0.6, 0.6). The initial coordinates of the spots are x1 = (0.8, 0.6), x2 = (0.6, 0.8), x3 = (0.4, 0.6), and x4 = (0.6, 0.4). For this domain and parameter set, the numerical computation of the non- linear algebraic system for the source strengths give S1 = S2 = 2.82, and S3 = S4 = 5.69 > Σ2. Based on our stability criterion, the two spots with the larger source strengths are predicted to undergo splitting. The full nu- merical results show that the spots x1 and x2 begin to deform at t = 11, undergo self-replication after t = 21, and then ultimately approach a steady state 6−spot pattern when t ≈ 581. This numerical experiment confirms our asymptotic spot-replication criterion. Many further numerical experiments are done in §3.7. In Chapter §4, we consider two different types of instabilities of multi- spot quasi-equilibrium solutions to the GS model that occur in different parameter regions. In contrast to the spot-splitting instability occurring in the high feed-rate regime, as considered in Chapter §3, in Chapter §4 we formulate a certain eigenvalue problem associated with the angular mode m = 0 in order to study competition and oscillatory profile instabilities of multi-spot patterns in the low feed-rate regime A = O(ε(− ln ε)1/2). Recall that in the low feed-rate regime A = O(ε1/2) in a one-dimensional domain unstable eigenvalues can enter the right half-plane either along the Im(λ) = 0 axis or through a Hopf-bifurcation, resulting in either a competition insta- bility or an oscillatory profile instability of the spike amplitudes, respectively (cf. [54] and [86]). Some theoretical instability thresholds for the 2-D GS model for the low feed-rate regime A = O(ε(− ln ε)1/2) have been derived previously from a rigorous analysis of a nonlocal eigenvalue problem in [99], [101], and [100]. With a different scaling of the parameters, and to leading order in ν, this previous NLEP theory relies on certain properties of the solution of the radi- ally symmetric scalar ground state problem w′′+ρ−1w′−w+w2 = 0 for the fast variable from (1.7b), together with a constant leading order approxima- tion for the slow variable u in the inner region. With this approximation for 25 1.6. Objectives And Outline Figure 1.4: With A = 2.35, D = 1, ε = 0.02, in a square domain [0, 1]×[0, 1], we compute numerical solutions to (1.7) from an initial 4-spot pattern with spots initially equally spaced on a ring of radius 0.2 centered at (0.6, 0.6). The asymptotic theory yields the source strengths S1 = S2 = 2.82 (two dull spots), and S3 = S4 = 5.69 > Σ2 (two bright spots). The two spots with larger source strengths are predicted to split. The numerical solutions are plotted as time evolves. At t = 11 the two spots begin to deform, and undergo replication after t = 21. When t = 581, a steady state 6−spot pattern is approached. The numerical simulation confirms our asymptotic spot-replication criterion. 26 1.6. Objectives And Outline the inner solution near each spot, a nonlocal eigenvalue problem is derived and analyzed to leading order accuracy in ν. This NLEP theory provides a qualitative way to understand the mechanisms for competition and oscilla- tory instabilities. However, since the instability thresholds from this NLEP theory are only accurate to leading order in ν, our higher order stability the- ory based on accounting for all logarithmic correction terms in ν is essential for providing accurate instability thresholds for moderately small values of ε. By studying our eigenvalue problem form = 0 numerically we show that, similar to the one-dimensional case, eigenvalues can enter the right half- plane Re(λ) > 0 through either a Hopf bifurcation, leading to an oscillatory profile instability, or along the real axis Im(λ) = 0, leading to a competi- tion instability. The eigenvalue problem corresponding to the mode m = 0 consists of k separate local eigenvalue problems that are coupled together through a homogeneous linear algebraic system and a nonlinear algebraic system for the source strengths. A numerical method to compute instability thresholds from this non-standard eigenvalue formulation is proposed and implemented. Finally, by combining the profile instability results with the self-replication threshold results of Chapter §3, we plot phase diagrams for certain multi-spot patterns in simple domains that illustrate specific param- eter ranges where each instability occurs. The predictions from these phase diagrams are compared with full numerical simulations of the GS model. The outline of Chapter §4 is as follows. In §4.1, we briefly review the analysis of Chapter §3 that constructed a quasi-equilibrium k-spot pattern with an error smaller than any power of ν = −1/ ln ε. In addition, we state the eigenvalue problem corresponding to m = 0, and discuss the two dis- tinct instability types associated with this problem. Then, we develop a numerical algorithm that efficiently calculates the threshold conditions for the two types of profile instabilities. In §4.2, some analytical results from [99] and [101] for both one-spot and multi-spot patterns, which were obtained from a leading order in ν = −1/ ln ε NLEP theory, are summarized. From a perturbation calculation for ν ≪ 1, we show that to leading order our constructions of the quasi-equilibrium solution and the eigenvalue problem reproduce those results obtained previously in the NLEP theory of [99] and [101]. From our eigenvalue problem that accounts for all logarithmic correc- tion terms in ν, in §4.3 we numerically calculate the oscillatory instability threshold for a one-spot solution, Moreover, we also compute competition and oscillatory instability thresholds for two-spot patterns in R2. Then, in §4.4 and §4.5 we consider finite domains such as the unit circle and square for both one-spot or certain symmetric multi-spot patterns. The resulting 27 1.6. Objectives And Outline theoretical instability thresholds are again confirmed from full numerical simulations of the GS model (1.7). Finally in §4.6, we conclude with a short discussion and a list of some open problems. 28 Chapter 2 Dynamics and Oscillatory Instabilities of Spikes in the One-Dimensional Gray-Scott Model 2.1 The Dynamics of k-Spike Quasi-Equilibria: A Stefan Problem In this section we first derive the ODE-PDE Stefan-type problem that de- termines the time-dependent trajectories of the spike locations for (1.6) in the intermediate regime O(1)≪ A≪ O(ε−1/2). To do so we first motivate the scalings of u, v, and the slow time that are needed for describing spike dynamics when O(1) ≪ A ≪ O(ε−1/2). In the inner region, we introduce a scaling such that Auv = O(1), with v = O(A) as suggested by the equilibrium theory of §4 of [54] We also introduce the slow time σ = σ0t, with σ0 ≪ 1. With v = Avj, u = uj/A2, and y = ε−1 [x− xj(σ)], where σ = σ0t, (1.6) becomes −ε−1σ0dxj dσ v′j = v ′′ j − vj + ujv2j , (2.1a) −σ0τA2 ( dxj dσ ) u′j = D εA2u ′′ j + ε ( 1− ujA2 ) − ujv2j , (2.1b) where the primes on uj and vj indicate derivatives in y. Multiplying both sides of (2.1b) by εA2, we have the leading order equation Du′′j0 = 0, and the next order term is εA2ujv2j with ujv2j = O(1), which suggests that uj = uj0+O(εA2), where uj0 is a constant. This enforces from the equation for vj that vj = vj0 +O(εA2), and σ0 = ε2A2. Since uj is independent of y to leading order, the left-hand side of the equation for uj can be neglected only when σ0τεA2/A2 ≪ 1. When σ0 = ε2A2, this condition is satisfied only when τε3A2 ≪ 1. 29 2.1. The Dynamics of k-Spike Quasi-Equilibria This simple scaling argument suggests that we expand the solution in the jth inner region as u = 1 A2 [ u0j + εA2u1j + · · · ] , v = A [v0j + εA2v1j + · · · ] , (2.2a) where umj = umj(y, σ) and vmj(y, σ) depend on the inner variable y, and the slow time σ is defined by y ≡ ε−1 [x− xj(σ)] , σ ≡ ε2A2t . (2.2b) Upon substituting (2.2) into (1.6), and assuming that τε3A2 ≪ 1, we obtain the leading-order problem v′′0j − v0j + u0jv20j = 0 , u′′0j = 0 , −∞ < y <∞ . (2.3) At next order, we obtain on −∞ < y <∞ that v′′1j − v1j + 2u0jv0jv1j = −u1jv20j − x′jv′0j , Du′′1j = u0jv20j . (2.4) From (2.3) we take u0j to be independent of y. The leading-order solution is then written as u0j = 1 γj , v0j = γjw , (2.5) where γj = γj(σ), referred to as the amplitude of the j th spike, is to be found. Here w(y) on −∞ < y <∞ satisfies w′′ − w + w2 = 0 ; w(±∞) = 0 , w(0) > 0 , w′(0) = 0 . (2.6) The unique homoclinic of (2.6) is w = 32sech 2(y/2). Therefore, from (2.4), we get on −∞ < y <∞ that Lv1j ≡ v′′1j − v1j + 2wv1j = −γ2jw2u1j − x′jγjw′ Du′′1j = γjw2 . (2.7) Since Lw′ = 0, the solvability condition for vij in (2.7) yields that x′j ∫ ∞ −∞ ( w′ )2 dy = −γj 3 ∫ ∞ −∞ u1j d dy ( w3 ) dy . (2.8) Upon integrating (2.8) by parts twice, and noting that w and u′′1j are even functions of y, we obtain x′j ∫ ∞ −∞ ( w′ )2 dy = γj 6 [ u′1j(+∞) + u′1j(−∞) ] ∫ ∞ −∞ w3 dy . (2.9) 30 2.1. The Dynamics of k-Spike Quasi-Equilibria The explicit form of w is then used to evaluate the integrals in (2.9) to get x′j = γj [ u′1j(+∞) + u′1j(−∞) ] , j = 1, · · · , k . (2.10) Now consider the outer region defined where |x − xj| = O(1) for j = 1, · · · , k. Since the term ε−1uv2 in (1.6) is localized near each x = xj , its effect on the outer solution for u can be calculated in the sense of dis- tributions. From (2.2a), and ∫∞ −∞w 2 dy = 6, we obtain that ε−1uv2 →∑k j=1 γj (∫∞ −∞w 2 dy ) δ(x − xj) = ∑k j=1 6γjδ(x − xj). This leads to the outer problem for u(x, σ) given by τε2A2uσ = Duxx + (1− u)− k∑ j=1 6γj δ(x− xj) , |x| ≤ 1 , with ux = 0 at x = ±1. The matching condition between the inner and outer solutions for u yields u(xj(σ), σ) = 1 γjA2 ≪ 1 , ux(x ± j (σ), σ) = u ′ 1j(±∞) . (2.11) We summarize our asymptotic construction as follows: Principal Result 2.1: Assume that τε3A2 ≪ 1. Then, in the intermediate regime O(1) ≪ A ≪ O(ε−1/2), the GS model (1.6) can be reduced to the coupled ODE-PDE Stefan-type problem τε2A2uσ = Duxx + (1− u)− k∑ j=1 6γj δ(x− xj) , |x| ≤ 1 , (2.12a) dxj dσ = γj [ ux(x + j , σ) + ux(x − j , σ) ] , j = 1, · · · , k , (2.12b) u(xj(σ), σ) = 1 γjA2 , j = 1, · · · , k , (2.12c) with ux(±1, σ) = 0. Here σ = ε2A2t is the slow time scale, and xj(σ) is the center of the jth spike. We observe that (2.12a) is a heat equation with singular Dirac source terms, and that (2.12b) become explicit ODE’s only when we can deter- mine ux(x ± j , σ) analytically. Finally, the constraints in (2.12c) implicitly determine the spike amplitudes γj(σ), for j = 1, · · · , k. As a remark, the interface motion in (2.12b) resulting from the average of the flux of ux across the interface differs from that of the classic Stefan moving boundary prob- lem (cf. [74]) from mathematical theories of phase change that involve the difference in the flux of ux across the interface. 31 2.1. The Dynamics of k-Spike Quasi-Equilibria 2.1.1 Multi-Spike Quasi-Equilibria: The Quasi-Steady Limit τε2A2 ≪ 1 We suppose that τ ≪ ε−2A−2 so that u in (2.12a) is quasi-steady. With this assumption, we readily calculate from (2.12a), together with ux = 0 at x = ±1, that u =   1− g1 cosh(θ(x+1))cosh(θ(x1+1)) , −1 ≤ x < x1 1− gj sinh(θ(xj+1−x))sinh(θ(xj+1−xj)) − gj+1 sinh(θ(x−xj)) sinh(θ(xj+1−xj)) , xj ≤ x ≤ xj+1 , 1− gk cosh(θ(1−x))cosh(θ(1−xk)) , xk < x ≤ 1 . (2.13) In the middle expression above the index j ranges from j = 1, . . . , k − 1. Here θ ≡ D−1/2 and u = 1 − gj at x = xj . From this solution we calculate ux as x → x±j , and we impose the required jump conditions [Dux]j = 6γj for j = 1, · · · , k as seen from (2.12a). Here [ζ]j ≡ ζ(x+j )− ζ(x−j ). This leads to the matrix problem B g = 6√ D Γ e , (2.14) where the matrices B and Γ are defined by B ≡   c1 d1 0 · · · 0 d1 c2 d2 · · · 0 ... . . . . . . . . . ... 0 · · · dk−2 ck−1 dk−1 0 · · · 0 dk−1 ck   , Γ ≡   γ1 · · · 0 ... . . . ... 0 · · · γk   . (2.15) The vectors g and e are gT ≡ (g1, · · · , gk), eT ≡ (1, · · · , 1). The matrix entries of B are given explicitly by c1 = coth(θ(x2 − x1)) + tanh(θ(1 + x1)) , ck = coth(θ(xk − xk−1)) + tanh(θ(1− xk)) , (2.16) cj = coth(θ(xj+1 − xj)) + coth(θ(xj − xj−1)) , j = 2, · · · , k − 1 , dj = −csch(θ(xj+1 − xj)) , j = 1, · · · , k − 1 . Next, we write the constraint (2.12c) in matrix form and combine it with (2.14) to get g = e− 1A2Γ −1e . γ = √ D 6 ( Be− 1A2BΓ −1e ) , (2.17) 32 2.1. The Dynamics of k-Spike Quasi-Equilibria where γ ≡ (γ1, · · · , γk)t. Since Γ−1 depends on γ, (2.17) is a nonlinear algebraic system for the spike amplitudes γj for j = 1, · · · , k. For A ≫ 1, (2.17) reduces asymptotically to the two-term result γ = √ D 6 B ( I − 6A2√DΓ −1 0 ) e+O(A−4) , (2.18) where I is the identity matrix. Here Γ−10 is the inverse of the diagonal matrix Γ0 defined by Γ0 ≡   (Be)1 · · · 0 ... . . . ... 0 · · · (Be)k   , (2.19) where (Be)j denotes the jth component of the vector Be. Upon using the identity cothµ− cschµ = tanh(µ/2), we can write (2.18) component-wise as γj = √ D 6 (Be)j ( 1− 6A2√Drj ) +O(A−4) , rj ≡ (BΓ−10 e)j (Be)j , (2.20a) where (Be) =   tanh ( θ 2(x2 − x1) ) + tanh (θ(1 + x1)) ... tanh ( θ 2 (xj − xj−1) ) + tanh ( θ 2(xj+1 − xj) ) ... tanh ( θ 2(xk − xk−1) ) + tanh (θ(1− xk))   . (2.20b) With this approximation, and since Γ−1 ∼ 6√ D Γ−10 for A ≫ 1, (2.17) for g reduces asymptotically to g = e− 6A2√D Γ −1 0 e+O(A−4) . (2.21) This asymptotically determines the unknown constants gj for j = 1, · · · , k in (2.13) with an error O(A−4). Next, we derive an explicit form for the ODE’s in (2.12b). By calculating ux(x ± j ) from (2.13), we readily derive that x ′ = (x′1, · · · , x′k)t satisfies x′ = − 1√ D ΓQg , (2.22) 33 2.1. The Dynamics of k-Spike Quasi-Equilibria where Q is the tridiagonal matrix defined by Q ≡   e1 f1 0 · · · 0 −f1 e2 f2 · · · 0 ... . . . . . . . . . ... 0 · · · −fk−2 ek−1 fk−1 0 · · · 0 −fk−1 ek   , (2.23a) with matrix entries e1 = − coth(θ(x2 − x1)) + tanh(θ(1 + x1)) , ek = coth(θ(xk − xk−1))− tanh(θ(1− xk)) , (2.23b) ej = − coth(θ(xj+1 − xj)) + coth(θ(xj − xj−1)) , j = 2, · · · , k − 1 , fj = csch(θ(xj+1 − xj)) , j = 1, · · · , k − 1 . Finally, by combining (2.22) and (2.17), we obtain the following asymp- totic result for the dynamics of k-spike quasi-equilibria: Principal Result 2.2: Assume that O(1)≪ A≪ O(ε−1/2) and that τ sat- isfies τε2A2 ≪ 1. Then, the quasi-equilibrium solution for v(x, σ) satisfies v(x, σ) ∼ A k∑ j=1 γjw [ ε−1(x− xj(σ)) ] , (2.24) where w(y) = 32sech 2 (y/2). The corresponding outer approximation for u is given in (2.13) where the coefficients gj in (2.13) for j = 1, · · · , k are given asymptotically in (2.21). For A ≫ 1, the spike amplitudes γj(σ) are given asymptotically in terms of the instantaneous spike locations xj by (2.20a). Finally, the vector of spike locations satisfy the ODE system dx dσ ∼ − 1√ D ΓQ ( I − 6A2√DΓ −1 0 ) e , (2.25) with σ = ε2A2t. Here Γ0 is defined in (2.19) and Γ is defined in (2.15). The equilibrium positions of the spikes satisfy ΓQg = 0. In equilibrium, γj is a constant independent of j, and hence g = ce where c is some constant. Consequently, the equilibrium spike locations satisfy Qe = 0. By using (2.23), this leads to xj = −1 + (2j − 1)/k for j = 1, · · · , k. The leading-order approximation for the ODE’s in (2.25) is obtained by using Γ ∼ √ D 6 Γ0+O(A−2), and g ∼ e+O(A−2). With this approximation, 34 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria which neglects the O(A2) terms, (2.25) reduces to x′1 ∼ − 1 6 ( tanh2 [θ(1 + x1)]− tanh2 [ θ 2 (x2 − x1) ]) , x′j ∼ − 1 6 ( tanh2 [ θ 2 (xj − xj−1) ] − tanh2 [ θ 2 (xj+1 − xj) ]) , for j = 1, · · · , k − 1 , (2.26) x′k ∼ − 1 6 ( tanh2 [ θ 2 (xk − xk−1) ] − tanh2 [θ(1− xk)] ) . 2.2 Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria We now analyze the stability of the k-spike quasi-equilibrium solution of §2.1 to instabilities occurring on a fast O(1) time scale. Since the spike locations drift slowly with speed O(ε2A2) (cf. (2.25)), in our stability analysis we make the asymptotic approximation that the spikes are at some fixed locations x1, · · · , xk. This asymptotic separation of time scales with “frozen” spike locations allows for the derivation of an NLEP governing fast instabilities. Let ue and ve be the quasi-equilibrium solution constructed in §2.1. We substitute u(x, t) = ue + e λtη(x) and v(x, t) = ve + e λtφ(x) into (1.6), and then linearize to obtain ε2φxx − φ+ 2Aueveφ+Av2eη = λφ , |x| ≤ 1 ; φx(±1) = 0 , (2.27a) Dηxx − (1 + τλ) η − ε−1v2eη = 2ε−1ueveφ , |x| ≤ 1 ; ηx(±1) = 0 . (2.27b) We then look for a localized eigenfunction for φ in the form φ(x) = k∑ j=1 φj [ ε−1(x− xj) ] , (2.28) with φj → 0 as |y| → ∞. Since ve ∼ Aγjw and ue ∼ 1/(γjA2) near x = xj , we obtain from (2.27a) that φj(y) satisfies φ′′j − φj + 2wφj +A3γ2jw2η(xj) = λφj , −∞ < y <∞ . (2.29) Next, we consider (2.27b). Since φj is localized near each spike, the spatially inhomogeneous coefficients in (2.27b) can be approximated by 35 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria Dirac masses. In this way, and by using ve ∼ Aγjw, ue ∼ 1/(γjA2), and∫∞ −∞w 2 dy = 6, we obtain for x near xj that, in a distributional sense, 2ε−1ueveφ ∼ 2A (∫∞ −∞wφj dy ) δ(x−xj) and ε−1v2e ∼ 6A2γ2j δ(x−xj). There- fore, the outer problem for η in (2.27b) becomes Dηxx −  1 + τλ+ 6A2 k∑ j=1 γ2j δ(x− xj)   η = k∑ j=1 2 A (∫ +∞ −∞ wφj dy ) δ(x − xj) , (2.30) with ηx = 0 at x = ±1. Defining [ξ]j by [ξ]j ≡ ξ(xj+) − ξ(xj−), we obtain the following equivalent problem for η: Dηxx − (1 + τλ) η = 0 , |x| ≤ 1 ; ηx(±1) = 0 , (2.31a) [η]j = 0 , [Dηx]j = −6ωj + 6A2γ2j η(xj) , j = 1, .., k ; ωj ≡ − 2A (∫∞ −∞wφj dy∫∞ −∞w 2 dy ) . (2.31b) Next, we calculate ηj ≡ η(xj) from (2.31), which is needed in (2.29). To do so, we solve (2.31a) on each subinterval and then use the jump conditions in (2.31b). This leads to the matrix problem Eλη = 6 [D(1 + τλ)]1/2 ω , (2.32) where the vectors ω and η are defined by ωt = (ω1, · · · , ωk) and ηt = (η1, · · · , ηk), and where t denotes transpose. The matrix Eλ in (2.32) is defined in terms of a tridiagonal matrix Bλ by Eλ ≡ Bλ + 6A 2√ D(1 + τλ) Γ2 , Bλ ≡   c1,λ d1,λ 0 · · · 0 d1,λ . . . . . . . . . ... 0 . . . . . . . . . 0 ... . . . . . . . . . dk−1,λ 0 · · · 0 dk−1,λ ck,λ   . (2.33) 36 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria Here Γ is the diagonal matrix of spike amplitudes defined in (2.15). The matrix entries of Bλ are c1,λ = coth(θλ(x2 − x1)) + tanh(θλ(1 + x1)) , ck,λ = coth(θλ(xk − xk−1) + tanh(θλ(1− xk)) , (2.34) cj,λ = coth (θλ(xj+1 − xj)) + coth (θλ(xj − xj−1)) , j = 2, · · · , k − 1 , dj,λ = −csch (θλ(xj+1 − xj)) , j = 1, · · · , k − 1 . In (2.34), θλ is the principal branch of the square root for θλ ≡ θ0 √ 1 + τλ, with θ0 ≡ D−1/2. Notice that when λ = 0, Bλ = B, where B was defined in (2.15). Next, we invert (2.32) to obtain η = 1 [D(1 + τλ)]1/2 E−1λ ω = − 12 A √ D(1 + τλ) (∫∞ −∞w (E−1λ φ) dy∫∞ −∞w 2 dy ) , (2.35) where φt = (φ1, · · · , φk). Substituting (2.35) into (2.29), we obtain the following NLEP in terms of a new matrix Cλ: φ′′ −φ+ 2wφ−w2 (∫∞ −∞w (Cλφ) dy∫∞ −∞w 2 dy ) = λφ ; Cλ ≡ 12A 2√ D(1 + τλ) Γ2E−1λ . (2.36) Since Γ is a positive definite diagonal matrix and Eλ is symmetric, we can decompose Cλ as Cλ = SXS−1 , (2.37) where X is the diagonal matrix of eigenvalues χj of Cλ and S is the matrix of its eigenvectors sj. We then set ψ = S−1φ to diagonalize (2.36), and we rewrite Cλ in terms of Bλ by using (2.33). This leads to the following stability criterion for O(1) time scale instabilities: Principal Result 2.3: The k-spike quasi-equilibrium solution of §2.1 with fixed spike locations is stable on a fast O(1) time scale if the following k NLEP’s on −∞ < y <∞, given by ψ′′−ψ+2wψ−χj(τλ)w2 (∫∞ −∞wψ dy∫∞ −∞w 2 dy ) = λψ ; ψ → 0 as |y| → ∞ , (2.38) only have eigenvalues with Re(λ) < 0. Here χj(τλ) for j = 1, · · · , k are the eigenvalues of the matrix Cλ defined by Cλ = 2 [ I + √ D(1 + τλ) 6A2 BλΓ −2 ]−1 . (2.39) 37 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria We now analyze (2.38). Let τ = O(1) and assume that the inter-spike distances dj = xj+1 − xj for j = 0, · · · , k are O(1). Here, for j = 0 and j = k we define d0 = x1 + 1 and dk = 1 − xk, respectively. Then, for A ≫ 1, (2.39) yields Cλ ∼ 2I so that χj ∼ 2 for j = 1, · · · , k. Since χj is a constant with χj > 1 in this limit, the result of Theorem 1.4 of [98] proves that Re(λ) < 0. This guarantees stability for A ≫ 1 when τ = O(1) and dj = xj+1 − xj = O(1). Next, we consider the limit A ≫ 1, τ = O(1), but we will allow for the inter-spike distances dj = xj+1− xj to be small but with dj ≫ O(ε). Recall that the analysis above and in §2.1 leading to Principal Result 2.3 required that dj ≫ O(ε). We will show that for the scaling regime dj = O(A−2/3)≫ O(ε) of closely spaced spikes, and with O(1) ≪ A ≪ O(ε−1/2), the NLEP (2.38) can have an unstable eigenvalue for any τ > 0. For simplicity we will only consider a closely spaced equilibrium configuration with j = 1, · · · , k− 1, and d0 = dk = L/2. In this limit it is readily shown from (2.34) that Bλ ∼ 1 θλL B0 , B0 ≡   1 −1 0 · · · 0 −1 2 −1 · · · 0 ... . . . . . . . . . ... 0 · · · −1 2 −1 0 · · · 0 −1 1   . (2.40) Moreover, for A ≫ 1 and with spikes of a uniform spacing L≪ 1, we obtain from (2.20) that the spikes have a common amplitude given by γj ∼ √ D 6 (Be)j = √ D 3 tanh ( θ0L 2 ) ∼ L 6 . (2.41) Substituting (2.40) and (2.41) into (2.39), we obtain that the eigenvalues χj of Cλ are given by χj ∼ 2 [ 1 + √ D(1 + τλ) 6A2 ( 36 L2 )( rj θλL )]−1 = 2 [ 1 + 6Drj A2L3 ]−1 , (2.42) where rj for j = 1, · · · , k are the eigenvalues of the matrix B0 defined in (2.40). Appendix E of [40] and Theorem 1.4 of [98] prove that when χj is constant, then Re(λ) > 0 if and only if χj < 1 for j = 1, · · · , k. The stability threshold is determined by χm ≡ minj=1,··· ,k(χj) = 1, which depends on the largest eigenvalue rm ≡ maxj=1,···k(rj) of B0. By calculating rm and setting χm = 1, we obtain that the k-spike equilibrium 38 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria solution is unstable for any τ > 0 on the regime O(1) ≪ A ≪ O(ε−1/2) when L < L∗, where L∗ ≡ ( 6Drm A2 )1/3 , rm ≡ 2 [1 + cos(π/k)] . (2.43) The result (2.43) shows that competition instabilities in the intermediate regime O(1) ≪ A ≪ O(ε−1/2) can only occur if the spikes are sufficiently close with inter-spike separation O(A−2/3). If we did set A = O(1) in (2.43), then L∗ = O(1) is consistent with the scalings in [54] and [86] for competition instabilities of equilibrium and quasi-equilibrium spike patterns in the low feed-rate regime A = O(1). Next, we obtain our main instability result that oscillatory instabilities of k-spike quasi-equilibria can occur when τ = O(A4) ≫ 1 and with O(1) inter-spike separation distances. From the choice of the principal value of the square root, we obtain that Re(θλ) = θ0Re( √ 1 + τλ)→ +∞ on |arg(λ)| < π as τ → ∞. Therefore, for τ ≫ 1, and for xj+1 − xj = O(1), we calculate from the matrix entries of Bλ in (2.34) that Bλ → 2I in |arg(λ)| < π. Therefore, with τ = τ̃A4, where τ̃ = O(1), we get from (2.39) that Cλ and its eigenvalues χj are given asymptotically by Cλ ∼ 2 [ I + √ Dτ̃λ 3 Γ−2 ]−1 ; χj ∼ 2 [ 1 + √ Dτ̃λ 3 γ−2j ]−1 j = 1, · · · , k . (2.44) This limiting expression for χj suggests that we define τH by τ̃ = 9D −1γ4j τH , so that C ∼ 2 [1 +√τHλ ]−1 I. Therefore, in the scaling regime τ = O(A4), the stability of k-spike quasi-equilibria is determined by the NLEP (2.38) with the single multiplier χ∞ = 2 1 + √ τHλ , when τ = 9A4τHγ4j D ≫ 1 . (2.45) This limiting NLEP was derived for equilibrium spike solutions in [19], [18], [63], and [54] in the intermediate regime. From the rigorous analysis of [18] and [54], this NLEP has a Hopf bifurcation at some τH = τH0. From numerical computations, τH0 ≈ 1.748, and Re(λ) < 0 only when τH < τH0 (cf. [19], [54]). Although this limiting NLEP has a continuous spectrum on the non-negative real axis Re(λ) ≤ 0, there are no edge bifurcations arising from the end-point λ = 0 (cf. [18], [54]). The scaling law for τ in (2.45) shows that there are k distinct values of τ where complex conjugate eigenvalues appear on the imaginary axis. The 39 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria minimum of these values determines the instability threshold for τ . Finally, by using (2.20) to calculate the spike amplitudes asymptotically for A ≫ 1, we obtain the following main instability result: Principal Result 2.4: Let τ = O(A4) with xj+1 − xj = O(1) and O(1)≪ A ≪ O(ε−1/2). Then, the frozen k-spike quasi-equilibrium solution of §2.1 develops an oscillatory instability due to a Hopf bifurcation when τ increases past τh, where τh = min j=1,··· ,k (τhj) , τhj ∼ DA 4 144 τH0 (Be)4j ( 1− 6A2√Drj )4 , j = 1, · · · , k . (2.46) Here τH0 ≈ 1.748, while (Be)j and rj are defined in terms of the spike locations by (2.20). For an equilibrium k-spike pattern where xj = −1 + (2j − 1)/k for j = 1, · · · , k, we calculate from (2.20b) that (Be)j = 2 tanh (θ0/k) and rj = [2 tanh (θ0/k)] −1 for j = 1, · · · , k. This leads to the following equilibrium stability threshold, which agrees asymptotically with the following result given in Proposition 4.3 of [54]: τh ∼ DA 4 9 τH0 tanh 4 ( θ0 k )( 1− 3A2√D tanh (θ0/k) )4 . (2.47) An important remark concerns the range of validity of Principal Result 2.4 with respect to τ . The derivation of (2.46) assumed that τε2A2 ≪ 1, while the scaling law for instabilities predicts that τ = O(A4). Therefore, (2.46) only holds in the subregimeO(1)≪ A≪ O(ε−1/3) of the intermediate regime. 2.2.1 Numerical Experiments of Quasi-Equilibrium Spike Layer Motion We now give two examples illustrating our results of §2.1 for slow spike dynamics. Experiment 2.1: A two-spike evolution Consider a two-spike evolution with parameter valuesD = 1.0, A = 10.0, τ = 5.0, and ε = 0.005. The initial spike locations are x1(0) = −0.20 and x2(0) = 0.80. In Fig 2.1(a) we plot the spike layer trajectories versus the slow time σ, computed numerically from (2.25), which shows the gradual ap- proach to the equilibrium values at x1 = −1/2 and x2 = 1/2. In Fig. 2.1(c) 40 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria we plot the spike amplitudes versus σ. A plot of the quasi-equilibrium solu- tion at several instants in time is shown in Fig. 2.1(d). In Fig. 2.1(b) we plot the two Hopf bifurcation values τhj, for j = 1, 2, versus σ calculated from (2.46). In these figures the solid curves were obtained when computing for the spike amplitudes from the nonlinear algebraic system (2.17), while the dotted curves correspond to using the two-term approximation (2.18). For the spike trajectories shown in Fig. 2.1(a) these two curves are indistinguish- able. In the plots of the spike locations and spike amplitudes in Fig. 2.1(a) and Fig. 2.1(c) the marked points are the full numerical results computed from (1.6) using the adaptive-grid software of [8]. From these figures we observe that the asymptotic and full numerical results agree very closely. An important observation from Fig. 2.1(b) is that the stability threshold τh(σ), defined by τh(σ) ≡ minj=1,2 [τhj(σ)], is an increasing function of σ with τh(0) ≈ 12.5 and τh(∞) ≈ 67.7. This monotone behavior of τh(σ), together with the initial value τh(0) > τ = 5.0, implies that there is no dy- namic triggering of an oscillatory instability in the spike amplitudes before the equilibrium two-spike pattern is reached. More generally, for a k-spike pattern, we have been unable to prove an- alytically from the DAE system (2.25) and (2.18) that the stability thresh- old for τ , given by τh(σ) ≡ minj=1,··· ,k(τhj(σ)) of (2.46), must always be an increasing function of σ. However, we have performed many numeri- cal experiments to examine this condition, and in each case we have found numerically that τh(σ) is a monotonically increasing function of σ. This leads to the conjecture that there are no dynamically triggered oscilla- tory instabilities of multi-spike quasi-equilibria in the intermediate regime O(1) ≪ A ≪ O(ε−1/2). In other words, τ < τh(0) is a sufficient condition for stability of the quasi-equilibrium multi-spike pattern for all σ > 0. In contrast, for the simple case of symmetric two-spike quasi-equilibria in the low feed-rate regime A = O(1), dynamically triggered instabilities due to either Hopf bifurcations or from the creation of a real positive eigenvalue were established analytically in [86]. Experiment 2.2: A three-spike evolution Next, we consider a three-spike evolution with parameter values D = 1.0, A = 10.0, τ = 2.0, and ε = 0.005. The initial spike locations are x1(0) = −0.50, x2(0) = 0.20, and x3(0) = 0.80. In Fig 2.2(a) an Fig 2.2(c) we plot the spike layer trajectories and spike amplitudes, respectively, computed from (2.25) and the two-term approximation (2.18). Snapshots of the quasi- equilibrium solution are shown in Fig. 2.2(d). In Fig. 2.2(b) we plot the three Hopf bifurcation values τhj, for j = 1, · · · , 3, versus σ calculated from 41 2.2. Oscillatory Profile Instabilities of k-Spike Quasi-Equilibria 1.00 0.75 0.50 0.25 0.00 −0.25 −0.50 −0.75 −1.00 4035302520151050 xj σ ×××××××× × × ×× × × × × × × × × ×××××××× × × ×× × × × × × × × × (a) xj vs. σ 240 200 160 120 80 40 0 403020100 τhj σ (b) τhj vs. σ 0.21 0.18 0.15 0.12 0.09 0.06 40200 γj σ ××××× × × × × × × × × × × × ××× ×× ×× × × × × × × × × × × × × ××× (c) γj vs. σ 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1.000.750.500.250.00−0.25−0.50−0.75−1.00 v x (d) v vs. x Figure 2.1: Experiment 2.1: Slow evolution of a two-spike quasi-equilibrium solution for ε = 0.005, A = 10, and D = 1.0, with x1(0) = −0.2 and x2(0) = 0.8. Top Left: xj vs. σ. Top Right: τhj vs. σ. Bottom Left: γj vs. σ. Bottom Right: v vs. x at σ = 0 (heavy solid curve), σ = 10 (solid curve), and σ = 50 (dotted curve). Except for the plot of v vs. x, the solid curves correspond to computing the spike amplitudes from (2.17), while the dotted curves correspond to the two-term approximation (2.18). The crosses in the plots of xj and γj vs. σ are the full numerical results computed from (1.6). 