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An analysis of localized patterns in some novel reaction diffusion models Gomez, Daniel 2020

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An Analysis of Localized Patterns in Some Novel ReactionDiffusion ModelsbyDaniel GomezB.E., University of Saskatchewan, 2014B.Sc., University of Saskatchewan, 2014M.Sc., University of British Columbia, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)November 2020© Daniel Gomez, 2020The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:An Analysis of Localized Patterns in Some Novel Reaction DiffusionModelssubmitted by Daniel Gomez in partial fulfillment of the requirements for the de-gree of Doctor of Philosophy in Mathematics.Examining Committee:Michael J. Ward, MathematicsSupervisorJuncheng Wei, MathematicsCo-supervisorNeil Balmforth, MathematicsUniversity ExaminerJiahua Chen, StatisticsUniversity ExaminerAdditional Supervisory Committee Members:Colin B. Macdonald, MathematicsSupervisory Committee MemberWayne Nagata, MathematicsSupervisory Committee MemberiiAbstractIn this thesis we investigate strongly localized solutions to systems of singularlyperturbed reaction-diffusion equations arising in several new contexts. The firstsuch context is that of bulk-membrane-coupled reaction diffusion systems in whichreaction-diffusion systems posed on the boundary and interior of a domain are cou-pled. In particular we analyze the consequences of introducing bulk-membrane-coupling on the behaviour of strongly localized solutions to the singularly per-turbed Gierer-Meinhardt model posed on the one-dimensional boundary of a flatdisk and the singularly perturbed Brusselator model posed on the two-dimensionalunit sphere. Using formal asymptotic methods we derive hybrid numerical-asymptoticequations governing the structure, linear stability, and slow dynamics of stronglylocalized solutions consisting of multiple spikes. By numerically calculating sta-bility thresholds we illustrate that bulk-membrane coupling can lead to both thestabilization and the destabilization of strongly localized solutions based on intri-cate relationships between the bulk-membrane-coupling parameters.The remainder of the thesis focuses exclusively on the singularly perturbedGierer-Meinhardt model in two new contexts. First, the introduction of an in-homogeneous activator boundary flux to the classically studied one-dimensionalGierer-Meinhardt model is considered. Using the method of matched asymptoticexpansions we determine the emergence of shifted boundary-bound spikes. By lin-earizing about such a shifted boundary-spike solution we derive a class of shiftednonlocal eigenvalue problems parametrized by a shift parameter. We rigorouslyprove partial stability results and by considering explicit examples we illustratenovel phenomena introduced by the inhomogeneous boundary fluxed. In the sec-ond and final context we consider the Gierer-Meinhardt model in three-dimensionsiiifor which we use formal asymptotic methods to study the structure, stability, anddynamics of strongly localized solutions. Most importantly we determine two dis-tinguished parameter regimes in which strongly localized solutions exist. This isin contrast to previous studies of strongly localized solutions in three-dimensionswhere such solutions are found to exist in only one parameter regime. We tracethis distinction back to the far-field behaviour of certain core problems and formu-late an appropriate conjecture whose resolution will be key in the rigorous study ofstrongly localized solutions in three-dimensional domains.ivLay SummaryThe formation of patterns in biological models is often described by systems ofreaction-diffusion equations describing ways in which diffusing chemicals inter-act with one another. A detailed description of the patterns that emerge in thesesystems is often unattainable without extensive computer simulations. However,when one of the chemicals is assumed to diffuse very slowly, patterns described bystrongly localized solutions emerge. Such solutions are characterized by the con-centration of chemicals at discrete points and can be accurately described using avariety of mathematical techniques. In this thesis we apply and extend these tech-niques to analyze the structure and behaviour of localized solutions in several newmodels.vPrefaceThe content of this thesis is based on the original research conducted by the author,Daniel Gomez, under the supervision of Dr. Michael J. Ward and Dr. JunchengWei. The introduction found in Chapter 1 is original unpublished work of which Iam the sole author.A version of Chapter 2 has been published as a paper titled The linear stabilityof symmetric spike patterns for a bulk-membrane coupled Gierer-Meinhardt modelby Michael Ward, Juncheng Wei, and myself in Volume 18, Issue 2 of the SIAMJournal of Applied Dynamical Systems. The project conception is due to MichaelWard and I conducted derivations and numerical computations. Juncheng Wei wasinvolved in technical discussions and provided valuable feedback throughout thisproject.The content of Chapter 3 is unpublished original work of which I am the soleauthor. A version of this chapter is currently in preparation for publication.Chapter 4 is based on joint work with Juncheng Wei and a version of this chap-ter has been submitted for publication. The rigorous results of §4.3 are primarilydue to Juncheng Wei whereas the derivations, analysis, and numerical simulationsare primarily due to myself.Chapter 5 is based on joint work with Michael Ward and Juncheng Wei. Aversion of this chapter has been submitted for publication. The conceptualizationfor this project is in part due previous work by Michael Ward on D=O(1) regime,and in part due to ongoing research by Juncheng Wei on rigorous existence resultsfor ground states of the core problem. I performed derivations in the D = O(ε−1)regime, performed a detailed analysis of the resulting quasi-equilibrium solutionsand their stability, and performed all numerical simulations.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Classical Linear Stability Analysis . . . . . . . . . . . . . . . . . 41.2 Bulk-Membrane Coupled Models . . . . . . . . . . . . . . . . . 81.3 Strongly Localized Patterns . . . . . . . . . . . . . . . . . . . . . 121.4 Main Contributions and Thesis Outline . . . . . . . . . . . . . . . 152 The Linear Stability of Symmetric Spike Patterns for a Bulk-MembraneCoupled Gierer-Meinhardt Model . . . . . . . . . . . . . . . . . . . 202.2 Spike Equilibrium and its Linear Stability: General AsymptoticTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Asymptotic Construction of N-Spike Equilibrium Solution 242.2.2 Linear Stability of N-Spike Equilibrium Solution . . . . . 28vii2.2.3 Reduction of NLEP to an Algebraic Equation and an Ex-plicitly Solvable Case . . . . . . . . . . . . . . . . . . . 312.3 Symmetric N-Spike Patterns: Equilibrium Solutions and their Sta-bility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1 NLEP Multipliers for the Well-Mixed Limit . . . . . . . . 362.3.2 NLEP Multipliers for the Disk . . . . . . . . . . . . . . . 372.3.3 Synchronous Instabilities . . . . . . . . . . . . . . . . . . 382.3.4 Asynchronous Instabilities . . . . . . . . . . . . . . . . . 452.3.5 Numerical Support of the Asymptotic Theory . . . . . . . 472.4 The Effect of Boundary Perturbations on Asynchronous Instabilities 542.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Localized Spot Patterns in a Bulk-Membrane Coupled BrusselatorModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1 Asymptotic Construction of N-Spot Quasi-Equilibrium . . . . . . 663.1.1 Geometric Preliminaries: Local Geodesic Normal Coordi-nates on the Unit Sphere . . . . . . . . . . . . . . . . . . 673.1.2 Matched Asymptotic Expansions and the Nonlinear Alge-braic System . . . . . . . . . . . . . . . . . . . . . . . . 703.2 Linear Stability on an O(1) Timescale . . . . . . . . . . . . . . . 783.2.1 The m≥ 2 Mode Instabilities . . . . . . . . . . . . . . . . 793.2.2 The m = 0 Mode Instabilities . . . . . . . . . . . . . . . . 813.2.3 Instability Thresholds for Symmetric N-Spot Patterns . . . 833.3 Slow-Spot Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 843.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.4.1 One-Spot Pattern . . . . . . . . . . . . . . . . . . . . . . 933.4.2 Two-Spot Patterns . . . . . . . . . . . . . . . . . . . . . 933.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014 The Singularly PerturbedOne-Dimensional Gierer-MeinhardtModelwith Non-Zero Activator Boundary Flux . . . . . . . . . . . . . . . 1034.1 Quasi-Equilibrium Multi-Spike Solutions and their Slow Dynamics 1064.1.1 Equilibrium Multi-Spike Solutions by the Gluing Method 111viii4.2 Linear Stability of Multi-Spike Pattern . . . . . . . . . . . . . . . 1134.2.1 Reduction of NLEP to an Algebraic System . . . . . . . . 1154.2.2 Zero-Eigenvalues of the NLEP and the Consistency Con-dition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3 Rigorous Stability and Instability Results for the Shifted NLEP . . 1184.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4.1 Example 1: One Boundary Spike at x = 0 with A > 0 andB = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.4.2 Example 2: Two Boundary Spikes with A = B≥ 0 . . . . 1304.4.3 Example 3: Two Boundary Spikes with a One Sided Flux(A≥ 0 and B = 0) . . . . . . . . . . . . . . . . . . . . . 1364.4.4 Example 4: One Boundary and Interior Spike with One-Sided Feed (A≥ 0, B = 0) . . . . . . . . . . . . . . . . . 1404.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455 Localized Patterns in the 3D GMModel . . . . . . . . . . . . . . . . 1485.1 Asymptotic Construction of an N-Spot Quasi-Equilibrium Solution 1505.1.1 The Core Problem . . . . . . . . . . . . . . . . . . . . . 1515.1.2 Derivation of the Nonlinear Algebraic System (NAS) . . . 1535.1.3 Symmetric and Asymmetric N-Spot Quasi-Equilibrium . . 1555.2 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.2.1 Competition and Synchronous Instabilities for the l = 0Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.3 Slow Spot Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 1725.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 1785.5 The Weak Interaction Limit D = O(ε2) . . . . . . . . . . . . . . 1795.6 General Gierer-Meinhardt Exponents . . . . . . . . . . . . . . . . 1835.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189A Bulk-Membrane-Coupled Reaction-Diffusion-Systems are a Lead-ing Order Approximation . . . . . . . . . . . . . . . . . . . . . . . . 204A.1 Geometric Preliminaries . . . . . . . . . . . . . . . . . . . . . . 204ixA.2 Derivation of Bulk-Membrane-Coupled Reaction-Diffusion System 207B Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 212B.1 Green’s Functions in the Well-Mixed Limit and for the Disk . . . 212B.1.1 Uncoupled Membrane Green’s Function . . . . . . . . . . 212B.1.2 Bulk and Membrane Green’s functions in the Well-MixedLimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213B.1.3 Bulk and Membrane Green’s functions in the Disk . . . . 