42 2.3. Oscillatory Drift Instabilities of k-Spike Patterns (2.46). In the plots of τhj, γj and xj versus σ the solid curves correspond to computing the spike amplitudes from the nonlinear algebraic system (2.17), while the dotted curves correspond to the two-term approximation (2.18). In the plots of xj and γj the marked points are the full numerical results computed from (1.6) using the software of [8]. As in Experiment 2.1, the asymptotic and full numerical results agree rather closely. From Fig. 2.1(b) we again observe that the stability threshold for τ , given by τh(σ) ≡ minj=1,2,3 [τhj(σ)], is an increasing function of σ with τh(0) ≈ 2.6 and τh(∞) ≈ 14.0. This monotonicity of τh(σ), together with τh(0) > τ = 2.0, precludes the occurrence of a dynamically triggered Hopf bifurcation in the spike amplitudes. 2.3 Oscillatory Drift Instabilities of k-Spike Patterns In this section we compute numerical solutions to the Stefan problem (2.12) when τε2A2 = O(1) and O(ε−1/3) ≪ A ≪ O(ε−1/2). For notational conve- nience, so as not to confuse spike locations with gridpoints of the discretiza- tion, we re-label the spike trajectories as ξj(σ), for j = 1, · · · , k. Then, defining τ0 ≡ ε−2A−2τ = O(1) to be the key bifurcation parameter, (2.12) on |x| ≤ 1 becomes τ0uσ = Duxx + (1− u)− k∑ j=1 6γj δ(x − ξj) ; ux(±1, σ) = 0 , (2.48a) ξ′j = γj [ ux(ξ + j , σ) + ux(ξ − j , σ) ] , j = 1, · · · , k , (2.48b) u(ξj(σ), σ) = 1/(γjA2) , j = 1, · · · , k . (2.48c) By using a SLEP approach, which is related to the method pioneered in [69], it was shown in [53] that a one-spike equilibrium solution will be- come unstable at some critical value of τ0 to an oscillatory drift instability associated with a pure imaginary small eigenvalue λ ≪ 1 of the lineariza- tion. As τ0 increases above this critical value a complex conjugate pair of eigenvalues crosses into the right half-plane Re(λ) > 0, leading to the initi- ation of a small-scale oscillatory motion of the spike location. This oscilla- tory drift instability is the dominant instability mechanism in the subregime O(ε−1/3)≪ A≪ O(ε−1/2) of the intermediate regime. 43 2.3. Oscillatory Drift Instabilities of k-Spike Patterns 1.00 0.75 0.50 0.25 0.00 −0.25 −0.50 −0.75 −1.00 200180160140120100806040200 xj σ ××××××××××××××××××××× ××× ×× × × × × × × × ××××××××××××××××××××× ××× ×× × × × × × × × ××××××××××××××××××××× ××× ×× × × × × × × × (a) xj vs. σ 240 200 160 120 80 40 0 200180160140120100806040200 τhj σ (b) τhj vs. σ 0.15 0.12 0.09 0.06 200180160140120100806040200 γj σ ×××××××××××××××× × × × × × × × × × ×××××××××××××××××× × × × × ×××××× ××× ×××× ××× × × × × × × (c) γj vs. σ 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1.000.750.500.250.00−0.25−0.50−0.75−1.00 v x (d) v vs. x Figure 2.2: Experiment 2.2: Slow evolution of a three-spike quasi-equilibrium solution for ε = 0.005, A = 10, and D = 1.0, with x1(0) = −0.5, x2(0) = 0.2, and x3(0) = 0.80, obtained from the two-term asymptotic result (2.18) for the spike amplitudes. Top Left: xj vs. σ. Top Right: τhj vs. σ. Bottom Left: γj vs. σ. Bottom Right: v vs. x at σ = 0 (heavy solid curve), σ = 40 (solid curve), and σ = 140. Except for the plot of v vs. x, the solid curves correspond to computing the spike amplitudes from (2.17), while the dotted curves correspond to the two-term approximation (2.18). The crosses in the plots of xj and γj vs. σ are the full numerical results computed from (1.6). 44 2.3. Oscillatory Drift Instabilities of k-Spike Patterns For the one-spike case where k = 1 we now show that the condition for the stability of the equilibrium solution to the Stefan problem (2.48) is equivalent to that obtained in [53] using the SLEP method. We let U ≡ U(x) and γ1 = γ1e denote the equilibrium solution to (2.48) with ξ1 = 0. Then, U satisfies DUxx + (1− U) = 0 , |x| < 1 ; Ux(±1) = 0 , (2.49a) [DUx]0 = 6γ1e ; U(0) = 1 γeA2 . (2.49b) Here we have defined [v]0 ≡ v(0+)− v(0−). We readily calculate that U = { 1−B cosh(θ0(1+x))cosh θ0 , −1 ≤ x < 0 ; 1−B cosh(θ0(1−x))cosh θ0 , 0 < x ≤ 1 , (2.50a) where θ0 ≡ D−1/2, and γ1e ≡ B √ D 3 tanh θ0 , B ≡ 1− 1 γ1eA2 . (2.50b) By solving the quadratic equation for γ1e we get γ1e = √ D 6 tanh θ0 [ 1 + √ 1− A 2 1e A2 ] , A1e ≡ √ 12θ0 tanh θ0 . (2.50c) For A ≫ 1, this expression reduces to γ1e = √ D 3 tanh θ0 − 1A2 +O(A −4) , B = 1− 3A2√D tanh θ0 +O(A−4) . (2.50d) For δ ≪ 1, we perturb the equilibrium solution as u(x, σ) = U(x) + δeλσφ(x) , γ1 = γ1e + δβe λσ , ξ1 = δe λσ . (2.51) We substitute (2.51) into (2.48a), and collect O(δ) terms to get Dφxx− (1+ τ0λ)φ = 0 on |x| ≤ 1. From the linearization of the ODE (2.48b), we obtain δλeλσ = ( γ1e + δβe λσ )( Ux(0 +) + Ux(0 −) + δeλσ[ φx(0 +) + φx(0 −) + Uxx(0+) + Uxx(0−) ] ) . (2.52a) 45 2.3. Oscillatory Drift Instabilities of k-Spike Patterns In addition, the conditions [Dux]ξ1 = 6γ1 and u(ξ1) = 1/(γ1A2) become D ( Ux(0 +)− Ux(0−) + δeλσ [ Uxx(0 +)− Uxx(0+) + φx(0+)− φx(0−) ]) = 6 ( γ1e + εδe λσ ) , (2.52b) U(0±) + δeλt ( Ux(0 ±) + φ(0±) ) = 1 γ1eA2 − δβ γ21eA2 eλσ . (2.52c) Since Ux(0 +) + Ux(0 −) = 0, Uxx(0+) = Uxx(0−), and Ux(0+) = ±3γ1e/D, we collect the O(δ) terms in (2.52) to obtain the following problem for φ: Dφxx − (1 + τ0λ)φ = 0 , |x| ≤ 1 ; φx(±1) = 0 , (2.53a) [Dφx]0 = 6β ; φ(0 ±) = ∓3γ1e D − β γ21eA2 , (2.53b) λ = 2γ1e ( Uxx(0) + 1 2 [ φx(0 +) + φx(0 −) ]) . (2.53c) The solution to (2.53a) satisfying the prescribed values for φ(0±) is φ = C− cosh(θλ(x+ 1)) cosh θλ , −1 ≤ x < 0 ; (2.54a) φ = C+ cosh(θλ(1− x)) cosh θλ , 0 < x ≤ 1 , (2.54b) where θλ ≡ θ0 √ 1 + τ0λ, and C± = ∓3γ1e D − β γ21eA2 . (2.55) The jump condition [Dφx]0 = 6β, then yields that β = 0. Then, we substi- tute values for φx(0 ±) and Uxx(0) = −D−1 ( 1− 1/(γ1eA2) ) into (2.53c) to obtain λ = 2γ1e [ 3γ1e D θλ tanh θλ − 1 D ( 1− 1 γ1eA2 )] . (2.56) Finally, by using (2.50c) for γ1e, we readily derive that λ satisfies the tran- scendental equation λ = µ [√ 1 + τ0λ tanh θ0 tanh θλ − 1 ] , µ ≡ θ0 6 tanh θ0 [ 1 + √ 1− A 2 1e A2 ]2 . (2.57) 46 2.3. Oscillatory Drift Instabilities of k-Spike Patterns Here θλ ≡ θ0 √ 1 + τ0λ, θ0 ≡ D−1/2, while A1e is defined in (2.50c). Equa- tion (2.57) is identical to that derived in equation (5.5) of [53] from using the SLEP method based on a linearization of (1.6) around a one-spike equi- librium solution. We introduce ω, τd, and ζ, by λ = µω, τ0 = τd/µ, and ζ = τdω. Then, from (2.57), ζ is a root of F (ζ) ≡ ζ τd −G(ζ) = 0 , G(ζ) ≡ √ 1 + ζ tanh θ0 tanh [ θ0 √ 1 + ζ ] − 1 . (2.58) By examining the roots of (2.58) in the complex ζ plane, the following result was proved in §5 of [53]. Proposition 2.5: There is a complex conjugate pair of pure imaginary eigenvalues to (2.57) at some unique value τd = τdh depending on D. For any τd > τdh there are exactly two eigenvalues in the right half-plane. These eigenvalues have nonzero imaginary parts when τdh < τd < τdm, and they merge onto the positive real axis at τd = τdm. They remain on the positive real axis for all τd > τdm. The numerically computed Hopf bifurcation value τdh is plotted versus D in Fig. 2.3(a). In Fig. 2.3(a) we also plot the magnitude |ωh| of the pure imaginary eigenvalues ωh = ±i|ωh| at τd = τdh. Since τ0 = τd/µ and τ = ε−2A−2τ0, the Hopf bifurcation value for the onset of an oscillatory drift instability for a one-spike solution is τtw ∼ ε−2A−2τ0d , τ0d ≡ τdh µ , µ ≡ θ0 6 tanh θ0 [ 1 + √ 1− A 2 1e A2 ]2 . (2.59) Recall from (2.47) that there is an oscillatory instability in the amplitude of an equilibrium spike when τ = τh = O(A4). Thus, τh ≪ τtw when O(1) ≪ A≪ O(ε−1/3) and τtw ≪ τh when O(ε−1/3)≪ A≪ O(ε−1/2). In Fig. 2.3(b) we plot log10(τtw) and log10(τh) versus A, log10(τh) versus A, showing the exchange in the intermediate regime. For the equilibrium problem on the infinite line this exchange in the dominant instability mechanism was also noted in [65] and [19]. As a remark, we notice that ωh, representing the (scaled) temporal frequency of small oscillations, satisfies ωh → 0 in the infinitely long domain limitD → 0. Therefore, the existence of an oscillatory instability requires a finite domain. 47 2.3. Oscillatory Drift Instabilities of k-Spike Patterns 4.0 3.0 2.0 1.0 0.0 1.501.251.000.750.500.25 τdh, |ωh| D (a) τdh and |ωh| vs. D 4.0 3.0 2.0 1.0 0.0 12.011.010.09.08.07.06.05.0 log10(τ) A (b) log10(τtw) and log10(τh) vs. A Figure 2.3: Left Figure: The drift instability thresholds τdh (heavy solid curve) and |ωh| (solid curve) versus D. Right figure: Plots of log10(τh) (heavy solid curve) from (2.47) with k = 1 and log10(τtw) (solid curve) from (2.59) versus A for D = 0.75 and ε = 0.015. This shows the exchange in the dominant instability mechanism in the intermediate regime. 2.3.1 Numerical Solution of the Coupled ODE-PDE Stefan Problem To compute numerical solutions to the Stefan problem (2.48) we adapt the numerical method of [89] for computing solutions to a heat equation with a moving singular Dirac source term. We use a Crank-Nicolson scheme to discretize (2.48a) together with a forward Euler scheme for the ODE (2.48b). For simplicity we first consider (2.48a) for one spike where k = 1. Let h and ∆t denote the space step and time step, respectively. We set ∆t = h2/2 to satisfy the stability requirement of the forward Euler scheme. The number of grid points is M = 1 + 2/h, and we label the jth grid point by xj = −1 + (j − 1)h for j = 1, · · · ,M . The total time period is Tm, and we label σn = n∆t for n = 1, · · · , Tm/∆t. At time σn, the solution u(xj , σn) at the jth grid point is approximated by Unj , and we label U n = (Un1 , · · · , UnM )T . Also, the spike location ξ1(σn) and spike amplitude γ1(σn) are approximated by ξn1 and γ n 1 , respectively. As in [89] we approximate the delta function δ(x − ξ1) in (2.48a) with the discrete delta function d(1)(x− ξ1), where d(1)(y) is given by d(1)(y) = { (2h− |y|)/(4h2) , |y| ≤ 2h , 0 , otherwise . (2.60) 48 2.3. Oscillatory Drift Instabilities of k-Spike Patterns The numerical approximation d(1) of the delta function in (2.48a) spreads the singular force to the neighboring grid points within its support. Next, we write the constraint of (2.48c), given by u(ξ1(σ), σ) = 1/(γ1A2), in the form u(ξ1(σ), σ) = ∫ ∞ −∞ u(x, σ)δ(x − ξ1(σ)) dx = 1 γ1A2 . (2.61) With Unj ≈ u(xj, σn), ξn1 ≈ ξ1(σn), and γn1 ≈ γ1(σn), the discrete version of (2.61) is u(ξn1 , σn) ≈ h ∑ j Unj d (2)(xj − ξn1 ) = 1 γn1A2 , (2.62) where d(2)(y) is a further numerical approximation to the delta function given explicitly by d(2)(y) = 1 h   1− (y/h)2 , |y| ≤ h , 2− 3|y|/h + (y/h)2 , h ≤ |y| ≤ 2h , 0 , otherwise . (2.63) The function d(2) interpolates grid values Unj to approximate the requirement u(ξn1 , σn) = 1/(γ n 1A2) at the spike location. A rough outline of the algorithm for the numerical solution to (2.48) to advance from the nth to the (n+1)th time-step is as follows: We first use (2.48b) to take a forward Euler step to get the new location ξn+11 given U n and γn1 . Then, we use the approximate condition u(ξn+11 , σn+1) ≈ 1 2A2 ( 1 γn1 + 1 γn+11 ) , (2.64) on the right-hand side of (2.62) together with the Crank-Nicolson discretiza- tion of (2.48a) to solve for γn+11 . Then, we calculate U n+1 from the dis- cretization of (2.48a) using the value of γn+11 . Finally, we set n = n+1, and repeat the iteration. In order to obtain higher accuracy, we use a fourth order accurate scheme as suggested in [89] to approximate the second order differential operator uxx as uxx(xj, σn) ≈ Dn2j , where Dn2j ≡   1 12h2 (Unj+4 − 6Unj+3 + 14Unj+2 − 4Unj+1 − 15Unj + 10Unj−1) , j = 1 , 1 12h2 (−Unj+2 + 16Unj+1 − 30Unj + 16Unj−1 − Unj−2) , 2 ≤ j ≤ N , 1 12h2 (Unj−4 − 6Unj−3 + 14Unj−2 − 4Unj−1 − 15Unj + 10Unj+1) , j =M . 49 2.3. Oscillatory Drift Instabilities of k-Spike Patterns Here Un0 and U n M+1 are ghost points outside each end of the boundaries at x = ±1. The Neumann condition ux(±1, t) = 0 is also discretized as ux(x1, σn) ≈ (−3Un0 − 10Un1 + 18Un2 − 6Un3 + Un4 )/(12h) , ux(xM , σn) ≈ (3UnM+1 + 10UnM − 18UnM−1 + 6UnM−2 − UnM−3)/(12h) . Eliminating the ghost points from the discretization of uxx, we obtain the M ×M coefficient matrix T = T0/(12h2), where T0 ≡   −145/3 56 −6 −8/3 1 0 0 0 · · · 0 58/3 −36 18 −4/3 0 0 0 0 · · · 0 −1 16 −30 16 −1 0 0 0 · · · 0 0 −1 16 −30 16 −1 0 0 · · · 0 0 0 −1 16 −30 16 −1 0 · · · 0 . . . . . . . . . . . . . . . 0 · · · 0 0 −1 16 −30 16 −1 0 0 · · · 0 0 0 −1 16 −30 16 −1 0 · · · 0 0 0 0 −4/3 18 −36 58/3 0 · · · 0 0 0 1 −8/3 −6 56 −145/3   . Next, we outline how we calculate the fluxes in (2.48b). For convenience, in the notation of this paragraph we omit the superscript n at time σn. In order to approximate the one-sided first order derivatives ux(ξ ± 1 , σn) in (2.48b), we first use the solution values at gridpoints on each side of ξ1 to in- terpolate the values at points that are equally distributed near ξ1. Then, we approximate ux(ξ ± 1 , σn) by differencing values at these points. For instance, assuming that the closest grid point on the left-hand side of the spike is xj , we use six point values (ū, Uj , Uj−1, Uj−2, Uj−3, Uj−4) with ū = u(ξ1, σn) = 1/(γ1A2) to interpolate the solution at x = (ξ1− 3h, ξ1− 2h, ξ1−h, ξ1). De- noting the results to be (U−1 , U − 2 , U − 3 , U − 4 ) with U − 4 = u(ξ1, σn) = 1/(γ1A2), we can use them for estimating ux(ξ − 1 , σn) as ux(ξ − 1 , t) ≈ (−3U−3 + 3 2 U−2 − 1 3 U−1 )/h . (2.65a) Similarly, we obtain (U+1 , U + 2 , U + 3 , U + 4 ) at x = (ξ1, ξ1 + h, ξ1 + 2h, ξ1 + 3h) by interpolation, and we then discretize to obtain ux(ξ + 1 , t) ≈ (3U+2 − 3 2 U+3 + 1 3 U+4 )/h . (2.65b) 50 2.3. Oscillatory Drift Instabilities of k-Spike Patterns Next, we discretize (2.48a) by the Crank-Nicolson scheme to get [ (1 + 1 2 µ)I− µD 2 T ] Un+1 = [ (1− 1 2 µ)I+ µD 2 T ] Un − 3µ (dn+1γn+11 + dnγn1 )+ p , (2.66) where µ ≡ ∆t/τ0 and pt ≡ (µ, µ, · · · , µ). Here dn+1 denotes the discretiza- tion of the delta function by d(1) at time σn+1. Denoting the coefficient matrix of Un+1 to be A and that of Un to be B, the system above can be written in the simpler form AUn+1 = BUn − 3µ (dn+1γn+11 + dnγn1 )+ p . (2.67) In (2.67), only the boldface variables are matrices and vectors. We then approximate (2.48c) by the discrete form (2.62). We write this interpolation as a dot product Nn+1Un+1, where Nn+1Un+1 = 1 2A2 ( 1 γn1 + 1 γn+11 ) . (2.68) Since A has full rank, we then left multiply (2.67) by Nn+1A−1 to get a quadratic equation for γn+11 given by 3µNn+1A−1dn+1γn+11 − [ Nn+1A−1(BUn − 3µdnγn1 + p) − 1 2A2 1 γn1 ] + 1 2A2 1 γn+11 = 0 . (2.69) SinceA≫ 1, we can solve (2.69) for the correct root γn+11 . We then calculate Un+1 using (2.67) and start the next iteration by using a forward Euler step on (2.48b) to advance the spike location. This numerical method can be readily extended to treat (2.48) for two or more spikes. For instance, consider the case of two spikes. With the same Crank-Nicolson scheme, we let d1 and d2 be the discrete approximations of δ(x−ξ1) and δ(x−ξ2) in (2.48a), respectively, with the discrete delta function d(1). Define the row vectors Nn+11 and N n+1 2 to the discrete versions of d (2) near ξ1 and ξ2, respectively. The conditions u(ξ n+1 j , σ n+1) = 1/(γn+1j A2), can then be approximated by the two discrete constraints Nn+1j U n+1 = 1 2A2 ( 1 γnj + 1 γn+1j ) , j = 1, 2 . (2.70) 51 2.3. Oscillatory Drift Instabilities of k-Spike Patterns The Crank-Nicolson discretization of (2.48a) with k = 2 is written as AUn+1 = BUn−3µ(dn+11 γn+11 +dn+12 γn+12 )−3µ(dn1γn1 +dn2γn2 )+p . (2.71) Upon imposing the constraints (2.70) we obtain the following system for γn+11 and γ n+1 2 : 3µNn+1j A −1(dn+11 γ n+1 1 + d n+1 2 γ n+1 2 ) + 1 2A2 1 γn+1j − [ Nn+1j A −1(BUn − 3µ(dn1γn1 + dn2γn2 ) + p)− 1 2A2 1 γnj ] = 0 , j = 1, 2 . (2.72) After calculating γn+11 and γ n+1 2 from this system, we then compute U n+1 from (2.72), and update the locations of the spikes. 2.3.2 Numerical Experiments of Oscillatory Drift Instabilities We now perform some numerical experiments on (2.48) to verify the drift instability threshold of a one-spike solution given in (2.59). We also use the numerical scheme of §2.3.1 to compute large-scale oscillatory motion for some one- and two-spike patterns. For each of the numerical experiments below we give initial spike lo- cations ξj(0) = ξj0, for j = 1, · · · , k, and we choose the initial condition u(x, 0) for (2.48) to be the quasi-equilibrium solution (2.13) with gj = 1 and xj = ξj0 for j = 1, · · · , k in (2.13). In all of the numerical computations below we chose the mesh size h = 0.01 and the diffusivity D = 0.75. For D = 0.75, the stability threshold from Proposition 2.5 is computed numeri- cally as τdh = 2.617. Experiment 2.3: A one-spike solution near equilibrium: Testing the insta- bility threshold Let D = 0.75 and A = 28.5. Then, from (2.59) we calculate τ0d ≈ 4.19. For an initial spike location at ξ1(0) = 0.1, in Fig 2.4 we plot the numerically computed spike layer trajectory ξ1 versus σ for six values of τ0. Comparing Fig. 2.4(d) with Fig. 2.4(e) we see that the amplitude of the oscillation grows slowly when τ0 = 4.22, and decays slowly when τ0 = 4.20. From Fig. 2.4(f) we observe that the drift Hopf bifurcation value is τ0d ≈ 4.21, which compares favorably with the theoretical prediction of τ0d ≈ 4.19 from (2.59). With regards to the convergence of the numerical scheme for (2.48), we estimate numerically that τ0d ≈ 4.23 with the coarser meshsize h = 0.02, 52 2.3. Oscillatory Drift Instabilities of k-Spike Patterns 0 5 10 15 20 25 30 35 40 0 0.02 0.04 0.06 0.08 0.1 The spike location (a) ξ1 vs. σ for τ0 = 1.0 0 20 40 60 80 100 120 140 160 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 The spike location (b) ξ1 vs. σ for τ0 = 4.0 0 5 10 15 20 25 30 35 40 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 The spike location (c) ξ1 vs. σ for τ0 = 4.4 0 20 40 60 80 100 120 140 160 180 200 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 The spike location (d) ξ1 vs. σ for τ0 = 4.20 0 20 40 60 80 100 120 140 160 180 200 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 The spike location (e) ξ1 vs. σ for τ0 = 4.22 0 20 40 60 80 100 120 140 160 180 200 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 The spike location (f) ξ1 vs. σ for τ0 = 4.21 Figure 2.4: Experiment 2.3: Testing the stability threshold for initial values near the equilibrium value. Plots of ξ1 versus σ with ξ1(0) = 0.1, D = 0.75, and A = 28.5, for different values of τ0. The numerical threshold τ0d ≈ 4.21 in the plot (f) compares well with the theoretical prediction τ0d ≈ 4.19. 53 2.3. Oscillatory Drift Instabilities of k-Spike Patterns and that τ0d ≈ 4.20 for the very fine meshsize h = 0.005. As a trade-off between accuracy and computational expense, we chose h = 0.01 in our computations. Experiment 2.4: A one-spike solution: Large-scale oscillatory dynamics We again let D = 0.75 and A = 28.5. However, we now choose the initial spike location ξ1(0) = 0.3, which is not near its equilibrium value. In Fig. 2.5 we plot the numerically computed spike layer trajectory ξ1 versus σ for τ0 = 4.20 and τ0 = 4.22. For the smaller value of τ0 the spike slowly approaches its steady state at ξ1 = 0. However, for τ0 = 4.22, the spike location exhibits a sustained large-scale oscillation. 0 20 40 60 80 100 120 140 160 180 200 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 The spike location (a) ξ1 vs. σ for τ0 = 4.20 0 5 10 15 20 25 30 35 40 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 The spike location (b) ξ1 vs. σ for τ0 = 4.22 Figure 2.5: Experiment 2.4: Plots of ξ1 versus σ with ξ1(0) = 0.3, D = 0.75, and A = 28.5. (a) τ0 = 4.20; (b) τ0 = 4.22. Experiment 2.5: A two-spike pattern: Oscillatory drift instabilities and the breather mode We now compute two-spike solutions. For D = 0.75 and A = 63.6, in Fig. 2.6(a) and Fig. 2.6(b) we plot the numerically computed spike trajecto- ries with initial values ξ1(0) = −0.38 and ξ2(0) = 0.42 for two values of τ0. For τ0 = 16 the spike trajectories develop a large-scale sustained oscillatory motion. In each case, the observed oscillation is of breather type, charac- terized by a 180◦ out of phase oscillation in the spike locations. A breather instability was shown in [38] to be the dominant instability mode for insta- bilities of equilibrium transition layer patterns for a Fitzhugh-Nagumo type model. It is also the dominant instability mode for equilibrium spike pat- terns of the GS model in the high feed-rate regime A = O(ε−1/2) (see [55]). 54 2.3. Oscillatory Drift Instabilities of k-Spike Patterns 0 5 10 15 20 25 30 35 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 The spike location (a) ξj vs. σ for τ0 = 15 0 5 10 15 20 25 30 35 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 The spike location (b) ξj vs. σ for τ0 = 16 0 10 20 30 40 50 60 70 80 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 (c) ξj vs. σ for τ0 = 15 0 10 20 30 40 50 60 70 80 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 The spike location (d) ξj vs. σ for τ0 = 17 Figure 2.6: Experiment 2.5: Spike trajectories ξj versus σ for a two-spike pattern with D = 0.75 and A = 63.6. Top row: the initial values ξ1(0) = −0.38 and ξ2(0) = 0.42 with τ0 = 15 and τ0 = 16. Bottom Row: the initial values ξ1(0) = −0.55 and ξ2(0) = 0.45 with τ0 = 15 and τ0 = 17. For the same values of D and A, in Fig. 2.6(c) and Fig. 2.6(d) we plot the numerically computed spike trajectories with initial values ξ1(0) = −0.55 and ξ2(0) = 0.45 for τ0 = 15 and τ0 = 17, respectively. 2.3.3 Integral Equation Formulation In this subsection we show how to cast the Stefan problem (2.48) for a one-spike solution into an equivalent integral equation formulation. We let K(x, σ; η, s) to be the Green’s function satisfying τ0Kσ = DKxx−K + δ(x− η)δ(σ− s) , |x| < 1 ; Kx(±1, σ) = 0 , (2.73) 55 2.3. Oscillatory Drift Instabilities of k-Spike Patterns with initial value K = 0 for 0 < σ < s. The solution for K is represented as an eigenfunction expansion as K(x, σ; η, s) = H(σ − s) τ0 e−(σ−s)/τ0 [1 2 + ∞∑ n=1 exp ( −Dn 2π2 4 (σ − s) τ0 ) cos (nπ 2 (ξ + 1) ) cos (nπ 2 (η + 1) ) ] . (2.74) By using Green’s identity, a straightforward calculation shows that the solution u(x, σ) to (2.48) with one spike can be written in terms of K as u(x, σ) = 1 + τ0 ∫ 1 −1 [u(η, 0) − 1]K(x, σ; η, 0) dη − 6 ∫ σ 0 γ1(s)K(x, σ; ξ1(s), s) ds , (2.75) where ξ1(s) is the spike trajectory. By imposing the constraint condition u = 1/(γ1A2) at x = ξ1(σ), we obtain an integral equation for γ1(σ) in terms of the unknown spike location as 1 γ1(σ)A2 = 1 + τ0 ∫ 1 −1 [u(η, 0) − 1]K(ξ1(σ), σ; η, 0) dη − 6 ∫ σ 0 γ1(s)K(ξ1(σ), σ; ξ1(s), s) ds . (2.76) Then, by calculating ux as x → ξ1(σ) from above and from below, we calculate ux(ξ ± 1 (σ), σ) = τ0 ∫ 1 −1 [u(η, 0) − 1]Kx(ξ±1 (σ), σ; η, 0) dη − 6 ∫ σ 0 γ1(s)Kx(ξ ± 1 (σ), σ; ξ1(s), s) ds . (2.77) Finally, by substituting (2.77) into (2.48c), we obtain the integro-differential equation dξ1 dσ = γ1(σ) ( τ0 ∫ 1 −1 [u(η, 0) − 1] (Kx(ξ+1 (σ), σ; η, 0) +Kx(ξ−1 (σ), σ; η, 0)) dη −6 ∫ σ 0 γ1(s) ( Kx(ξ + 1 (σ), σ; ξ1(s), s) +Kx(ξ − 1 (σ), σ; ξ1(s), s) ) ds ) . (2.78) 56 2.4. Discussion Here u(η, 0) is the initial condition for (2.48). This shows that the speed of the spike layer at time σ depends on its entire past history. Therefore, ξ1(σ) is determined, essentially, by a continuously distributed delay model. Such problems typically lead to oscillatory behavior when τ0 is large enough. This suggests the mechanism underlying the oscillatory dynamics computed in §2.3.2. We remark that as a result of the infinite series representation of K in (2.74), together with the constraint (2.76) for γ1, the integro-differential equation (2.78) is significantly more complicated in form than a related integro-differential equation modeling a flame-front on the infinite line that was studied numerically in [76]. 2.4 Discussion We have studied the dynamics and oscillatory instabilities of spike solu- tions to the one-dimensional GS model (1.6) in the intermediate regime O(1) ≪ A ≪ O(ε−1/2) of the feed-rate parameter A. In the subregime O(1) ≪ A ≪ O(ε−1/3), and for τ ≪ O(ε−2A2), we have derived an ex- plicit DAE system for the spike trajectories from the quasi-steady limit of the Stefan problem (2.12). From the analysis of a certain nonlocal eigen- value problem, it was shown that the instantaneous spike pattern in this regime will become unstable to a Hopf bifurcation in the spike amplitudes at some critical value critical value τ = τH = O(A4). Alternatively, in the subregime O(ε−1/3)≪ A≪ O(ε−1/2), and with τ = O(ε−2A−2), oscillatory drift instabilities for the spike locations were computed numerically from time-dependent Stefan problem (2.48) with moving sources. In this sub- regime, the onset of such drift instabilities occur at occur at some critical value τ = τTW = O(ε −2A−2) be calculated analytically. An open technical problem is to study the codimension-two bifurcation problem that occurs when A = O(ε−1/3) where both types of oscillatory instability can occur simultaneously at some critical value of τ = O(ε−4/3). There are two key open problems. The first problem is to extend the nu- merical approach of [76], used for a flame-front integro-differential equation model, to compute long-time solutions to the integro-differential equation formulation (2.78) of the Stefan problem (2.48). With such an approach, one could compute numerical solutions to (2.48) over very long time in- tervals to study whether irregular, or even chaotic, large-scale oscillatory drift behavior of the spike trajectories is possible for the GS model in the subregime O(ε−1/3) ≪ A ≪ O(ε−1/2). A second direction of inquiry is to 57 2.4. Discussion derive Stefan-type problems with moving sources from an asymptotic re- duction of related reaction-diffusion systems with localized solutions such as the Gierer-Meinhardt model with saturation [50], the combustion-reaction model of [60], the Brusselator model of [49], and the GS model in two spa- tial dimensions. The derivation of such Stefan problems would allow for the study of large-scale drift instabilities of localized structures for parameter values far from their bifurcation values at which a normal form reduction, such as in [25] and [24], can be applied. 58 Chapter 3 Dynamics and the Spot-Replication Instability for the Two-Dimensional Gray-Scott Model 3.1 k-spot Quasi-Equilibrium Solutions We first construct a quasi-equilibrium solution to (1.7) for a k-spot pattern in a two-dimensional domain. The stability of this quasi-equilibrium solution with respect to spot self-replication is then studied. For each j = 1, . . . , k, we denote the center of the jth spot by xj = (xj , yj) where xj ∈ Ω. We assume that the spots are well-separated in the sense that dist(xi,xj) = O(1) for i 6= j and dist(xi, ∂Ω) = O(1) for i = 1, . . . , k. We then use the method of matched asymptotic expansions to construct the quasi-equilibrium solution. In an O(ε) neighborhood near each spot, the solution v is large and has a sharp gradient. In this inner region for the jth spot, we introduce the new local variables Uj , Vj, and y by u = ε A √ D Uj , v = √ D ε Vj , y = ε −1(x− xj) . In terms of these local variables, (1.7) transforms to leading order to ∆yVj − Vj + UjV 2j = 0 , ∆yUj − UjV 2j = 0 , y ∈ R2 . (3.1) This leading order inner problem, referred to as the core problem, is the same as that derived in [56] for the Schnakenburg model. We look for a radially symmetric solution to this core problem of the form Uj = Uj(ρ) and Vj = Vj(ρ), where ρ ≡ |y| and ∆y = ∂ρρ + ρ−1∂ρ denotes the radially symmetric part of the Laplacian. The radially symmetric core problem is to 59 3.1. k-spot Quasi-Equilibrium Solutions solve the coupled BVP system Ujρρ + 1 ρ Ujρ − UjV 2j = 0 , 0 < ρ <∞ (3.2a) Vjρρ + 1 ρ Vjρ − Vj + UjV 2j = 0 , 0 < ρ <∞ , (3.2b) subject to the the boundary conditions V ′j (0) = 0 , U ′ j(0) = 0 , (3.2c) Vj(ρ)→ 0 , Uj(ρ) ∼ Sj ln ρ+ χ(Sj) + o(1) , as ρ→∞ . (3.2d) We refer to Sj as the source strength of the j th spot. From the divergence theorem, it follows from the Uj equation that Sj = ∫∞ 0 UjV 2 j ρ dρ > 0. In the far-field behavior (3.2d) for Uj, the constant χ is a nonlinear function of the source strength Sj . This function must be computed from the solution to (3.2). The solution to (3.2) is calculated numerically for a range of values of Sj > 0 by using the BVP solver COLSYS (cf. [2]). In Fig. 3.1, we plot χ = χ(Sj) in the left subfigure, Vj(0) vs. Sj in the middle subfigure, and Vj(ρ) for different values of Sj in the right subfigure. For Sj > Sv ≈ 4.78, the profile Vj(ρ) develops a volcano shape, whereby the maximum of Vj occurs at some ρ > 0. These computations give numerical evidence to support the claim that there is a unique solution to (3.2) for each Sj > 0. 1 2 3 4 5 6 7 −5 0 5 10 15 20 Sj χ (a) χ vs. Sj 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 Sj V j(0 ) (b) Vj(0) vs. Sj 0 5 10 15 0 0.2 0.4 0.6 0.8 ρ V j (c) Vj vs. ρ Figure 3.1: Numerical results for the core problem (3.2) computed by COL- SYS ([2]). (a) The function χ vs. Sj ; (b) Vj(0) vs. Sj ; (c) The spot profile Vj(ρ) for Sj = 0.94, 1.45, 2.79 (solid curves from bottom to top at ρ = 0), and the volcano profile Vj(ρ) for Sj = 4.79, 5.73, 6.29 (dotted curves from top to bottom at ρ = 0). 60 3.1. k-spot Quasi-Equilibrium Solutions Since v is localized near each xj for j = 1, . . . , k, and is exponentially small in the outer region away from the spot centers, the effect of the non- linear term uv2 in the outer region can be calculated in the sense of distri- butions as uv2 ∼ ε2 k∑ j=1 (∫ R2 √ D Aε UjV 2 j dy ) δ(x− xj) ∼ 2πε √ D A k∑ j=1 Sj δ(x − xj) . Therefore, in the quasi-steady limit, the outer problem for u is D∆u+ (1− u) = 2π √ D ε A k∑ j=1 Sj δ(x− xj) , x ∈ Ω , (3.3a) ∂nu = 0 , x ∈ ∂Ω , (3.3b) u ∼ ε A √ D ( Sj ln |x− xj | − Sj ln ε+ χ(Sj) ) , as x→ xj , j = 1 . . . , k . (3.3c) The singularity condition above as x → xj was derived by matching the outer solution for u to the far-field behavior (3.2d) of the core solution and by recalling u = ε/(A √ D)Uj . The problem (3.3) suggests that we introduce new variables A = O(1) and ν ≪ 1, defined by ν = −1/ ln ε , A = νA √ D/ε . (3.4) In terms of these new variables, (3.3) transforms to ∆u+ (1− u) D = 2πν A k∑ j=1 Sj δ(x− xj) , x ∈ Ω , (3.5a) ∂nu = 0 , x ∈ ∂Ω , (3.5b) u ∼ 1A ( Sjν ln |x− xj |+ Sj + χ(Sj)ν ) , as x→ xj , j = 1 . . . , k . (3.5c) It is important to emphasize that the singularity behavior in (3.5c) specifies both the strength of the logarithmic singularity for u and the regular, or non-singular, part of this behavior. This pre-specification of the regular part of this singularity behavior at each xj will yield a nonlinear algebraic system for the source strengths S1, . . . , Sk. To solve (3.5), we introduce the reduced-wave Green’s function G(x;xj), which satisfies ∆G− 1 D G = −δ(x − xj) , x ∈ Ω ; ∂nG = 0 , x ∈ ∂Ω . (3.6) 61 3.1. k-spot Quasi-Equilibrium Solutions As x→ xj , this Green’s function has the local behavior G(x;xj) ∼ − 1 2π ln |x− xj |+Rjj + o(1) , as x→ xj , (3.7) where Rjj is referred to as the regular part of G at x = xj. In terms of G(x;xj) we can represent the outer solution for u in (3.5) as u = 1− k∑ i=1 2πνSi A G(x;xi) . Finally, by expanding this representation for u as x→ xj and equating the resulting expression with the required singularity behavior (3.5c), we obtain the following nonlinear algebraic system for S1, . . . , Sk: A = Sj(1+2πνRjj)+νχ(Sj)+2πν k∑ i=1,i6=j SiG(xi;xj) , j = 1, . . . , k . (3.8) Since ν = −1/ ln ε ≪ 1, it follows that the geometry of the domain Ω and the spatial configuration (x1, . . . ,xk) of the spot pattern has a weak, but not insignificant, influence on the source strengths S1, . . . , Sk through the Green’s function and its regular part. The nonlinear algebraic system (3.8) incorporates all logarithmic correction terms in ν involved in the determina- tion of the source strengths. Given values for the GS parameters A, ε and D, we first calculate A and ν from (3.4) and then solve (3.8) numerically for the source strengths S1, . . . , Sk, which determines the quasi-equilibrium in each inner region. The result is summarized as follows: Principal Result 3.1: Assume that ε ≪ 1 and define ν = −1/ ln ε and A = νA√D/ε, with A = O(1). Then, the solution v and the outer solution for u corresponding to a k-spot quasi-equilibrium solution of (1.7) are given by u ∼ 1− 2πνA k∑ j=1 SjG(x;xj) , v ∼ √ D ε k∑ j=1 Vj ( ε−1|x− xj | ) , (3.9) where (x1, · · · ,xk) are the centers of the spots, and G(x;xj) is the reduced- wave Green’s function satisfying (3.6). In (3.9), each spot profile Vj(ρ) for j = 1, . . . , k satisfies the BVP system (3.2) where the source strength Sj in (3.2d) is to be calculated from the nonlinear algebraic system (3.8). 62 3.1. k-spot Quasi-Equilibrium Solutions We emphasize that the nonlinear algebraic system (3.8) is the mechanism through which spots interact with each other and sense the presence of the domain Ω. This global coupling mechanism is rather significant since ν = −1/ ln ε is not very small unless ε is extremely small. We briefly illustrate this construction of a quasi-equilibrium pattern for the special case of a one-spot solution centered at the midpoint of the unit square. Many further illustrations of the theory are given in subsequent sections. 0 1 2 3 4 5 6 7 8 2 4 6 8 10 12 14 16 S1 A (a) A vs. S1 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 A u (x 1 ) (b) u(x1) vs. A Figure 3.2: Experiment 3.1: Let Ω = [0, 1] × [0, 1], and set ε = 0.02,D = 1, and x1 = (0.5, 0.5). (a) A vs. S1; the square marks the volcano threshold Sv = 4.78, and the circle marks the fold point Af = 3.3756 corresponding to Sf = 0.7499. (b) u(x1) vs. A, the square marks Sv = 4.78, and the circle marks the fold point Sf = 0.7499. The upper branch is for S1 < Sf , while the lower branch is for S1 > Sf . Experiment 3.1: One spot in a square: The fold point for A To illustrate the asymptotic theory, suppose that there is one spot cen- tered at x1 with source strength S1 inside the unit square Ω = [0, 1]× [0, 1]. Then, (3.8) reduces to the scalar nonlinear algebraic equation A = S1 (1 + 2πνR11) + νχ(S1). (3.10) We fix ε = 0.02 and D = 1, so that ν = 0.2556, and we let x1 = (0.5, 0.5). Then, by using (3.59) for the Green’s function as given below in §3.6, we obtain that R11 = 0.7876. In the left subfigure of Fig. 3.2 we plot A vs. S1, showing the existence of a fold point at Af = 3.3756 corresponding to Sf = 0.7499. Thus, A ≥ Af is required for the existence of a one-spot quasi-equilibrium solution centered at the midpoint of the square. In the 63 3.1. k-spot Quasi-Equilibrium Solutions right subfigure of Fig. 3.2 we plot u(x1) = ν U(0)/A with respect to A. For A > Af , u has two solution branches. The upper branch corresponds to S1 < Sf , while the lower branch is for the range S1 > Sf . In the remainder of this section we derive analytical approximations for the solution to the nonlinear algebraic system (3.8) for the source strengths. To do so, it is convenient to first re-write (3.8) in matrix form. For a k-spot pattern we define the Green’s matrix G, the vector of source strengths s, and the identity vector e by G ≡   R11 G12 G13 · · · G1k G21 R22 G23 · · · G2k G31 G32 R33 · · · G3k ... ... ... . . . ... Gk1 Gk2 Gk3 · · · Rkk   , s ≡   S1 S2 S3 ... Sk   , e ≡   1 1 1 ... 1   . (3.11) Here Gij ≡ G(xi;xj), and Gij = Gji by reciprocity, so that G is a symmetric matrix. Then, (3.8) is written in matrix form as A e = s+ 2πνG s+ νχ(s) , (3.12) where χ(s) is defined by χ(s) = (χ(S1), . . . , χ(Sk)) t and t denotes transpose. We will consider (3.12) for two ranges of D. For D = O(1), we can obtain a two-term approximation of the source strengths in terms of ν ≪ 1 by expanding s = s0 + νs1 + · · · . Upon substituting this expansion into (3.12) and collecting powers of ν, we readily obtain the two-term expansion s = Ae− ν [2πAG e+ χ(A)e] +O(ν2) . (3.13) This shows that, for ν ≪ 1, the leading order approximation for s is the same for all of the spots, but that the next order term depends on the spot locations and the domain geometry. An interesting special case for multi-spot patterns is when the spatial configuration (x1, . . . ,xk) of spots is such that the Green’s matrix G is a circulant matrix. For instance, this occurs when k spots are equally spaced on a circular ring that is concentric within a circular disk. When G is circulant, then it has the eigenpair Ge = θe, where θ = k−1∑ki=1∑kj=1Gij . For this special case, (3.12) shows that the spots have a common source strength Sj = Sc for j = 1, . . . , k to all orders in ν, where Sc is the solution to the single nonlinear algebraic equation A = Sc + 2πνθSc + νχ(Sc) . (3.14) 64 3.1. k-spot Quasi-Equilibrium Solutions For ν ≪ 1, a two-term approximation for this common source strength is Sc = A− ν [2πθA+ χ(A)] +O(ν2) . Next, we consider (3.12) for the distinguished limit where D is loga- rithmically large in ε, given by D = D0/ν with D0 = O(1). Since the reduced-wave Green’s function depends on D, we first must approximate it for D large. Assuming that Ω is a bounded domain, we expand G and its regular part R for D ≫ 1 as G ∼ DG−1 +G0 + 1 D G1 + · · · , R ∼ DR−1 +R0 + 1 D R1 + · · · . Substituting this expansion into (3.6), and collecting powers of D, we obtain that G−1 is constant and that G0 satisfies ∆G0 = G−1 − δ(x − xj) , x ∈ Ω ; ∂nG0 = 0 , x ∈ ∂Ω . The divergence theorem then shows that G−1 = |Ω|−1, where |Ω| denotes the area of Ω. In addition, the divergence theorem imposed on the G1 problem enforces that ∫ ΩG0 dx = 0, which makes G0 unique. Next, from the singularity condition (3.7) for G we obtain for x ≈ xj that DG−1 +G0(x;xj) + · · · ∼ − 1 2π ln |x− xj |+DR−1 +R0(x;xj) + · · · . Since G−1 = |Ω|−1, we conclude that R−1 = G−1 = 1/|Ω|. In this way, we obtain the following two-term expansion for the reduced-wave Green’s function and its regular part in the limit D ≫ 1: Rjj ∼ D|Ω| +R (N) jj + · · · , G(x;xj) ∼ D |Ω| +G (N)(x;xj) + · · · . (3.15) Here G(N)(x;xj) is the Neumann Green’s function with regular part R (N) jj , determined from the unique solution to ∆G(N) = 1 |Ω| − δ(x − xj) , x ∈ Ω , (3.16a) ∂nG (N) = 0 , x ∈ ∂Ω ; ∫ Ω G(N) dx = 0 , (3.16b) G(N)(x;xj) ∼ − 1 2π ln |x− xj |+R(N)jj , as x→ xj . (3.16c) 65 3.1. k-spot Quasi-Equilibrium Solutions We now use this approximation to find approximate solutions to the nonlinear algebraic system (3.12) in the limit D = D0/ν ≫ 1 with D0 = O(1) and ν ≪ 1. Substituting (3.15) into (3.12) we obtain Ae = s+ 2πD0|Ω| e e ts+ 2πνG(N)s+ νχ(s) , where G(N) is the Green’s matrix associated with the Neumann Green’s function, i.e. G(N)ij = G(N)(xi;xj) for i 6= j, and G(N)jj = R(N)jj . By expanding s as s = s0 + νs1 + · · · for ν ≪ 1, we then obtain that s0 and s1 satisfy( I + 2πD0|Ω| ee t ) s0 = Ae , (3.17a)( I + 2πD0|Ω| ee t ) s1 = −2πG(N)s0 − χ(s0) , (3.17b) where I is the k × k identity matrix. Since ete = k, the leading order approximation s0 shows that the source strengths have an asymptotically common value Sc given by s0 = Sce , Sc ≡ A [ 1 + 2πkD0 |Ω| ]−1 . This shows that decreasing the area of the domain or increasing the diffusion parameter D0 both have the same effect of decreasing Sc. The next order approximation s1 from (3.17) yields s1 = − ( I + 2πD0|Ω| ee t )−1 ( 2πScG(N) + χ(Sc) ) e . (3.18) Since the matrix I + 2piD0|Ω| eet is a rank-one perturbation of the identity, its inverse is readily calculated explicitly from the Shermann-Woodbury- Morrison formula as (I + µeet)−1 = I − µee t 1 + µete , µ = 2πD0 |Ω| , which determines s1 from (3.18). In this way, we obtain for D = D0/ν and ν ≪ 1 that the source strengths have the two-term expansion s = Sce− ( χ(Sc) 1 + µk e+ 2πA 1 + µk ( G(N) − µF 1 + µk I ) e ) ν +O(ν2) . (3.19) 66 3.2. The Spot-Splitting Instability Here Sc ≡ A/(1 + µk), µ ≡ 2πD0/|Ω|, and F ((x1, . . . ,xk) is a scalar func- tion denoting the sum of all of the entries of G(N), i.e. F (x1, . . . ,xk) = e tG(N) e = K∑ i=1 k∑ j=1 G(N)ij . (3.20) The k-components of the O(ν) correction to s have, in general, different values owing to the term involving G(N) e. A simple calculation shows that the sum of s is ets ∼ Ak 1 + µk − ν [ k χ(Sc) 1 + µk + 2πAF (1 + µk)2 ] . (3.21) From this expression, we note that ets is minimized when F (x1, . . . ,xk) is maximized. Finally, we remark that if the Neumann Green’s matrix G(N) is a circu- lant matrix, then it must have the eigenpair G(N)e = θ(N)e, where θ(N) = F/k. In this case, (3.19) shows that the spots have the common source strength given asymptotically by s = [ A 1 + µk − ν ( χ(Sc) 1 + µk + 2πA 1 + µk ( F k − µF 1 + µk ))] e+O(ν2) . (3.22) Owing to the rather simple structure of the nonlinear algebraic system (3.12) when the Green’s matrix is circulant, in many of our numerical ex- periments in the sections below we will focus on k-spot quasi-equilibrium patterns that lead to this special matrix structure. 3.2 The Spot-Splitting Instability We now study the stability of a quasi-equilibrium spot pattern to a spot self-replicating instability that can occur on a fast O(1) time-scale relative to the slow motion of speed O(ε2) of the spot locations. The slow motion of spots is studied later in Section 3.3. The stability analysis in this section is similar to that done in [56] for the Schnakenburg model. We first perturb the quasi-equilibrium solution ue, ve as u(x, t) = ue + e λ tη(x) , v(x, t) = ve + e λ tφ(x) , for a given and fixed spatial configuration (x1, . . . ,xk) of spots. Substituting this perturbation into (1.7) and linearizing, we obtain the eigenvalue problem 67 3.2. The Spot-Splitting Instability ε2∆φ− (1 + λ)φ+ 2Aueveφ+Av2eη = 0 , (3.23a) D∆η − (1 + τλ)η − 2ueveφ− v2eη = 0 . (3.23b) In the inner region near the jth spot, we recall that ue ∼ εA√DUj and ve ∼ √ D ε Vj , where Uj, Vj is the solution of the core problem (3.2). We then let y = ε−1(x − xj), and define Nj and Φj by η = εA√DNj and φ = √ D ε Φj. Then, upon neglecting algebraic terms in ε, (3.23) reduces to ∆yΦj − (1 + λ)Φj + 2UjVjΦj + V 2j Nj = 0 , (3.24a) ∆yNj − V 2j Nj − 2UjVjΦj = 0 , (3.24b) under the assumption that τ ≪ O(ε−2). Next we look for angular perturba- tions of the form Φj = e imψΦ̂j(ρ) , Nj = e imψN̂j(ρ), where Φ̂j and N̂j are radially symmetric in the inner variable ρ = |y|. Then, from (3.24), N̂j(ρ) and Φ̂j(ρ) on 0 < ρ <∞ satisfy Φ̂jρρ + 1 ρ Φ̂jρ − m 2 ρ2 Φ̂j − (1 + λ)Φ̂j + 2UjVjΦ̂j + V 2j N̂j = 0 , (3.25a) N̂jρρ + 1 ρ N̂jρ − m 2 ρ2 N̂j − V 2j N̂j − 2UjVjΦ̂j = 0 , (3.25b) with boundary conditions Φ̂′j(0) = 0, N̂ ′ j(0) = 0, Φ̂j(ρ)→ 0, as ρ→∞ . (3.25c) The specific form for the far-field behavior of N̂j will be different for the two cases m = 0 and m ≥ 2. For the case m = 1, which corresponds to translation invariance, it follows trivially that λ = 0. Since this case m = 1 is accounted for by the analysis of Section 3.3 for the slow spot dynamics it is not considered here. An instability of (3.25) for the mode m = 2 is associated with the ini- tiation of a peanut-splitting instability. This initial instability is found nu- merically to lead to a spot self-replication event, whereby a single spot splits into two identical spots. Instabilities for the higher modes m ≥ 3 suggest the possibility of more spatially intricate spot self-replication events. Thus, the eigenvalue problem (3.25) with angular mode m ≥ 2 determines spot- splitting instabilities. For this range of m, the linear operator for Nj in (3.25b) allows for decay as ρ→∞ owing to the m2N̂j/ρ2 term. As such, for 68 3.2. The Spot-Splitting Instability the modes m ≥ 2 we impose the far-field boundary condition that N̂ ′j → 0 as ρ→∞. The eigenvalue problem (3.25) is coupled to the core problem (3.2) for Uj and Vj , and can only be solved numerically. To do so, we first solve (3.2) numerically by using COLSYS (cf. [2]). Then, we discretize (3.25) by a centered difference scheme to obtain a matrix eigenvalue problem. By using the linear algebra package LAPACK [1] to compute the spectrum of this matrix eigenvalue problem, we estimate the eigenvalue λ0 of (3.25) with the largest real part as a function of the source strength Sj for different angular modes m ≥ 2. The instability threshold occurs when Re(λ0) = 0. We find numerically that λ0 is real when Sj is large enough. In the left subfigure of Fig. 3.3, we plot Re(λ0) as a function of the source strength Sj for m = 2, 3, 4. Our computational results show that the instability threshold for the modes m ≥ 2 occurs at Sj = Σm, where Σ2 ≈ 4.31, Σ3 ≈ 5.44, and Σ4 ≈ 6.14. In the right subfigure of Fig. 3.3, we plot the eigenfunction (Φ̂j, N̂j) corresponding to λ0 = 0 with m = 2 at Sj = Σ2. In this subfigure we have scaled the maximum value of Φ̂j to unity. 1 2 3 4 5 6 7 −1 −0.5 0 0.5 Sj R e( λ 0 ) (a) Re(λ0) vs. Sj 2 4 6 8 10 12 14 −1.5 −1 −0.5 0 0.5 1 ρ (Φ j, N j) (b) (Φ̂j , N̂j) vs. ρ Figure 3.3: Numerical results for the principal eigenvalue of (3.25) with mode m ≥ 2. (a) Re(λ0) vs. Sj; heavy solid curve is for m = 2 with Σ2 = 4.31, the solid curve is for m = 3 with Σ3 = 5.44, and the dashed curve is for m = 4 with Σ4 = 6.14. (b) For m = 2, the eigenfunctions (Φ̂j(ρ), N̂j(ρ)) near λ0 = 0 with Sj = Σ2 ≈ 4.31 are shown. The solid curve is Φ̂j(ρ), and the dashed curve is N̂j(ρ). Principal Result 3.2: Consider the 2-D Gray Scott model (1.7) with given parameters A,D, ε, and τ ≪ O(ε−2), on a fixed domain. Calculate A and ν from (3.4). In terms of A, D, and ν, calculate the source strengths S1, . . . , Sk 69 3.3. The Slow Dynamics of Spots for a k-spot quasi-equilibrium pattern from the nonlinear algebraic system (3.8). Then, if Sj < Σ2 ≈ 4.31, the jth spot is linearly stable to a spot deformation instability for modes m ≥ 2. Alternatively, for Sj > Σ2, it is linearly unstable to the mode m = 2 associated with peanut-splitting. We remark that in our stability analysis we linearized the GS model around a quasi-equilibrium solution, where the spots were assumed to be at fixed locations x1, . . . ,xk, independent of time. However, as shown below in Section 3.3, the spots have a speed O(ε2), so that the source strengths Sj for j = 1, . . . , k also vary slowly in time. As a result, there can be a triggered, or dynamically induced, spot-replication instability due to the slow evolution of Sj . More specifically, for a spot pattern that is initially stable to self-replication at t = 0 in the sense that Sj < Σ2 at t = 0 for j = 1, . . . , k, it is possible that as the Jth spot moves toward its equilibrium location in the domain that SJ > Σ2 after a sufficiently long time. Thus, the source strength of a particular spot (i.e. the Jth spot) may drift above the instability threshold for peanut-splitting as a result of the intrinsic slow dynamics of the collection of spots. This scenario is similar to other ODE and PDE problems that have triggered instabilities generated from slowly varying external bifurcation, or control, parameters. The key difference here, is that the spot self-replication instability can be triggered from the intrinsic motion of the collection of spots, and is not due to an externally imposed control parameter. Our second remark is that the eigenvalue problem (3.25) with mode m = 0 corresponds to instabilities in the amplitudes of the spots. For this mode, we cannot impose that the eigenfunction N̂j is bounded at infinity. Instead, we must impose that N̂j → Cj ln ρ + Bj as ρ → ∞. This type of spot profile instability is discussed in detail in Chapter §4. 3.3 The Slow Dynamics of Spots Next we derive the slow dynamics for the spot locations for a k-spot quasi- equilibrium solution to the GS model. The analysis is similar to that in [56]. In the inner region near the jth spot, we introduce y = ε−1(x−xj) and we expand u = ε A √ D (U0j(ρ) + εU1j(y) + · · · ) , v = √ D ε (V0j(ρ) + εV1j(y) + · · · ) . 70 3.3. The Slow Dynamics of Spots Note that the subscript 0 in U0j , V0j denotes the order of expansion, and j denotes that the variables are near the jth spot. In the analysis below we omit the subscript j if there is no confusion in the notation. The expansions above for u and v yield the same core problem for U0j and V0j as in (3.2). At next order, we get that U1 and V1 satisfy ∆yV1 − V1 + 2U0V0V1 + V 20 U1 = V ′0 d dς |x− xj | , ∆yU1 − 2U0V0V1 − V 20 U1 = 0 , (3.26) where we have introduced ς = ε2t as the slow time variable. We define θ to be the polar angle for the vector (x − xj), with xj = xj(ε2t), so that d|x−xj | dς = −x′j · eθ, where eθ = (cos θ, sin θ)t, xj = (x1, x2)t, and t denotes transpose. The system (3.26) for U1 and V1 can be written in matrix form as ∆ywj +Mjwj = fj , y ∈ R2, (3.27) where Mj = ( −1 + 2U0V0 V 20 −2U0V0 −V 20 ) , wj = ( V1 U1 ) , fj = ( −V ′0 x′j · eθ 0 ) . (3.28) Recall that the outer solution for u is u = 1− k∑ j=1 2πνSj A G(x;xj) . The local behavior of the reduced-wave Green’s function as x→ xj, written in terms of the inner variable y, is G(x;xj) ∼ − 1 2π ln |x− xj |+Rjj + ε∇R(xj ;xj) · y + · · · , where ∇f denotes the gradient of the function f . Upon matching the inner and outer solutions for u near the jth spot, we obtain the far-field behavior of the inner solution U1j given by Uij ∼ −2π [ Sj∇R(xj;xj) + k∑ i=1, i6=j Si∇G(xi;xj) ] · y , as y → ∞ . 71 3.3. The Slow Dynamics of Spots Therefore, the required far-field behavior for (3.27) is that wj → ( 0 −~αj · y ) , as y → ∞ , (3.29a) ~αj = ( α1 α2 ) ≡ 2π(Sj∇Rjj + k∑ i=1, i6=j Si∇Gij) . (3.29b) Here α1, α2 are the first and second components of the vector ~αj , respectively. We define P̂ ∗j (ρ) = (φ̂ ∗ j (ρ), ψ̂ ∗ j (ρ)) t to be the radially symmetric solution of the adjoint problem ∆ρP̂ ∗ j +MtjP̂ ∗j = 0 , (3.30) subject to the far-field condition that P̂ ∗j → (0, 1/ρ)t as ρ → ∞, where ∆ρ ≡ ∂ρρ + ρ−1∂ρ − ρ−2. We look for solutions P cj and P sj of the adjoint problem ∆yPj +MtjPj = 0 in the form P cj = P̂ ∗j cos θ and P sj = P̂ ∗j sin θ, where P̂ ∗j is a radially symmetric function. In terms of the adjoint solution P cj , the solvability condition for (3.27), subject to (3.29a), is that lim σ→∞ ∫ Bσ P cj · f dy = limσ→∞ ∫ ∂Bσ [ P cj · ∂ρwj −wj · ∂ρP cj ]∣∣∣ ρ=σ dρ . (3.31) Here Bσ is a ball of radius σ, i.e |y| = σ. Upon using the far-field condition (3.29a), we reduce (3.31) to x ′ 1 ∫ 2pi 0 ∫ ∞ 0 φ̂∗jV ′ 0 cos 2 θρ dρ dθ − x′2 ∫ 2pi 0 ∫ ∞ 0 φ̂∗jV ′ 0 cos θ sin θρ dρ dθ = lim σ→∞ ∫ 2pi 0 [ cos θ σ ~αj · eθ − σ~αj · eθ ( − 1 σ2 ) cos θ ] σ dθ . (3.32) Since ∫ 2pi 0 cos θ sin θ dθ = 0, we obtain ∂x1 ∂ς = 2α1∫∞ 0 φ̂ ∗ jV ′ 0ρ dρ . (3.33) Similarly, the solvability condition for (3.27), subject to (3.29a), with respect to the homogeneous adjoint solution P sj , yields ∂x2 ∂ς = 2α2∫∞ 0 φ̂ ∗ jV ′ 0ρdρ . (3.34) 72 3.3. The Slow Dynamics of Spots We summarize the result for the dynamics as follows: Principal Result 3.3: Consider the 2-D GS model (1.7) for ε → 0 with τ ≪ O(ε−2). Then, the slow dynamics of a collection (x1, . . . ,xk) of spots that are stable to spot profile instabilities are governed by the differential- algebraic (DAE) system dxj dt ∼ −2πε2γ(Sj)  Sj∇R(xj;xj) + k∑ i=1, i6=j Si∇G(xj ;xi)   , j = 1, . . . , k , (3.35) where γ(Sj) ≡ ( −2/ ∫∞0 φ̂∗jV ′0ρ dρ). Here the source strengths Sj, for j = 1, . . . , k, are determined in terms of the instantaneous spot locations and the parameters A, D, and ν = −1/ ln ε by the nonlinear algebraic system (3.8). In addition, V0 satisfies the core problem (3.2), and φ̂ ∗ j is the radially symmetric solution of the adjoint problem (3.30). The ODE system (3.35) coupled to the nonlinear algebraic system (3.8) comprises a DAE system for the time-dependent spot locations xj and source strengths Sj for j = 1, . . . , k. These collective coordinates evolve slowly over a long time-scale of order t = O(ε−2). The function γ(Sj) in (3.35) was previously computed numerically in Fig. 3 of [56], where it was shown that γ(Sj) is positive for a wide range of Sj. This plot is re-computed below in Fig. 3.4(f). In Section §3.7, we will compare the dynamics (3.35) with full numerical results for different patterns. It is important to emphasize that the DAE system in Principal Result 3.3 for the slow spot evolution is only valid if each spot profile is stable to any spot profile instability that occurs on a fast O(1) time-scale. Such instabilities are the spot-splitting instability studied in Section §3.2 and the competition and oscillatory profile instabilities studied later in Chapter 4. The equilibrium or steady-state locations of a collection (x1, . . . ,xk) of spots satisfies dxj/dt = 0 for j = 1, . . . , k, where the dynamics was given as in Principal Result 3.3. This leads to the following equilibrium result: Principal Result 3.4: Consider the 2-D GS model (1.7) with given pa- rameters A,D, ε. Then, the equilibrium spot locations xje and corresponding equilibrium source strengths Sje for j = 1, . . . , k satisfy Sj∇Rjj + k∑ i=1, i6=j Si∇Gij = 0 , j = 1, . . . , k , (3.36) 73 3.4. The Direction of Splitting subject to the nonlinear algebraic system (3.8), which relates the source strengths to the spot locations. In (3.36) we have labeled ∇ = (∂xj , ∂yj ). In general it is intractable to determine analytically all possible equilib- rium solution branches for k-spot patterns in an arbitrary 2-D domain as the parameters A and D are varied. However, some partial results are obtained in Section 3.7 for the special case of k spots equally spaced on a circular ring that lies within and is concentric with a circular disk domain. For this special case, the Green’s matrix is circulant, and the equilibrium problem is reduced to determining the equilibrium ring radius for the pattern. 3.4 The Direction of Splitting Next we determine the direction in which a spot splits, relative to the direc- tion of its motion, if its source strength is slightly above the threshold value Σ2 ≈ 4.31. When Sj = Σ2, the eigenspace for λ = 0 is four-dimensional; there are two eigenfunctions associated with the peanut-splitting mode m = 2, and there are two translation eigenfunctions associated with the mode m = 1. For Sj − Σ2 = O(ε), we have λ = 0(ε). Therefore, we will expand λ = ελ1 + · · · , and calculate both λ1 and its unstable eigenfunction, which will determine the direction for which a spot will split. The calculation below is similar to that in [56]. To analyze the direction of splitting, we must calculate a two-term ex- pansion for the solution of the core problem in addition to its associated eigenvalue problem near the jth spot. Throughout the calculation below, we omit the index j if there is no confusion. We first introduce U = U0(ρ) + εU1(y) + · · · , V = V0(ρ) + εV1(y) + · · · , ρ = |y| . Then, as in the analysis of the slow dynamics in Section §3.3, the perturba- tion w = (V1, U1) t satisfies (3.27) with x′j = −γ(Sj)~αj , ~αj = 2π ( Sj∇Rjj + k∑ i=1, i6=j Si∇Gij ) . Next, we introduce a complex constant g defined by g ≡ α1 − iα2 , so that in (3.28) the vector f can be written as f = γ(Sj) ( Re(geiθ)V ′0 0 ) , 74 3.4. The Direction of Splitting where ~αj = (α1, α2) t, and Re(A) is the real part of A. Then, w(y) can be written in terms of a real-valued radially symmetric function ŵ(ρ) as w(y) = 1 2 geiθŵ(ρ) + c.c., ŵ(ρ) = (V̂1(ρ), Û1(ρ)) t, where c.c. denotes the complex conjugate, and ŵ(ρ) on 0 ≤ ρ <∞ satisfies ∆ρŵ +Mŵ = γ(Sj) ( V ′0 0 ) , ŵ→ ( 0 −ρ ) as ρ→∞ . (3.37) For the eigenvalue problem (3.24), we expand Φ = Φ̂0(ρ) + εΦ1(y) + · · · , N = N̂0(ρ) + εN1(y) + · · · , λ = ελ1 + · · · . Upon substituting these expansions into (3.24), we obtain the leading order approximation P0 = (Φ̂0, N̂0) t as before, and obtain an equation for the next order terms P1 = (Φ1,N1) t. In this way, we get ∆ρP0 +MP0 = 0 , (3.38a) ∆yP1 +MP1 = Λ1P0 −M1P0 −FP0 , (3.38b) where M andM1 are defined by M = ( −1 + 2U0V0 V 20 −2U0V0 −V 20 ) , M1 = ( 2U1V0 + 2U0V1 2V0V1 −U1V0 − 2U0V1 −2V0V1 ) . In addition, we have defined Λ1 and F by Λ1 = ( λ1 0 0 0 ) F = ( x′j · ∇ 0 0 0 ) . Since x′j · ∇ = −γ(Sj)Re(geiθ)∂ρ, we can write M1 and F as M1 = 1 2 geiθM̂1 + c.c. , F = − 1 2 γ(Sj)(ge iθ + c.c.)F̂ , F̂ = ( ∂ρ 0 0 0 ) , where M̂1 is obtained by replacing V1, U1 with the radially symmetric so- lution V̂1, Û1 in every component of M1. For the leading order equation (3.38a), the solution consists of the eigenfunctions associated with the zero eigenvalue, so that P0 = AP̂01(ρ)e iθ +BP̂02(ρ)e 2iθ + c.c. 75 3.4. The Direction of Splitting For m = 1, the eigenfunctions are P̂01 = (V ′ 0 , U ′ 0) t, and for m = 2 with Sj ≈ Σ2, the eigenfunctions P̂02 are plotted in Fig. 3.3(b). The correspond- ing homogeneous adjoint problem has four independent solutions, which we write in complex form as P∗11 ≡ P̂∗1(ρ)eiθ + c.c. , P∗12 ≡ P̂∗1(ρ) ieiθ + c.c. , P∗21 ≡ P̂∗2(ρ)e2iθ + c.c. , P∗22 ≡ P̂∗2(ρ) i e2iθ + c.c. , (3.39) where P̂∗m(ρ) = (Φ̂∗m(ρ), N̂∗m(ρ))t for m = 1, 2 is the radially symmetric solution on 0 < ρ <∞ satisfying (P̂∗m)ρρ + 1 ρ (P̂∗m)ρ − m2 ρ2 P̂∗m +MtP̂∗m = 0 , P̂∗m → 0 as ρ→∞. (3.40) Then, the four solvability conditions for (3.38b) give the four equations∫ R2 (P∗ml) t ( Λ1 −M1 −F ) P0 dy = 0 , l,m = 1, 2 . (3.41) We introduce the integrals Jml, Iml, and Fml, defined by Jml = ∫ R2 (P∗ml) tΛ1P0 dy , Iml = ∫ R2 (P∗ml) tM1P0 dy , Fml = ∫ R2 (P∗ml) tFP0 dy . To calculate the integral I11 we write it as I11 = 1 2 ∫ R2 ( P̂∗1(ρ)e iθ + c.c. )t( geiθM̂1 + c.c. ) ( AP̂01(ρ)e iθ +BP̂02(ρ)e 2iθ + c.c. ) dy , = 1 2 ∫ 2pi 0 (eiθ + e−iθ)(geiθ + geiθ)(Aeiθ +Aeiθ) dθ ∫ ∞ 0 (P̂∗1) tM̂1P̂01ρ dρ + 1 2 ∫ 2pi 0 (eiθ + e−iθ)(geiθ + geiθ)(Be2iθ +Be2iθ) dθ ∫ ∞ 0 (P̂∗1) tM̂1P̂02ρ dρ , where ḡ is the complex conjugate of g. Since ∫ 2pi 0 e ijθdθ = 0 for any integer j 6= 0, only a constant integrand in the integral over θ gives a nonzero contribution to I11. The other integrals are calculated similarly. In this 76 3.4. The Direction of Splitting way, we obtain I11 = 2πRe(gB̄) ∫ ∞ 0 (P̂∗1) tM̂1P̂02ρdρ , I12 = 2πRe(igB̄) ∫ ∞ 0 (P̂∗1) tM̂1P̂02ρ dρ , I21 = 2πRe(gA) ∫ ∞ 0 (P̂∗2) tM̂1P̂01ρ dρ , I22 = 2πRe(−igA) ∫ ∞ 0 (P̂∗2) tM̂1P̂01ρ dρ . In a similar way, we calculate J11 = 2πRe(A)λ1 ∫ ∞ 0 Φ̂∗1Φ̂01ρ dρ , J12 = 2πRe(iĀ)λ1 ∫ ∞ 0 Φ̂∗1Φ̂01ρ dρ , J21 = 2πRe(B)λ1 ∫ ∞ 0 Φ̂∗2Φ̂02ρ dρ , J22 = 2πRe(iB̄)λ1 ∫ ∞ 0 Φ̂∗2Φ̂02ρ dρ , and F11 = 2πRe(gB̄)γ(Sj) ∫ ∞ 0 (P̂∗1) tF̂P̂02ρ dρ , F12 = 2πRe(igB̄)γ(Sj) ∫ ∞ 0 (P̂∗1) tF̂P̂02ρ dρ , F21 = 2πRe(gA)γ(Sj) ∫ ∞ 0 (P̂∗2) tF̂P̂01ρ dρ , F22 = 2πRe(−igA)γ(Sj) ∫ ∞ 0 (P̂∗2) tF̂P̂01ρ dρ . Substituting these formulae into the solvability conditions (3.41), we obtain λ1Re(A)κ1 = Re(gB̄) , λ1Re(iĀ)κ1 = Re(igB̄) , λ1Re(B)κ2 = Re(gA) , λ2Re(iB̄)κ2 = Re(−igA) , (3.42) where the constants κ1 and κ2 are defined by κ1 = ∫∞ 0 Φ̂ ∗ 1Φ̂01ρdρ∫∞ 0 (P̂ ∗ 1) t[M̂1 − γ(Sj)F̂ ]P̂02ρ dρ , κ2 = ∫∞ 0 Φ̂ ∗ 2Φ̂02ρdρ∫∞ 0 (P̂ ∗ 2) t[M̂1 − γ(Sj)F̂ ]P̂01ρ dρ . (3.43) 77 3.4. The Direction of Splitting For Sj = 4.31, we calculate that γ(Sj) ≈ 1.703. The solutions P̂∗m of the adjoint problem (3.40) are plotted in Fig. 5 of [56]. Numerical values for the constants κ1 and κ2 are also computed in §2.4 of [56] as κ1 = −0.926 and κ2 = −1.800. We define A = Ar + iAi, B = Br + iBi, and recall that g = α1 − iα2. Then, from (3.42) we obtain that  0 0 α1κ1 − α2 κ1 0 0 α2κ1 α1 κ1 α1 κ2 α2 κ2 0 0 − α2κ2 α1 κ2 0 0     Ar Ai Br Bi   = λ1   Ar Ai Br Bi   . (3.44) The eigenvalues λ1 of this problem are degenerate and satisfy λ21 = ( α1 κ1 α1 κ2 + α2 κ1 α2 κ2 ) ± i ( α1 κ1 α2 κ2 − α1 κ2 α2 κ1 ) = α21 + α 2 2 κ1 κ2 . Since κ1 < 0 and κ2 < 0, the eigenvalues and eigenvectors are, respectively, λ±1 = ± √ α21 + α 2 2√ κ1 κ2 , ( Br Bi ) = λ±1 κ1 α1 ± iα2 ( Ar Ai ) . (3.45) The unstable eigenvalue is λ+1 and its corresponding eigenvector is λ+1 = √ α21 + α 2 2√ κ1 κ2 , B = − √ κ1 κ2 α1 − iα2√ α21 + α 2 2 A = − √ κ1 κ2 g |g| A. Therefore, we conclude that the leading order eigenfunction P0 has the form P0 = 2A ( cos θP̂01 + κ Re(ge2iθ) |g| P̂02 ) , κ = − √ κ1 κ2 . The first term corresponds to the direction of motion of the jth spot, which is (α1, α2), while the second term controls the direction of the splitting of the spot. Since κ < 0 this splitting direction is orthogonal to the direction of spot motion. We remark that this analysis of the direction of spot-splitting is, largely, a local analysis based on a linearization of the core problem and the eigenvalue problem associated with the inner region. The only coupling to the other spots is through the determination of the source strength Sj from (3.8) and the vector ~αj, defined in (3.29a), representing the “force” exerted by the other spots and the domain geometry. As a consequence, we conclude that for any spot that slightly exceeds the instability threshold condition Sj = Σ2 it will split in a direction orthogonal to the direction of its motion. 