214B.1.4 A Useful Summation Formula for the Disk Green’s Functions214B.2 Derivation of Membrane Green’s Function for the Perturbed Disk 215C Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 223C.1 Derivation of Lemma 3.3.1 . . . . . . . . . . . . . . . . . . . . . 223C.2 Sign of Dynamic Terms . . . . . . . . . . . . . . . . . . . . . . . 225C.3 Linear Stability of the Common Angle Solution . . . . . . . . . . 226D Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 229D.1 Large λI Asymptotics ofFy0(iλI) . . . . . . . . . . . . . . . . . 229D.2 Numerical Support for Stability Conjecture . . . . . . . . . . . . 231D.3 Stability of Asymmetric Two-Boundary Spike Pattern when A = 0 231xList of TablesTable 2.1 K and Dv values at the sampled points in the two panels ofFig. 2.9: (a) Left panel: (τs,τb) = (0.2,2), and (b) Right panel:(τs,τb) = (0.6,2). Table (d) shows the K and Dv values at thesampled points for the disk appearing in the left panel of Fig. 2.12. 52Table 5.1 Core problems and inhibitor decay behaviour for some RD sys-tems. In each case the activator decays exponentially. . . . . . 187xiList of FiguresFigure 1.1 Solid lines indicate values of δ (µ) versus µ for the GM ex-ponents (p,q,m,s) = (2,1,2,0), with Dv = 2, τ = 0.25, andspecified values of Du > 0. Markers indicate the values of themodes µk = pi2k2 for which δ (µk)< 0 and the spatially homo-geneous pattern becomes susceptible to Turing instabilities. . 7Figure 1.2 Snapshots of numerically computed solutions of the one-dimensionalsingularly perturbed Gierer-Meinhardt model with exponents(p,q,m,s) = (2,1,2,0) when τ = 0.25, Dv = 2 and Du = 5×10−2 (top) and Du = 5×10−5 (bottom). . . . . . . . . . . . . 8Figure 2.1 Snapshots of the numerically computed solution of (2.1) start-ing from a 2-spike equilibrium for the unit disk with ε = 0.05,Db = 10, τs = 0.6, τb = 0.1, K = 2, and Dv = 10 (this corre-sponds to point 2 in the left panel of Figure 2.12). The bulk in-hibitor is shown as the colour map, whereas the lines along theboundary indicate the activator (blue) and inhibitor (orange)membrane concentrations. The results show a competition in-stability, leading to the annihilation of a spike. . . . . . . . . . 22xiiFigure 2.2 Snapshots of the numerically computed solution of (2.1) start-ing from a 2-spike equilibrium for the unit disk with ε = 0.05,Db = 10, τs = 0.6, τb = 0.1, K = 0.025, and Dv = 1.8 (thiscorresponds to point 5 in the left panel of Figure 2.12). Thebulk inhibitor is shown as the colour map, whereas the linesalong the boundary indicate the activator (blue) and inhibitor(orange) membrane concentrations. The results show a syn-chronous oscillatory instability of the spike amplitudes. . . . . 23Figure 2.3 Level sets ofM (τb,K) for Gierer-Meinhardt exponents (p,q,m,s)=(3,1,3,0) (left) and (p,q,m,s) = (2,1,2,0) (right). In bothcases the level set value corresponds to a value of τs =M (τb,K).Note also the contours tending to a vertical asymptote, andthe emergence of a horizontal asymptote as τs exceeds somethreshold. Geometric parameters are L = 2pi and A = pi . . . . 41Figure 2.4 Colour map of the synchronous instability threshold D?v in theK versus τb parameter plane for the well-mixed explicitly solv-able case for various values of τs with L = 2pi and A = pi . Thedashed vertical lines indicate the asymptotic predictions for thelarge K threshold branch, while the dashed horizontal linesindicate the asymptotic predictions for the small K thresholdbranch. The unshaded regions correspond to those parametervalues for which synchronous instabilities are absent. . . . . . 42Figure 2.5 Synchronous instability threshold D?v versus K for two pairsof (τs,τb) for a one-spike steady-state (N = 1) in the unit disk(R = 1). The quality of the well-mixed approximation rapidlyimproves as Db is increased. The labels for Db in the rightpanel also apply to the left and middle panels. . . . . . . . . . 44Figure 2.6 Asynchronous instability thresholds Dv versus the coupling Kin the well-mixed limit for different values of L, different (N,k)pairs, and for domain areas A = 3.142 (solid), 1.571 (dashed),and 0.785 (dotted). . . . . . . . . . . . . . . . . . . . . . . . 45xiiiFigure 2.7 Asynchronous instability thresholds Dv versus the coupling Kfor the unit disk with Gierer-Meinhardt exponents (3,1,3,0),and for different Db. The dashed lines show the correspondingthresholds for the well-mixed limit. The legend in the bottomright plot applies to each plot. . . . . . . . . . . . . . . . . . 46Figure 2.8 Comparison between numerical and asymptotic synchronousinstability threshold for N = 1 with L = 2pi , A = pi , τs = 0.6,and τb = 0.01. Notice that, as expected, the agreement im-proves as ε decreases. . . . . . . . . . . . . . . . . . . . . . 49Figure 2.9 Synchronous (solid) and asynchronous (dashed) instability thresh-olds in the Dv versus K parameter plane in the well-mixed limitfor N = 1 (blue), N = 2 (orange), and N = 3 (green). At thetop of each of the three panels a different pair (τs,τb) is speci-fied. See Table 2.1 for Dv and K values at the numbered pointsin each panel. Figures 2.10 and 2.11 show the correspondingspike dynamics from full PDE simulations of (2.1) at the indi-cated points. . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 2.10 Numerically computed spike heights (vertical axis) versus time(horizontal axis) from full PDE simulations of (2.1) for τs =0.2 and τb = 2 at the points indicated in the left panel of Fig-ure 2.9. Distinct spike heights are distinguished by line types(solid, dashed, and dotted). . . . . . . . . . . . . . . . . . . . 53Figure 2.11 Numerically computed spike heights (vertical axis) versus time(horizontal axis) from full PDE simulations of (2.1) for τs =0.6 and τb = 2 at the points indicated in the middle panel ofFigure 2.9. Distinct spike heights are distinguished by linetypes (solid, dashed, and dotted). . . . . . . . . . . . . . . . . 53xivFigure 2.12 Left panel (a): Synchronous (solid) and asynchronous (dashed)instability thresholds in the Dv versus K parameter plane forthe unit disk with Db = 10 and (τs,τb) = (0.6,0.1). N = 1spike and N = 2 spikes correspond to the (blue) and (orange)curves, respectively. The faint grey dotted lines are the corre-sponding well-mixed thresholds. Right panel (b): Numericallycomputed spike heights (vertical axis) versus time (horizontalaxis) from full PDE simulations of (2.1) at the points indicatedin the left panel for N = 1 and N = 2 spikes. For videos of thePDE simulations please see the supplementary materials. . . . 54Figure 2.13 The effect of boundary perturbations on the asynchronous sta-bility of symmetric N-spike patterns for the unit disk. The toprow shows the multiplier MN,k, defined in (2.7), as a functionof K while the bottom row shows the leading order correctionto the asynchronous instability threshold, with the dashed lineindicating the unperturbed threshold. Each column correspondto a choice of Db = 50 or Db = 5 with Gierer-Meinhardt ex-ponents of (p,q,m,s) = (3,1,3,0) or (p,q,m,s) = (2,1,2,0).In the second row the boundary perturbation has parametersξ = 1 (indicating an outward bulge at the spike locations), andδ = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 3.1 Example of geodesic normal coordinates (ζ1,ζ2,ζ3) at xi ∈∂Ω. The blue (resp. orange) curves indicate geodesics ob-tained by varying −pi/2 < ζ1 < pi/2 (resp. −pi < ζ2 < pi) andfixing ζ3 = 0 and ζ2 = 0 (resp. ζ1 = 0). . . . . . . . . . . . . 68Figure 3.2 Numerically calculated (a) far-field constant χ(S, f ), (b) m-mode instability threshold Σm( f ) for 2 ≤ m ≤ 4, and (c) slowdynamics multiplier γ(S, f ). The solid circles in (c) indicatevalues of γ(S, f ) at the splitting instability thresholds Σ2( f ).These functions depend only on the local structure of each spotand are therefore independent of bulk-membrane coupling. . . 72xvFigure 3.3 Plots of θc/pi versus K and f for fixed values of Dw = 1 (toprow) and Dw = 10 (bottom row) and η = 0.3,0.6,0.9 from leftto right. The solid lines with in-line text are contours indicatingfixed values of θc/pi . . . . . . . . . . . . . . . . . . . . . . . 95Figure 3.4 (a) Plot of C′E(cosθc) at K = 0 versus 0 < η < 1 and Dw > 0with the solid orange line indicating values where C′E(cosθc)=0 and demarcating regions where C′E(cosθc)> 0 and C′E(cosθc)<0. (b)-(c) Schematics showing the functions θ ?1 (θ) and θ ?2 (θ)in the absence and presence of a tilted solution. In the lattercase θ1 = θt and θ2 = θ ?1 (θt) are the polar angles of the tiltedsolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 3.5 (a)-(c) Plots of competition, splitting, and tilt instability thresh-olds as Dv/E20 versus K for select of Dw > 0 with η = 0.4,f = 0.6, and ν = 5×10−3. (d)-(f) Plots of the common angleθc, and tilted angles θ1 < θc < θ2 versus K at Dv/E20 = 0.1and with remaining parameters equal to those from (a)-(c) re-spectively. The solid (resp. dashed) line indicates the stability(resp. instability) of the common angle solution with respectto tilt instabilities. . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 4.1 (a) Examples of shifted one-spike solution concentrated at x =0 for various values of A ≥ 0 and with ε = 0.05 and D = 5.(b) Evolution of solution to GM problem with D = 0.6, ε =.005, τ = 0.1, and A = B = 0.08. The initial condition is anunstable two-spike equilibrium where both spikes concentrateat the boundaries. A competition instability predicted by ourasymptotic results in Figure 4.5a is triggered and leads to thesolution settling at an asymmetric pattern. . . . . . . . . . . . 105xviFigure 4.2 (a) Plot of the numerically computed principal and second eigen-values of the operator Ly0 . The dashed vertical line corre-sponds to y0 = y0c. (b) Plot of the stability thresholds µ1 and µ2as functions of y0. The dashed vertical and horizontal lines cor-respond to y0 = y0c and µ = 2 respectively. The NLEP (4.44)has been rigorously demonstrated to be stable in the regionbounded by the curves µ1 and µ2. Note that µ1 and µ2 are in-terchanged as y0 passes through y0c. (c) Plot of µ1(y0)−µc(y0)for 0 ≤ y0 < y0c. The NLEP is unstable for µ < µc and stablefor µ1 < µ < µ2 when 0 ≤ y0 < y0c. It is conjectured that theNLEP is stable for µ > µc. . . . . . . . . . . . . . . . . . . . 124Figure 4.3 Hopf bifurcation threshold and accompanying eigenvalue fora single boundary-spike solution with one-sided boundary fluxA ≥ 0 in (a) the shadow limit, D→ ∞, and (b and c) for finiteD > 0 at select values of 0 ≤ A < q0c. In (a) the dashed verti-cal line corresponds to the threshold A = q0c beyond which noHopf bifurcations occur. . . . . . . . . . . . . . . . . . . . . 127Figure 4.4 Plots of u(0, t) for a one boundary-spike solution with one-sided boundary flux x = 0 (i.e. A ≥ 0 and B = 0) with ε =0.005. Note that increasing the boundary flux A stabilizes thesingle boundary-spike solution for fixed values of D and τ . . . 129Figure 4.5 Plots of (a) A = A(D, l) and (b) thresholds for the existence ofzero, one, or two asymmetric two-boundary-spike solutions inthe presence of equal boundary fluxes considered in Example 2. 