78 3.5. Two-Spot Patterns in an Infinite Domain 3.5 Two-Spot Patterns in an Infinite Domain We first illustrate our asymptotic theory for the simple case of a two-spot pattern in R2 when D = O(1). For this infinite plane, the reduced-wave Green’s function and its regular part are readily available. Therefore, for this infinite domain problem it is rather straightforward to explicitly determine the spot dynamics and to characterize the self-replication instability. Without loss of generality, we assume that the centers of the two spots are at x1 = (α, 0) and x2 = (−α, 0) where α > 0 and α≫ O(ε). In addition, by a coordinate re-scaling it is readily shown that we can set D = 1 without loss of generality. The reduced-wave Green’s function for the infinite plane satisfies ∆G − G = −δ(x − xj), with G → 0 as |x| → ∞. The solution is simply G(x;xj) = 1 2π K0 (|x− xj |) , where K0(r) is the modified Bessel function of the second kind of order zero. The asymptotic behavior of K0(r) as r → 0 is K0(r) ∼ −(ln r − ln 2 + γe), where γe ≈ 0.5772 is Euler’s constant. Therefore, the entries of the Green’s matrix, consisting of G12 ≡ G(x1;x2) and its regular part Rjj = R(xj;xj), are simply G12 = G21 = 1 2π K0 (2α) , R11 = R22 = 1 2π (ln 2− γe) . Thus, the Green’s matrix G in (3.12) is a symmetric circulant matrix. Con- sequently, it follows from (3.14) of Section 3.1, that there is a common source strength Sc = S1 = S2, which satisfies the scalar nonlinear algebraic equation A = Sc [1 + ν(ln 2− γe) + νK0 (2α)] + ν χ(Sc) , ν = −1/ ln ε . (3.46) From this equation, we now numerically determine the relationship between Sc, the half-distance α between the two spots on the x-axis, and the feed-rate parameter A. Experiment 3.2: Two spots in the infinite plane We fix ε = 0.02 in the computational results below. In Fig. 3.4(a) we plot A versus Sc for three different values of α. From this plot we observe that the graph of A versus Sc has a fold point Af , which depends on α. This fold point Af (α) is shown in Fig. 3.4(b) by the dotted curve. The solid curve in Fig. 3.4(b) is the corresponding two-term asymptotic result for Af obtained by expanding (3.46) in powers of ν. This result shows that a 79 3.5. Two-Spot Patterns in an Infinite Domain quasi-equilibrium two-spot pattern exists for a spot separation 2α only when the feed-rate parameter A is larger than the existence threshold Af (α), i.e. A > Af (α). Next, we discuss the possibility of self-replicating instabilities for an inter-spot separation of 2α. In Fig. 3.4(c), we plot Sc vs. α for A = 3.5 (solid curve), A = 4.0 (dashed curve) and A = 4.5 (heavy solid curve). Notice that for a fixed A > Af , there could be two solutions of Sc for a given α. As such, we only plot the large solution branch for Sc in this figure. This figure shows that for a fixed distance 2α between the spots, the source strength Sc is an increasing function of the feed-rate A. In addition, for a fixed feed-rate A, the source strength Sc increases as α increases. For the solid curve in Fig. 3.4(c) with A = 3.5, the source strength Sc is always below the spot-replication threshold Σ2, so that the spots never split for any inter-separation distance. Alternatively, the dashed curve for A = 4.0 and the heavy solid curve for A = 4.5 in Fig. 3.4(c) intersect the spot self- replication threshold Sc = Σ2 at α ≈ 1.81 and α ≈ 0.46, respectively. This threshold initiates a spot-replication event. In Fig. 3.4(d), we plot A vs. α corresponding to the different spot- deformation thresholds Sc = Σ2 ≈ 4.31 (solid curve), Sc = Σ3 ≈ 5.44 (dashed curve) and Sc = Σ4 ≈ 6.14 (heavy solid curve). For the solid curve in this figure, which corresponds to Sc = Σ2, we observe that A decreases with α but asymptotes to its minimum value As = 3.98 as α → ∞. This minimum value is obtained by setting Sc = Σ2 and K0(2α) → 0 in (3.46). We conclude that for any A > As, a two-spot pattern in an infinite domain will be linearly unstable to a peanut-splitting instability if the inter-spot separation distance 2α is taken to be sufficiently large. Next we derive the ODE system that determines the slow dynamics of the two-spot pattern when Sc < Σ2. A simple application of the general result in Principal Result 3.3 shows that αc ≡ dα dς = −4ε2γ(Sc)ScK ′0 (2α) , γ(Sc) ≡ −2∫∞ 0 φ ∗V ′0ρ dρ, , ς = ε2t . (3.47) Since K ′0(r) < 0 for any r > 0 and |K ′0(r)| a decreasing function of r, it follows that the two spots repel each other along the x-axis for all time and that their common speed αc = | dαdς | is a decreasing function of inter- separation distance 2α. This is shown in Fig. 3.4(e) for three different values of A. In Fig. 3.4(f) we plot the curve γ(Sc) versus Sc used in the numerical computation of the ODE (3.47). Since the two spots are repelling and Sc is a monotonically increasing 80 3.5. Two-Spot Patterns in an Infinite Domain 0 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 S c A (a) A vs. Sc 0 0.5 1 1.5 2 2 2.5 3 3.5 4 α A f (b) Af vs. α 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 2.5 3 3.5 4 4.5 5 5.5 α S c (c) Sc vs. α 0.5 1 1.5 2 2.5 3 4 5 6 7 8 α A (d) A vs. α 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 60 α α c (e) αc vs. α 0 1 2 3 4 5 6 0 2 4 6 8 S γ (f) γ vs. S Figure 3.4: Experiment 3.2: Two spots at (±α, 0) in the infinite plane, with ε = 0.02. (a) A vs. Sc for α = 0.1 (solid curve), α = 0.5 (dashed curve) and α = 1.0 (heavy solid curve). (b) The fold point Af vs. α; two-term asymptotic result (solid curve), and numerical result from (3.46) (dotted curve). (c) Sc vs. α for A = 3.5 (solid curve), A = 4.0 (dashed curve) and A = 4.5 (heavy solid curve). (d) A vs. α with Sc = Σ2 (solid curve), Sc = Σ3 (dashed curve) and Sc = Σ4 (heavy solid curve). (e) αc vs. α for A = 3.5 (solid curve), A = 4.0 (dashed curve) and A = 4.5 (heavy solid curve). (f) γ vs. S defined in the ODE (3.47) . 81 3.6. The Reduced-Wave and Neumann Green’s Functions function of α it follows that a dynamically triggered spot-replicating insta- bility will occur when A > As ≈ 3.98. In particular, suppose that A = 4.0. Then, the graph of Sc versus α is shown by the dashed curve in Fig. 3.4(c). This curve shows that Sc > Σ2 if and only if α ≥ 1.81. Now consider an initial two-spot quasi-equilibrium solution with α = 1 at t = 0. For this initial value of α, A exceeds the existence threshold Af of Fig. 3.4(b). Then, since the ODE (3.47) predicts that α→∞ as t→∞, it follows that α = 1.81 at some sufficiently long time t = O(ε−2), which initiates the spot self-replication instability. Based on the theory in Section 3.2, the spots should split in a direction perpendicular to the x-axis. In summary, two spots on the infinite plane can readily undergo dynamically triggered spot self-replication events that are induced by their collective motion. 3.6 The Reduced-Wave and Neumann Green’s Functions In order to implement the asymptotic theory in the previous sections we re- quire some detailed analytical results for the reduced-wave Green’s function for a given domain. In addition, for D = O(ν−1)≫ 1, this Green’s function can be approximated by the Neumann Green’s function G(N), as was dis- cussed in Section 3.1. In this section we give analytical formulae for both the reduced-wave and Neumann Green’s functions for the unit disk and the rectangle for a given source point at x0. 3.6.1 Green’s Function for a Unit Disk Let Ω be the unit disk Ω = {x : |x| ≤ 1}. Introducing polar coordinates, (x0, y0) = (ρ0 cos θ0, ρ0 sin θ0), the reduced-wave Green’s function satisfies Gρρ + 1 ρ Gρ + 1 ρ2 Gθθ − 1 D G = − 1 ρ δ(ρ − ρ0)δ(θ − θ0) , in Ω , (3.48) with boundary conditions G(ρ, θ + 2π) = G(ρ, θ) and Gρ(1, θ) = 0. To determine G we extract the θ dependence in a complex Fourier series by introducing G(ρ, ρ0; θ, θ0) = 1 2π ∞∑ n=−∞ G̃n(ρ, ρ0; θ0)e −inθ , (3.49) so that G̃n(ρ, ρ0; θ0) = ∫ 2pi 0 einθG(ρ, ρ0; θ, θ0) dθ . 82 3.6. The Reduced-Wave and Neumann Green’s Functions In the usual way, we derive from (3.48) that G̃nρρ + 1 ρ G̃nρ − n 2 ρ2 G̃nθθ − 1 D G̃n = −δ(ρ− ρ0) ρ einθ0 , 0 < ρ < 1 , with boundary conditions G̃nρ(1, ρ0; θ0) = 0 , G̃n(0, ρ0; θ0) <∞ . The solution to this problem is given by G̃n =   A1In ( ρ√ D ) , 0 < ρ < ρ0 , A2 [ In ( ρ√ D ) − I ′ n “ 1√ D ” K ′n “ 1√ D ”Kn ( ρ√ D )] , ρ0 < ρ < 1 , (3.50) where In(r) and Kn(r) are modified Bessel functions of the second kind of order n. In (3.50), the coefficients A1 and A2 are obtained by making G̃n continuous at ρ0 and by requiring that the jump in the derivative of G̃n at ρ0 is [G̃nρ]ρ0 = − e inθ0 ρ0 . Upon using the Wronskian determinant Kn ( ρ0√ D ) I ′n ( ρ0√ D ) −K ′n ( ρ0√ D ) In ( ρ0√ D ) = √ D ρ0 , we calculate A1 and A2 as A1 = −einθ0  K ′n ( 1√ D ) I ′n ( 1√ D ) In ( ρ0√ D ) −Kn ( ρ0√ D ) , A2 = −einθ0 In ( ρ0√ D ) K ′n ( 1√D ) I ′n ( 1√ D ) . Upon substituting these expressions into (3.50), we obtain G̃n =   [ Kn ( ρ0√ D ) − K ′ n “ 1√ D ” I′n “ 1√ D ” In ( ρ0√ D )] einθ0In ( ρ√ D ) , 0 < ρ < ρ0[ Kn ( ρ√ D ) − K ′ n “ 1√ D ” I′n “ 1√ D ” In ( ρ√ D )] einθ0In ( ρ0√ D ) , ρ0 < ρ < 1. (3.51) 83 3.6. The Reduced-Wave and Neumann Green’s Functions Therefore, the Fourier expansion for the reduced-wave Green’s function for (3.48) is G(ρ, ρ0; θ, θ0) = 1 2π ∞∑ n=−∞ e−in(θ−θ0)Fn(ρ) , 0 < ρ < ρ0 , (3.52a) Fn(ρ) ≡  Kn ( ρ0√ D ) − K ′n ( 1√ D ) I ′n ( 1√ D ) In ( ρ0√ D ) In ( ρ√ D ) . (3.52b) Alternatively, for the range ρ0 < ρ < 1 we can simply interchange ρ and ρ0 in (3.52). Next, we identify the free-space Green’s function Gf , which is given by Gf (ρ, ρ0; θ, θ0) = 1 2π ∞∑ n=−∞ e−in(θ−θ0)Kn ( ρ0√ D ) In ( ρ√ D ) , 0 < ρ < ρ0 . (3.53) For the range ρ0 < ρ < ∞ we interchange ρ and ρ0 in the formula above. Then, by the well-known addition formula K0 ( R√ D ) = ∞∑ n=−∞ e−in(θ−θ0)Kn ( ρ0√ D ) In ( ρ√ D ) , (3.54) with R = √ ρ2 + ρ20 − 2ρρ0 cos(θ − θ0), we can decompose (3.52) as G = 1 2π K0 ( R√ D ) − 1 2π ∞∑ n=−∞ e−in(θ−θ0) K ′n ( 1√ D ) I ′n ( 1√ D ) In ( ρ0√ D ) In ( ρ√ D ) . (3.55) In (3.55), the first term arises from the source at (ρ0, θ0), while the Fourier series term represents the effects of the boundary conditions. Since n is an integer, we can use K−n(z) = Kn(z) and I−n(z) = In(z), to further simplify (3.55) to G = 1 2π K0 ( R√ D ) − 1 2π K ′0 ( 1√ D ) I ′0 ( 1√ D ) I0 ( ρ0√ D ) I0 ( ρ√ D ) + δM − 1 π M∑ n=1 cos(n(θ − θ0)) K ′n ( 1√ D ) I ′n ( 1√ D ) In ( ρ0√ D ) In ( ρ√ D ) . (3.56) 84 3.6. The Reduced-Wave and Neumann Green’s Functions Since K ′n ( 1√ D ) < 0, and I ′n ( 1√ D ) > 0 for any integer n ≥ 0, then δM is a small error term bounded by PM , where |δM | ≤ PM ≡ − ∞∑ n=M+1 K ′n ( 1√ D ) I ′n ( 1√ D ) In ( ρ0√ D ) In ( ρ√ D ) . The expression (3.56) for G is the one used in the numerical simulations in Section 3.7. From (3.56) we can readily extract the regular part R of the reduced-wave Green’s function. The gradients of R and G that are needed in Principal Result 3.3 to determine the motion of a collection of spots is determined numerically from (3.56). In Table 3.1, we give the number M of Fourier terms in (3.56) that are required in order to calculate the Green’s function within a tolerance of 10−8. The series for PM is found to converge fairly fast, especially when the spots are not close to the boundary of the unit circle and D is not too large. ρ0 ρ M PM 0.8 0.8 31 8.1737e-009 0.8 0.5 16 6.7405e-009 0.8 0.2 8 9.5562e-009 0.5 0.5 11 4.3285e-009 Table 3.1: The number of Fourier terms needed to determine the reduced- wave Green’s function within a tolerance 10−8 for D = 1. 3.6.2 Neumann Green’s Function for a Unit Disk Next for D = O(ν−1)≫ O(1), we recall from (3.15) that the reduced-wave Green’s function and its regular part can be approximated by the Neu- mann Green’s function G(N)(x;x0) and its regular part R (N)(x;x0) satisfy- ing (3.16). From equation (4.3) of [51], this Neumann Green’s function and its regular part for the unit disk are given in terms of Cartesian coordinates 85 3.6. The Reduced-Wave and Neumann Green’s Functions by G(N) = 1 2π ( − ln |x− x0| − ln ∣∣∣∣x|x0| − x0|x0| ∣∣∣∣+ 12(|x|2 + |x0|2)− 34 ) , (3.57) R(N) = 1 2π ( − ln ∣∣∣∣x0|x0| − x0|x0| ∣∣∣∣+ |x0|2 − 34 ) . (3.58) The gradients of these quantities can be obtained by a simple calculation as ∇G(N)(x;x0) = − 1 2π [ − x− x0|x− x0|2 + |x0|2 x̄|x0|2 − x̄0 − x ] , ∇R(N)(x0;x0) = 1 2π ( 2− |x0|2 1− |x0|2 ) x0 . These results allow us to explicitly determine the dynamics of a collection of spots when D is large, as described in Principal Result 3.3. 3.6.3 Green’s Function for a Rectangle Next, we calculate the reduced-wave Green’s function G(x, y;x0, y0) in the rectangular domain Ω = [0, L] × [0, d]. The free-space Green’s function Gf is given by Gf = 1 2π K0 ( |x− x0|√ D ) . Therefore, using the method of images, we can write the Green’s function satisfying (3.6) as G(x;x0) = 1 2π ∞∑ n=−∞ ∞∑ m=−∞ 4∑ l=1 K0 ( |x− x(l)mn|√ D ) , (3.59) where x(1)mn = (x0 + 2nL, y0 + 2md) , x (2) mn = (−x0 + 2nL, y0 + 2md) , x(3)mn = (x0 + 2nL,−y0 + 2md) , x(4)mn = (−x0 + 2nL,−y0 + 2md) . The modified Bessel function K0(r) decays exponentially as r → ∞. For example when r > 22, then K0(r) < 7.5×10−11. Therefore, provided that D is not too large, the infinite series in (3.59) converges fairly fast. The regular part of the reduced-wave Green’s function together with the gradients of G and R, as required in Principal Result 3.3, are readily evaluated numerically from (3.59). 86 3.7. Comparison of Theory with Numerical Experiments 3.6.4 Neumann Green’s Function for a Rectangle If D = O(ν−1) is large, we can approximate the reduced-wave Green’s func- tion and its regular part by the corresponding Neumann Green’s function G(N) and its regular part R(N), as given by the expansion in (3.15). This latter Green’s function was given analytically in formula (4.13) of [56] as G(N)(x;x0) = − 1 2π ln |x− x0|+R(N)(x;x0) , where R(N)(x;x0) is given explicitly by R(N)(x;x0) = − 1 2π ∞∑ n=0 ln(|1− qnz+,+||1− qnz+,−||1− qnz−,+||1− qnζ+,+| |1− qnζ+,−||1− qnζ−,+||1− qnζ−,−|) + L d [ 1 3 − x L + 1 2 ( x20 + x 2 L2 )] − 1 2π ln ( |1− z−,−| |r−,−| ) − 1 2π ∞∑ n=1 ln |1− qnz−,−| . (3.60) Here points in the rectangle are written as complex coordinates. In (3.60), q ≡ e−2Lpi/d, while z±,± and ζ±,± are given by z+,± ≡ exp(µ(−|x+ x0|+ i(y ± y0))/2) , z−,± ≡ exp(µ(−|x− x0|+ i(y ± y0))/2) , ζ+,± ≡ exp(µ(|x+ x0| − 2L+ i(y ± y0))/2) , ζ−,± ≡ exp(µ(|x− x0| − 2L+ i(y ± y0))/2) , where µ is defined by µ = 2π/d. 3.7 Comparison of Asymptotic Results and Full Numerical Results In this section we compare our asymptotic results for spot dynamics, the spot self-replication threshold, and the equilibrium positions of spot patterns with full numerical simulations of (1.7). The comparisons are made for the unit disk and square, since for these two specific domains the required Green’s functions are analytically known as given in the previous section. In all of the numerical simulations below we have set ε = 0.02 so that ν = −1/ ln ε = 0.2556. In each numerical experiment, we first fix the parameters 87 3.7. Comparison of Theory with Numerical Experiments A and D. Then, for an initial configuration of spot locations (x1, . . . ,xk), we compute the source strengths (S1, . . . , Sk) from the nonlinear algebraic system (3.8). The initial condition for the full numerical simulations is taken to be the quasi-equilibrium solution (3.9) with the values for Vj(0) as plotted in Fig. 3.1(b), corresponding to the computed values of Sj. Since this initial condition provides a decent, but not sufficiently precise, initial k−spot pattern, we only begin to track the spot locations from the full numerical simulations after the completion of a short transient period. To numerically identify the locations of the spots at any time, we determine all local maxima of the computed solution v by identifying the maximal grid values. This simple procedure is done since it is expensive to interpolate the grid values to obtain more accurate spot locations every time step. In this way, the trajectories of the spots are obtained from the full numerical results, and are then compared with the asymptotic dynamics of the spots as predicted by Principal Result 3.3. 3.7.1 The Unit Square In this subsection, we consider the unit square domain Ω = [0, 1] × [0, 1]. The numerical simulations in this domain are done by solving (1.7) using the PDE solver VLUGR (cf. [7]), with a minimum step size 0.005. In each of the experiments below, we plot the contour map of the numerical solution v in a gray-scale form. The bright (white) regions correspond to the spot regions where v has a large amplitude, while the dark region is where v is asymptotically zero. Experiment 3.3: A three-spot pattern: Slowly drifting spots Let A = 20 and D = 1, and consider an initial three-spot pattern with spots equally spaced on a ring of radius r = 0.2 centered at (0.5, 0.5). The initial coordinates for the spots locations are at xj, yj, as given in Table 3.2. For this pattern, the Green’s matrix G is not strictly circulant, but from (3.12) we compute that all three spots share a common source strength Sc ≈ 3.71. Since Sc < Σ2, we predict that there is no spot self-replication initiated at t = 0. In Fig. 3.5 we plot v at t = 1, 21, 106, 581, where we observe that all three spots drift outwards, and approach equilibrium locations xjn, yjn for j = 1, . . . , 3 after a sufficiently long time. During this evolution, the source strengths never exceed the threshold Σ2, and so there is no triggered dynamical spot-replication instability. In Fig. 3.6 the solid curves are the spot trajectories obtained from the asymptotic result of Principal Result 3.3. The green, blue, and red curves, in 88 3.7. Comparison of Theory with Numerical Experiments this figure correspond, respectively, to the 1st, 2nd, and 3rd spots. Since the initial locations of the 1st and 2nd spots are exactly symmetric, it follows that G13 = G23, R11 = R22, and G12 = G21 in the Green’s matrix. There- fore, from Principal Result 3.3, these two spots move at the same speed and their trajectories in the x−direction exactly overlap, as plotted by the green curve in Fig. 3.6(a). The spot locations as predicted from the full numerical solution are indicated by circles in Fig. 3.6. We observe that the asymptotic and numerical results for the spot dynamics agree very closely, which confirms that the asymptotic result in Principal Result 3.3 closely predicts the slow evolution of spots when there is no spot self-replication. In addition, from the asymptotic result in Principal Result 3.4, we calcu- late the equilibrium location xje, yje and the equilibrium source strength Sje for j = 1, . . . , 3. These values are given in Table 3.2, and our equilibrium result is found to closely predict the final equilibrium state xjn, yjn, Sjn obtained by numerically computing solutions to the full system (1.7) for a sufficiently long time. j xj yj Sj Vj(0) xjn yjn Sjn xje yje Sje 1 0.40 0.67 3.71 34.2 0.33 0.77 3.68 0.33 0.77 3.68 2 0.40 0.33 3.71 34.2 0.33 0.23 3.68 0.33 0.23 3.68 3 0.70 0.50 3.71 34.2 0.79 0.50 3.95 0.79 0.50 3.95 Table 3.2: Data for Experiment 3.3: The columns of xj , yj , Sj and Vj(0) correspond to the initial condition. The remaining data is with regards to the asymptotic and numerical results for the equilibrium state. Figure 3.5: Experiment 3.3: Fix A = 20 and D = 1, and consider a three- spot initial pattern with spots equally spaced on a ring of radius r = 0.2 centered at (0.5, 0.5). The initial common source strength is Sc = 3.71, as given in Table. 3.2. The numerical solution v is shown at t = 1, 21, 106, 581, and all spots drift slowly outwards. 89 3.7. Comparison of Theory with Numerical Experiments 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 t x j (a) xj vs. t 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 t y j (b) yj vs. t Figure 3.6: Experiment 3.3: Fix A = 20 and D = 1. We compare the dynamics of a 3−spot pattern (solid curves) with full numerical results (dis- crete markers) starting from t = 10. (a) xj vs. t; (b) yj vs. t. Experiment 3.4: A three-spot pattern: Spot-splitting and dynamics after splitting We fix A = 20 and D = 1, and consider an initial three-spot pattern on a ring of radius r = 0.3 centered at (0.4, 0.4). Note that this experiment has the same parameters as in the previous experiment, except now the initial spot configuration is different. The initial data are given on the left side of Table 3.3. Since the source strength for the 3rd spot now exceeds the spot-splitting threshold, i.e. S3 > Σ2, the asymptotic theory predicts that the 3rd spot will undergo splitting beginning at t = 0. In Fig. 3.7, we plot the numerical solution v at t = 1, 10, 31, 46, 61, 91, 121, 421. From this figure we observe that the 3rd spot (the brightest one at t = 1) deforms into a peanut shape at t = 31, and then splits into two spots at t = 46 in a direction perpendicular to its motion. Subsequently, the four spots interact and slowly drift to a symmetric four-spot equilibrium state when t = 421. After the splitting event, we use the new 4−spot pattern xjt, yjt for j = 1, . . . , 4 at t = 61, as given in Table 3.3, as the initial conditions for the asymptotic ODE’s in Principal Result 3.3. At t = 61, the source strengths S1t, · · · , S4t are below the splitting threshold, and so there is no further spot- splitting at this time. In Fig. 3.8, the subsequent asymptotic trajectories of the spots for t ≥ 61 are shown by the solid curves. The x−coordinates of the 1st and 2nd spots, as well as the 3rd and 4th spots, essentially overlap in Fig. 3.8(b), as a result of their almost symmetric locations. The discrete 90 3.7. Comparison of Theory with Numerical Experiments markers on the spot trajectory curves in Fig. 3.8 are the full numerical results as computed from the GS model (1.7). From this figure, we observe that there is a very close agreement between the asymptotic and full numerical results for the dynamics. Thus, Principal Result 3.3 can also be used to predict the motion of a collection of spots after a spot self-replication event. From Principal Result 3.4, we calculate the asymptotic result for the symmetric equilibrium locations xje, yje and the equilibrium source strengths Sje, for j = 1, . . . , 4. These results are given in Table. 3.3, and compare al- most exactly with the results from the full numerical simulations at t = 421 (not shown). j xj yj Sj Vj(0) xjt yjt Sjt xje yje Sje 1 0.25 0.66 4.05 32.54 0.30 0.75 3.44 0.25 0.75 2.87 2 0.25 0.14 2.37 35.48 0.28 0.20 3.01 0.25 0.25 2.87 3 0.70 0.40 4.79 27.65 0.76 0.36 2.40 0.75 0.25 2.87 4 - - - - 0.77 0.59 2.55 0.75 0.75 2.87 Table 3.3: Data for Experiment 3.4: The columns of xj , yj , Sj and Vj(0) correspond to the initial condition. The data xjt, yjt, and Sjt correspond to the initial conditions used for the asymptotic dynamics at t = 61 after the spot-splitting event. The final columns are the equilibrium results for the 4-spot pattern obtained from the asymptotic theory. Experiment 3.5: An asymmetric four-spot pattern We fix A = 30 and D = 1, and consider an initial 4−spot pattern with equally spaced spots on a ring of radius 0.2 centered at (0.6, 0.6). The initial spot locations xj , yj for j = 1, . . . , 4 and initial source strengths are given on the left side of Table 3.4. For this case, the two source strengths S3 and S4 satisfy S3 > Σ2 and S4 > Σ2, so that our asymptotic theory predicts that the 3rd and 4th spots undergo spot-splitting events starting at t = 0. The full numerical results for this pattern are given in Fig. 3.9, where we observe that the 3rd and 4th spots split in a direction perpendicular to their motion. The resulting 6−spot pattern, with initial locations xjt, yjt for j = 1, . . . , 6 at t = 21, as given in Table 3.4, are used as initial conditions for the asymptotic ODE’s in Principal Result 3.3. These six spots evolve outwards as t increases and eventually approach their equilibrium states xje, yje, and Sje, for j = 1, . . . , 6, as given in Table 3.4, when t = 581. The asymptotic result obtained from Principal Result 3.4 for the 6−spot equilibrium state is found to compare very favorably with the full numerical 91 3.7. Comparison of Theory with Numerical Experiments Figure 3.7: Experiment 3.4: Fix A = 20 and D = 1, and consider a three- spot initial pattern with spots equally spaced on a ring of radius r = 0.3 centered at (0.4, 0.4). The initial source strengths Sj for j = 1, . . . , 3 are given in Table. 3.3, for which S3 > Σ2. We plot the full numerical solution v as time evolves, and observe that the 3rd spot undergoes self-replication. Subsequently, all four spots drift slowly towards a symmetric equilibrium 4−spot pattern. 50 100 150 200 250 300 350 400 450 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t x j (a) xj vs. t 50 100 150 200 250 300 350 400 450 0 0.2 0.4 0.6 0.8 1 t y j (b) yj vs. t Figure 3.8: Experiment 3.4: Fix A = 20 and D = 1. Starting from t = 61 after the spot-replication event, we compare the dynamics of the 4−spot pattern from asymptotic analysis (solid curves) with full numerical results (discrete markers). (a) xj vs. t; (b) yj vs. t. 92 3.7. Comparison of Theory with Numerical Experiments results at t = 581. We remark that this parameter set corresponds to the numerical experiment given in Chapter 1 in Fig. 1.4. A very favorable comparison of analytical (solid curves) and full numer- ical results (discrete markers) for the spot locations after t = 21 is shown in Fig. 3.10. This agreement further validates our main asymptotic result in Principal Result 3.3. j xj yj Sj Vj xjt yjt Sjt xje yje Sje 1 0.80 0.60 2.82 36.21 0.84 0.58 3.13 0.83 0.61 2.91 2 0.60 0.80 2.82 36.21 0.58 0.84 3.13 0.61 0.83 2.91 3 0.40 0.60 5.69 19.07 0.25 0.49 2.93 0.17 0.39 2.91 4 0.60 0.40 5.69 19.07 0.26 0.66 2.73 0.21 0.79 3.07 5 - - - - 0.49 0.25 2.93 0.39 0.17 2.91 6 - - - - 0.66 0.26 2.73 0.79 0.21 3.07 Table 3.4: Data for Experiment 3.5: The columns of xj , yj , Sj and Vj(0) correspond to the initial condition. The data xjt, yjt, and Sjt correspond to the initial conditions used for the asymptotic dynamics at t = 21 after the two spot-splitting events. The final columns are the equilibrium results for the 6-spot pattern. 3.7.2 The Unit Disk In this subsection we compare results from our asymptotic theory for spot patterns in the unit disk Ω = {x : |x| ≤ 1}, with corresponding full numeri- cal results. For the unit disk, numerical solutions to the GS model (1.7) are computed by using the finite element code developed by W. Sun in [50]. We first consider the special case when k spots are equally distributed on a ring of radius r centered at the origin of the unit disk. The centers of the spots are at xj = re 2piij/k, j = 1, 2, · · · , k, where i = √−1. For this symmetric arrangement of spots, the Green’s matrix in (3.12) is circulant and is a function of the ring radius r. Hence, it follows that Ge = θe with e = (1, . . . , 1)t, where θ = pk(r)/k and pk(r) is defined by pk(r) ≡ ∑k i=1 ∑k j=1 Gij . Since the Green’s matrix G is circulant, the source strengths Sj for j = 1, . . . , k have a common value Sc. From (3.14) of Section §3.1, it follows that the source strength Sc satisfies the 93 3.7. Comparison of Theory with Numerical Experiments Figure 3.9: Experiment 3.5: Fix A = 30 and D = 1, and consider a four-spot initial pattern with spots equally spaced on a ring of radius r = 0.2 centered at (0.6, 0.6). The initial source strengths Sj for j = 1, . . . , 4 are given in Table 3.4, for which S3 > Σ2 and S4 > Σ2. We plot the full numerical solution v as time evolves, and observe that the 3rd and 4th spots undergo self-replication. Subsequently, all six spots move towards a symmetric 6−spot equilibrium pattern. 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 t x j (a) xj vs. t 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 t y j (b) yj vs. t Figure 3.10: Experiment 3.5: Fix A = 30 and D = 1. Starting from t = 21 after the two spot-replication events, we compare the asymptotic dynamics of the 6−spot (solid curves) with full numerical results (discrete markers). (a) xj vs. t; (b) yj vs. t. 94 3.7. Comparison of Theory with Numerical Experiments scalar nonlinear algebraic equation A = Sc ( 1 + 2πν k pk(r) ) + νχ(Sc) . (3.61) To determine the dynamics of the k spots, we calculate ~α in (3.29b) as ~αj = 2π Sc k ∇F (x1, . . . ,xk) , ∇F (x1, · · · ,xk) = p′k(r) e2piij/k . Then, the dynamics (3.35) of Principal Result 3.3 is reduced to an ODE for the ring radius r(t), given by dr dt = −2πε 2 k γ(Sc)Sc p ′ k(r) . (3.62) Therefore, we predict that the k spots remain equally distributed on a slowly evolving ring. The function pk(r) ≡ ∑k i=1 ∑k j=1 Gij cannot be calculated analytically when G is the reduced-wave Green’s function. Instead, we calculate it nu- merically from the explicit form for G given in (3.56). However, for D ≫ 1, we can analytically calculate a two-term expansion for pk(r) in terms of D by using the two-term expansion (3.15) for the reduced-wave Green’s func- tion in terms of the Neumann Green’s function, together with the simple explicit form for the Neumann Green’s function in (3.58). We obtain for D ≫ 1 that pk(r) ∼ k2D/|Ω|+ ∑k i=1 ∑k j=1 G(N)ij , where the double sum here was calculated in Proposition 4.3 of [51]. In this way, for D ≫ 1, we obtain that pk(r) ∼ k 2D π + 1 2π [ −k ln(krk−1)− k ln(1− r2k) + r2k2 − 3 4 k2 ] . (3.63) For D ≫ 1, the equilibrium ring radius re satisfies p′k(r) = 0, which yields the transcendental equation k − 1 2k − r2 = r 2k 1− r2k . It is readily shown that there is only one root re to this equation for any k ≥ 2 in the interval r ∈ (0, 1). Hence, for D ≫ 1, there is unique equilibrium ring radius for k spots uniformly distributed on a ring. It is also readily shown from (3.63) that p ′′ k(re) > 0. Hence, for D ≫ 1, pk(r) has a minimum at the equilibrium radius re, so that re is a stable equilibrium point for the ODE 95 3.7. Comparison of Theory with Numerical Experiments (3.62) when D ≫ 1. From (3.61), this condition also implies that Sc(r) has a maximum value at r = re when D ≫ 1. In Fig. 3.11(a), we fix D = 3.912 and plot the source strength Sc as a function of the ring radius r with k = 3, A = 30 (dotted curve), k = 4, A = 40 (solid curve) and k = 5, A = 48 (heavy solid curve). For these three patterns, the equilibrium states are respectively (re1, Sce1) = (0.55, 4.57), (re2, Sce2) = (0.60, 4.70), and (re3, Sce3) = (0.63, 4.58), which are marked by the solid circles in this figure. The square markers correspond to the spot-replicating threshold Σ2 ≈ 4.31, which occurs at rs1 = 0.17, rs2 = 0.14, and rs3 = 0.22, respectively. Any portion of these curves above the square markers correspond to ring radii where simultaneous spot self-replication is predicted to occur. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 4 4.2 4.4 4.6 4.8 r S c (a) Sc vs. r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 3.6 3.8 4 4.2 4.4 4.6 4.8 5 r S c (b) Sc vs. r Figure 3.11: Fix D = 3.912. (a) Sc vs. r with k = 3, A = 30 (dotted curve), k = 4, A = 40 (solid curve) and k = 5, A = 48 (heavy solid curve). (b) Fix k = 4, we plot Sc vs. r with A = 36 (dotted curve), A = 37.5 (solid curve), and A = 40 (heavy solid curve). The square markers indicate spot-splitting thresholds, and the solid circles mark the equilibrium states. In Fig. 3.11(b), we fix k = 4 andD = 3.912 and plot Sc vs. r withA = 36 (dotted curve), A = 37.5 (solid curve), and A = 40 (heavy solid curve). The equilibria are at (re4, Sce4) = (0.60, 4.21), (re5, Sce5) = (0.60, 4.40) and (re2, Sce2) = (0.60, 4.70), respectively. In this figure we mark the spot- replicating thresholds for each case by the square markers. Note that for k spots on a ring, the maximum value of the source strength Sc is attained at the equilibrium location. Therefore, for A = 36 and k = 4, since the source strength at the equilibrium state is Sce5 = 4.21, which is below the splitting threshold Σ2 ≈ 4.31, this 4-spot pattern is always stable to spot-splitting for all values of the ring radius. In contrast, on the solid curve for A = 37.5 and 96 3.7. Comparison of Theory with Numerical Experiments k = 4 if the ring radius is in the interval between the two square markers, i.e. 0.35 < r < 0.84, we predict that all four spots will split. From the heavy solid curve for A = 40 and k = 4 in Fig. 3.11(b), we predict that the four spots will split simultaneously when r > rs2 = 0.14. We now confirm this prediction with full numerical results. Experiment 3.6: One ring pattern with D ≫ O(1): Spot splitting We fix the parameters as A = 40 and D = 3.912 and consider an initial 4−spot pattern with equally spaced spots lying on an initial ring of radius r = 0.5 > rs2. Since the initial common source strength for this pattern is Sc = 4.69, as obtained from (3.61), we predict that all four spots will begin to split simultaneously at t = 0. The full numerical solution v at t = 4.6, 70, 93, 381, computed from the GS model (1.7), is plotted in Fig. 3.12. The numerical computations show that the four spots undergo spot-splitting in the direction perpendicular to their motion, and generate a pattern of eight equally spaced spots on a ring. This confirms our prediction based on the asymptotic analysis. -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -5 0 123 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -5 0 12 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -5 0 1234 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -5 0 123 Figure 3.12: Experiment 3.6: Fix D = 3.912, A = 40.0. We start from an initial 4-spot pattern with spots equally spaced on a ring of radius r = 0.5. The common source strength is Sc = 4.69. We plot the solution v at t = 4.6, 70, 93, 381 respectively, where we observe that all four spots split. This leads to an equilibrium eight-spot pattern on a ring. Next, we consider a different pattern whereby k − 1 spots are equally spaced on a ring of radius r, with an additional spot located at the center of the unit disk. The k − 1 spots on the ring have a common source strength Sc, while the k th spot at the origin has a source strength denoted by Sk. For this pattern, the spot coordinates in complex form are xj = re 2piij/(k−1) , j = 1, 2, . . . , k − 1 ; xk = 0 . We will analyze the dynamics of spots and the possibility of spot-splitting for this pattern for the limiting case D ≫ 1, for which the reduced-wave 97 3.7. Comparison of Theory with Numerical Experiments Green’s function can be approximated by the Neumann Green’s function. For this special configuration, Principal Result 3.3 shows that the kth spot at the center of the disk will not move. Alternatively, the motion of the remaining k − 1 spots on the ring satisfy x′j = −2πε2γ(Sc) [( Sc 2 ) ∇xjF (x1, . . . ,xk−1) + Sk∇xjG(xj ; 0) ] , (3.64) for j = 1, · · · , k − 1, where ∇xjF (x1, . . . ,xk−1) = 1 k − 1p ′ k−1(r) e 2piij/(k−1) , j = 1, · · · , k − 1 , ∇xjG(xj ; 0) = 1 2π ( r − 1 r ) e2piij/(k−1) . For D ≫ 1, the function pk−1(r) is given in (3.63). In order to calculate the equilibrium radius rke for this pattern, we need to solve a coupled system involving the source strengths Sk and Sc. We first denote µ = 2πν, and then define α and β for D ≫ 1 by α ≡ G(r; 0) ∼ D|Ω| − 1 2π ln r + r2 4π − 3 8π , β ≡ R(0; 0) = D|Ω| − 3 8π . We substitute these formulae into the nonlinear algebraic system (3.8) and set x′j = 0 in (3.64) for j = 1, . . . , k − 1. This leads to a three-dimensional nonlinear algebraic system for the equilibrium radius rke and the source strengths Sk and Sc given by Sk Sc + k − 2 2 − r2 ( k − 1 + Sk Sc ) − (k − 1) r 2k−2 1− r2k−2 = 0 , (3.65a) A = Sc ( 1 + µpk−1(r) k − 1 ) + νχ(Sc) + µαSk , (3.65b) A = (k − 1)µαSc + νχ(Sk) + Sk (1 + µβ)) . (3.65c) In Fig. 3.13(a) and Fig. 3.13(b), we fix D = 3.912 and plot the source strengths Sc and Sk versus the ring radius r for k = 3,A = 22 (solid curves) and k = 4,A = 38 (heavy solid curves). For each pattern, there are two equilibrium states, which are marked by the solid circles. The two equilibrium ring radii for k = 3, A = 22 are re1 = 0.62 with corre- sponding source strengths (Sc1, Sk1) = (3.22, 3.31), and re2 = 0.49 with (Sc2, Sk2) = (4.50, 0.56). For k = 4, A = 38, they are re1 = 0.65 with (Sc1, Sk1) = (4.56, 4.04) , and re2 = 0.57 with (Sc2, Sk2) = (5.70, 0.51). The 98 3.7. Comparison of Theory with Numerical Experiments spot-splitting threshold Σ2 is also marked on the curves in Fig. 3.13(a) and Fig. 3.13(b) by the square markers. On the solid curve for k = 3, they are at (r, Sc) = (0.34, 4.31), (r, Sc) = (0.84, 4.31) and (r, Sk) = (0.87, 4.31). On the heavy solid curve for k = 4, they are at (r, Sc) = (0.74, 4.31) and (r, Sk) = (0.68, 4.31). 0.2 0.4 0.6 0.8 1 2 3 4 5 6 r S c (a) Sc vs. r 0.2 0.4 0.6 0.8 1 0 2 4 6 8 r S k (b) Sk vs. r 0.5 0.6 0.7 0.8 0.9 1 2.5 3 3.5 4 4.5 r S c (c) Sc vs. r 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 r S k (d) Sk vs. r Figure 3.13: Fix D = 3.912. (a) Sc vs. r; (b) Sk vs. r. The solid curves correspond to A = 22, k = 3, the heavy solid curves correspond to A = 38, k = 4, the square markers correspond to spot-splitting thresholds, and the solid circles correspond to equilibrium ring radii. (c) Sc vs. r; (d) Sk vs. r. The heavy solid curves correspond to A = 82, k = 10 and the solid curves correspond to A = 60, k = 10, the square markers correspond to spot-splitting thresholds, and the solid circles correspond to equilibrium ring radii. We now show how Fig. 3.13 can readily be used to predict spot-splitting 99 3.7. Comparison of Theory with Numerical Experiments behavior. Experiment 3.7: A one-ring pattern with a center-spot for D ≫ O(1): Dif- ferent types of spot-splitting We fix A = 38 and D = 3.912 and consider an initial pattern of one center-spot and three equally spaced spots on a ring of initial radius r = 0.8. We can then predict the dynamical behavior of this pattern by appealing to the heavy solid curves in Fig. 3.13(a) and Fig. 3.13(b). From the data used to plot Fig. 3.13(a) and Fig. 3.13(b), we obtain that Sk = 5.38 > Σ2 and Sc < Σ2 at t = 0. Therefore, we predict that only the center-spot will undergo spot-splitting, while the other three spots remain on a ring whose radius decreases slowly in time. For this parameter set, we show the full numerical results computed from (1.7) in Fig. 3.14 . These numerical results confirm the prediction based on the asymptotic theory. -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 123 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 12 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 1234 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 1234 Figure 3.14: Experiment 3.7: Fix A = 38,D = 3.912. The numerical results computed from (1.7) for the initial 4-spot pattern are shown for initial ring radius r = 0.8. The plots (from left to right) are at times t = 4.6, 46, 102, 298. Alternatively, suppose that the initial ring radius is decreased from r = 0.8 to r = 0.5. Then, from the heavy solid curves in the top row of Fig. 3.13(a) and Fig. 3.13(b), we now calculate that Sc = 4.92 > Σ2 and Sk < Σ2 at t = 0. Thus, we predict that the three spots on the ring undergo simultaneous spot-splitting starting at t = 0, while the center-spot does not split. The full numerical results computed from (1.7), as shown in Fig. 3.15, fully support this asymptotic prediction. Experiment 3.8: A one-ring pattern with a center-spot for D ≫ O(1): A triggered dynamical instability In Fig. 3.13(c) and Fig. 3.13(d), we fix k = 10 and D = 3.912 and plot Sc and Sk versus r for A = 82 (heavy solid curve), and for A = 60 (solid curve). In both figures, solid circles and square markers denote the equilibrium states 100 3.8. Discussion -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 1234 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -5 0 123 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -5 0 123 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -5 0 123 Figure 3.15: Experiment 3.7: Fix A = 38,D = 3.912. The numerical results computed from (1.7) for the initial 4-spot pattern are shown for initial ring radius r = 0.5. The plots (from left to right) are at times t = 4.6, 56, 140, 298. and the spot-splitting thresholds, respectively. On the solid curve for A = 60, we have the equilibria at re1 = 0.71 with corresponding source strengths (Sc1, Sk1) = (2.86, 3.41), and re2 = 0.68 with (Sc2, Sk2) = (3.17, 0.55). In this case, only the center-spot could split, and the minimum ring radius that leads to spot-splitting is r = 0.80. Alternatively, on the heavy solid curve for A = 82, we have the equilibria at re1 = 0.71 with corresponding source strengths (Sc1, Sk1) = (3.92, 4.81), and re2 = 0.68 with (Sc2, Sk2) = (4.40, 4.32). In this case, the splitting criteria are (r, Sc) = (0.52, 4.31) and (r, Sk) = (0.68, 4.31). Consider an initial ring radius of r = 0.62. Then, from the heavy solid curve in Fig. 3.13(c) and Fig. 3.13(d), we calculate the initial source strengths as Sc = 4.03 and Sk = 3.38 at t = 0. Therefore, we predict that spot-splitting is not initiated at t = 0. However, as the ring expands slowly to approach its equilibrium state at re1 = 0.71, the spot-splitting threshold is achieved at the ring radius r = 0.68 when Sk begins to exceed Σ2, Thus, the asymptotic theory predicts that a dynamical spot-splitting instability will be triggered at this point. The predicted dynamical instability phenomenon is confirmed by the full numerical results shown in Fig. 3.16 computed from the GS model (1.7). From this figure, we observe that this pattern is stable to spot-splitting for t < 60, but that the center-spot eventually undergoes a dynamical spot-splitting event. 3.8 Discussion In this chapter we have characterized the dynamics of a collection of spots for the GS model (1.7) and have provided an explicit criterion, based on a linearized stability analysis, for the initiation of a spot-splitting instabil- 101 3.8. Discussion -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 12 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 123 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 123 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 12 Figure 3.16: Experiment 3.8: Fix A = 82,D = 3.912. The full numerical results computed from (1.7) for an initial pattern with nine spots equally spaced on a ring of initial radius r = 0.62 together with a center-spot. The plots are shown at times t = 60, 93, 144, 214. ity. The analysis used in this chapter has been a combination of a formal asymptotic analysis to construct quasi-equilibrium spot patterns and to de- termine their dynamics, together with a numerical study of a linearized eigenvalue problem associated with a spot self-replication instability. In Principal Result 3.1, a quasi-equilibrium k-spot pattern was constructed by matched asymptotic analysis. The collective coordinates characterizing this pattern are the spatial configuration (x1, . . . ,xk) of spots and their corre- sponding source strengths (S1, . . . , Sk), which measure the strengths of the logarithmic growth of the local, or core, solution for u near each spot. The slow motion, with speed O(ε2), of the spots are characterized by the DAE system (3.35), where the source strengths are determined in terms of the instantaneous spatial configuration of spots by the nonlinear algebraic sys- tem (3.8). This system incorporates the interaction between the spots and the geometry of the domain mediated by the reduced-wave Green’s function and its regular part, and also accounts for all logarithmic correction terms in ν = −1/ ln ε. The asymptotic results for slow spot motion and the predic- tion of the initiation of spot self-replication have been successfully validated through some detailed comparisons with full numerical results computed from the GS model (1.7) for the unit square and the unit disk. There are a few problems related to spot dynamics and spot-replication behavior for the GS model that should be studied. The first problem is to show that a linear instability of a spot to peanut-splitting does in fact lead to a spot self-replication event. This problem could perhaps be analyzed by performing a weakly nonlinear analysis of a localized spot to show that the initial peanut-splitting instability does not saturate in the nonlinear regime to a large, but steady-state, deformation of a single spot. The second prob- lem is to use the equilibrium problem in Principal Result 3.4 to calculate 102 3.8. Discussion detailed bifurcation diagrams in terms of A and D of steady-state k-spot patterns in arbitrary domains. The third open problem is to study the stabil- ity of the equilibrium points of the ODE’s in Principal Result 3.3 describing slow spot motion. Any unstable eigenvalue of this system is small of or- der O(ε2) and, consequently, represents a weak, or translational instability. Based on the study of [36] of discrete interacting particles, this instability should be manifested for the unit disk when the number of equally spaced spots on a concentric ring exceeds some critical value. 103 Chapter 4 Competition and Oscillatory Profile Instabilities for the Two-dimensional Gray-Scott Model In §3.2 of Chapter 3 we formulated the eigenvalue problem (3.25) to deter- mine the stability of the core solution near each spot to localized pertur- bations with angular mode m ≥ 2, which are associated with spot-splitting instabilities. In this chapter, we focus on the stability of the spot profile to the radially symmetric m = 0 mode, which is associated with instabilities in the amplitude of a spot. As shown below, there are two types of such instabilities; oscillatory instabilities of the spot amplitude, and competition, or overcrowding, instabilities that lead to the annihilation of a spot. 4.1 Eigenvalue Problem for the Mode m = 0 To determine the stability of the quasi-equilibrium solution ue and ve in (3.9), we introduce the perturbation u(x, t) = ue + e λ tη(x) , v(x, t) = ve + e λ tφ(x) , in (1.7). This leads to the eigenvalue problem ε2∆φ− (1 + λ)φ+ 2Aueveφ+Av2eη = 0 , x ∈ Ω , (4.1a) D∆η − (1 + τλ)η − 2ueveφ− v2eη = 0 , x ∈ Ω , (4.1b) ∂nφ = ∂nη = 0 , x ∈ ∂Ω . (4.1c) In the inner region near the jth spot centered at xj, we introduce ρ, Nj(ρ) and Φj(ρ) in (4.1) defined by ρ = ε −1|x − xj |, η = εA√DNj(ρ), and φ = 104 4.1. Eigenvalue Problem for the Mode m = 0 √ D ε Φj(ρ). Then, when τ satisfies τλ≪ O(ε−2), we obtain that (4.1) trans- forms to the following radially symmetric eigenvalue problem on 0 < ρ <∞: Φ′′j + 1 ρ Φ′j − Φj + 2UjVjΦj + V 2j Nj = λΦj , (4.2a) N ′′j + 1 ρ N ′j − V 2j Nj − 2UjVjΦj = 0 , (4.2b) where Uj and Vj is the solution of the radially symmetric core problem (3.2) for the jth spot. The boundary conditions for this system are Φ′j(0) = N ′ j(0) = 0 ; Φj(ρ)→ 0 , Nj(ρ)→ Cj ln ρ+Bj as ρ→∞. (4.2c) We emphasize that as ρ→∞ we must, in general, allow Nj to grow log- arithmically. This behavior is in contrast to that imposed for the eigenvalue problem (3.25) with mode m ≥ 2 studied in Section 3.2, where we required that Nj → 0 as ρ → ∞. By applying the divergence theorem to (4.2b), we can identify Cj as Cj ≡ ∫ ∞ 0 (V 2j Nj + 2UjVjΦj)ρ dρ. To formulate our eigenvalue problem we must match the far-field loga- rithmic growth of Nj with a global outer solution for η. This matching has the effect of globally coupling the local eigenvalue problems near each spot. Next, we determine the outer problem for η that couples the local prob- lems near each spot. Since ve is localized near xj for j = 1, . . . , k, we can use ue ∼ εA√DUj , and ve ∼ √ D ε Vj to represent the last two terms in (4.1b) in the sense of distributions as 2ueveφ+ v 2 eη ∼ 2πε √ D A k∑ j=1 (∫ ∞ 0 (2UjVjΦj + V 2 j Nj) ρ dρ ) δ(x − xj) . Here δ(x − xj) is the Dirac delta function concentrated at x = xj . Upon identifying the integral in the expression above as Cj, we then obtain that (4.1b) in the outer region becomes ∆η − (1 + τλ) D η = 2πε A √ D k∑ j=1 Cjδ(x− xj) , x ∈ Ω , (4.3) 105 4.1. Eigenvalue Problem for the Mode m = 0 with ∂nη = 0 for x ∈ ∂Ω. This outer solution can be represented in terms of a λ-dependent Green’s function as η = − 2πε A √ D k∑ j=1 CjGλ(x;xj) . (4.4) Here this Green’s function Gλ(x;xj) satisfies ∆Gλ− (1 + τλ) D Gλ = −δ(x−xj) , x ∈ Ω ; ∂nGλ = 0 , x ∈ ∂Ω , (4.5a) with the following singularity behavior as x→ xj: Gλ(x;xj) ∼ − 1 2π ln |x− xj |+Rλ jj + o(1) , as x→ xj . (4.5b) We remark that the regular part Rλjj depends on xj , λ, τ , and D. The matching condition between the outer solution (4.4) for η as x→ xj and the far-field behavior (4.2c) as ρ→∞ of the inner solution Nj near the jth spot yields that − 2πε A √ D [ Cj ( − 1 2π ln |x− xj |+Rλ jj )] − 2πε A √ D k∑ i6=j CiGλ ij ∼ ε A √ D [ Cj ln |x− xj|+ Cj ν +Bj ] , (4.6) where Gλ ij ≡ Gλ(xi;xj) and ν = −1/ ln ε. This matching condition yields the k equations Cj (1 + 2πνRλ jj) + νBj + 2πν k∑ i6=j CiGλ ij = 0 , j = 1, . . . , k . (4.7) We remark that the constants τ and D appear in the operator of the λ- dependent Green’s function defined by (4.5). Suppose that τ = O(1) and D = O(1). Then, since ν ≪ 1, we obtain from (4.7) that to leading-order Cj = 0 for j = 1, · · · , k. This implies that the eigenfunction Nj(ρ) in (4.2) is bounded as ρ→∞, and so we can impose thatN ′j(ρ)→ 0 as ρ→∞. Therefore, when τ = O(1) andD = O(1), then to leading order in ν the eigenvalue problems (4.2) for j = 1, . . . , k are coupled together only through the determination of the source strengths S1, . . . , Sk from the nonlinear algebraic system (3.8) of Chapter 3. 106 4.1. Eigenvalue Problem for the Mode m = 0 Therefore, to leading order in ν, we must compute the real part of the principal eigenvalue λ0 of (4.2) as a function of Sj , subject to the condition that N ′j(ρ) → 0 as ρ → ∞. This computation is done by discretizing (4.2) by finite differences and then calculating the spectrum of the resulting matrix eigenvalue problem using LAPACK (cf. [1]). The plot of Re(λ0) versus Sj is shown in Fig. 4.1 for the range Sj < 7.5, which includes the value Sj = Σ2 ≈ 4.31 corresponding to the peanut-splitting instability. We conclude that Re(λ0) < 0 for Sj < 7.5. In Fig. 4.2(a) and Fig. 4.2(b) we plot the corresponding eigenfunctions Φj(ρ) and Nj(ρ), respectively, for Sj = 3.0 (solid curve) and Sj = 6.0 (dashed curve). -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1  0  0  1  2  3  4  5  6  7  8 Re  λ S Figure 4.1: The real part of the largest eigenvalue Re(λ0) of (4.2) vs. Sj subject to the condition that Nj is bounded as ρ→∞. However, since ν = −1/ ln ε decreases only very slowly as ε decreases, the leading-order approximation Cj = 0 for j = 1, . . . , k to (4.7) is not very accurate. Consequently, we must examine the effect of the coupling in (4.7). Since (4.2) is a linear homogeneous problem, we introduce B̂j by Bj = B̂jCj , and we define the new variables Φ̂j and N̂j by Φj = CjΦ̂j and Nj = CjN̂j . Then, the eigenvalue problem (4.2) on 0 < ρ <∞ becomes Φ̂′′j + 1 ρ Φ̂′j − Φ̂j + 2UjVjΦ̂j + V 2j N̂j = λΦ̂j , (4.8a) N̂ ′′j + 1 ρ N̂ ′j − V 2j N̂j − 2UjVjΦ̂j = 0 , (4.8b) with boundary conditions Φ̂′j(0) = N̂ ′ j(0) = 0 ; Φ̂j(ρ)→ 0 , N̂j(ρ)→ ln ρ+ B̂j , as ρ→∞ . (4.8c) 107 4.1. Eigenvalue Problem for the Mode m = 0 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02  0  0.02  0.04  0  2  4  6  8  10  12  14  16 Φ ρ (a) Φj vs. ρ -0.3 -0.2 -0.1  0  0.1  0.2  0.3  0  2  4  6  8  10  12  14  16 N ρ (b) Nj vs. ρ Figure 4.2: Eigenfunctions Φj and Nj of (4.2) with Nj bounded at infinity. Left figure: Φj vs. ρ for Sj = 3.0 (solid curve) and Sj = 6.0 (dashed curve). Right figure: Nj vs. ρ for Sj = 3.0 (solid curve) and Sj = 6.0 (dashed curve). The constant B̂j in (4.8c) is a function of the, as yet unknown, complex- valued eigenvalue parameter λ. It also depends on the source strength Sj through the core solution Uj and Vj , which satisfies (3.2). In terms of B̂j, the problem (4.7) transforms to the homogeneous linear system for C1, . . . , Ck given by Cj(1 + 2πνRλ jj + νB̂j) + 2πν k∑ i6=j CiGλ ij = 0 , j = 1, . . . , k . (4.9) It is convenient to express (4.9) in matrix form as M c = 0 M≡ I + νB + 2πνGλ , (4.10a) where I is the k × k identity matrix and c ≡ (C1, . . . , Ck)t. In (4.10a), B is a diagonal matrix and Gλ is the λ-dependent Green’s matrix defined by Gλ ≡   Rλ 11 Gλ 12 · · · Gλ 1k Gλ 21 Rλ 22 · · · Gλ 2k ... ... ... ... Gλ k1 Gλ k2 · · · Gλ kk   , B ≡   B̂1 0 · · · 0 0 B̂2, · · · 0 ... ... ... ... 0 0 · · · B̂k   . (4.10b) Since Gλ is a symmetric matrix, then so is the matrix M. 108 4.1. Eigenvalue Problem for the Mode m = 0 The condition that determines the eigenvalue λ is that there is a nontriv- ial solution c 6= 0 to (4.10a). Hence, λ is determined by the condition that M is a singular matrix, i.e. det(M) = 0. This condition leads, effectively, to a transcendental equation for λ. The roots of this equation determine the discrete eigenvalues governing the stability of the k-spot quasi-equilibria to locally radially symmetric perturbations near each spot. We summarize our formulation of the eigenvalue problem as follows: Principal Result 4.1: Consider a k−spot quasi-equilibrium solution to the two-dimensional Gray-Scott model (1.7). For ε → 0, with D = O(1) and τ = O(1), the stability of this pattern to locally radially symmetric per- turbations near each spot is determined by the condition det(M) = 0, where M is defined in (4.10). The diagonal matrix B in (4.10) is determined in terms of Sj and λ by the local problems (4.8) for j = 1, . . . , k. If the principal eigenvalue λ0 of this problem is such that Re(λ0) < 0, then the k−spot quasi- equilibrium solution is stable to locally radially symmetric perturbations near each spot. It is unstable if Re(λ0) > 0. We remark that (4.10) couples the local spot solutions in two different ways. First, the λ-dependent terms Rλ jj and Gλ ij in the Green’s matrix Gλ in (4.10) depend on τ , D, and λ, as well as the spatial configuration (x1, . . . ,xk) of the spot locations.and the geometry of Ω. Secondly, the con- stant B̂j in the matrix B depends on the eigenvalue λ and on Sj. Recall that the source strengths S1, . . . , Sk are coupled through the nonlinear algebraic system (3.8) of Chapter 3, which involves A, D, the reduced-wave Green’s function, and the spatial configuration (x1, . . . ,xk) of spot locations. From our study of this eigenvalue problem below, there are two mech- anisms through which a k-spot quasi-equilibrium pattern can lose stability. Firstly, for k ≥ 1 there can be a complex conjugate pair of eigenvalues λ0 that crosses into the unstable half-plane Re(λ0) > 0. This instability as a result of a Hopf bifurcation results in an oscillatory profile instability, whereby the spot amplitudes undergo temporal oscillations. Such an insta- bility typically occurs if τ is sufficiently large. Alternatively, for k ≥ 2, the principal eigenvalue λ0 can be real and enter the unstable right half-plane Re(λ0) > 0 along the real axis Im(λ0) = 0. This instability, due to the cre- ation of a positive real eigenvalue, gives rise to an unstable sign-fluctuating perturbation of the spot amplitudes and initiates a competition instability, leading to spot annihilation events. This instability can be triggered if D is sufficiently large or, equivalently, if the inter-spot separation is too small. The eigenvalue problem is rather challenging to investigate in full gen- 109 4.1. Eigenvalue Problem for the Mode m = 0 erality owing to the complexity of the two different coupling mechanisms in (4.10). However, for the special case when the spot configuration (x1, . . . ,xk) is such that G is a circulant matrix, then the complexity of our eigenvalue problem reduces considerably. For this special arrangement of spot loca- tions, the spots have a common source strength Sc = Sj for j = 1, . . . , k, as was discussed in Section 3.1. Hence, the inner problem (4.8) is the same for each spot. This enforces that B̂j ≡ B̂c for j = 1, . . . , k. where B̂c = B̂c(λ, S) is a function of Sc and λ. Therefore, we can write B = B̂cI in (4.10). More- over, let v be an eigenvector of the λ-dependent Green’s matrix Gλ with eigenvalue ωλ, i.e. Gλv = ωλv. This matrix is also circulant when G is circu- lant. Then, the condition that M in (4.10) is a singular matrix reduces to k transcendental equations in λ. We summarize the result as follows: Principal Result 4.2: Under the conditions of Principal Result 4.1, sup- pose that the spot locations (x1, . . . ,xk) are such that G, and consequently Gλ, are circulant matrices. Then, the eigenvalue condition det(M) = 0 for (4.10) reduces to the k nonlinear algebraic equations that 1 + νB̂c + 2πνωλ = 0 , (4.11) where ωλ is any eigenvalue of the matrix Gλ. The eigenvectors of Gλ deter- mine the choices for c. Owing to the considerable reduction in complexity of the eigenvalue problem when G and Gλ are circulant matrices, we will study in detail the stability of such symmetric quasi-equilibrium spot patterns in Section §4.5. 4.1.1 Numerical Methods for the Eigenvalue Problem The eigenvalue problem, consisting of (4.8) coupled with (4.10a), can only be solved numerically. One numerical approach would be to calculate the eigenvalue λ directly for a given set of parameters A, D, τ , and ε and for a given configuration of spots. The outline of a fixed-point iterative type method is as follows: 1. Given fixed parameter values τ, ε,A,D, A = νε−1A√D, we first calcu- late Sj, j = 1, · · · , k from the algebraic system (3.8), and then compute the core solution Uj , Vj from (3.2). 2. For the nth iteration, starting from a known eigenvalue λn, we could solve (4.8) for B̂j for j = 1, . . . , k. 110 4.1. Eigenvalue Problem for the Mode m = 0 3. Next, with B̂j known for j = 1, . . . , k, we could update the value of λ by requiring that the λ-dependent Green’s function and its regular part be such that the matrix M in (4.10a) is a singular matrix. This step requires an additional inner iteration. The new approximation to the eigenvalue is denoted by λn+1. 