132Figure 4.6 Results of numerical simulation of (4.2) using FlexPDE 6 [1]with ε = 0.005, τ = 0.1, and select values of D and A. Ineach plot the solid (resp. dashed) lines correspond tot he spikeheight at x = 0 (resp. x = 1). Both the asymptotically con-structed symmetric (l = 1/2) and asymmetric 0 < l < 1/2 so-lutions were used as initial conditions. See Figure 4.5a forposition of parameter values relative to existence and stabilitythresholds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135xviiFigure 4.7 Plot of A = A(D, l) for Example 3 obtained by solving (4.88)when (a) 0< l < 1/2 and (b) 1/2 < l < 1. The solid curves l =ltopmax(D) and lbotmax(D) indicate the values of l at which A(D, l) ismaximized as well as the competition instability threshold inthe l > 1/2 and l < 1/2 regions respectively. The correspond-ing existence thresholds of A versus D are plotted against D in(c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Figure 4.8 Numerical simulations for Example 2 when ξL > ξR. (a) Out-come of numerical simulation of (4.2) starting from the asym-metric two-boundary-spike pattern constructed using the indi-cated values of D, l, and A. Blue and orange markers indicatethe two-boundary spike pattern settled to the stable two-spikepattern (i.e. with l < ltopmax(D)) or collapsed to a single spikepattern respectively. Black dots indicate values of D, A, andl for which the spike heights are plotted over time in Figures(b) and (c). The left and right dashed vertical lines indicateD = 0.054 and D = 0.204 respectively. In (b) and (c) we plotspike heights at x = 0 (solid) and x = 1 (dashed) at given val-ues of D and A and with initial condition specified by indicatedvalue of l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Figure 4.9 Numerical simulations for Example 2 when ξL < ξR. (a) Out-come of numerical simulation of (4.2) starting from the asym-metric two-boundary-spike pattern constructed using the indi-cated values of D, l, and A. Blue and orange markers indicatethe two-boundary spike pattern settled to the stable two-spikepattern (i.e. with l < lbotmax(D)) or collapsed to a single spikepattern respectively. Black dots indicate values of D, A, andl for which the spike heights are plotted over time in Figures(b) and (c). The left and right dashed vertical lines indicateD = 0.592 and D = 1.508 respectively. In (b) and (c) we plotspike heights at x = 0 (solid) and x = 1 (dashed) at given val-ues of D and A and with initial condition specified by indicatedvalue of l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140xviiiFigure 4.10 Plot of A = A(D, l) obtained by solving (4.88) when (a) 0 <l < 1/3 and (b) 1/3 < l < 1. The curves l = ltopmax(D) andlbotmax(D) indicate the values of l at which A(D, l) is maximizedwhile the curves ltopcomp(D), lbotcomp,1(D), and lbotcomp,2(D) indicatethe competition instability thresholds. The corresponding ex-istence thresholds, Atopmax(D) and Abotmax(D) are plotted in (c). . . 142Figure 5.1 Plots of numerical solutions of the core problem (5.2): (a) µ(S)versus S, as well as the (b) activator V and (c) inhibitor U , ata few select values of S. The value S = S? ≈ 0.23865 corre-sponds to the root of µ(S) = 0. . . . . . . . . . . . . . . . . . 151Figure 5.2 Plots of (a) Sl(Sr) and (b) S′l(Sr) for the construction of asym-metric N-spot patterns. (c) Plots of f (S,θ) for select valuesof θ ≡ n/N. For 0 < θ < 0.5 the function f (S,θ) attains aninterior minimum in Scrit < S < S?. . . . . . . . . . . . . . . . 157Figure 5.3 (a) Illustration of solutions to (5.18) as the intersection be-tween µ(S) and κS. There is a unique solution if κ < κc1 ≡µ(Scrit)/Scrit. (b) Illustration of solutions to (5.23) as the in-tersection between µ(S) and κ f (S,θ) where θ = n/N denotesthe fraction of large spots in an asymmetric pattern. Note thatwhen θ = 0.2 < 0.5 and κ > κc1 ≈ 0.64619 there exist two so-lutions. (c) Plot of κc2−κc1 versus n/N. Observe that κc2−κc1increases as the fraction of large spots decreases. . . . . . . . 159Figure 5.4 Bifurcation diagram illustrating the dependence on κ of thecommon spot strength Sc as well as the asymmetric spot strengthsSr and Sl or S˜r and S˜l . In (a) and (b) we have n/N < 0.5 so thatthere are more small spots than large spots in an asymmetricpattern. As a result, we observe that there can be two types ofasymmetric patterns with strengths Sr and Sl or S˜r and S˜l . In(c) the number of large spots exceeds that of small spots andonly one type of asymmetric pattern is possible. . . . . . . . . 160xixFigure 5.5 (a) Spectrum of the operatorMl defined in (5.35b). The dashedblue line indicates the eigenvalue with second largest real partfor l = 0. Notice that the dominant eigenvalue of M0 is zerowhen S = Scrit ≈ 0.04993, corresponding to the maximum ofµ(S) (see Figure 5.1a). (b) Plot of B(λ ,S). The dashed lineblack indicates the largest positive eigenvalue of M0(S) andalso corresponds to the contour B(λ ,S) = 0. We observe thatB(λ ,S) is both continuous and negative for S > Scrit ≈ 0.04993. 165Figure 5.6 Leading order (a) Hopf bifurcation threshold τh(κ) and (b)critical eigenvalue λ = iλh for a symmetric N-spot pattern ascalculated by solving (5.48) numerically. The leading ordertheory assumes ε|1+ τλ |/D0  1 and is independent of thespot locations. We calculate the higher order Hopf bifurca-tion threshold for an N = 1 spot pattern centred at the originof the unit ball with ε = 0.01 by solving (5.42) directly (noteκ = 3D0). In (c) we see that although the leading order Hopfbifurcation threshold diverges as κ → κc1, going to higher or-der demonstrates that a large but finite threshold persists. . . . 170Figure 5.7 Plot of the numerically-computed multiplier γ(S) as defined inthe slow gradient flow dynamics (5.69). . . . . . . . . . . . . 177Figure 5.8 (a) Leading order Hopf bifurcation threshold for a one-spotpattern. (b) Plots of the spot height v(0, t) from numericallysolving (5.1) using FlexPDE6 [1] in the unit ball with ε = 0.05at the indicated τ and D0 values. . . . . . . . . . . . . . . . . 178Figure 5.9 (a) Plots of the spot heights (solid and dashed lines) in a two-spot symmetric pattern at the indicated values of D0. Resultswere obtained by using FlexPDE6 [1] to solve (5.1) in the unitball with ε = 0.05 and τ = 0.2. (b) plot of three-dimensionalcontours of v(x, t) for D0 = 0.112, with contours chosen at v=0.1,0.2,0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179xxFigure 5.10 (a) Bifurcation diagram for solutions to the core problem (5.70)in the D = ε2D0 regime. (b) Dominant eigenvalue of the lin-earization of the core problem for each mode l = 0,2,3,4, ascomputed numerically from (5.74). . . . . . . . . . . . . . . 180Figure 5.11 Snapshots of FlexPDE6 [1] simulation of (5.1) in the unit ballwith ε = 0.05, D = 16ε2, and τ = 1 and with initial conditiongiven by a single spot solution in the weak interaction limitcalculated from (5.70) with V (0) = 5. The snapshots showcontour plots of the activator v(x, t) at different times where foreach spot the outermost, middle, and innermost contours corre-spond to values of 0.006, 0.009, and 0.012 respectively. Notethat the asymptotic theory predicts a maximum peak height ofv∼ ε2V (0)≈ 0.0125. . . . . . . . . . . . . . . . . . . . . . 182Figure 5.12 Left panel: Plot of µ(S), computed from the generalized GMcore problem (5.76), for the indicated exponent sets (p,q,m,s).Right panel: µ(S) for exponent sets (p,1, p,0) with p= 2,3,4.For each set, there is a unique S= S? for which µ(S?) = 0. Theproperties of µ(S) in Conjecture 5.1.1 for the prototypical set(2,1,2,0) still hold. . . . . . . . . . . . . . . . . . . . . . . . 185Figure 5.13 Plots of the far-field constant behaviour for the (a) Gierer-Meinhardt with saturation, (b) Schnakenberg or Gray-Scott,and (c) Brusselator models. See Table 5.1 for the explicit formof the kinetics F(v,u) and G(v,u) for each model. A zero-crossing of µ(S) at some S > 0 occurs only for the GMS model. 186Figure D.1 (a) Plot of Fy0(0) versus the shift parameter y0. (b) and (c)Real and imaginary parts ofFy0(iλI) for select values of y0 ≥0. The dashed lines indicate the λI  1 asymptotics. . . . . . 230xxiFigure D.2 (a) Plot of the real part of the dominant eigenvalue of the shiftedNLEP (4.44) versus shift parameter y0 and multiplier µ . Thedotted red line corresponds to the critical threshold µc definedin (4.56) and the solid dark line is the zero-contour of Reλ0.(b) Plot of the difference between dominant eigenvalues ofLy0and the NLEP (4.44). . . . . . . . . . . . . . . . . . . . . . . 232Figure D.3 (a) NLEP multipliers for a boundary-boundary and boundary-interior configuration. (b) Plot of l versus D determining asym-metric boundary-boundary and boundary-interior spike patterns.Solid (resp.) dashed lines indicate the two-spike pattern is lin-early stable (resp. unstable) with respect to competition insta-bilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233xxiiAcknowledgementsFirst and foremost, I want to thank my supervisors Michael Ward and JunchengWei for their continued guidance and support throughout the completion of thisthesis. Your boundless enthusiasm for research has and always will be a source ofinspiration and motivation.I would also like to thank the people that I am fortunate to call my supportsystem. For their openness and enthusiasm in talking about life and research Iwant to thank my friends and colleagues at UBC: Aaron, Thomas, Frédéric, Sarafa,and Tony. I want to thank my family for always supporting and encouraging methroughout all of my studies. And last but not least I want to thank Kirin for herlove and unwavering support when I needed it most.My decision to pursue graduate studies was largely influenced by the earlymentorship I received from James Brooke, Jacek Szmigielski, and Alexey Shevyakovat the University of Saskatchewan and for this I will always be grateful. Thank you.Finally I am grateful for the financial support I have received from the NaturalSciences and Engineering Research Council of Canada which first gave me theopportunity to pursue research as an undergraduate student and then allowed me tofocus on research as a graduate student.xxiiiChapter 1IntroductionIn 1952, Alan M. Turing’s paper The Chemical Basis of Morphogenesis [93] laiddown the foundations for a rich and insightful direction of biological and mathe-matical enquiry that continues to this day. Early embryonic development, Turinghypothesized, is guided by pre-patterns of biochemical morphogens that undergopassive diffusion with prescribed reaction kinetics. Turing’s key insight was that,under certain conditions on the morphogens’ diffusivities and their reaction kinet-ics, a spatially homogeneous distribution of morphogens could undergo a symme-try breaking bifurcation resulting in the formation of morphogen patterns. Sur-prisingly, this insight implies that diffusion, which is typically assumed to have asmoothing and stabilizing effect, could instead have a structured coarsening effectleading to the formation of spatial patterns.While the biological implications of Turing’s original theory of morphogenesiscontinue to be influential, it is Turing’s mathematical approach that is most relevantfor this thesis. In particular, to illustrate his insights Turing considered a systeminvolving two morphogens whose concentrations, u and v, satisfy a two-componentreaction-diffusion (RD) system of the formut = Du∆u+ f (u,v), vt = Dv∆v+g(u,v), in