4. We repeat step 2 using the updated λn+1 until it converges to λ, with an error that is less than a given tolerance Tol = 10−6. Then λ is the eigenvalue of the coupled problem. However, this iteration approach for λ converges very slowly, and very much relies on the initial guess. In addition, we need to study the stability for a range of values of the parameter τ , in order to determine the Hopf bi- furcation threshold. Therefore, the computational cost of this simple scheme will be very large. Instead of the former approach, we used an alternative numerical method that is based on the assumption that an instability occurs at some critical values of the parameters. For instance, in order to compute the Hopf bifur- cation threshold τH , we assume that there is a complex conjugate pair of pure imaginary eigenvalues occurring at τ = τH for which λr ≡ Re(λ) = 0 and λi ≡ Im(λ) 6= 0. Instead of solving for λ for each given τ , we fix λr = 0, and solve for the two variables τH and λi by Newton’s method. A rough outline of our algorithm is as follows: 1. Given fixed parameter values ε,A,D, we first calculate the source strength Sj from (3.8) and then compute the corresponding core solu- tion Uj , Vj from (3.2). 2. Fix λr = 0. Starting from the initial guess τ, λi, we calculate Gλ ij , Rλ jj and their partial derivatives with respect to τ, λi numerically. We also calculate B̂j and its derivative ∂λB̂j from (4.8). 3. In (4.10a), we use Newton’s method on the condition that det(M) = 0 in (4.10a), to update τ, λi. We then go to step 2 and iterate further until reaching the specified tolerance. Although this method is theoretically appealing, the determination of the constants τ, λi such that det(M) = 0, with M defined in (4.10a), is rather challenging. As such we will mainly focus on studying the instabilities of a one-spot solution and the k−spot symmetric patterns with k ≥ 2, for which the Green’s matrices are circulant. For this special case, Principal Result 111 4.2. Nonlocal Eigenvalue Analysis 4.2 applies. The Hopf bifurcation threshold τH for each of the different spot patterns considered in §4.3, §4.4, and §4.5, below were calculated by the algorithm above. Moreover, to calculate the competition instability threshold, we fix the parameters ε,D and set λ = 0, since the principal eigenvalue enters the unstable right half-plane Re(λ) > 0 along the real axis. At some critical parameter value of D in terms of A, there could be a zero eigenvalue. The numerical computation of this critical value of D is done from the following rough algorithm: 1. Given fixed parameters ε,A, and an initial guess for D, we first calcu- late Sj from (3.8) and then Uj, Vj from (3.2). 2. Fix λ = 0. Starting from the initial guess for D, we calculate Gλ ij , Rλ jj and their partial derivatives with respect to D numerically. We also calculate B̂j and its derivative ∂DB̂j from (4.8). 3. In (4.10a), we use Newton’s method to update D by requiring that det(M) = 0. We then iterate until reaching the tolerance. These numerical algorithms enable us to calculate the threshold param- eter values for oscillatory profile instabilities and competition instabilities. Below in §4.3, §4.4, and §4.5 we demonstrate our theory by numerically com- puting these instability thresholds for various spot configurations in both finite and infinite domains. 4.2 Nonlocal Eigenvalue Analysis In the previous section, we formulated a coupled eigenvalue problem associ- ated with competition and oscillatory profile instabilities. However, in [99] and [101] a different analytical approach based on an analysis of a nonlocal eigenvalue problem (NLEP) was used to obtain a leading-order asymptotic theory for the existence and stability of spot patterns to the GS model (1.7), with an error of order O(−1/ ln ε). In this section we summarize this theory and compare it with our quasi-steady state analysis in §3.1 and with our eigenvalue formulation in §4.1. 4.2.1 A One-spot Solution in an Infinite domain In [99], the existence and stability of a one-spot pattern in the infinite plane R 2 was studied. In the infinite plane, we can eliminate the parameter D by 112 4.2. Nonlocal Eigenvalue Analysis introducing x̃ and ε̃ by x̃ = x√ D , ε̃ = ε√ D . (4.12) Then, in terms of x̃ ∈ R2, (1.7) becomes vt = ε̃ 2∆v − v +Auv2 , τut = ∆u+ (1− u)− uv2 . (4.13) In the inner region near the spot centered at x0, a different scaling was used in [99] to construct the local solution near each spot. Instead of a coupled BVP system for the core problem as in (3.2), the leading-order scaling in [99] showed that the slow variable u satisfies u ∼ u0, where u0 is locally constant near each spot. Moreover, from [99], the fast variable v was found to satisfy v ∼ v0 = 1Au0w(ρ), where w(ρ) is the radially symmetric ground- state solution on 0 < ρ <∞ of the scalar equation ∆ρw − w + w2 = 0 ; w(0) > 0 , w′(0) = 0 ; w→ 0 , as ρ→∞ . (4.14) The analysis in [99] showed, asymptotically for ε → 0, that u0 satisfies the quadratic equation 1− u0 ∼ L u0 , L ≡ ε̃ 2 ν̃ A2 ∫ ∞ 0 w2(ρ)ρ dρ , ν̃ = −1/ ln ε̃ . (4.15) Therefore, the existence threshold for a one-spot equilibrium solution is that L ≤ 1 4 ⇒ A ≥ Af ≡ 2ε̃ √ b0 ν̃ , b0 ≡ ∫ ∞ 0 w2(ρ)ρ dρ . (4.16) Define L0 and γ by L0 = limε̃→0 L = O(1) and γ ≡ ν̃ ln τ = O(1), so that τ = O(ε̃−γ). Then, the following nonlocal eigenvalue problem (NLEP) was derived in [99]: ∆yψ0−ψ0+2wψ0 − 2(1− u0)(2− γ) 2u0 + (1− u0)(2− γ) w 2 ∫∞ 0 wψ0 ρ dρ∫∞ 0 w 2 ρ dρ = λψ0 , (4.17) with ψ0 → 0 as ρ → ∞. An analysis of this NLEP in [99] led to Theorem 2.2 of [99], which we summarize as follows: 1. There exists a saddle node bifurcation at L0 = 1 4 , such that there are two equilibrium solutions u±0 given by u±0 = 1 2 ( 1± √ 1− 4L0 ) , for L0 < 1 4 , and no equilibrium solutions when L0 > 1 4 . 113 4.2. Nonlocal Eigenvalue Analysis 2. Assume that 0 ≤ γ < 2 and L0 < 14 . Then, the solution branch (u+0 , v + 0 ) is linearly unstable. 3. Assume that 0 ≤ γ < 2. Then, the other solution branch (u−0 , v−0 ) is linearly unstable if L0 > 1 4 [ 1− ( γ 4 + γ )2] , ⇒ A < As ≡ Af [ 1− ( − γ 4 + γ )2]−1/2 . (4.18) 4. Assume that γ = 0 and L0 < 1 4 . Then, (u − 0 , v − 0 ) is linearly stable. 5. If 0 ≤ γ < 2, the stability of (u−0 , v−0 ) is unknown for A > As. We remark that the saddle-node bifurcation value of A, obtained from (4.15), has the scaling A = O ( ε̃[− ln ε̃]1/2) as ε̃ → 0. The stability results listed above from [99] also relate only to this range in A. We remark that the quasi-equilibrium spot solution constructed in §3.1 and the spot self- replication threshold of §3.2 occurs for the slightly larger value of A, with scaling A = O (ε̃[− ln ε̃]) (see (3.4) of §3.1). Consequently, the occurrence of spot self-replication behavior for the GS model (1.7) was not observed in the scaling regime for A considered in [99]. 4.2.2 Multi-Spot Patterns in a Finite Domain In [101] a related analytical approach based on the analysis of other NLEP’s was used to obtain a leading-order asymptotic theory for the existence and stability of k-spot patterns to the GS model (1.7) in a bounded two- dimensional domain. In the inner region near the jth spot centered at xj , the leading-order scaling in [101] showed that the slow variable u sat- isfies u ∼ uj , where uj is a constant, while the the fast variable v satisfies v ∼ vj ≡ 1Aujw(ρ), where w(ρ) is the radially symmetric ground-state solu- tion of (4.14). For a k-spot pattern, it was shown in [101] that the uj for j = 1, . . . , k satisfy the nonlinear algebraic system 1− uj − ηεLε uj ∼ k∑ i=1 Lε ui , (4.19) where Lε and ηε are defined by Lε = ε22πb0 A2|Ω| , ηε = |Ω| 2πDν , b0 = ∫ ∞ 0 w2(ρ)ρ dρ , ν = − 1 ln ε . (4.20) 114 4.2. Nonlocal Eigenvalue Analysis Here |Ω| is the area of domain. Assuming that uj = u0 has a common value for all of the spots, then (4.19) is a quadratic equation. In this way, the following existence criteria for k-spot quasi-equilibria were obtained in [101] for different ranges of the parameter ηε:  4kL0 < 1, for ηε → 0 ⇔ D ≫ O(ν−1), 4ηεLε < 1, for ηε →∞ ⇔ D = O(1), 4(k + η0)L0 < 1, for ηε → η0 ⇔ D = O(ν−1) . (4.21) Here L0 = limε→0 Lε = O(1) and η0 = limε→0 ηε = O(1). From (4.20), it follows that the second line in (4.21) is for the parameter range A = O ( ε[− ln ε]1/2) withD = O(1), while the third line in (4.21) is for the range A = O (ε) with D = O(ν−1) and ν = −1/ ln ε. We emphasize that the asymptotic construction of k-spot quasi-equilibria in §3.1 is for the range D = O(1) and A = O (ε[− ln ε]) (see (3.4) and (3.12)). In addition, equation (3.19) derived in §3.1 pertains to the range A = O (ε[− ln ε]) with D = O(ν−1). Consequently, the k-spot quasi-equilibrium patterns constructed in [101] do not correspond to the range A = O (ε[− ln ε]) where spot self- replication occurs, as considered in our analysis. As shown in [101], the leading order quasi-equilibrium solution uj ∼ u0 in the inner region, together with the global representation for the v, have the form u±0 ∼   1 2 ( 1±√1− 4kL0 ) , for D ≫ O(ν−1) , 1 2 ( 1±√1− 4ηεLε ) for D = O(1) , 1 2 ( 1±√1− 4(k + η0)L0) , for D = O(ν−1) , (4.22a) v±0 ∼ k∑ j=1 1 Au±0 w ( ε−1|x− xj | ) . (4.22b) The stability analysis in [101] was largely based on the derivation and anal- ysis of the nonlocal eigenvalue problem ∆yψ − ψ + 2wψ − γ w2 ∫∞ 0 wψρdρ∫∞ 0 w 2ρ dρ = λψ , 0 < ρ <∞ , (4.23) with ψ → 0 as ρ→∞. In (4.23) there are two choices for γ, given by γ = 2ηεLε(1 + τλ) + kLε (u20 + Lεηε)(1 + τλ) + kLε , or γ = 2ηεLε u20 + Lεηε . (4.24) 115 4.2. Nonlocal Eigenvalue Analysis In (4.24) it was assumed in [101] that τ = O(1). Theorem 2.3 of [101] gives a stability result for k-spot equilibrium solu- tion based on an analysis of the NLEP (4.23) together with certain properties of the multi-variable function F(x1, . . . ,xk) defined by F(x1, . . . ,xk) = − k∑ i=1 k∑ j=1 Gij . (4.25) Here G is the Green’s matrix of (3.11) defined in terms of the reduced-wave Green’s function. It was shown in [101] that the equilibrium spot locations for an equilibrium k-spot pattern must be at a critical point of F(x1, . . . ,xk). We denote by H0 the Hessian of F at this critical point. Suppose that 4(ηε+k)Lε < 1/4. Then, Theorem 2.3 of [101] proves that the large solutions (u+, v+) are all linearly unstable. For the small solutions (u−, v−), the following rigorous results of [101] hold for different ranges of ηε: 1. Assume that ηε → 0, so that D ≫ O(ν−1). Then, • If k = 1, and all the eigenvalues of the Hessian H0 are negative, then there exists a unique τ1 > 0, such that for τ < τ1, (u −, v−) is linearly stable, while for τ > τ1, it is linearly unstable. • If k > 1, (u−, v−) is linearly unstable for any τ ≥ 0. • If the Hessian H0 has a strictly positive eigenvalue, (u−, v−) is linearly unstable for any τ ≥ 0. 2. Assume that ηε →∞, so that D ≪ O(ν−1). Then, • If all the eigenvalues of the Hessian H0 are negative, then (u−, v−) is linearly stable for any τ > 0. • If the Hessian H0 has a strictly positive eigenvalue, (u−, v−) is linearly unstable for any τ ≥ 0. 3. Assume that ηε → η0, so that D = O(ν−1). Then, • If L0 < η0(2η0+k)2 , and all eigenvalues of H0 are negative, then (u−, v−) is linearly stable for τ = O(1) sufficiently small or τ = O(1) sufficiently large. • If k = 1, L0 > η0(2η0+k)2 , and all the eigenvalues ofH0 are negative, then there exists τ2 > 0, τ3 > 0, such that for τ < τ1, (u −, v−) is linearly stable, while for τ > τ3, it is linearly unstable. 116 4.2. Nonlocal Eigenvalue Analysis • If k > 1 and L0 > η0(2η0+k)2 , then (u−, v−) is linearly unstable for any τ > 0. • If the Hessian H0 has a strictly positive eigenvalue, (u−, v−) is linearly unstable for any τ ≥ 0. We close this section with several remarks relating our analysis with that in [101]. Firstly, the condition of [101] that an equilibrium k-spot configu- ration (x1, . . . ,xk) must be at a critical point of F in (4.25) is equivalent to the equilibrium result in Principal Result 3.4 of §3.3 provided that we replace the source strengths Sj in (3.36) by their leading-order asymptoti- cally common value Sc ∼ A, as given in (3.13). The stability condition of [101] expressed in terms of the sign of the eigenvalues of the Hessian H0 of F is equivalent to the statement that the equilibrium spot configuration (x1, . . . ,xk) is stable with respect to the slow motion ODE dynamics (3.35) in Principal Result 3.3 when we use the leading-order approximation Sj ∼ Sc for j = 1, . . . , k in (3.35). Therefore, the condition on the Hessian H0 in [101] relates to the small eigenvalues of order O(ε2) that are reflected in the stability of equilibrium points under the ODE dynamics (3.35). Next, we relate our asymptotic parameter ranges considered for A and D with those considered in [101]. The second result of Theorem 2.3 of [101] for ηε →∞, as written above, includes the range D = O(1). The condition 4(ηε + k)Lε < 1/4, together with (4.20) relating Lε to A, shows this second result of [101] holds for the range A = O ( ε [− ln ε]1/2 ) . Therefore, for this range of A and D, the analysis in [101] predicts that there is an equilib- rium solution branch whose stability depends only on the eigenvalues of the Hessian matrix H0. Likewise, it is readily seen that the third statement in Theorem 2.3 of [101], as written above, pertains to the range D = O(ν−1) with A = O(ε). Finally, we remark that, with the exception of the condi- tion on the Hessian matrix of F , the stability criteria developed in [101] can depend on the number k of spots, but does not depend on the configuration (x1, . . . ,xk) of spot locations within the domain. In our theoretical framework for the stability of k-spot quasi-equilibria solutions, as expressed in Principal Results 4.1 and 4.2, we considered the parameter range A = O (ε [− ln ε]) and D = O(1). To leading-order in ν, where the k spots can be decoupled, we showed from a numerical com- putation of the two-component eigenvalue problem (4.2) with N ′ j → 0 as ρ → ∞, that a k-spot quasi-equilibrium solution is always stable to O(1) time-scale instabilities. This leading-order result is similar to that proved in the second statement of Theorem 2.3 of [101] for the nearby parameter 117 4.2. Nonlocal Eigenvalue Analysis range A = O ( ε [− ln ε]1/2 ) and D = O(1). Our goal in the later sections §4.3, §4.4, and §4.5 is to go beyond this simple leading-order theory in ν and to show, specifically, that the spatial configuration (x1, . . . ,xk) does have a rather significant effect on the stability of quasi-equilibrium k-spot patterns. In addition, in our stability formulation, we will are able to consider a single parameter range for A and D, i.e. A = O (ε [− ln ε]) and D = O(1), where three separate instability mechanisms can occur: spot self-replication, oscil- latory instabilities of the spot amplitudes, and competition instabilities of the spot amplitudes. 4.2.3 Comparison of the Quasi-Equilibrium Solutions In this sub-section we show that our asymptotic construction of quasi- equilibrium k-spot solutions, as given in Principal Result 3.1 of §3.1, can be reduced to that given in [101] when A = O(ν1/2) in (3.8). Note that when A = O(ν1/2), then A = O ( ε [− ln ε]1/2 ) from (3.4), which is the key parameter regime of [101]. For A = O(ν1/2), the nonlinear algebraic system (3.8) together with the core problem (3.2), suggest that we expand Uj , Vj, Sj, and χ, in (3.2) as χ ∼ ν−1/2(χ0j + νχ1j + · · · ) , Uj ∼ ν−1/2(U0j + νU1j + · · · ) , Sj ∼ ν1/2(S0j + νS1j + · · · ) , Vj ∼ ν1/2(V0j + νV1j + · · · ) . (4.26) Upon substituting (4.26) into (3.2), and collecting powers of ν, we obtain the leading-order problem ∆U0j = 0 , U0j → χ0j as ρ→∞ , (4.27a) ∆V0j − V0j + U0jV 20j = 0 , V0j → 0 as ρ→∞ . (4.27b) At next order, we obtain that ∆U1j = U0jV 2 0j , U1j → S0j ln ρ+ χ1j as ρ→∞ , (4.28a) ∆V1j − V1j + 2U0jV0jV1j = −U1jV 20j , V1j → 0 as ρ→∞ . (4.28b) At one higher order, we find that U2j satisfies ∆U2j = 2U0jV0jV1j + V 2 0jU1j , U2j → S1j ln ρ+ χ2j as ρ→∞ . (4.29) The solution to (4.27) is simply U0j = χ0j , and V0j = w/χ0j , where w(ρ) is the radially symmetric ground-state solution of the scalar nonlinear 118 4.2. Nonlocal Eigenvalue Analysis problem (4.14). Upon applying the divergence theorem to (4.28a), and using U0j = χ0j and V0j = w/χ0j , we calculate S0j in terms of χ0j as S0j = ∫ ∞ 0 U0jV 2 0j ρ dρ = b0 χ0j , b0 ≡ ∫ ∞ 0 w2 ρ dρ . (4.30) Next, we decompose the solution to (4.28) in the form U1j = 1 χ0j ( χ0jχ1j + Û1j ) , V1j = 1 χ30j ( − χ0jχ1j w + V̂1j ) , where Û1j and V̂1j are the unique solutions of ∆ρÛ1j = w 2 , Û1j − b0 ln ρ→ 0 as ρ→∞ , (4.31a) ∆ρV̂1j − V̂1j + 2wV̂1j = −Û1j w2 , V̂1j → 0 as ρ→∞ . (4.31b) Similarly, by applying the divergence theorem to (4.29), we calculate S1j as S1j = − b0χ1j χ20j + b1 χ30j , b1 ≡ ∫ ∞ 0 (w2Û1j + 2wV̂1j) ρ dρ . (4.32) The BVP solver COLSYS (cf. [2]) is used to numerically compute the ground state solution w(ρ) of (4.14) and the solutions Û1j and V̂1j of (4.31). Then, by using a simple numerical quadrature, we calculate b0 = 4.9347 and b1 = 0.8706.. Finally, upon substituting (4.30) and (4.32) into the nonlinear algebraic system (3.8) of §3.1 we obtain that A = O(ν1/2), where A = ν1/2 ( χ0j + b0 χ0j ) + ν3/2 [ b1 χ30j + ( 1− b0 χ20j ) χ1j + 2π   b0 χ0j Rjj + k∑ i6=j b0 χ0i Gij  ] . (4.33) For a prescribed A with A = A0ν1/2 + A1ν3/2 + · · · , we could then cal- culate χ0j and χ1j from (4.33). Since A = O(ν1/2) corresponds to A = O ( ε [− ln ε]1/2 ) , we conclude that it is in this parameter range that the coupled core problem (3.2) reduces to the scalar ground-state problem for the fast variable v, as was studied in [101]. 119 4.2. Nonlocal Eigenvalue Analysis Finally, we show that (4.33) reduces to the expression (4.19) used in [101]. To see this, we note that A = A √ Dν ε , uj = ε A √ D Uj ≈ ε A √ Dν1/2 χ0j . Moreover, for D ≫ O(1), we use Rjj = D|Ω| +O(1) and Gij = D|Ω| +O(1) for i 6= j. Then, from (4.33), we obtain A √ Dν ε ∼ A √ Dν ε uj+ b0ε A √ Duj + 2πνD |Ω|   b0ε A √ Duj + k∑ i6=j b0ε A √ Dui   . (4.34) Upon using the definitions of Lε and ηε in (4.20), we can write (4.34) as 1− uj = b0ε 2 A2Dν uj + 2π |Ω| b0ε 2 A2 k∑ i=1 1 ui = Lεηε uj + k∑ i=1 Lε ui , (4.35) which is exactly the same as in (4.19). 4.2.4 Comparison of the Eigenvalue Problem Next, we show for A = O(ν1/2) and D = O(ν−1) that our eigenvalue prob- lem, consisting of (4.2) coupled to (4.7), can be reduced to the nonlocal eigenvalue problem (4.23) considered in [101]. To show this, we expand Φj, Nj , Bj , and Cj , in (4.2) as Bj = Bj0 + νBj1 + · · · , Cj = ν(Cj0 + νCj1 + · · · ) , Nj = Nj0 + νNj1 + · · · , Φj = νψj + · · · . Upon substituting these expansions, together with Uj ∼ ν−1/2χ0j and Vj ∼ ν1/2w/χ0j , into (4.2b), we obtain that Nj0 = Bj0, and that Nj1 satisfies ∆ρNj1 = w2 χ20j Bj0 + 2wψj , Nj1 → Cj0 ln ρ+Bj1 as ρ→∞ . (4.36) Upon applying the divergence theorem to (4.36) we obtain that Cj0 = b0 χ20j Bj0 + 2 ∫ ∞ 0 wψjρ dρ , b0 ≡ ∫ ∞ 0 w2ρ dρ . (4.37) Then, we let D = D0/ν ≫ 1 where D0 = O(1), and we assume that χ0j = χ0 so that to leading-order each local spot solution is the same. For 120 4.2. Nonlocal Eigenvalue Analysis D ≫ 1, the λ-dependent Green’s function Gλ(xi;xj) and its regular part, Rλ jj are given by Gλ ij ∼ D|Ω|(1 + τλ) +O(1) , i 6= j ; Rλ jj ∼ D |Ω|(1 + τλ) +O(1) , We substitute this leading-order behavior for Gλ ij and Rλjj, together with Bj ∼ Bj0, Cj ∼ νCj0, and D = D0/ν, into (4.7). In this way, we obtain to leading-order that (4.7) reduces asymptotically to Cj0 + 2πD0 (1 + τλ)|Ω| k∑ i=1 Ci0 +Bj0 = 0 . Upon substituting (4.37) for Cj0 into this equation, we obtain that Bj for j = 1, . . . , k satisfies ( 1 + b0 χ20 ) Bj0 + 2πD0 (1 + τλ)|Ω| b0 χ20 k∑ i=1 Bi0 = −2 ∫ ∞ 0 wψj ρ dρ− 4πD0 (1 + τλ)|Ω| k∑ i=1 ∫ ∞ 0 wψi ρ dρ . (4.38) Next, we define the vectors p0 and Ψ by p=0 (B10, . . . , Bk0) t , Ψ = (∫ ∞ 0 ωψ1 ρ dρ , . . . , ∫ ∞ 0 ωψk ρ dρ )t . This allows us to re-write (4.38) in matrix form as[( 1 + b0 χ20 ) I + 2πD0 (1 + τλ)|Ω| b0 χ20 eet ] p0 = − ( 2I + 4πD0 (1 + τλ)|Ω|ee t ) Ψ , where e = (1, . . . , 1)t. The matrix multiplying p0 is an invertible rank-one perturbation of the identity matrix. Hence, we calculate that p0 = DΨ , (4.39a) where the matrix D is defined by D ≡ − [( 1 + b0 χ20 ) I + 2πD0 (1 + τλ)|Ω| b0 χ20 eet ]−1( 2I + 4πD0 (1 + τλ)|Ω|ee t ) . (4.39b) 121 4.3. Instabilities in an Unbounded Domain We then rewrite (4.39b) in terms of Lε and ηε, as defined in (4.20), to get b0 χ20 = ηεLε u20 , 2πD0 (1 + τλ)|Ω| = 1 ηε 1 1 + τλ . Then, since the eigenvalues of matrix eet are either k or 0, it readily follows that the two distinct eigenvalues r1 and r2 of the matrix D are r1 = −2χ 2 0 b0 Lεηε + kLε 1+τλ (u20 + Lεηε) + kLε 1+τλ , r2 = −2χ 2 0 b0 Lεηε u20 + Lεηε . (4.40) Finally, to obtain a nonlocal eigenvalue problem we substitute Uj ∼ ν−1/2χ0, Vj ∼ ν1/2w/χ0, Φj ∼ νψj, and Nj0 ∼ Bj0, into (4.2a) to obtain the vector NLEP ∆yψ − ψ + 2wψ + w 2b0 χ20 ∫∞ 0 ρwDψ dρ∫∞ 0 ρw 2 dρ = λψ , where ψ ≡ (ψ1, . . . , ψk)t. By diagonalizing this vector NLEP by using the eigenpairs of D, we obtain the scalar NLEP problem ∆yψc − ψc + 2wψc + w2 ( rib0 χ20 ) ∫∞ 0 ρwψc dρ∫∞ 0 ρw 2 dρ = λψc , i = 1, 2 , (4.41) where r1 and r2 are the eigenvalues of D given in (4.40). Finally, by using (4.40) for r1 and r2, it readily follows that (4.41) reduces to the NLEP (4.23) studied rigorously in [101]. Therefore, we conclude that our eigenvalue formulation in §4.1 reduces to leading-order in ν to the NLEP problem of [101] when A = O ( ε [− ln ε]1/2 ) and D = O(ν−1). 4.3 Instabilities in an Unbounded Domain In this section, we calculate stability thresholds for the case of the infinite planar domain R2. For this domain, we can eliminate the diffusivity D by scaling x and ε with √ D as in (4.12). Therefore, without loss of generality, we take D = 1 in this section. For D 6= 1, we simply note that ε and x must be re-defined as in (4.12). For the infinite-domain problem with Ω = R2, the reduced-wave Green’s function and its regular part satisfying (3.6) with D = 1 is simply G(x;xj) = 1 2π K0 |x− xj| , Rjj = 1 2π (ln 2− γe) . (4.42) 122 4.3. Instabilities in an Unbounded Domain Here γe is Euler’s constant, and K0(r) is the modified Bessel function of the second kind of order zero. The construction of a k-spot quasi-equilibrium solution with spots at xj ∈ R2 for j = 1, . . . , k proceeds by the analysis in §3.1. This quasi- equilibrium solution is given in (3.9) of Principal Result 3.1, where from (3.8) and (4.42), the source strengths S1, . . . , Sk satisfy the nonlinear algebraic system A = Sj ( 1 + ν(ln 2− γe) ) + ν k∑ i6=j SiK0 |xi − xj|+ νχ(Sj) , j = 1, . . . , k , (4.43a) A = ε−1νA , ν = −1/ ln ε . (4.43b) To determine the stability of the k-spot quasi-equilibrium solution to locally radially symmetric perturbations, we must find the eigenvalues of the coupled system, consisting of (4.8) for j = 1, . . . , k together with the homogeneous linear system (4.9). The λ-dependent Green’s function and its regular part, satisfying (4.5) with D = 1 in R2, are given by Gλ(x;xj) = 1 2π K0 (√ 1 + τλ |x− xj| ) , (4.44a) Rλ jj = 1 2π ( ln 2− γe − log √ 1 + τλ ) . (4.44b) Since λ can be complex, we must take log z to be the principal branch of the logarithm function and choose for z = √ 1 + τλ the principal branch of the square root, in order that K0(z) decays when z →∞ along the positive real axis. In terms of these Green’s functions, the homogeneous linear system (4.9) becomes Cj ( 1 + ν(ln 2− γe − log √ 1 + τλ+ B̂j) ) + ν k∑ i6=j CiK0 (√ 1 + τλ |xi − xj | ) = 0 , j = 1, . . . , k . (4.45) From this formulation, in the next two subsections we consider the spe- cific cases of one- and two-spot solutions in R2. 123 4.3. Instabilities in an Unbounded Domain 4.3.1 A One-Spot Solution For a one-spot pattern with spot at the origin, (4.43) reduces to the scalar nonlinear algebraic equation A = S1 + νS1(ln 2− γe) + νχ(S1) , A = ε−1νA , ν = −1/ ln ε . (4.46) In order that C1 6= 0 in (4.45), we require that λ satisfy B̂1 + 1 ν + ( ln 2− γe − log √ 1 + τλ ) = 0 . (4.47) Here B̂1 = B̂1(λ, S1) is to be computed from the inner problem (4.8) in terms of the core solution U1 and V1. Therefore, (4.47) is a transcendental equation for the eigenvalue λ. We now use our numerical procedure, outlined in §4.1.1, to compute λ and to determine the threshold conditions for instability of the one-spot profile. Experiment 4.1: One-spot solution in the infinite domain: A Hopf bifurca- tion For a one-spot solution in R2, our computational results lead to a phase diagram in the A versus τ parameter-plane characterizing the stability of a one-spot solution. In the numerical results below we have fixed ε = 0.02. Firstly, from (4.46) we compute that there is no quasi-equilibrium one- spot solution when A ≥ Af = 0.1766. This minimum value of A for the existence of such a solution was obtained by varying the source strength S1 in (4.46) in the interval S1 ∈ [0.22, 7.41]. This value A = Af is the lower horizontal dot-dashed line in Fig. 4.3(a). The peanut-splitting threshold for A, obtained from the theory in Chapter 3, is obtained by using the spot- splitting criterion S1 = Σ2 ≈ 4.3071 and χ(Σ2) ≈ −1.783 in (4.46), which yields A2 = 0.311. This threshold value for spot self-replication is shown by the heavy dotted horizontal line in Fig. 4.3(a). Then, we use the numerical algorithm in §4.1.1 to compute a Hopf bifurcation threshold τ = τH for values of A in the interval A ∈ [0.1766, 0.298]. At this Hopf bifurcation value, a complex complex conjugate pair of eigenvalues first enters the right half-plane. In Fig. 4.3(a), we plot the Hopf bifurcation value A vs. τH by the solid curve. The resulting parameter-plane as shown in Fig. 4.3(a) is divided into four distinct regions with different solution behavior. The behavior of the one-spot solution in the four different regions of Fig. 4.3(a) is as follows. In Regime σ with A < Af , the quasi-equilibrium solution does not exist; in Regime β enclosed by the dot-dashed line Af = 124 4.3. Instabilities in an Unbounded Domain 0.4 0.3114 0.1766 0.25 0.2 0.15 316.25250200150100508 A τ σ θ β γ (a) A vs. τ -0.3 -0.2 -0.1  0  0.1  0.2  0.3 -0.04 -0.03 -0.02 -0.01  0  0.01  0.02 Im  λ Re λ τ increasing (b) Im(λ) vs. Re(λ) Figure 4.3: Experiment 4.1: One-spot solution in R2. (a) Phase-diagram in the parameter plane A vs. τ . The solid curve plots the Hopf bifurca- tion threshold τH , the lower horizontal dot-dashed line plots the existence condition Af = 0.1766 and the upper heavy dotted horizontal line indicates the spot-splitting threshold A2 = 0.311. Regime σ: No quasi-equilibrium solution exists; Regime β: Oscillations in the spot amplitude; Regime γ: Stable one-spot solution; Regime θ: spot self-replication regime. (b) Fix A = 0.18, we plot the spectrum in the complex λ plane for τ ∈ [30, 40]. At τ = 30, λ = −0.026 ± 0.296 i. At τ = 36.05, λ = −0.000049 ± 0.285 i. At τ = 40, λ = 0.015 ± 0.278 i. Here i ≡ √−1. As τ increases a complex conjugate pair of eigenvalues enters the right half-plane. 125 4.3. Instabilities in an Unbounded Domain 0.1766 and the solid Hopf bifurcation curve, the quasi-equilibrium solution is unstable to an oscillatory profile instability; in Regime γ it is stable; and in Regime θ, it is unstable to spot self-replication. In Fig. 4.3(b), we fix A = 0.18 and plot the eigenvalue λ in the complex plane for the range τ ∈ [30, 40]. Note that for A = 0.18, the Hopf bifurcation threshold τH = 36.06 is where the complex conjugate pair of eigenvalues intersect the imaginary axis. As τ increases, we find numerically that Re(λ) increases. 0.185 0.182 0.18 0.1766 0.175  0.5  0.6  0.7  0.8  0.9  1 A γ Figure 4.4: One-spot solution in R2. In the parameter plane A vs. γ ≡ − ln τ/ ln ε, we plot the existence threshold Af = 0.1766 by the thin dashed line, our Hopf bifurcation threshold A by the dot-dashed curve, and the sta- bility threshold Ãs from the NLEP theory by the thick solid curve. In Fig. 4.4, we compare the stability threshold for A obtained from a nu- merical computation of the coupled eigenvalue formulation (4.8) and (4.47) with that obtained from the NLEP analysis of [99], summarized in §4.2.1. Recall from (4.16) and (4.18) that there are two threshold values Ãf and Ãs predicted from the leading-order (in ν) NLEP theory of [99]. For A < Ãf a one-spot quasi-equilibrium solution does not exist. For A < Ãs this solution is unstable. These threshold values, with Ãs depending on τ , are Ãf ≡ 2ε √ b0 ν , b0 ≡ ∫ ∞ 0 w2(ρ)ρ dρ ≈ 4.9347 , (4.48a) Ãs ≡ Ãf [ 1− ( − γ 4 + γ )2]−1/2 , γ ≡ −ln τ/ ln ε . (4.48b) For ε = 0.02, we compute that Ãf ≈ 0.1757, which agrees very well with our existence threshold value Af ≈ 0.1766. In Fig. 4.4 we compare the 126 4.3. Instabilities in an Unbounded Domain NLEP stability threshold Ãs versus γ = − ln τ/ ln ε with the Hopf bifurca- tion threshold A versus γ = − ln τH/ ln ε, which corresponds to the solid curve of Fig. 4.3(a). We recall from §4.2.1 that the NLEP theory predicts instability when A < Ãs for 0 ≤ τ ≤ ε−2, but has no conclusion regarding stability or instability if A > Ãs. Our Hopf bifurcation threshold predicts that stability changes when we cross the heavy solid curve in Fig. 4.4. There- fore, we conclude that our stability result and that predicted by the NLEP theory are only in disagreement for τ sufficiently large, after the two curves in Fig. 4.4 have crossed. This disagreement likely results from neglecting quantitatively significant logarithmic correction terms in ν = −1/ ln ε in the leading-order NLEP theory. 4.3.2 A Two-Spot Solution Without loss of generality, we assume that the two spots are centered at (α, 0) and (−α, 0) with α > 0 along the horizontal axis. The distance between the two spots is 2α. Since the domain is the infinite plane, the two spots always drift apart along the horizontal axis with speed O(ε2) as t increases, and there is no final equilibrium solution. However, this motion is slow compared to the possibility of O(1) time-scale oscillatory or competition instabilities of the spot profile. We now numerically determine the range of parameters for these instabilities. For this two-spot quasi-equilibrium solution, it was shown in (3.46) that the spots have a common source strength Sc, which satisfies A = Sc [1 + ν(ln 2− γe) + νK0 (2α)]+ ν χ(Sc) , A = ε−1νA , ν = − 1 ln ε , (4.49) where γe is Euler’s constant. The λ-dependent Green’s function is given in (4.44) and the λ-dependent Green’s matrix Gλ is circulant. Therefore, Principal Result 4.2 applies. The eigenvectors Vi for i = 1, 2 and eigenvalues ωλ i for Gλ are V1 ≡ (1, 1)t and V2 ≡ (1,−1)t, where ωλ 1 = Gλ12 +Rλ11 = K0(2α √ 1 + τλ) + (ln 2− γe − log √ 1 + τλ) , for V1 , ωλ 2 = Gλ12 −Rλ11 = K0(2α √ 1 + τλ)− (ln 2− γe − log √ 1 + τλ) , for V2 . From (4.11) of Principal Result 4.2, our stability formulation is to find the roots of the two transcendental equations for λ, given by 1 + νB̂c + 2πνωλ j = 0 , j = 1, 2 , (4.50) where B̂c(λ, Sc) is to the common value for B̂j for j = 1, 2 in (4.8). 127 4.3. Instabilities in an Unbounded Domain We will show numerically below that in some region of the A versus α parameter plane the two-spot solution becomes unstable when τ is increased due to the creation of an unstable eigenvalue λ from ωλ 1 as a result of a Hopf bifurcation. Since this oscillatory instability is associated with the eigenvector V1 = (1, 1) t = (C1, C2) t, it leads to the initiation of a simulta- neous in-phase oscillation in the amplitudes of the two spots. In another region of the A versus α parameter plane we will also show numerically that an unstable eigenvalue λ for ωλ 2 can occur even when τ = 0 if the inter-spot separation distance 2α is below some threshold. This instability results from the creation of a real positive eigenvalue. Since this instability is associated with the eigenvector V2 = (1,−1)t = (C1, C2)t, it leads to the initiation of a sign-fluctuating instability in the amplitudes of the two spots, which leads ultimately to the annihilation of one of the two spots. Since this instability is triggered when 2α is sufficiently small, it is also referred to either as a competition or an overcrowding instability. Experiment 4.2: Two-spot solution in the infinite plane: Oscillatory profile instabilities and competition instabilities We fix ε = 0.02 in the computations below. In Fig. 4.5(a) the two-spot existence threshold Af as a function of α ∈ [0.02, 2.02] is shown by the lower solid curve, while the spot self-replication threshold A2 vs. α is shown by the top solid curve. To obtain Af we determined the minimum value of A as Sc is varied in (4.49), while A2 is obtained by setting Sc = Σ2 ≈ 4.31 in (4.49). In Fig. 4.5(a) the middle dotted curve is the threshold for the onset of a competition instability. These three curves separate the A versus α parameter-plane of Fig. 4.5(a) into four distinct regions. In Regime σ, the quasi-equilibrium two-spot solution does not exist. In Regime β, the two-spot quasi-equilibrium solution undergoes a competition instability for any τ ≥ 0. This instability ultimately leads to the annihilation of one of the spots. We note that competition threshold approaches the existence threshold when the inter-separation distance α becomes large. This is be- cause when α is sufficiently large, the spots essentially do not interact, and there is no competition instability for a one-spot solution in R2. In Regime ζ, the two-spot pattern is stable to a competition instability, but becomes unstable to an oscillatory profile instability only when τ is large enough. As τ increases above a certain threshold τH , a Hopf bifurcation occurs and the two-spot solution is unstable to either in-phase or out-of-phase spot ampli- tude oscillations. The in-phase, or synchronous, oscillatory instability is the dominant of these two types. In Regime θ the two-spot quasi-equilibrium solution is unstable to spot self-replication for any τ ≥ 0. 128 4.3. Instabilities in an Unbounded Domain  0.15  0.2  0.25  0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.5  1  1.5  2 A α σ β ζ θ (a) A vs. α  0  50  100  150  200  250  0  0.5  1  1.5  2 τ α (b) τ vs. α  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2  2.2 Im  λ α (c) λi vs. α -0.25 -0.2 -0.15 -0.1 -0.05  0  0.05  0.1  0.15  0.2  0.25 -0.08 -0.06 -0.04 -0.02  0  0.02  0.04  0.06 Im  λ Re λ τ increasing (d) Im(λ) vs. Re(λ) Figure 4.5: Experiment 4.2: Two-spot solution in R2. (a) We plot the exis- tence threshold Af vs. α by the lower curve, and the spot-splitting threshold A2 vs. α by the top curve. (b) Hopf bifurcation threshold τH1 associated with in-phase oscillations (bottom of each pair of curves) and τH2 associated with out-of-phase oscillations (top of each pair of curves), are plotted as a func- tion of α for A = 0.18 (bottom pair), A = 0.2 (middle pair), and A = 0.22 (top pair). The dotted portions on τH1 correspond to the parameter regime β in subfigure (a) where competition instabilities occur for any τ . (c) Imag- inary part λi of λ versus α at the Hopf bifurcations thresholds for A = 0.18 (bottom), A = 0.2 (middle), and A = 0.22 (top). The heavy solid curves are the eigenvalues associated with in-phase oscillations, and the thin solid curves are the eigenvalues for out-of-phase oscillations. The dotted portions lie on τH1 where a competition instability occurs. (d) Eigenvalue path in the complex plane for A = 0.18 and α = 1 on the range τ ∈ [10, 20]. The eigenvalues associated with in-phase and out-of-phase oscillations enter the right half plane at τH1 and τH2, respectively. 129 4.3. Instabilities in an Unbounded Domain A a function of α and for three values of A, in Fig. 4.5(b) we plot the Hopf bifurcation thresholds τH1, associated with the eigenvector V1 = (1, 1) t (the bottom curve in each pair of curves), and τH2, associated with eigenvector V2 = (1,−1)T (the top curve in each pair of curves). From the bottom pair of curves to the top pair in Fig. 4.5(b), the values of A are A = 0.18, A = 0.2 and A = 0.22, respectively. We observe that for each fixed A, the two thresholds τH1 and τH2 essentially overlap when α is large enough. This is because when the two spots are far away, the interaction between them is negligible. As a consequence, when α→∞, each of the two spots can be treated as a single isolated spot in the infinite plane, so that both τH1 and τH2 approximate the Hopf bifurcation threshold τH of an isolated one-spot solution. From Fig. 4.5(b) the Hopf bifurcation curves for τH1, which each have a dotted portion, end at the values αf = 0.888, 0.206, 0.068 for the curves A = 0.18, 0.2, 0.22, respectively. At these critical values of α, the quasi- equilibrium existence condition A > Af (α) is no longer satisfied. Moreover, the Hopf bifurcation threshold τH2 terminates at another set of critical val- ues αc = 0.904, 0.28, 0.16 for the curves A = 0.18, 0.2, 0.22, respectively. The reason for this disappearance is seen in Fig. 4.5(c), where we plot the imagi- nary part of the eigenvalues at the Hopf bifurcation thresholds for V1 (curves with dotted portions) and V2 (curves without dotted portions). From this figure it is clear that as α → αc from above, the complex conjugate pair of eigenvalues associated with τH2 merge onto the real axis at the origin, which is precisely the crossing point for the onset of a competition instability. When the spots are two close in the sense that αf < α < αc, then there is a positive real eigenvalue in the unstable right half-plane for any τ ≥ 0. If in addition, τ < τH1, it is the only unstable eigenvalue. This eigenvalue is the one associated with a competition instability. In Fig. 4.5(b) and Fig. 4.5(c), the dotted segments of the curves correspond to the range of α where this competition instability occurs. Next, we observe from Fig. 4.5(b) that τH1 ≤ τH2. Therefore, the in- phase synchronous oscillatory instability of the spot amplitudes is always the dominant instability mechanism for each α > αc as τ is increased. Moreover, we observe that the Hopf bifurcation threshold τH1 is an increasing function of spot inter-separation distance. Recall that the slow dynamics of the two spots is repulsive in the sense that α increases as t increases, with speed O(ε2) (see (3.47) of Section §3.5).. The monotone increasing nature of τH1 with α eliminates any possibility of a dynamically triggered oscillatory instability for the two-spot solution. Namely, consider an initial two-spot quasi-equilibrium with initial inter-separation distance 2α0 and with a fixed 130 4.4. A One-Spot Solution in a Finite Domain τ with τ < τH1(α0), where α0 > αc. Then, since α increases as time t increases, and τH1 is monotone increasing in α, we conclude that τ < τH1(α) for any t > 0. Therefore, there are no dynamically triggered oscillatory instabilities for two-spot patterns on the infinite plane.. We remark that the phase diagram obtained for the two-spot quasi- equilibrium solution in R2 is qualitatively rather similar to the case of a two-spike solution to the one-dimensional GS model on the infinite real line in the low reed-rate regime, as was studied in [86]. Finally, we show a plot of the eigenvalue in the complex plane as τ is increased. We take A = 0.18 and α = 1.0, and we vary τ ∈ [10, 20]. The two Hopf bifurcation values are, respectively, τH1 = 13.1 and τH2 = 15.6. The paths of the eigenvalue in the complex plane as τ increases are shown in Fig. 4.5(d) for the eigenvectors V1 (solid curves) and V2 (dotted curves) For each of these eigenvectors, the real part of both eigenvalues λr increases as τ increases. 4.4 A One-Spot Solution in a Finite Domain Next, we study the stability of a one-spot quasi-equilibrium solution in a finite domain. We will only consider the unit disk and the unit square, since it is for these domains that the reduced-wave Green’s functions and its regular part, satisfying (3.6), are analytically known (see Section 3.6). We remark that for these special domains, the λ-dependent Green’s function and its regular part, satisfying (4.5), can be readily obtained by replacing D with D/(1 + τλ) in the formulae for the reduced-wave Green’s functions given in Section 3.6. Since λ is in general complex, we require that the software for the necessary special function evaluations in Section 3.6 can treat the case of complex arguments. 4.4.1 A One-Spot Solution in the Unit Disk Consider the unit disk Ω = {x : |x| ≤ 1}, and suppose that a one-spot quasi-equilibrium solution is centered at x0 inside the unit disk. Experiment 4.3: One-spot solution in the unit disk: Dynamical profile in- stability Let r = |x0 denote the distance from the spot to the center of the unit disk, so that 0 ≤ r ≤ 1. The equilibrium location for the slow dynamics of the spot is at the center r = 0, as seen from Principal Result 3.3. We choose a value for the feed-rate parameter A such that A exceeds the existence 131 4.4. A One-Spot Solution in a Finite Domain threshold Af for the entire range 0 < r < 1. In Fig. 4.6(a), we plot the Hopf bifurcation threshold τH versus r for fixed A = 0.16, but for different values of the diffusivity D. The two solid curves in this figure correspond D = 4, 5, with the lower solid curve in this figure corresponding to D = 4. For these values of D, we observe from Fig. 4.6(a) that the maximum of τH = τH(r) is obtained at the equilibrium location r = 0, and that τH decreases monotonically as r increases. Next, we plot by the dashed curves, the Hopf bifurcation thresholds for the larger values of D given by D = 6, 7, 8. Notice that for these larger values of D, the curves of τH = τH(r) are concave down near r = 0, which yields a local minimum for τH at r = 0. Therefore, we conclude that when D is sufficiently large, we can obtain a dynamically triggered oscillatory instability in the spot amplitude. To illustrate this, suppose that D = 8. Then, we calculate τH(0) ≈ 3.73 and τH(0.612) ≈ 3.88. Suppose that we take τ = 3.8 with the initial spot location at time t = 0 at r = 0.612. Then, since τ < τH(0.612), the spot is stable at t = 0. However, since the motion of the spot is towards the origin and τ > τH(0), it follows that a dynamically triggered oscillatory profile instability will occur before the spot reaches the center of the disk. We did a similar numerical experiment for A = 0.18. For this value of A, we plot τH versus r in Fig. 4.6(b). The solid curves correspond to D = 2, 3, for which τH(r) has a maximum at r = 0. For the value D = 4.3, 5, 6, as indicated by the dashed curves, τH(r) has a local minimum at the equilibrium location r = 0. For these larger values of D, we can choose a value of τ and an initial spot location that will lead to a dynamical instability. Experiment 4.4: One-spot solution in the unit disk: A profile instability observed by full numerical simulation In this experiment we compare our numerically computed Hopf bifur- cation threshold obtained from the coupled eigenvalue problem with full numerical results for the threshold as computed from the Gray-Scott PDE model (1.7). The full system was computed numerically by an explicit time- step finite element code, which was first developed and used in [50]. Given a parameter set ε, τ,A,D and an initial condition, we evolve the solution to the PDE (1.7) in time, and record the amplitude vm = max{v(x) : x ∈ Ω} of the solution component v at each time step. In Fig. 4.7, we plot vm vs. t with fixed parameters ε = 0.02, A = 0.16, and D = 4 for three values of τ . This figure shows that the unstable oscillation occurs near τ = 3.2, which roughly agrees with the Hopf bifurcation threshold τH ≈ 3.5 computed from our coupled eigenvalue formulation (see Fig. 4.6(a)). We remark that if 132 4.4. A One-Spot Solution in a Finite Domain  1.5  2  2.5  3  3.5  4  0  0.2  0.4  0.6  0.8  1 τ r (a) τH vs. r A = 0.16  0  5  10  15  20  25  30  35  40  0  0.2  0.4  0.6  0.8  1 τ r (b) τH vs. r A = 0.18 Figure 4.6: Experiment 4.3: One-spot solution in the unit disk. Plot of τH vs. r for different A and D. (a) Fix A = 0.16. The solid curves are for D = 4, 5, arranged from lower to upper y−intercepts, respectively. The dashed curves are for D = 6, 7, 8, arranged from upper to lower y-intercepts, respectively. (b) Fix A = 0.18. The solid curves are for D = 2, 3 arranged from lower to upper y−intercepts, respectively. The dashed curves are for D = 4.3, 5, 6, arranged from upper to lower y−intercepts, respectively. we changed the parameters to be A = 0.18, and D = 6, then τH ≈ 10.3 (see Fig. 4.6(b)), as predicted by our coupled eigenvalue formulation. This compares reasonably well with the result τ ≈ 9 as observed in Fig. 4.8 and computed from the full numerical computations of (1.7).  37  38  39  40  41  42  43  44  45  46  47  48  0  10  20  30  40  50  60  70 (a) τ = 3.0  36  38  40  42  44  46  48  50  0  10  20  30  40  50  60  70 (b) τ = 3.2  25  30  35  40  45  50  55  60  65  0  10  20  30  40  50  60  70 (c) τ = 3.4 Figure 4.7: Experiment 4.4: One-spot solution with spot at the center of the unit disk. We fix ε = 0.02, A = 0.16,D = 4, and plot the spot amplitude vm vs. t for (a) τ = 3; (b) τ = 3.2; (c) τ = 3.4. 133 4.4. A One-Spot Solution in a Finite Domain  55  56  57  58  59  60  61  62  63  64  65  0  10  20  30  40  50  60  70 (a) τ = 8.5  55  56  57  58  59  60  61  62  63  64  65  0  10  20  30  40  50  60  70 (b) τ = 9.0  54  56  58  60  62  64  66  68  0  10  20  30  40  50  60  70 (c) τ = 9.5 Figure 4.8: Experiment 4.4: One spot solution with spot at the center of the unit disk. We fix ε = 0.02, A = 0.18,D = 6, and plot the spot amplitude vm vs. t for (a) τ = 8.5; (b) τ = 9.0; (c) τ = 9.5. 4.4.2 A One-Spot Solution in a Square In this subsection we compute stability thresholds for a one-spot quasi- equilibrium solution in the unit square Ω = [0, 1] × [0, 1]. Experiment 4.5: One-spot solution in a square: A dynamical profile insta- bility compared with full numerical simulations We fix ε = 0.02, and we let the spot be centered on the diagonal line (x, x). From Principal Result 3.3 it follows that the spot will move slowly along this diagonal until it approaches its equilibrium location at (0.5, 0.5). In Fig. 4.9(a), we fix A = 0.28, and plot τH vs. x for D = 1 (heavy solid curve), D = 1.5 (solid curve), D = 2 (dotted curve) and D = 2.5 (dashed curve) as computed from our eigenvalue problem. The last two curves show the existence of a dynamical instability, since as the spot drifts towards its equilibrium location along the diagonal of the unit square the Hopf bifur- cation threshold decreases. In Fig. 4.9(b), we fix D = 1, and plot τH vs. x for four values of A. In Fig. 4.10, we plot the numerically computed spot amplitude vm = vm(t) for a spot centered at the midpoint of the unit square. This spot amplitude was computed for three values of τ from the full GS model (1.7) by using the fortran software package VLUGR2 (cf. [7]). The estimated Hopf bifurcation value is τ ≈ 3.3, which compares very favorably with our prediction of τH as seen in Fig. 4.9(a). 134 4.4. A One-Spot Solution in a Finite Domain  1.4  1.6  1.8  2  2.2  2.4  2.6  2.8  3  3.2  3.4  3.6  0.5  0.55  0.6  0.65  0.7  0.75  0.8  0.85  0.9 τ x (a) τH vs. x  0  20  40  60  80  100  120  140  160  180  0.5  0.55  0.6  0.65  0.7  0.75  0.8  0.85  0.9  0.95 τ x (b) τH vs. x Figure 4.9: Experiment 4.5: One-spot solution centered at (x, x) on a diago- nal of the unit square [0, 1]× [0, 1] (a) Fixing A = 0.28, we plot τH vs. x for D = 1 (lower solid curve), D = 1.5 (upper solid curve), D = 2 (dotted curve) and D = 2.5 (dashed curve). The last two curves for larger values of D show the existence of a dynamical oscillatory instability in the spot amplitude since as the spot drifts towards its equilibrium location along the diagonal, the Hopf bifurcation threshold decreases and has its minimum at the center of the square. (b) Fixing D = 1, we plot τH vs. x for A = 0.28, 0.3, 0.35, 0.4, arranged from lower to upper y−intercepts, respectively. 0 10 20 30 40 50 60 70 80 22 22.5 23 23.5 24 24.5 25 25.5 26 (a) τ = 3.2 0 10 20 30 40 50 60 70 80 22 22.5 23 23.5 24 24.5 25 25.5 26 26.5 (b) τ = 3.3 0 10 20 30 40 50 60 70 80 21 22 23 24 25 26 27 28 (c) τ = 3.4 Figure 4.10: Experiment 4.5: One-spot solution in a square. We fix ε = 0.02, A = 0.28,D = 1, and let the spot be located at its equilibrium location x = (0.5, 0.5). From the full GS model we plot the spot amplitude vm vs. t for (a) τ = 3.2; (b) τ = 3.3; (c) τ = 3.4. 135 4.5. Symmetric k−Spot Patterns in a Finite Domain 4.5 Symmetric k−Spot Patterns in a Finite Domain Next, we consider multi-spot patterns in a finite domain. It is computa- tionally rather intensive to numerically study our eigenvalue formulation for the general case where k spots are arbitrarily located within the domain. Therefore, we will assume that the k spots are arranged to form certain symmetric patterns. We define a spot pattern to be symmetric if and only if the Green’s matrix G in (3.11) is circulant, which immediately implies that the λ-dependent Green’s matrix Gλ in (4.10) is also a circulant matrix. With this special structure, our coupled eigenvalue problem reduces to that given in Principal Result 4.2. One special case where a circulant Green’s matrix arises is when there are k-spots equally spaced on a ring that is concentric with the unit disk. 4.5.1 A Symmetric Circulant Matrix We assume that the k−spot pattern is symmetric. We denote the first row of G by the row vector a = (a1, . . . , ak). Since G is circulant it follows that all of the other rows of G can be obtained by rotating the components of the vector a. In addition, since G is also necessarily a symmetric matrix it follows that a2 = ak, a3 = ak−1, . . ., and aj = ak+2−j for j = 2, . . . , [k/2]. Next, we calculate the eigenvectors and eigenvalues of the λ-dependent Green’s matrix Gλ as needed in Principal Result 4.2. We recall that if a matrix of size k × k is circulant, its eigenvectors Vj consist of the kth roots of unity, given by κj = k−1∑ m=0 am+1e 2pi(j−1)m/k , Vj = (1, e2pi(j−1)/k , . . . , e2pi(j−1)(k−1)/k)t . Here κj is the eigenvalue of Gλ associated with Vj for j = 1, . . . , k, and am is the mth component of the row vector a. For illustration, let k = 3. Then, the eigenpairs are V1 = (1, 1, 1) with κ1 = a1 + a2 + a3, V2 = (1, e 2pii/3, e4pii/3) with κ2 = a1 + a2e 2pii/3 + a3e 4pii/3, and V3 = (1, e 4pii/3, e2pii/3) with κ2 = a1+a2e 4pii/3+a3e 2pii/3. Then, since the matrix is also symmetric, we have a2 = a3, so that the last two eigenvalues are equal, i.e. κ2 = κ3 = a1−a2, yielding one eigenvalue with multiplicity 2. Then, since any linear combination of V2 and V3 is also an eigenvector, we take the real part of V2 as one such vector, and the imaginary part of V2 as the other. In summary, we can take V1 = (1, 1, 1) t, V2 = (1,−0.5,−0.5)t and 136 4.5. Symmetric k−Spot Patterns in a Finite Domain V3 = (0, √ 3/2,−√3/2)t as the three eigenvectors of Gλ for the symmetric 3-spot pattern. In general, the symmetry of Gλ implies that aj = ak+2−j and Vj = Vk+2−j, and Vjm = Vj(k+2−m) for m = 2, . . . , [k/2] and j = 2, . . . , [k/2]. Since the eigenvalue is κj = Vj · a, we have κj = κk+2−j . This generates [k/2]−1 eigenvalues with multiplicity 2, whose eigenvectors can be obtained by taking any linear combination of two complex conjugate eigenvectors. For instance, for the jth eigenvalue κj the corresponding two eigenvectors can be taken as the real and imaginary parts of Vj , respectively. In summary, we know that all of the eigenvectors can be chosen to be real, but the eigenvalues in general will not be. This leads to the following general result for the spectrum of a k × k symmetric and circulant matrix whose first row vector is a = (a1, . . . , am):  κ1 = ∑k m=1 am, V t 1 = (1, . . . , 1), κj = ∑k−1 m=0 cos ( 2pi(j−1)m k ) am+1, eigenvalues with multiplicity 2, V tj = ( 1, cos ( 2pi(j−1) k ) , . . . , cos ( 2pi(j−1)(k−1) k )) , V tk+2−j = ( 0, sin ( 2pi(j−1) k ) , . . . , sin ( 2pi(j−1)(k−1) k )) , j = 2, . . . , [k/2] + 1 . (4.51) Note that if k is even, then κ[k/2]+1 = ∑k m=1(−1)m−1am is a simple eigen- value with eigenvector (1,−1, · · · , 1,−1)t. For a symmetric spot pattern, we can simply substitute (4.51) into (4.11) of Principal Result 4.2 and derive the eigenvalue problem associated with the jth eigenvector Vj . This yields, 1 + νRe(B̂c) + 2πν k∑ m=1 Re(am)Vjm = 0 , (4.52a) νIm(B̂c) + 2πν k∑ m=1 Im(am)Vjm = 0 , j = 1, . . . , [k/2] + 1 . (4.52b) Here the row vector a has components a1 = Rλ 11, and am = Gλ 1m for m = 2, . . . , k, where Gλ 1m and Rλ 11 are the λ-dependent Green’s function and its regular part satisfying (4.5). 137 4.5. Symmetric k−Spot Patterns in a Finite Domain 4.5.2 A Ring of Spots in the Unit Disk In this subsection, we assume that all k spots are equally spaced on a ring of radius r concentric with the unit disk. The location of the spots are at xj = re 2piij/k , j = 1, . . . , k . For this special arrangement of spot locations, the Green’s matrices G and Gλ are both symmetric and circulant. Our goal is to compute the Hopf bifurcation threshold τH as a function of the ring radius r for various one-ring patterns, and to compare our stability threshold results with those computed from full numerical computations of the GS model (1.7). Experiment 4.6: A two-spot solution in the unit disk: Dynamical oscillatory profile instability and dynamical competition instability Suppose that there are two spots equally spaced on a ring of radius r, with 0 < r < 1, concentric with the unit disk. The equilibrium ring radii for different values of D, representing the equilibrium point for the ODE in Principal Result 3.3, are given in Table 4.1. These equilibria do not depend on the values of A. D re D re 0.8 0.45779 2.0 0.45540 1.0 0.45703 3.0 0.45483 1.3 0.45630 4.0 0.45454 1.5 0.45596 5.0 0.45436 Table 4.1: Equilibrium ring radius re for a two-spot symmetric pattern in the unit disk for different values of D. In Fig. 4.11, we fix the feed-rate parameter at A = 0.26, and denote by the solid curve the Hopf bifurcation threshold τH with V1 = (1, 1) t. The dotted segment is the portion of this curve where a competition instability occurs. In Fig. 4.11(a) we plot τH versus r for D = 0.8, 1.0, 1.3, 1.5, and in Fig. 4.11(b), we plot τH versus r for D = 2, 3, 4, 5. At r = 0.45, the lowest curve corresponds to the largest value of D. We conclude that for D ≥ 2, there could be a dynamical oscillatory profile instability for a two- spot symmetric pattern with A = 0.26. Another observation is that for D = 5, a competition instability occurs when r ≈ 0.568. Upon recalling from Table 4.1 that the equilibrium ring radius is re = 0.455, we conclude that there could be a dynamical competition instability. To illustrate this, 138 4.5. Symmetric k−Spot Patterns in a Finite Domain let D = 5 and τ = 1 and suppose that we have an initial two-spot symmetric pattern with initial value r = 0.7 at t = 0. Then, as the two spots move toward each other, a dynamical competition instability will be triggered before the ring reaches its equilibrium ring radius at re ≈ 0.455.  0  5  10  15  20  25  30  35  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1 τ r (a) τH vs. r  0  10  20  30  40  50  60  70  80  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1 τ r (b) τH vs. r Figure 4.11: Experiment 4.6: Two spots in the unit disk. (a) Fix A = 0.26. We plot τH vs. r for small values of D = 0.8, 1.0, 1.3, 1.5, with the lower curves corresponding to smaller values of D. The solid curves correspond to regions in r where an oscillatory profile instability occurs, and the dotted portions of these curves correspond to where a competition instability occurs. (b) τH vs. r for the larger values of D given by D = 2, 3, 4, 5. The lower curves at r = 0.6 correspond to larger values of D. Next we fix A = 0.26 and D = 3. In Fig. 4.12(a) we plot the Hopf bifur- cation thresholds τH1 for in-phase oscillation (1, 1) (lower solid curve) and τH2 for out-of-phase oscillation (1,−1) (upper solid curve). The dotted por- tion on τH1 shows where a competition instability occurs for any τ ≥ 0. In Fig. 4.12(b), we fix two spots on the equilibrium ring of radius re = 0.45483, and we plot the path of the complex conjugate pair of eigenvalues in the complex plane for the range τ ∈ [3.7, 36]. The solid curve is the eigenvalue path for in-phase oscillations with eigenvector (1, 1), while the dotted curve is the eigenvalue path for out-of-phase oscillations with eigenvector (1,−1). We note that as τ increases, the real parts of both pairs of eigenvalues in- crease. The eigenvalue for (1, 1) enters the right half-plane at τ ≈ 11.5. Note that the imaginary eigenvalue of (1,−1) becomes very close to the real axis for τ < 3.7, and our numerical computation breaks down because our assumption that λr = 0, λi 6= 0 does not hold in this case. 139 4.5. Symmetric k−Spot Patterns in a Finite Domain  10  20  30  40  50  60  70  80  90  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1 τ H r (a) τH vs. r -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05  0  0.05  0.1 Im  λ Re λ τ increasing (b) Im(λ) vs. Re(λ) Figure 4.12: Experiment 4.6: Two spots in the unit disk. Fix A = 0.26 and D = 3. (a) τH vs. r for in-phase oscillation (1, 1) (lower solid curve), and τH2 for out-of-phase oscillation (1,−1) (upper solid curve). The dotted portions on τH1 show where a competition instability occurs for any tau ≥ 0. (b) Two spots equally spaced on an equilibrium ring of radius re = 0.45483. Plot of λ in the complex plane on the range τ ∈ [3.7, 36]. The solid curve is for (1, 1), while the dotted curve is for (1,−1). 8 7 6 5 4 2 3.3 1.16  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1 D r (a) Dc vs. r -0.4 -0.3 -0.2 -0.1  0  0.1  0.2  0.3  0.4 -0.1 -0.05  0  0.05  0.1  0.15  0.2  0.25 Im  λ Re λ (b) Im(λ) vs. Re(λ) Figure 4.13: Experiment 4.7: Two spots in the unit disk. Fix A = 0.26 and ε = 0.02 (a) Plot of Dc vs. r (solid curve), where Dc is the critical value of D at which a competition instability is initiated. The region 1.1583 < D < 3.3030 between the two horizontal lines is where the leading-order NLEP theory predicts that no competition instability occurs. (b) For τ = 30 and r = 0.3, plot the spectrum in the complex plane by varying D in the range [0.8, 3]. The solid curve corresponds to the eigenvalues with (1,−1), and the dotted curve corresponds to the eigenvalues with (1, 1). The arrows show the path as D increases. 