Ω, (1.1)where Du and Dv denote the morphogens’ diffusivities and f (u,v) and g(u,v) theirrespective kinetics. With a simple choice of reaction-kinetics reflecting a chemi-1cal reaction, Turing determined conditions under which a spatially homogeneousequilibrium of (1.1) that is stable with respect to the reaction-kinetics may bifur-cate into a spatially heterogeneous equilibrium. Such a transition to a spatiallyheterogeneous equilibrium is the result of what is now commonly referred to as aTuring, or diffusion-driven, instability, while the resulting spatially heterogeneousstates are often referred to as Turing patterns. In addition to discovering this mech-anism for pattern formation, it is also important to highlight that by restrictinghis attention to a system of the form (1.1), Turing distilled the complex and mul-tifaceted problem of pattern formation in biological systems into an analyticallytractable model equation from which key insights about the pattern-forming poten-tial of two interacting mechanisms, mainly passive diffusion and reaction-kinetics,can be inferred. Such an approach is now ubiquitous in mathematical biology (see[68]) and has led to the ongoing application, extension, and refinement of Turing’soriginal ideas.Reaction-diffusion systems exhibiting pattern forming behaviour, often timesthrough a Turing, or Turing-like, instability mechanism, have been proposed in avariety of applications including animal coat markings [67], stripe patterns in themarine angelfish Pomacanthus [50], sea shell patterns [80], early limb develop-ment [64], morphogenesis in the fresh water polyp hydra [23, 62], plant growth[35, 36, 69], ecological models [86], and the formation of crime hot spots [8, 87]among many others (see [59] for a review of chemical and biological applications).These models typically prescribe reaction-kinetics that reflect specific chemical re-actions, such as in the Gray-Scott [30, 31] and Brusselator [79] models, or otherphenomenologically motivated interactions as in the activator-inhibitor model ofGierer and Meinhardt [23]. Originally motivated by the role of lateral inhibition invisual pattern recognition, the Gierer-Meinhardt (GM) model is particularly impor-tant in the literature for its recognition of the importance of autocatalytic feedbackloops and inhibitory reactions, commonly referred to as local self-activation andlong-range inhibition, in biological pattern formation.While some molecular candidates for morphogens have been identified [42](see also [62, 88]) the experimental establishment of a Turing mechanism drivingpattern formation in biological systems remains an area of ongoing research [61,88]. Moreover, theoretical shortcomings of the classical Turing mechanism such2as a sensitivity to initial conditions and a lack of robustness have been recognized[61]. Such shortcomings do not repudiate Turing’s original ideas however, butinstead serve as an opportunity to better understand pattern formation in general.One way this has been done is by incorporating additional mechanisms to reaction-diffusion systems of the form (1.1). For example, studies incorporating growth[10, 14, 25, 52, 78], mechanical feedback [7, 63], and bulk-membrane coupling[54, 58, 81] have shown favourable results. The latter, mainly the introduction ofbulk-membrane coupling, serves as the primary motivation for Chapters 2 and 3 ofthis thesis and is reviewed in more detail in §1.2 below.In addition to serving as a model equation for pattern forming systems, reaction-diffusion systems of the form (1.1) and its extensions discussed above have alsobeen the focus of the application and development of numerous mathematicalideas. While classical stability analysis predicts parameter thresholds beyond whicha spatially homogeneous equilibrium is destabilized, it does not adequately pre-dict the equilibrium, if one exists, to which it settles. Linear stability analysisis therefore often accompanied with full numerical simulations to explore systemdynamics beyond the onset of instabilities [77]. Alternatively, techniques from adynamical systems perspective such as centre manifold reductions [9] and weaklynonlinear analysis [76] can be used to further examine the system dynamics nearthe onset of linear instabilities. However, for parameter values far from the sta-bility thresholds predicted by the linear theory, diverse techniques need to be em-ployed (see [15, 72]). A particular class of reaction-diffusion systems of the form(1.1) for which substantial progress has been made in understanding pattern for-mation far from the onset of Turing instabilities occurs when the ratio betweenthe species, Du/Dv, is asymptotically small. Such reaction-diffusion systems oftenexhibit localized patterns that are characterized by large amplitude solutions com-pactly supported in asymptotically small spatial regions [100] (see also [44] foran overview of localization in general dissipative systems). The natural separationof spatial scales exhibited by these solutions makes them particularly amenable toanalysis by both asymptotic and rigorous reduction methods, both of which havebeen highly successful in characterizing the existence, structure, and stability ofthese solutions. These types of solutions, and especially the asymptotic methodsused to analyze them, underlie the bulk of this thesis and are outlined in §1.3.3In the preceding paragraphs we have outlined several of the theoretical andapplied aspects as well as recent modelling and mathematical developments ofTuring’s original theory of biological pattern formation. Two key aspects fromthis discussion constitute the primary motivation and core technical aspects of thisthesis. The first is the incorporation of bulk-membrane coupling into models ofintracellular processes. The resulting bulk-membrane-coupled (BMC), or bulk-surface-coupled, models explicitly incorporate the inherent compartmentalizationof cytosolic- and membrane- bound biochemical processes often found in intracel-lular processes. These models serve as the primary motivation for Chapters 2 and3 and are therefore reviewed in more detail in §1.2. The second aspect constitutesthe core technical component of this thesis and involves the asymptotic analysisof localized patterns in singularly perturbed reaction-diffusion systems and is re-viewed in more detail in §1.3 below. To more explicitly place both of these aspectsin the context of Turing instability driven pattern formation, we first provide anoutline of the calculation of Turing instability thresholds for the two-componentreaction-diffusion system (1.1) using classical linear stability analysis.1.1 Classical Linear Stability AnalysisThe starting point for the classical analysis of Turing instabilities is the assumptionthat (1.1) admits a spatially homogeneous steady state that is stable with respect tothe reaction-kinetics. This means that constants u? and v? can be found such thatf (u?,v?) = g(u?,v?) = 0, traceJ? < 0, det J? > 0, (1.2)where J? is the Jacobian evaluated at (u,v) = (u?,v?) and explicitly given byJ? =(f ?u f?vg?u g?v).We close (1.1) by imposing homogeneous Neumann boundary conditions so thatthe spatially homogeneous solution is an equilibrium of the closed system. Next welet µk ≥ 0 and φk(x) (k ≥ 0) denote the eigenvalues and accompanying eigenfunc-tions of −∆ in Ω with homogeneous Neumann boundary conditions on ∂Ω. We4recall that φ0 = 1 while the eigenvalues satisfy the ordering 0= µ0 < µ1≤ µ2≤ ·· · .Although we are assuming that (u?,v?) is stable with respect to the spatially homo-geneous k = 0 mode, it may become unstable with respect to k ≥ 1 modes and wetherefore consider perturbations of the formu = u?+ξeλ tφk(x), v = v?+ηeλ tφk(x), (k ≥ 1).Substituting int (1.1) and linearizing yields an eigenvalue problem with character-istic polynomialλ 2− τ(µk)λ +δ (µk) = 0,whereτ(µ) = trace J?− (Du+Dv)µ, δ (µ) = DuDvµ2− (Dug?v +Dv f ?u )µ+det J?,and for which we are interested in conditions under which an unstable root (i.e.with positive real part) can be found. Since µk > 0 and trace J? < 0 we deduce thatτ(µk)< 0 for all k≥ 1 and Turing instabilities are therefore triggered if δ (µk)< 0for some k ≥ 1. From the form of δ (µ) we deduce that a Turing instability willbe triggered provided that Dug?v +Dv f?u is sufficiently large. Indeed, by calculatingthe roots of δ (µ) we determine the necessary condition for a Turing instabilityDvDuf ?u −2√DvDudet J?+g?v > 0. (1.3)We emphasize that this inequality is a necessary condition and in practice the re-quirement that δ (µk) < 0 for some k ≥ 1 often requires that Dv/Du exceeds thethreshold predicted by (1.3). In some cases the requirement for a large diffusiv-ity ratio needed to trigger Turing instabilities is physically prohibitive which hasmotivated the exploration of extensions to the classical Turing mechanism. Mostpertinent to this thesis, and reviewed in more detail in §1.2 below, is the proposalthat bulk membrane coupling can lead to the formation of Turing-like patterns.Together with the assumption that (u?,v?) is stable with respect to k= 0 modes,(1.3) provides us with insights on the reaction-kinetics for which (1.1) admits Tur-ing instabilities. In particular, trace J? < 0 implies that at least one of f ?u or g?v5is negative. Without loss of generality we assume that g?v < 0, with which (1.3)implies f ?u > 0. Furthermore, the condition that det J? > 0 implies that f ?v and g?uare of opposite sign which characterizes activator-inhibitor and activator-substratemodels depending on whether fv > 0 and gu < 0 or fv < 0 and gu > 0 respectively.To directly illustrate the above analysis we consider the one-dimensional GMmodel posed on the unit interval with homogeneous Neumann boundary condi-tions. Specifically, we considerut = Duuxx−u+ upvq, τvt = Dvvxx− v+ umvs, 0 < x < 1, (1.4)with ux(0) = ux(1) = vx(0) = vx(1) = 0 and for which we impose the constraintsp > 1, q > 0, m > 0, and s≥ 0 on the GM exponents. The spatially homogeneoussteady state is then given by u? = v? = 1 with which the Jacobian is explicitly givenbyJ? =(p−1 −qmτ−1 −(1+ s)τ−1),and from which we deduce that (u,v) = (u?,v?) is linearly stable with respect tothe reaction-kinetics provided that0 < τ <1+ sp−1 , 0 <p−1q<m1+ s. (1.5)Calculatingδ (µ) = DuDvµ2−((p−1)Dv− 1+ sτ Du)µ+mq− (p−1)(1+ s)τ, (1.6)we then determine that (1.3) impliesDvDu>2mq(p−1)2τ(1− (1+ s)(p−1)2mq+√1− (1+ s)(p−1)mq). (1.7)Since µk = (pik)2 and φk(x) = cospikx (k ≥ 0), the system (1.4) is susceptible toTuring instabilities provided that δ (pi2k2)< 0 for some k ≥ 1.In the commonly studied case where the GM exponents are given by (p,q,m,s)=60 5 10 1542024Du=5×10 20 100 200100500Du=5×10 30 10000 2000010000500005000Du=5×10 5Figure 1.1: Solid lines indicate values of δ (µ) versus µ for the GM expo-nents (p,q,m,s) = (2,1,2,0), with Dv = 2, τ = 0.25, and specified val-ues of Du > 0. Markers indicate the values of the modes µk = pi2k2for which δ (µk) < 0 and the spatially homogeneous pattern becomessusceptible to Turing instabilities.