140 4.5. Symmetric k−Spot Patterns in a Finite Domain Experiment 4.7: A two-spot solution in unit disk: Competition instability Fix ε = 0.02, A = 0.26. In Theorem 2.3 of [101], as summarized in Section 4.2.2, the leading order NLEP analysis proves that if D = O(ν−1) and L0 < η0 (2η0+k)2 , then the small solution u−, v− of (4.2.2) is stable for any τ sufficiently small. In addition, the leading-order NLEP theory predicts that if L0 > η0 (2η0+k)2 , then this solution is unstable for any τ > 0. From (4.20), we recall that L0 = limε→0 ε 22pib0 A2|Ω| and η0 = limε→0 |Ω| 2piDν , where b0 = ∫∞ 0 w 2ρ dρ ≈ 4.9347 and w is the ground-state solution of (4.14). With |Ω| = π and ν = −1/ ln ε, and for the other parameter values as given, this stability bound yields 0.296 < Dν < 0.844, or equivalently 1.158 < D < 3.303. For this range of D, the leading order NLEP theory predicts that the solution u−, v− is stable for τ sufficiently small. Outside of this bound for D, the leading-order NLEP theory predicts that a competition instability.occurs for any τ > 0. We fix ε = 0.02 and A = 0.26. In Fig. 4.13(a), we plot Dc vs. r by the solid curve, where Dc denotes the critical value of D where a competition instability is initiated. To the left of this curve, oriented with respect to the direction of increasing D, we predict that a competition instability occurs for any τ > 0, while to the right of this curve we predict that the solution is stable when τ is small enough. The two horizontal dotted lines with D1 = 1.158 and D2 = 3.303 in this figure bound a region of stability as predicted by the leading-order NLEP theory of [101]. The leading-order in ν prediction from NLEP theory does not capture the dependence of the stability threshold on the ring radius r. Our stability formulation, which accounts for all orders in ν = −1/ ln ε, shows that Dc = Dc(r). For ε = 0.02, ν is certainly not very small, and this dependence of the threshold on r, not accounted for by NLEP theory, is indeed very significant. In Fig. 4.13(b), for A = 0.26, τ = 30, and r = 0.3, we plot the spectrum in the complex plane as D varies over the range [0.8, 3.0]. This range of D corresponds to taking a vertical slice at r = 0.3 in Fig. 4.13(a) that cuts across the stability boundary at two values of D. The arrows in this figure indicate the direction of the path of eigenvalues as D is increased. The loop of spectra, corresponding to the solid curves in this figure, plot the eigenvalues associated with the eigenvector (1,−1). From Fig. 4.13(a) we predict instability when D is near D = 0.8 or when D is near D = 3.0. This is clearly shown by the behavior of the closed loop of spectra in Fig. 4.13(b). With regards to the loop of spectra (solid curves) in Fig. 4.13(b) asso- 141 4.5. Symmetric k−Spot Patterns in a Finite Domain ciated with the (1,−1) eigenvector, our results show that for D = 0.8 the eigenvalue is on the positive real axis at λ = (0.218, 0). As D is increased above D = 0.8 the real eigenvalue moves along the horizontal axis to the left, and splits into a complex conjugate pair at D ≈ 0.84. At D = 0.84, then λ = (0.111, 0), while at D = 0.85 then λ = (0.0649, 0.0322). For D ≈ 1.0, λ enters the stable left half-plane Re(λ) < 0. For D = 2.89 the eigenvalues merge onto the negative real axis. One eigenvalue then enters the right half- plane at D ≈ 2.95. This is the value of D where a competition instability is initiated. For D > 2.95, there is a positive real eigenvalue associated with the (1,−1) eigenvector, which generates a competition instability. Alternatively, the dotted curves in Fig. 4.13(b) correspond to the path of eigenvalues associated with the (1, 1) eigenvector, representing synchronous instabilities. This path depends sensitively on the value chosen for τ . For τ = 30, our computational results show that asD is increased aboveD = 0.8, the real part of these eigenvalues first starts to decrease and then crosses into the stable left half-plane at D ≈ 1.33. The real part of these eigenvalues starts increasing when D ≈ 1.59 and the path then re-enters the unstable right half-plane at D = 1.89 where stability is lost at a Hopf bifurcation corresponding to synchronous oscillations of the two spot amplitudes. This path of eigenvalues remain in the unstable right half-plane for D > 1.89. We conclude that if 1.33 < D < 1.89, this two-spot symmetric pattern is stable to both competition and synchronous oscillatory instabilities. When D > 2.95 there is an unstable real eigenvalue in the right half-plane associ- ated with a competition instability in addition to a synchronous oscillatory instability associated with the complex conjugate pair of eigenvalues. On the range 1.89 < D < 2.95 there is only a synchronous oscillatory instabil- ity. We remark that if τ is taken to be considerably smaller than our chosen value of τ = 30, there would be only one unstable eigenvalue in Re(λ) > 0 (located on the positive real axis) when D > 2.95. We set A = 0.26, D = 3, τ = 30, and consider an initial two-spot sym- metric pattern with ring radius r = 0.3. With these parameter values and initial condition, in Fig. 4.14 we show results from a full numerical simula- tion of the GS model (1.7) at later times. For these parameter values, we conclude from Fig. 4.13(b) that there is an unstable real eigenvalue together with an unstable complex conjugate pair of eigenvalues. We remark that since we use the finite element method, the locations of the two spots is not exactly symmetric, which has essentially a similar effect to adding a small perturbation to the initial condition to initiate an instability. For the left spot, its amplitude is initially vm1 ≈ 37.5, while the initial amplitude of the right spot is vm2 ≈ 40.1. From the full numerical results shown in Fig. 4.14 142 4.5. Symmetric k−Spot Patterns in a Finite Domain -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 1234 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 12345 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  0 123456 -1 -0.5  0  0.5  1 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1  05 12345 Figure 4.14: Experiment 4.7: We fix D = 3.0, A = 0.26, τ = 30, and ε = 0.02. The initial condition is a symmetric two-spot pattern on a ring of radius 0.3. The solution v is plotted at times t = 0.5, 8, 10, 15, respectively. The spot on the left is rather quickly annihilated. at t = 0.5, 8, 10, 15, we observe that the amplitude of the left spot decays with time, and this spot disappears after t = 15. In Fig. 4.15 we plot the amplitude of the two spots. The amplitude vm1 vs. t of the left spot is shown by the dot-dashed curve, while the amplitude vm2 vs. t of the right spot is shown by the solid curve. These amplitudes were obtained by tracking the locations of spots in the numerical simulation at each time step. The amplitude of the left spot decays rapidly to zero before t = 13. Meanwhile, the amplitude of the right spot begins to oscillate, but the oscillation ceases after the left spot is annihilated. This is because after the left spot has been removed, the value τ = 30 is below the Hopf bifurcation value for a one-spot solution. Experiment 4.8: A two-spot solution in the unit disk: Phase diagram for different diffusion coefficients D In Experiment 4.8, we plot the phase diagram A vs. r for different diffusion coefficients D. In Fig. 4.16, we plot the existence threshold Af for a two-spot quasi-equilibrium solution by the solid curve. The critical value of A for the competition instability is shown by the dotted curve, and the peanut-splitting threshold A2 by the upper heavy solid curve. The solution behavior in the four different parameter regimes in this figure is as follows. In Regime σ the quasi-equilibrium solution does not exist. In Regime β the quasi-equilibrium solution exists but is unstable to competition. In Regime ζ the solution is unstable to an oscillatory profile instability when τ exceeds a Hopf bifurcation threshold τH . In region θ the solution is unstable to peanut-splitting. The subfigures from left to right are plotted for: (a) D = 0.2; (b) D = 1; (c) D = 5. Note that when D = 0.2 ≪ 1, the subfigure is qualitatively similar to the phase diagram shown in Fig. 4.5(a) 143 4.5. Symmetric k−Spot Patterns in a Finite Domain  0  10  20  30  40  50  60  70  0  5  10  15  20  25  30  35 v m t (a) vm vs. t Figure 4.15: Experiment 4.7: Two spots in the unit disk. Fix ε = 0.02, A = 0.26, D = 3, and τ = 20. The dot-dashed curve plots the amplitude of the left spot vm1 vs. t., the solid curve plots the amplitude of the right spot vm1 vs. t. for a two-spot solution in the infinite plane. Experiment 4.9 k-spot solution with k > 2 in the unit disk: Phase diagram Finally, we compute the stability phase diagram associated with k-spots that are equally spaced on a ring of radius r. We fix D = 0.2 and ε = 0.02 and plot this phase diagram in the parameter plane A versus the ring radius r for k = 2, 4, 8, 16. We are particularly interested in determining if there exists k−spot pattern with fixed parameters D and ε such that the peanut- splitting regime for k−spots intersects the competition regime for a pattern with 2k-spots. If this occurs then it would be possible that a k−spot pattern undergoes peanut-splitting first, followed by a competition instability with 2k spots, leading to the annihilation of k of them, with the cycle of self- replication followed by self-destruction repeating again. We assume that all k spots are equally distributed on a ring of radius r. Then using the techniques in Section 4.5.1 it follows readily that for 2k = 4, the eigenvectors are V1 = (1, 1, 1, 1) t, V2 = (1,−1, 1,−1)t, V3 = (1, 0,−1, 0)t and V4 = (0, 1, 0,−1)t. Note that the last three of these vectors have components with different signs. Moreover, the computational results show that the competition instability threshold associated with V2 is always above that of V3 or V4, and so it has a larger competition instability regime in the phase diagram, and this instability occurs more readily than with V3 144 4.6. Discussion  0.35  0.4  0.45  0.5  0.55  0.6  0.65  0.7  0.75  0.8  0.85  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 A r σ β ζ θ (a) D = 0.2  0.2  0.25  0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65  0.7  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 A r σ β ζ θ (b) D = 1.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 A r σ β ζ θ (c) D = 5.0 Figure 4.16: Experiment 4.8: Two spots in the unit disk. We plot the phase diagram A vs. r. The solid curve is the existence threshold Af for a two-spot quasi-equilibrium solution; the dotted curve is the critical value for a compe- tition instability; the heavy solid-curve is the peanut-splitting threshold A2. Regime σ: no solution; Regime β: the quasi-equilibrium solution is unstable to competition; Regime ζ the solution is unstable to a profile oscillation if τ > τH(r); Regime θ: the solution is unstable to peanut-splitting. Subfigures from left to right are plotted for: (a) D = 0.2; (b) D = 1; (c) D = 5. and V4. The profile oscillatory instability occurs first as τ is increased for the eigenvector V1. Similarly for k = 8, 16, the instability with eigenvector of the form V = (1,−1, 1,−1, . . . , 1,−1)t is always the dominant competition mechanism. Therefore for 2k−spot equally distributed on a ring, if the competition instability does occur, it will lead to the annihilation of one of the two adjacent spots. For 2k even, half the total number of total spots will disappear leaving a k−spot pattern. In Fig. 4.17, we plot the phase diagram in the A versus r plane for k = 2, 4, 8, 16. The computations are done for the fixed value D = 0.2. The thin solid curves on the bottom is the existence threshold Af for a quasi-equilibrium solution; the dotted curves are the critical value of A for a competition instability, and the heavy solid curves on top are the peanut- splitting threshold A2. In Regime σ the quasi-equilibrium solution does not exist. Regime β is where the quasi-equilibrium solution exists but is unstable to competition. In Regime ζ the solution is unstable to profile oscillation if τ > τH(r). In Regime θ there is instability to peanut-splitting. 4.6 Discussion In this chapter, we formulated an eigenvalue problem associated with in- stabilities in the amplitudes of spots. This eigenvalue problem, associated with locally radially symmetric perturbations, differs from the eigenvalue 145 4.6. Discussion  0.35  0.4  0.45  0.5  0.55  0.6  0.65  0.7  0.75  0.8  0.85  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 A r σ β ζ θ (a) k = 2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  1.1  1.2  1.3  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 A r β σ ζ θ (b) k = 4  0  0.5  1  1.5  2  2.5  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 A r σ β ζ θ (c) k = 8  0.5  1  1.5  2  2.5  3  3.5  4  4.5  5  5.5  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 A r β σ θ ζ (d) k = 16 Figure 4.17: Experiment 4.9: Symmetric k-spot pattern in the unit disk. We plot the phase diagram A vs. r with D = 0.2. The thin solid curves are the existence threshold Af for the quasi-equilibrium; the dotted curves are the critical values of A for a competition instability; the heavy solid curves is the peanut-splitting threshold A2. In Regime σ the quasi-equilibrium solu- tion does not exist. In Regime β the quasi-equilibrium solution exists but is unstable to competition. In Regime ζ the solution is unstable to a pro- file oscillation if τ exceeds a Hopf bifurcation threshold τH(r) In Regime θ the solution is unstable to peanut-splitting. Subfigures from left to right are plotted for: (a) k = 2; (b) k = 4; (c) k = 8; (d) k = 16. 146 4.6. Discussion problem of Chapter 3 in that the local eigenfunction Nj(ρ) grows logarith- mically as ρ → ∞. This logarithmic growth has the effect of introducing a strong inter-spot coupling effect in the formulation of our eigenvalue prob- lem governing spot amplitude instabilities. If in the complex λ-plane a real eigenvalue crosses into the unstable right half-plane Re(λ) > 0 along the real axis, then a competition instability is initiated. Alternatively, if there is a complex conjugate pair of eigenvalues entering the unstable right half-plane through a Hopf bifurcation then an oscillatory profile instability is initiated. By numerically computing discrete eigenvalues of the eigenvalue problem (4.8) and (4.10a) using the algorithms in §4.1.1, we are able to calculate both the Hopf bifurcation threshold and the critical parameter values for a competition instability. These computations allow us to plot phase dia- grams in parameter space where the three different types of instabilities can occur. These instabilities are spot self-replication, an oscillatory instability in the spot amplitudes, and a competition instability of the spot amplitudes. These phase diagrams are plotted for various spatial configuration of spot patterns and for both finite and infinite domains. There are several open problems that warrant investigation. One such problem is to derive the correction term to the leading-order NLEP the- ory in §4.2 to more accurately determine the threshold parameter values for spot stability in terms of the spatial configuration of spots. A second open problem is to consider a weakly nonlinear analysis around a multi-spot quasi-equilibrium solution for parameter values near the onset of either an oscillatory profile instability or a competition instability. With such a weakly nonlinear theory one could determine whether the oscillatory instability is subcritical or supercritical, and one could characterize in more detail a spot competition process in the weakly nonlinear regime. A third open problem is to investigate the stability thresholds associated with other, less symmet- ric, configurations of spots such as for the case of k−1 spots on a ring and a spot at center of the unit disk or the case of two spots arbitrarily spaced in a domain. Finally, the most interesting open problem is to try to determine a parameter range where a spot-replication \ spot-annihilation “loop” could occur. Such an attractor would consist of a continuously evolving pattern of self-organized spots undergoing repeated cycles of spot self-replication followed by spot-annihilation due to overcrowding instabilities. 147 Chapter 5 Conclusions and Future Work 5.1 Conclusions In this thesis, we have presented results on the dynamics and instabilities of one-dimensional spike patterns and two-dimensional spot patterns in differ- ent parameter regimes for the Gray-Scott model, based on a combination of asymptotic and numerical methods. With regards to the one-dimensional Gray-Scott model in the semi- strong interaction regime D = O(1), we have focused on analyzing two types of oscillatory instabilities of multi-spike patterns in a finite domain, identified in the intermediate feed rate regime O(1) ≪ A ≪ O(ε−1/2). The oscillatory profile instability resulting from a Hopf bifurcation of the large eigenvalues occurs in fast time t, with the spikes evolving on the slow time σ = ε2A2t in (2.12b). Therefore, we formulated a nonlocal eigen- value problem (2.38) associated with the profile instability, based on the quasi-equilibrium solution obtained by freezing the locations of spikes. The threshold τh = O(A4) for the Hopf bifurcation can be calculated explicitly by Principal Results 2.4, with the threshold depending on the instantaneous configuration of spikes in the interval. In contrast, the oscillatory drift in- stability resulting from a Hopf bifurcation of the small eigenvalue occurs on the slow time-scale σ. For this problem, we derive and compute solutions to a Stefan-type problem (2.12) with a moving singular source. This for- mulation allows us to study the behavior of multi-spike pattern away from bifurcation point τtw = O(ε −2A−2) in (2.59). In particular, in Fig. 2.4(c) we have computed a large-scale oscillation in the location of the spike. Accord- ing to the scalings of these two thresholds, we identified two subregimes: when O(1) ≪ A ≪ O(ε−1/3), τh ≪ τtw, the oscillatory profile instability dominates; when O(ε−1/3)≪ A ≪ O(ε−1/2), τtw ≪ τh, the oscillatory drift instability dominates. Therefore, in the semi-strong interaction regime, and with the feed rate A increasing, the primary instability mechanisms of multi- 148 5.1. Conclusions spike patterns are respectively in order: competition instability, oscillatory profile instability (τ > τh), oscillatory drift instability (τ > τtw) and then finally, pulse-splitting. An open problem here is to compute the long time evolution of the oscillatory drift solutions based on the integro-differential equation (2.78), by using a numerical method similar to that used in [76] for a related, but considerably simpler, integro-differential flame-front equation. It is of interest to determine if the spike trajectories can exhibit similar chaotic behavior as was observed in the flame-front model of [76]. For the Gray-Scott model in a two-dimensional domain in the semi- strong interaction regimeD = O(1), we construct the quasi-equilibrium spot patterns by matched asymptotic analysis, and derive a DAE system (3.35) governing the dynamics of a collection of spots. An existence condition A > Af = O(ε(− ln ε)1/2) for the quasi-equilibrium multi-spot solution is required, as a result of a saddle node bifurcation of the parameter A in (3.8). The asymptotic and numerical investigation of the eigenvalue problem (3.23) corresponding to different instability types has shown that, in the limit A = O(ε(− ln ε)1/2), competition and oscillatory profile instabilities can both occur. This regime is similar to the low-feed rate regime for one- dimensional spike patterns. If the feed rate is increased to A = O(ε(− ln ε)), the multi-spot pattern can be unstable to self-replication, which leads to the creation of more spots. Spot-replication criteria have been explicitly calculated in §3.2 in terms of the source strength Sj for each spot, with Σ2 ≈ 4.31, Σ3 ≈ 5.44, and Σ4 ≈ 6.14. Recall that in §3.1, there exists a critical value Sv ≈ 4.78 such that if Sj > Sv, the radially symmetric solution V (ρ) develops a volcano shape, as shown in Fig. 3.1(c). For example, given a large enough feed rate, if the source strength of jth spot satisfies Σ4 > Sj > Σ3 > Sv, it would first expand and form a ring shape, and then break up into three spots, which is similar to the breakup instability discussed in [57] in the same semistrong interaction regime D = O(1) with larger feed rate A. In [57], the stability of stripe and ring patterns with respect to zigzag and breakup instabilities was studied. In the regimes O(ε1/2) ≪ A ≪ O(1) and A = O(1), equilibrium solutions of homoclinic stripes and rings exist. For these solutions, it was shown that a band of unstable modes always occurs for both breakup and zigzag instabilities. These instabilities cannot be prevented unless the domain is with O(ε)≪ O(1) thin. Note that the unstable bands for zigzag and breakup instabilities overlap in a way that a zigzag instability is always accompanied by a breakup instability. 149 5.1. Conclusions Therefore, a ring or stripe pattern always breaks up and leads to a spot pattern. Such spot patterns have much better stability properties. Similar to the one-dimensional case, we can categorize the equilibrium solutions and instability mechanism in terms of an ascending feed-rate parameter A as follows competition of spots, profile oscillation of spots (τ > τh), self- replication of spots, breakup of stripes or rings and zigzag instability of stripes or rings. One remark is that while the first two instabilities reduce the number of spots, spot-replication creates more spots. Moreover, if A≫ O(ε1/2), the equilibrium solution could be ring or stripe, although they are unstable to modes of a certain wave number associated with breakup and zigzag instabilities. This result agrees with our intuition from FIS reaction that with the reagents fed in a high rate, there would be more spots as a results of more intense reaction, and there would be a threshold above which larger localized regions such as stripes or rings could be formed from coagulation of spots. In contrast, in the weak interaction regime D = O(ε2), both u and v are localized variables, stripes and rings can be unstable to a zigzag instability without undergoing breakup instability. As a result, a labyrinth pattern can be generated in this regime. Numerical studies in [57] has shown that patterns in this regime depend very sensitively on the parameters, with spots, rings, stripes and labyrinths being formed through zigzag, breakup or spot-splitting instabilities. One open problem in two-dimensional Gray-Scott model is to compute the equilibrium bifurcation diagram for multi-spot pattern in a finite do- main. Multiple equilibrium locations are possible by calculating x′j(t) = 0, j = 1, · · · , k in (3.35). For each steady-state, we could try to determine the set of initial conditions leading to long-time behavior that approaches this state. This basin of attraction problem is an interesting open question. Another problem is to identify a parameter range where a spot-replication \ spot-annihilation “loop” could occur. Since this continuous process of spot birth by replication and death by competition has been observed numeri- cally in the weak interaction regime D = O(ε2) in [77]. Therefore, it would be interesting to determine whether it can be found, largely based on ana- lytical results, in the regime O(ε2) ≪ D ≪ O(1), by using results for the instability thresholds of spot-replication, competition and profile oscillation, for two-particle interactions. Finally, it would be interesting to study small eigenvalue instabilities associated with the motion of spots. In particular, what instability would occur if there are too many spots on one ring? Is there a Stefan-type problem similar to the one-dimensional case for some range of τ and A? 150 5.2. Application to Other Systems A long term goal is to investigate generic types of instabilities of particle- like solutions to reaction-diffusion systems that have non-variational struc- ture. We would also like to derive conditions on the kinetic F (A,H) and G(A,H) for a general two-component reaction-diffusion model (1.1). In the next section, we propose a few projects as potential future work. 5.2 Application to Other Systems It should be possible to apply our theoretical framework used for the two- dimensional GS model to other reaction-diffusion systems that have numer- ically been shown to exhibit spot self-replication phenomena. We anticipate that our theoretical approach will still be useful in a wide context, but may have to be tailored to each particular new RD model. We now propose a few examples of possible future work in this direction. 5.2.1 Localized Patterns in Cardiovascular Calcification Yochelis et. al.(cf. [105]) have recently modeled cardiovascular calcification by a Gierer-Mainhardt (GM) type system with saturated autocatalytic ki- netics for the activator and an inhibition source term S. In dimensionless form this model is ut = D∆u− u+ u 2v−1 1 + u2 , vt = ∆v − Ev + S +Gu2 , (5.1) where D = 0.005, G = 1, E = 2, and the constant inhibition source S is the control parameter. This problem is motivated by the results of in vitro experiments that vascular-derived mesenchymal stem cells can display self-organized calcified patterns such as labyrinths and spots. In these experiments, the bone mor- phogenetic protein 2 (BMP-2) acts as an activator, and the matrix GLA protein (MGP) acts as the inhibitor, which can be altered by external addi- tion of MGP. In [105], a spatial dynamics approach together with numerical continuation was used to identify the primary instabilities and possible sta- ble and unstable one-dimensional solution branches as a function of the inhibition source S. Then these results were extended to calculate the sec- ondary instabilities and the parameter regions in S in which two-dimensional patterns, such as labyrinths, spots, stripes and mixtures of spots and stripes, can form. It has been shown numerically that labyrinthine patterns arise at lower concentrations of added MGP (corresponding to small S values), while a higher MGP concentration (larger S values) leads to spot patterns. 151 5.2. Application to Other Systems It would be very interesting to develop an analytical theory for the dy- namics and different types of instabilities of such localized patterns by an asymptotic method that is valid in the limit D ≪ O(1). With regards to the spot self-replication instability, the mechanism for this instability is likely to be essentially similar to that for the GS model, as studied in Chapter 3 of this thesis. Such a study would allow for a theoretical understanding of the different types of localized patterns in the GM model (5.1), and to determine the parameter range of the inhibition source S that leads to the formation of calcified patterns. 5.2.2 The Brusselator Model with Superdiffusion Golovin et. al. [34] have studied the effect of superdiffusion on pattern for- mation in a generalized Brusselator model, formulated as ut = ∇αu+ (B − 1)u− U3 +Q2v + B Q u2 + 2Quv + u2v , (5.2a) η2vt = ∇βv −Bu−Q2v − B Q u2 − 2Quv − u2v . (5.2b) Here the superdiffusion term ∇β is defined by its action in Fourier space F [∇β ](k = −|k|βF [u](k). Different from normal diffusion that describes the random walk of molecules, superdiffusion models molecules that have a jump size distribution with infinite moments. Superdiffusive processes are common in plasma and turbulent flows, in surface diffusion, and in diffu- sion in porous media with fluid flow etc. In [34], a linear stability analysis has shown that a Turing instability can occur even when the diffusion of the inhibitor is slower than that of the activator. A set of coupled am- plitude equations was formulated from a weakly nonlinear analysis, which determined the selected hexagonal and striped patterns and their stabil- ity. Moreover, these theoretical results were confirmed by full numerical simulations near stability boundaries. In the computation, a new pattern, involving self replicating spots not described by the analysis, was observed in the parameter region where hexagons and stripes are unstable. It would be interesting to try to extend our theoretical framework developed for the GS model to investigate the self-replicating spot patterns observed in the numerical computations of [34] for (5.2). 5.2.3 A Three-Component Reaction-Diffusion System It would also be interesting to consider three-component reaction-diffusion systems, which presumably allow for more complex spatio-temporal patterns 152 5.2. Application to Other Systems than for two-component systems such as the GS or GM models. One such system was initially introduced by Schenk et. al. [80] to model a planar semiconductor gas-discharge system. The dimensionless system is Ut = ∆U + U − U3 − ε(αV + βW + γ) , (5.3a) τVt = 1 ε2 ∆V + U − V , (5.3b) θWt = D2 ε2 ∆W + U −W . (5.3c) Here 0 < ε ≪ 1, τ > 0, θ ≪ ε−3, and α, β, γ are O(1) constants. This system has two inhibitors V and W with fast diffusivities, which differ only in their reaction-time constants and diffusion scale. The activator U is only weakly coupled at O(ε) to V and W . In [5], the laboratory experiments on a planar gas discharge device have shown the occurrence of birth, death, and scattering of multiple spots in different parameter regimes. This three-component model (5.3) has been mostly studied by numerical simulations, and the dynamics exhibited by the localized patterns agree qualitatively with those seen in actual experiments. There has been some theoretical work for (5.3) for the case of a one- dimensional spatial domain. Doelman et. al. have investigated the exis- tence (cf. [23]) and stability (cf. [37]) of pulse and front patterns in the one-dimensional case, and they have developed an approach by which the existence and stability of front or pulse patterns of 3-component system can be established. Theses studies [23], [37] have provided a detailed mathe- matical analysis and obtained explicit results for various types of bifurca- tions such as saddle-node bifurcations and Hopf bifurcations, which result in breathing oscillations. However, there has been no comprehensive analysis for (5.3) in a two-dimensional spatial domain. It would be interesting to extend our basic approach for the GS model to asymptotically study the ex- istence of spot solutions, the dynamics of spots (spot scattering), and their various instabilities. Based on the instabilities observed from [5], we antici- pate that there will be two types of instabilities; a spot-splitting instability (spot birth) and a competition instability (spot death). 5.2.4 A General Class of Reaction-Diffusion Models An interesting extension of this thesis would be to extend the analysis to consider localized spot patterns arising in a class of singularly perturbed reaction-diffusion models in a two-dimensional domain. For the case of a one- dimensional spatial domain, Doelman and Kaper (cf. [20]) have developed 153 5.2. Application to Other Systems theories for the stability and dynamics of pulse solutions in the semi-strong interaction limit for a class of singularly perturbed RD system. By adopting the theoretical framework of this thesis it should be possible to extend the results of [20] to the case of spot patterns in a two-dimensional domain. 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