(2,1,2,0) we calculate that τ < 1 in order for the spatially homogeneous solutionto be linearly stable with respect to the reaction kinetics. On the other hand, af-ter fixing τ = 0.25, (1.7) implies that Dv/Du > 23.4 is a necessary condition forTuring instabilities. Fixing Dv = 2 we then find that as Du is decreased the k = 1mode is the first to go unstable at a value of Du ≈ 0.067 for which Dv/Du ≈ 29.8.In Figure 1.1 we plot δ (µ) for values of Du = 5× 10−2,5× 10−3,5× 10−5 forwhich we note that, respectively, the k = 1, 1 ≤ k ≤ 4, and 1 ≤ k ≤ 45 modes arelinearly unstable. In Figure 1.2 we plot the solution at discrete times obtained bynumerically solving (1.4) when Du = 5× 10−2 and Du = 5× 10−5. In the formercase only the k = 1 mode is linearly unstable and this is qualitatively reflected inthe solution’s time evolution. Using the proximity of the diffusivity ratio Dv/Du tothe Turing instability threshold a weakly nonlinear analysis could be used to trackthe sole unstable mode and effectively characterize the resulting equilibrium solu-tion. On the other hand, when Du = 5× 10−5 our previous calculations indicatethat 45 distinct spatial modes are linearly unstable. These modes interact nonlin-early eventually leading to the formation of the spiky solution shown in Figure 1.2.In particular, this illustrates that perturbation techniques using the closeness of thediffusivity ratio to the Turing instability threshold fail to hold when Du  Dv, inpart due to the emergence of a large band of unstable spatial modes. However, theresulting numerically computed solution illustrates that this large band of unstablemodes leads to a localized solution and suggests that perturbation methods that70.0 0.5 1.0x0.000.250.500.751.00t=0uv0.0 0.5 1.0x0.000.250.500.751.00t=5uv0.0 0.5 1.0x0.00.51.01.52.0t=50uv0.0 0.5 1.0x0.000.250.500.751.00t=0uv0.0 0.5 1.0x0510t=5uv0.0 0.5 1.0x0102030t=50uvFigure 1.2: Snapshots of numerically computed solutions of the one-dimensional singularly perturbed Gierer-Meinhardt model with expo-nents (p,q,m,s) = (2,1,2,0) when τ = 0.25, Dv = 2 and Du = 5×10−2(top) and Du = 5×10−5 (bottom).exploit large disparities between spatial scales can instead be used approximatethe resulting solution. Indeed, a combination of both rigorous and formal asymp-totic methods have been used to study such localized solutions and we provide anoutline of these developments in §1.3 below.1.2 Bulk-Membrane Coupled ModelsOne of the key biological motivation for the recent interest in mathematical modelsincorporating bulk-membrane coupling is the naturally occurring compartmental-ization of cytosol- and membrane-bound processes found in complex intracellularself-organizing processes. One classic example of such compartmentalization isthe Min System in which the dynamics of cytosol- and membrane-bound MinC,MinD, and MinE are believed to drive the positioning and localization of the con-tractile Z-ring ultimately leading to cell septation [37]. Another classic examplecan be found in the establishment of cell polarity by the Rho family of guanosine-triphosphate binding proteins (GTPases) for which GTPases undergo activationand inactivation on the cell membrane in addition to cell membrane attachment8and detachment [65, 81]. While early mathematical models of these intracellu-lar processes recognized the compartmentalization of bulk- and membrane-boundbiochemical agents by separately modelling their respective concentrations andreaction kinetics, the resulting mathematical models did not explicit model cellmembrane attachment and detachment [37, 38, 65]. However in 2005 Levine andRappel studied the formation of Turing-like patterns in a bulk-membrane-coupled(BMC) reaction-diffusion (RD) system and found that the details of the membraneattachment and detachment process can have a pronounced effect on pattern for-mation [54]. This discovery has motivated numerous additional investigations ofpattern formation in a new class of BMC RD systems. In particular, the systematicincorporation of bulk-membrane-coupling has been introduced into new models ofthe Min System [5, 6, 33] and GTPase driven cell polarization [16, 17, 29, 75, 81].More generally Halatek et. al. [34] have proposed that the key mechanism under-lying intracellular pattern formation involves the mass-conserving redistributionof proteins by cytosolic diffusion together with the cycling of proteins betweenbulk- and membrane-bound states. Such a restriction to mass-conserved BMC RDsystems however neglects the synthesis of proteins or other signalling moleculeswithin the cytosol. In this context it remains pertinent to understand pattern for-mation in more general BMC RD systems incorporating cytosolic synthesis byintroducing, for example, cytosol-bound reaction-kinetics [58] or cytosol-boundspatial inhomogeneities (see Chapter 3).In the remainder of this section we outline several key recent developmentsin the analysis of Turing-like patterns in BMC RD systems. First however weintroduce the general mathematical description of BMC models. Letting Ω be abounded domain in RN (N ≥ 2) a BMC RD system governing the concentrationsU = (U1, ...,Un)T and u = (u1, ...,um)T of n bulk-bound and m membrane-boundspecies respectively is given byτB∂tU = DB∆U +F (U ), in Ω, (1.8a)DB∂nU = γq(u,U ), on ∂Ω, (1.8b)τM∂tu = DM∆∂Ωu+ f (u)−q(u,U ), in ∂Ω, (1.8c)where τB and DB are matrices consisting of the time constants and diffusivities of9the bulk-bound species while τM and DM those of the membrane-bound species, Fand f reflect the bulk- and membrane-bound kinetics respectively, and q reflectsthe membrane attachment-detachment process while γ > 0 accounts for asymme-tries in the exchange. With some notable exceptions (see [16, 17]) authors studyingBMC-RD models have primarily modelled the membrane attachment-detachmentprocess as a Langmuir process (see §4 of [43]) which is reflected by choosing q as alinear function of the membrane- and bulk-bound concentrations. In particular thisleads to Robin boundary conditions and linear terms in the bulk-bound problemand membrane equations respectively. Unless otherwise specified, the models wediscuss in the remainder of this section use a Langmuir process to describe mem-brane attachment-detachment. Finally we note that (1.8) is to be understood as aleading order approximation of a system of two N-dimensional RD systems, oneposed inside of the bulk and the other posed on a thin external protrusion, in thelimit of the external protrusions thickness going to zero. The systematic derivationof this leading order approximation is presented in Appendix A.In the two-component BMC-RD system considered by Levine and Rappel [54]the membrane-bound diffusivities were assumed to be identically zero in order toreflect experimental observations that indicate membrane-bound diffusivities aremuch smaller than bulk-bound diffusivities [55]. In addition the authors assumedthat within the bulk the two species only diffuse and decay but do not react, withreactions being instead restricted to the membrane. Under these assumptions thetwo-component BMC-RD system reduces to a system coupling two bulk-boundlinear partial differential equations (PDEs) to two membrane-bound nonlinear or-dinary differential equations (ODEs). For this simplified model Levine and Rappelperformed a linear stability analysis about an equilibrium that is spatially homoge-neous on the boundary and demonstrated that even if the bulk-bound diffusivitiesof two species are equal, bulk-membrane-coupling can lead to pattern formation onthe membrane. Thus, bulk-membrane-coupling provides an attractive extension toclassically studied models exhibiting Turing instabilities which otherwise requirelarge differences in diffusivities to initiate pattern formation.Motivated both by the natural compartmentalization between bulk- and membrane-bound processes and by the weakened restrictions for Turing-like instabilities illus-trated by Levine and Rappel [54], numerous additional studies have explored the10effects of bulk-membrane-coupling on pattern formation. In the context of sym-metry breaking in cell signalling networks, Rätz and Röger have studied the for-mation of Turing-like patterns in a two-component BMC-RD system with nonzeromembrane-bound diffusion in which only one species detaches into the bulk whereit undergoes passive diffusion and bulk-decay [81, 83]. Similarly, Madzvamuseet. al. [57, 58] studied the formation of Turing-like patterns in a general two-component BMC-RD system with diffusion and reactions in both the bulk and themembrane. In both the studies by Rätz and Rögers and those of Madzvamuse et. al.,criteria for Turing-like instabilities were derived using an analogue of the classicallinear stability analysis reviewed in §1.1 above. Moreover the authors in both stud-ies used numerical methods, a phase-field approach in the former and a BMC finite-element method in the latter, to verify their linear stability predictions and explorethe resulting patterns formed beyond the onset of linear instabilities. The effectsof cell shape and diffusion barriers for two-dimensional BMC-RD systems wereinvestigated by Giese et. al. [24] through extensive in silico experimentation on thecell polarization models of Goryachev et. al. [29], for which a Turing-like instabil-ity drives polarization, as well as that of Mori et. al. [65], in which wave-pinningis the central polarizing mechanism. We conclude by noting the studies of Dieg-miller et. al.[17], as well as Cussedu et. al. [16], for which the wave-pinning modelof Mori et. al. [65] was extended to explicitly model bulk-membrane coupling inthree-dimensional domains. These latter models consist of a single-componentBMC-RD system for which membrane attachment-detachment is modelled by anonlinear Hill function. In contrast to the self-activation and lateral inhibition re-quired for Turing instabilities, the wave-pinning mechanism instead relies on theinterplay between bi-stability and mass conservation.With the exception of the analysis of the wave-pinning model found in [16, 17],the analysis of Turing-like patterns in the BMC-RD systems discussed above havebeen primarily limited to in silico experiments and linear stability analysis. Whileweakly nonlinear analysis and the derivation of amplitude equations have beenused to analyze dynamics beyond the onset of linear instabilities [73, 74] there areno studies analyzing of patterns formation far from equilibrium in BMC-RD sys-tems. This gap serves as the primary motivation for the contents of Chapters 2 and3 for which pattern formation is investigated in two singularly perturbed BMC-RD11systems. In particular, in these chapters we study the effects of incorporating bulk-membrane coupling into two well studied singularly perturbed RD systems: theone-dimensional GM model, and the two-dimensional Brusselator model posed onthe unit sphere. These two models are part of a large class of singularly perturbedRD models that exhibit strongly localized solutions for which substantial rigorousand asymptotic developments have been made in the past two decades and whichwe outline in more detail in the next section.1.3 Strongly Localized PatternsAlthough a detailed analysis of far-from-equilibrium solutions to general two-component reaction-diffusion systems of the form (1.1) is typically intractable,substantial progress has been made for a wide class of singularly perturbed reaction-diffusion systems of the formut = ε2∆u+ f (u,v), τvt = D∆v+g(u,v), x ∈Ω, (1.9)where Ω ⊂ Rd (d ≥ 1) and ε  1 is an asymptotically small parameter. Specifi-cally, when the reaction kinetics f (u,v) and g(u,v;ε) are of activator-inhibitor oractivator-substrate type (see §1.1) the system (1.9) exhibits strongly localized so-lutions in the sense that the activator u(x, t) is of a large amplitude in asymptoticallysmall spatial regions. This separation of spatial scales makes (1.9) particularly wellsuited to analysis by both formal and rigorous reduction methods. In this sectionwe outline some of the key developments in the analysis of strongly localized so-lutions to (1.9).Historically, the Gierer-Meinhardt (GM) model given byut = ε2∆u−u+ upvq, τvt = D∆v− v+ umvs, x ∈Ω⊂ Rd (1.10)∂nu = 0, ∂nv = 0, x ∈ ∂Ω, (1.11)where d ≥ 1 and the GM exponents (p,q,m,s) satisfy (1.5), has been one of theprimary models driving the development of both formal and rigorous techniquesin the study of strongly localized solutions. Early work on the GM model focused12on the shadow-limit obtained by letting D→ ∞ and for which v(x, t)→ ξ (t) isspatially constant. By an appropriate rescaling, steady state solutions to (1.10) inthe shadow limit can be found by solving the single equationε2∆u−u+up = 0, x ∈Ω, ∂nu = 0, x ∈ ∂Ω, (1.12)which has a variational structure with energy functionalJε [u]≡ˆΩ(ε22|∇u|2+ 12u2− 1p+1up+1+)dx, u+ ≡max(0,u).In this case both formal asymptotic [39, 40, 102] and rigorous [32, 51, 107] (seealso the review articles [71, 108] and book [112]) methods have been used toconstruct multi-spike equilibrium solutions and study their stability. Interestingly,while many multi-spike equilibrium solutions exist in the shadow limit, only thoseequilibrium solutions consisting of a single spike concentrating on the boundaryat a non-degenerate local maximum of the mean curvature are stable. In contrast,by using the method of matched asymptotic expansions when D > 0 is finite Ironet. al. [41] demonstrated that symmetric multi-spike equilibrium solutions to (1.10)when d = 1 are stable provided that D is sufficiently small. Similar methods werealso used by Ward and Wei [104] to construct asymmetric multi-spike solutions.In contrast to the shadow limit, the equilibrium system for (1.10) when D > 0is finite does not have a variational structure thereby limiting the availability ofrigorous techniques for studying the existence and stability of multi-spike equilib-rium solutions. Such rigorous results have nevertheless been established for d = 1[20, 90, 111] and d = 2 [45, 110] dimensional domains.While many of the formal asymptotic studies discussed above were in the con-text of the singularly perturbed GM model (1.10), the techniques used have beensuccessfully applied to study localized solutions in various d = 1, d = 2, andd = 3 dimensional singularly perturbed RD systems such as the one- and two-dimensional Gray-Scott [12, 47] and Brusselator [96, 97] models, the Brusselatorand Schnakenberg model on the surface of a sphere [84, 91] and torus [95] re-spectively, as well as the three-dimensional Schnakenberg model [98]. In each ofthese studies the method of matched asymptotic expansions is used to reduce the13problem of calculating an N-spike equilibrium solution to that of calculating theN spike locations as well as N parameters that determine the local spike profiles.In particular, the N spike profile parameters are found by solving a nonlinear alge-braic system arising from a leading order matching condition, whereas the N-spikelocations are found by calculating equilibrium configurations to an ODE systemarising from a higher order solvability condition and describing slow spike dynam-ics. While the method of matched asymptotic expansions typically proceeds in asimilar way for constructing localized spike solutions, differences in the choice ofreaction-kinetics and dimension of the domain lead to pronounced technical differ-ences in the details of the analysis. In particular the key role of the dimension ofthe domain arises through the free-space Green’s function satisfying∆G f =−δ (x−ξ ), x ∈ R f , (1.13)and the reduced wave Green’s function satisfying∆G−κ2G =−δ (x−ξ ), x ∈Ω⊂ Rd , (1.14)with appropriate boundary conditions if ∂Ω 6= /0, both of which are prominentlyfeatured in the method of matched asymptotic expansions with the former deter-mining the far-field behaviour of the inner solution and the latter playing a key rolein the construction of outer solutions. Moreover, the choice of reaction kineticsand order of D and τ with respect to ε  1 leads to pronounced qualitative andquantitative differences in the formulation of the appropriate inner problem.In addition to differences in the details of the method of matched asymptoticexpansions, different choices of reaction-kinetics and parameter values also leadto diverse dynamics of multi-spike solutions. In particular multi-spike solutionsmay undergo oscillatory, competition, and splitting instabilities depending on thechoice of parameter values and reaction-kinetics. Moreover, detailed thresholdsfor both Hopf and competition instabilities can be calculated by using the methodof matched asymptotic expansions to derive a Globally Coupled Eigenvalue Prob-lem (GCEP) which, for certain reaction-kinetics and parameter regimes, can befurther reduced to a Nonlocal Eigenvalue Problem (NLEP). While GCEPs are typ-14ically analytically intractable, a substantial collection of rigorous stability resultsfor various NLEPs have been established [99, 107, 112]. In cases where suchrigorous results are not applicable, instability thresholds can nevertheless be cal-culated by using a numerically-aided winding number argument [105] as well asstandard root-finding algorithms. Additionally, we remark that under appropriateconditions NLEPs can be explicitly solvable whereby they can be reduced to sim-pler algebraic equations [66, 70]. Finally, due to the fast decay of higher-modelocalized perturbations when d ≥ 2, splitting instabilities are independent of globalcontributions and instead depend only on the local spike profile. We note that thatsplitting instabilities when d = 1 arise through a different mechanism altogether[46]. While the instability thresholds calculated in these studies predict the onsetof linear instabilities, recent progress has been made to determine the criticality ofHopf [26, 28, 101] bifurcations and splitting [113] instabilities.1.4 Main Contributions and Thesis OutlineAs discussed in §1.2, BMC-RD systems provide an attractive extension to classicalTuring-instability driven pattern forming models. However, with the exception ofstudies performing a weakly nonlinear analysis near the onset of Turing-like insta-bilities, the majority of BMC-RD system studies have focused on the calculationof linear Turing-like instability thresholds predicting the onset of spatial instabili-ties near analogues to spatially homogeneous steady states. With patterns arisingfar from the onset of Turing-like instabilities in BMC-RD systems being primarilyexplored through in silico experiments there is a gap in our detailed understandingof such far-from-equilibrium patterns. By using the formal asymptotic methodsthat have been successfully used to develop a detailed understanding of stronglylocalized solutions to singularly perturbed RD-systems as outlined in §1.3 we aimto develop an analogous understanding of strongly localized patterns in singularlyperturbed BMC-RD systems in the first part of this thesis. Specifically, by in-troducing bulk-membrane-coupling to two well-studied singularly perturbed RDsystems, mainly the one-dimensional GM model and the Brusselator model posedon the unit sphere, we investigate the effects of bulk-membrane-coupling on boththe structure of multi-spike equilibrium solutions as well as their linear stability.15The analysis of these two singularly perturbed BMC-RD systems is pursued inChapters 2 and 3 with our main contributions outlined in more detail below.In the remainder of the thesis, mainly Chapters 4 and 5, we pursue the anal-ysis of strongly localized solutions for the GM model in two new context: ina one-dimensional domain with an inhomogeneous activator boundary flux, andin a three-dimensional domain. While numerous studies have considered multi-spike solutions for the one-dimensional GM system with homogeneous Neumann[41, 105] and Robin [60] boundary conditions, the effects of an inhomogeneousboundary flux have not yet been investigated. In 2018 Tzou et. al. [96] consideredthe effects of a non-zero boundary flux for the inhibitor, but a similar investigationfor the activator has not been pursued. With a growing interest in bulk-membranecoupling, understanding the effects of non-zero boundary fluxes for both activatorand inhibitor is increasingly important. By considering a non-zero activator bound-ary flux for the one-dimensional GM system in Chapter 4 we initiate this line ofinvestigation as outlined in more detail below. The final context in which we studystrongly localized solutions is the three-dimensional GM model. Although ouranalysis is heavily influenced by the work of Tzou et. al. [98] in which stronglylocalized solutions of the three-dimensional Schnakenberg model are analyzed,our analysis provides key new insights into the formation of strongly localizedsolutions in three-dimensional systems in general. Specifically, in contrast to theSchnakenberg model for which localized solutions can only be constructed in theD = O(ε−1) regime, we find that localized solutions can be constructed in boththe D = O(ε−1) and D = (1) regimes for the three-dimensional GM model. Wetrace this distinction back to the far-field behaviour of a particular core problemand by calculating this far-field behaviour numerically we formulate several keyconjectures. The formulation of these conjectures as well as the detailed asymp-totic analysis of localized patterns in the three-dimensional GM model is pursuedin Chapter 5 as outlined in more detail below.The detailed outlines of the remaining chapters of this thesis are as follows.In Chapter 2 we analyze a BMC PDE model in which a scalar linear two-dimensional bulk diffusion process for the inhibitor is coupled to the classicallystudied activator-inhibitor GM model posed on the domain boundary. In the sin-gularly perturbed limit of a long-range inhibition and short-range activation for the16membrane-bound species we use formal asymptotic methods to analyze the exis-tence of localized steady-state multi-spike membrane-bound patterns and to derivea nonlocal eigenvalue problem (NLEP) characterizing instabilities of these pat-terns. A novel feature of this NLEP is that it involves a membrane Green’s functionthat is coupled nonlocally to a bulk Green’s function. By considering two specialcases, mainly when the domain is a disk or when the bulk-bound inhibitor diffusiv-ity is infinitely large, we can calculate this membrane Green’s function explicitlywhich allows for the use of a hybrid analytical-numerical approach for determiningunstable spectra of the NLEP. This analysis reveals how bulk-membrane couplingmodifies the well-known linear stability properties of multi-spike equilibrium solu-tions to the singularly perturbed one-dimensional GM mode in the absence of bulk-membrane-coupling. In particular, bulk-membrane-coupling is shown to exhibitboth stabilization and destabilization with respect to either oscillatory instabilitiesdue to Hopf bifurcations or competition instabilities arising due to zero-eigenvaluecrossings. Moreover, in the case of oscillatory instabilities our analysis reveals anintricate dependence on the coupling parameters as well as the diffusivity and time-scale constant of the bulk-bound inhibitor. Finally, linear stability predictions fromthe NLEP analysis are confirmed with full numerical finite element simulations ofthe coupled PDE system. We remark that our approach is valid in more generalsettings than the disk or the well-mixed shadow system, with the key hurdle beingthe computation of the relevant Green’s functions. By restricting our detailed anal-ysis to these two specialized cases, we can bypass the computational challenges ofcalculating the Green’s functions and therefore focus instead on the novel effectsof coupling on the construction and stability of multi-spike solutions.In Chapter 3 we incorporate bulk-membrane-coupling to the Brusselator modelposed on the unit sphere by coupling it to a passive diffusion process with an in-homogeneous source for the activator within the bulk. Motivated by studies of theMin and GTPase systems, for which proteins and signalling molecules originatingin the bulk attach to the membrane, our model proposes a mechanism whereby abulk-bound activator source term is transported to the membrane by diffusion. Theresulting membrane-bound feed term substitutes the external feed term included inthe classically studied Brusselator model required for sustaining pattern formation.Our model therefore proposes a mechanism by which pattern sustaining feed terms17can be introduced in a self-contained manner. In the singularly perturbed limitwhere the membrane-bound activator diffusivity is asymptotically small, we useformal asymptotic methods to construct and study the stability and slow dynamicsof localized solutions. In particular we derive a nonlinear algebraic system, glob-ally coupled eigenvalue problem, and system of ODEs that determine, respectively,the structure, stability and slow dynamics of a multi-spot solution. Furthermore wehighlight the key differences introduced by bulk-membrane coupling in compari-son to previous studies of the uncoupled Brusselator model on the sphere [84, 91]and unit disk [96]. In particular, we find that changes to the linear stability dueto bulk-membrane-coupling primarily result from a recirculation mechanism forthe membrane-bound activator. This recirculation effect also introduces an attrac-tive term to the slow dynamics, but we show that it is weaker than the classicallyobserved coupling-independent repulsive term. Finally, analogously to results ob-tained for the Brusselator on a two-dimensional disk [96], we find that spots areattracted to local maximum points of the membrane-bound, bulk-originating, feedterm.In Chapter 4 we study the effects of an inhomogeneous activator boundary fluxon the existence, linear stability, and slow dynamics of multi-spike solutions to thesingularly perturbed GM model on the unit interval in the singularly perturbed limitof an asymptotically small activator diffusivity ε2  1. Specifically, we use themethod of matched asymptotic expansions to construct multi-spike solutions usingtwo classical methods pioneered in [39, 104]. One of the novel aspects introducedby assuming inhomogeneous Neumann boundary conditions for the activator is thatit necessitates the concentration of spikes at the boundaries. Furthermore thesespikes are parameterized by a shift parameters which plays a central role in thelinear stability of these boundary-bound spikes. Proceeding with standard methodspreviously used for the singularly perturbed one-dimensional GM model we derivea system of NLEPs governing the linear stability on an O(1) timescale as well asa system of ODEs governing slow spike dynamics on an O(ε−2) timescale. In thesimplest case of a single boundary-bound spike we formulate a scalar shifted NLEPfor which we establish partial stability results rigorously. Finally, by applying theasymptotically derived structure, linear stability, and slow dynamic results as wellas full numerical simulations to examples of two-spike configurations involving18both boundary and interior spike we highlight some of the novel phenomena thatarise.In Chapter 5 we study the existence, linear stability, and slow dynamics of lo-calized multi-spot solutions to the GM model in an arbitrary three-dimensional do-main in the singularly perturbed limit of an asymptotically small activator diffusiv-ity ε2 1. Using the method of matched asymptotic expansions we determine thatin the D =O(1) only symmetric multi-spike patterns can be constructed and theseare always linearly stable on an O(1) timescale. In contrast, in the D = O(ε−1)regime we find that both symmetric and asymmetric multi-spike patterns can beconstructed. However we show that the asymmetric patterns are always linearlyunstable on an O(1) timescale whereas the symmetric patterns may, upon ex-ceeding certain numerically computed thresholds, undergo oscillatory instabilitiesthrough a Hopf bifurcation or competition instabilities through a zero-eigenvaluebifurcation on an O(1) timescale. Both of these instability predictions are sup-ported by full numerical simulations of the three-dimensional GM model using thefinite-element software FlexPDE 6 [1]. Furthermore, the existence of multi-spotsolutions in both the D = O(1) and D = O(ε−1) regimes is traced back to thefar-field behaviour of a certain core problem which we may compute numericallyand from which we formulate several key conjectures. Additionally, we derivea system of ODEs governing the slow dynamics of multi-spot solutions over anO(ε−3) timescale in both the D =O(1) and D =O(ε−1) regimes. Finally, by per-forming a linear stability analysis we determine that multi-spike solutions in theweak-interaction D=O(ε2) regime may undergo peanut-splitting instabilities andwe numerically demonstrate that this leads to a cascade of self-replication events.19Chapter 2The Linear Stability ofSymmetric Spike Patterns for aBulk-Membrane CoupledGierer-Meinhardt ModelThe primary goal of this chapter is to initiate detailed asymptotic studies of stronglylocalized patterns in coupled bulk-surface RD systems. To this end, we introducesuch a PDE model in which a scalar linear 2-D bulk diffusion process is coupledthrough a linear Robin boundary condition to a two-component 1-D RD systemwith Gierer-Meinhardt (nonlinear) reaction kinetics defined on the domain bound-ary or “membrane”. Similar, but more complicated, coupled bulk-surface models,some with nonlinear bulk reaction kinetics and in higher space dimensions, havepreviously been formulated and studied through either full PDE simulations orfrom a Turing instability analysis around some patternless steady-state (cf. [81],[82], [83], [58], [58], [83], [56]). Our coupled model, formulated below, pro-vides the first analytically tractable PDE system with which to investigate howthe bulk diffusion process and the bulk-membrane coupling influences the exis-tence and linear stability of localized “far-from-equilibrium” (cf. [72]) steady-statespike patterns on the membrane. In the limit where the bulk and membrane are20uncoupled, our PDE system reduces to the well-studied 1-D Gierer-Meinhardt RDsystem on the membrane with periodic boundary conditions. The existence andlinear stability of steady-state spike patterns for this limiting uncoupled problem iswell understood (cf. [107], [41], [20], [21], [105]).Our model is formulated as follows: Given some 2-D bounded domain Ω wepose on its boundary an RD system with Gierer-Meinhardt kinetics∂tu = ε2∂ 2σu−u+up/vq, 0 < σ < L, t > 0, (2.1a)τs∂tv = Dv∂ 2σv− (1+K)v+KV + ε−1um/vs, 0 < σ < L, t > 0, (2.1b)where σ denotes arc length along the boundary of length L, and where both u andv are L-periodic. In Ω we consider the linear 2-D bulk diffusion processτb∂tV = Db∆V −V, x ∈Ω, Db∂nV +KV = Kv, x ∈ ∂Ω, (2.1c)where the coupling to the membrane is through a Robin condition. The Gierer-Meinhardt exponent set (p,q,m,s) is assumed to satisfy the usual conditions (cf. [41,107])p > 1, q > 0, m > 0, s≥ 0, 0 < p−1q<ms+1. (2.2)In this model τb and τs are time constants associated with the bulk and membranediffusion process, Db and Dv are the diffusivities of the bulk and membrane in-hibitor fields, and K > 0 is the bulk-membrane coupling parameter.The remainder of this chapter is organized as follows. In §2.2 we use themethod of matched asymptotic expansions to derive a nonlinear algebraic systemfor the spike locations and heights of a multi-spike steady-state pattern for themembrane-bound species. A singular perturbation analysis is then used to de-rive an NLEP characterizing the linear stability of these localized steady-states toO(1) time-scale instabilities. A more explicit analysis of both the nonlinear al-gebraic system and the NLEP requires the calculation of a novel 1-D membraneGreen’s function that is coupled nonlocally to a 2-D bulk Green’s function. Al-though intractable analytically in general domains, this Green’s function problem21is explicitly studied in two special cases: the well-mixed limit, Db 1, for the bulkdiffusion field in an arbitrary bounded 2-D domain with C2 boundary, and when Ωis a disk of radius R with finite Db.In §2.3 we restrict our steady-state and NLEP analysis to these two specialcases, and consider only symmetric N-spike patterns characterized by equally-spaced spikes on the 1-D membrane, for which the nonlinear algebraic systemis readily solved. In this restricted scenario, by using a hybrid analytical-numericalmethod on the NLEP we are then able to provide linear stability thresholds foreither synchronous or asynchronous perturbations of the steady-state spike ampli-tudes. More specifically, we provide phase diagrams in parameter space charac-terizing either oscillatory instabilities of the spike amplitudes, due to Hopf bifur-cations, or asynchronous (competition) instabilities, due to zero-eigenvalue cross-ings, that trigger spike annihilation events. These linear stability phase diagramsshow that the bulk-membrane coupling can have a diverse effect on the linear sta-bility of symmetric N-spike patterns. In each case we find that stability thresholdsare typically increased (making the system more stable) when the bulk-membranecoupling parameter K is relatively small, whereas the stability thresholds are de-creased as K continues to increase. This nontrivial effect is further complicatedFigure 2.1: Snapshots of the numerically computed solution of (2.1) startingfrom a 2-spike equilibrium for the unit disk with ε = 0.05, Db = 10,τs = 0.6, τb = 0.1, K = 2, and Dv = 10 (this corresponds to point 2 inthe left panel of Figure 2.12). The bulk inhibitor is shown as the colourmap, whereas the lines along the boundary indicate the activator (blue)and inhibitor (orange) membrane concentrations. The results show acompetition instability, leading to the annihilation of a spike.22Figure 2.2: Snapshots of the numerically computed solution of (2.1) startingfrom a 2-spike equilibrium for the unit disk with ε = 0.05, Db = 10, τs =0.6, τb = 0.1, K = 0.025, and Dv = 1.8 (this corresponds to point 5 inthe left panel of Figure 2.12). The bulk inhibitor is shown as the colourmap, whereas the lines along the boundary indicate the activator (blue)and inhibitor (orange) membrane concentrations. The results show asynchronous oscillatory instability of the spike amplitudes.when studying synchronous instabilities, for which there appears to be a complexinterplay between the membrane and bulk timescales, τs and τb, as well as with thecoupling K. At various specific points in these phase diagrams for both the well-mixed case (with Db infinite) and the case of the disk (with Db finite), our linearstability predictions are confirmed with full numerical finite-element simulationsof the coupled PDE system (2.1).As an illustration of spike dynamics resulting from full PDE simulations, inFigures 2.1 and 2.2 we show results computed for the unit disk with Db = 10,showing competition and oscillatory instabilities for a two-spike solution, respec-tively. The parameter values are given in the figure captions and correspond tospecific points in the linear stability phase diagram given in the left panel of Figure2.12.In §2.4 we use a regular perturbation analysis to show the effect on the asyn-chronous instability thresholds of introducing a small smooth perturbation of theboundary of the unit disk. This analysis, which requires a detailed calculation ofthe perturbed 1-D membrane Green’s function, shows that a two-spike pattern canbe stabilized by a small outward peanut-shaped deformation of a circular disk. Fi-nally, in §2.5 we briefly summarize our results and highlight some open problems23and directions for future research.2.2 Spike Equilibrium and its Linear Stability: GeneralAsymptotic Theory2.2.1 Asymptotic Construction of N-Spike Equilibrium SolutionIn this section we provide an asymptotic construction of an N-spike steady-statesolution to (2.1). Specifically, we consider the steady-state problem for the mem-brane speciesε2∂ 2σue−ue+upe/vqe = 0, 0 < σ < L, u is L-periodic, (2.1a)Dv∂ 2σve− (1+K)ve+KVe+ ε−1ume /vse = 0, 0 < σ < L, v is L-periodic,(2.1b)which is coupled to the steady-state bulk-diffusion process byDb∆Ve−Ve = 0, x ∈Ω ; Db∂nVe+KVe = Kve, x ∈ ∂Ω. (2.1c)From (2.1c), the bulk-inhibitor evaluated on the membrane is readily expressedin terms of a Green’s function asVe(σ) = Kˆ L0GΩ(σ , σ˜)ve(σ˜)dσ˜ , (2.2)where we have used arc-length to parameterize the boundary. Here, GΩ(σ , σ˜) isthe Green’s function satisfyingDb∆xGΩ(x, σ˜)−GΩ(x, σ˜) = 0, x ∈ΩDb∂nGΩ(σ , σ˜)+KGΩ(σ , σ˜) = δ (σ − σ˜), 0 < σ < L.(2.3)We remark that the values of the bulk-inhibitor field within the bulk can likewisebe obtained with a Green’s function whose source is in the interior. However, forour purposes it is only the restriction to the boundary that is important.24At this stage the steady-state membrane problem takes the formε2∂ 2σue−ue+upe/vqe = 0Dv∂ 2σve− (1+K)ve+K2ˆ L0GΩ(σ , σ˜)ve(σ˜)dσ˜ + ε−1ume /vse = 0,(2.4)for 0< σ < L and which differs from the problem studied in [41] for the uncoupled(K = 0) case only by the addition of the non-local term. This additional term leadsto difficulties in the construction of spike patterns. In particular, it complicates theconcept of a "symmetric" pattern since, in general, the non-local term will not betranslation invariant. Moreover, in the well-mixed and disk case, the constructionof asymmetric patterns is more intricate as a result of the non-local term.We now construct an N-spike steady-state pattern for (2.4) characterized by anactivator concentration that is localized at N distinct spike locations 0≤ σ1 < ... <σN < L to be determined. We assume that the spikes are well-separated in the sensethat |σ{(i+1) mod N}−σi mod L|  ε for i = 1, . . . ,N. Upon introducing stretchedcoordinates y = ε−1(σ −σ j), we deduce that the inhibitor field is asymptoticallyconstant near each spike, i. e.ve ∼ ve j ≡ ve(σ j). (2.5)In addition, the activator concentration is determined in terms of the unique solu-tion w(y) to the core problemw′′−w+wp = 0, y ∈ R,w′(0) = 0, w(0)> 0, w(y)→ 0 as |y| → ∞.(2.6)Since the solution to the core problem decays exponentially as y→±∞ we deducethatue(σ)∼N∑j=1vγe jw(ε−1[σ −σ j]), as ε → 0+, (2.7)where γ ≡ q/(p−1). The solution to (2.6) is given explicitly asw(y) =(p+12) 1p−1[sech(p−12y)] 2p−1. (2.8)25Next, since ue is localized, we have in the sense of distributions thatε−1ume /vse −→ ωmN∑j=1[ve(σ j)]γm−sδ (σ −σ j) as ε → 0+,where we have definedωm ≡ˆ ∞−∞[w(y)]m dy. (2.9)In this way, for ε → 0+, we obtain from (2.4) the following integro-differentialequation for the inhibitor field:Dv∂ 2σve− (1+K)ve+K2ˆ L0GΩ(σ , σ˜)ve(σ˜)dσ˜ =−ωmN∑j=1vγm−se j δ (σ −σ j).To conveniently represent the solution to this equation we introduce the Green’sfunction G∂Ω(σ ,ζ ) satisfyingDv∂ 2σG∂Ω(σ ,ζ )− (1+K)G∂Ω(σ ,ζ )+K2ˆ L0GΩ(σ , σ˜)G∂Ω(σ˜ ,ζ )dσ˜ =−δ (σ −ζ ),(2.10)for 0 < σ ,ζ < L. In terms of this Green’s function, the membrane inhibitor field isgiven byve(σ) = ωmN∑j=1vγm−se j G∂Ω(σ ,σ j). (2.11)Substituting σ = σi, and recalling the definition vei ≡ ve(σi), (2.11) yields the Nself-consistency conditionsvei−ωmN∑j=1vγm−se j G∂Ω(σi,σ j) = 0, i = 1, . . . ,N. (2.12)These conditions provide the first N algebraic equations for our overall system in2N unknowns to be completed below. The remaining N equations arise from solv-ability conditions when performing a higher-order matched asymptotic expansionanalysis of the steady-state solution.To this end, we again introduce stretched coordinates y = ε−1(σ −σ j), but we26now introduce a two-term inner expansion for the surface bound species for ε → 0asue(y)∼ vγe jw(y)+ εu1(y)+O(ε2),ve(y)∼ve j + εv1(y)+O(ε2), Ve ∼ O(1).(2.13)Upon substituting this expansion into (2.1), and collecting the O(ε) terms, we getL0u1 ≡u′′1−u1+ pwp−1u1 = qvγ−1e j wpv1,Dvv′′1 + vγm−se j wm = 0.(2.14)SinceL0w′ = 0, the solvability condition for the first equation yields thatqvγ−1e jˆ ∞−∞wpw′v1 dy = 0 ⇐⇒ˆ ∞−∞(wp+1)′v1 dy = 0.Then, we integrate by parts twice, use the exponential decay of w(y) as |y| → ∞,and substitute (2.14) for v′′1 . This yields thatIp(y)v′1(y)∣∣∣∣∞−∞+vγm−se jDvˆ ∞−∞Ip(y)[w(y)]m dy = 0,where we have defined Ip(y)≡´ y0 [w(z)]p+1dz. Since w is even, while Ip is odd, theintegral above vanishes, and we getv′1(+∞)+ v′1(−∞) = 0.In this way, a higher order matching process between the inner and outer solutionsyields the balance conditions,∂σve(σi+0)+∂σve(σi−0) = 0, i = 1, . . . ,N.By using (2.11) for ve, we can write these balance equations in terms of the Green’sfunction G∂Ω asvγm−sei[∂σG∂Ω(σi+0,σi)+∂σG∂Ω(σi−0,σi)]+2∑j 6=ivγm−se j ∂σG∂Ω(σi,σ j) = 0,27for i = 1, . . . ,N. We summarize the results of this formal asymptotic constructionin the following proposition:Proposition 2.2.1. As ε→ 0+ an N-spike steady-state solution to (2.1) with spikescentred at σ1, ...,σN is asymptotically given byue(σ)∼N∑j=1vγe jw(ε−1[σ −σ j]), ve(σ)∼ ωmN∑j=1vγm−se j G∂Ω(σ ,σ j), (2.15a)Ve(σ)∼ ωmKN∑j=1vγm−se jˆ L0GΩ(σ , σ˜)G∂Ω(σ˜ ,σ j)dσ˜ , (2.15b)whereωm≡´ ∞−∞[w(y)]m dy, γ ≡ q/(p−1), and GΩ and G∂Ω are the bulk and mem-brane Green’s functions satisfying (2.3) and (2.10) respectively. Here the steady-state spike locations σ1, ...,σN and ve1, ...,veN , which determine the heights of thespikes, are to be found from the following non-linear algebraic system:vei−ωmN∑j=1vγm−se j G∂Ω(σi,σ j) = 0, (2.16a)vγm−sei[∂σG∂Ω(σi+0,σi)+∂σG∂Ω(σi−0,σi)]+2∑j 6=ivγm−se j ∂σG∂Ω(σi,σ j) = 0, (2.16b)for each i = 1, . . . ,N.2.2.2 Linear Stability of N-Spike Equilibrium SolutionIn our linear stability analysis, given below, of N-spike equilibrium solutions wemake two simplifying assumptions. First, we focus exclusively on the case s = 0.Second, we consider only instabilities that arise on an O(1) timescale. Therefore,we do not consider very weak instabilities, occurring on asymptotically long time-scales in ε , that are due to any unstable small eigenvalue that tends to zero asε → 0.Let ue(σ), ve(σ), and Ve(x) denote the the steady-state constructed in §2.2.1.For λ ∈ C, we consider a perturbation of the formu(σ) = ue(σ)+ eλ tφ(σ), v(σ) = ve(σ)+ eλ tψ(σ), V (x) =Ve(x)+ eλ tη(x),where φ , ψ , and η are small. Upon substituting into (2.1) and linearizing, we28obtain the eigenvalue problemε2∂ 2σφ −φ + pup−1e v−qe φ −qupe v−(q+1)e ψ = λφ , 0 < σ < L, (2.17a)Dv∂ 2σψ−µ2sλψ+Kη =−mε−1um−1e φ , 0 < σ < L, (2.17b)Db∆η−µ2bλη = 0, x ∈Ω, (2.17c)Db∂nη+Kη = Kψ, x ∈ ∂Ω, (2.17d)where we have defined µsλ and µbλ byµsλ =√1+K+ τsλ , µbλ =√1+ τbλ . (2.18)The bulk inhibitor field evaluated on the boundary is represented asη(σ) = Kˆ L0GλΩ(σ , σ˜)ψ(σ˜)dσ˜ ,where GλΩ is the λ -dependent bulk Green’s function satisfyingDb∆xGλΩ(x, σ˜)−µ2bλGλΩ(x, σ˜) = 0, x ∈Ω,Db∂nGλΩ(σ , σ˜)+KGλΩ(σ , σ˜) = δ (σ − σ˜), 0 < σ < L.(2.19)Next, we seek a localized activator perturbation of the formφ(σ)∼N∑j=1φ j(ε−1[σ −σ j]), (2.20)where we impose that φ j(y)→ 0 as |y| → ∞. With this form, we evaluate in thesense of distributions thatε−1mum−1e φ −→ mN∑j=1vγ(m−1)e j(ˆ ∞−∞[w(y)]m−1φ j(y)dy)δ (σ −σ j) as ε → 0+.29By using this limiting result in (2.17b), the problem for ψ becomesDv∂ 2σψ−µ2sλψ+K2ˆ L0GλΩ(σ , σ˜)ψ(σ˜)dσ˜=−mN∑j=1vγ(m−1)e j(ˆ ∞−∞[w(y)]m−1φ j(y)dy)δ (σ −σ j).The solution to this problem is represented asψ(σ) = mN∑j=1vγ(m−1)e j Gλ∂Ω(σ ,σ j)ˆ ∞−∞[w(y)]m−1φ j(y)dy, (2.21)where Gλ∂Ω is the λ -dependent membrane Green’s function satisfyingDv∂ 2σGλ∂Ω(σ ,ζ )−µ2sλGλ∂Ω(σ ,ζ )+K2ˆ L0GλΩ(σ , σ˜)Gλ∂Ω(σ˜ ,ζ )dσ˜ =−δ (σ−ζ ), (2.22)for 0 < σ ,ζ < L.Next, it is convenient to re-scale ve asve(σ) = ω11−γmm vˆe(σ), ve j = ω11−γmm vˆe j. (2.23)In the stretched coordinates y = ε−1(σ −σ j), we use (2.21) to obtain that (2.17a)becomesφ ′′i −φi+ pwp−1φi−mqwpN∑j=1vˆγ−1ei Gλ∂Ω(σi,σ j)vˆγ(m−1)e j´ ∞−∞wm−1φ j dy´ ∞−∞wm dy= λφi.To recast this spectral problem in vector form we defineφ ≡φ1...φN , Vˆe ≡vˆe1 0. . .0 vˆeN , G λ∂Ω ≡Gλ∂Ω(σ1,σ1) · · · Gλ∂Ω(σ1,σN)· · · . . . ...Gλ∂Ω(σN ,σ1) · · · Gλ∂Ω(σN ,σN) , (2.24)and we introduce the matrix E byE ≡ Vˆ γ−1e G λ∂ΩVˆ γ(m−1)e . (2.25)30In this way, we deduce that φ must solve the vector nonlocal eigenvalue problem(NLEP) given byφ ′′(y)−φ (y)+ pwp−1φ (y)−mqwp´ ∞−∞[w(y)]m−1Eφ (y)dy´ ∞−∞[w(y)]m dy= λφ (y). (2.26)We can reduce this vector NLEP to a collection of scalar NLEPs by diagonalizingit. Specifically, we seek perturbations of the form φ = φc where c is an eigenvectorof E , that isE c = χ(λ )c. (2.27)Then, it readily follows that the vector NLEP (2.26) can be recast as the scalarNLEPL0φ −mqχ(λ )wp´ ∞−∞[w(y)]m−1φ(y)dy´ ∞−∞[w(y)]m dy= λφ , (2.28)where χ(λ ) is any eigenvalue of E . In (2.28), the operator L0, referred to as thelocal operator, is defined byL0φ ≡ φ ′′(y)−φ(y)+ pwp−1φ(y). (2.29)Notice that we obtain a (possibly) different NLEP for each eigenvalue χ(λ ) of E .Therefore, the spectrum of the matrix E will be central in the analysis below forclassifying the various types of instabilities that can occur.2.2.3 Reduction of NLEP to an Algebraic Equation and an ExplicitlySolvable CaseNext, we show how to reduce the determination of the spectrum of the NLEP (2.28)to a root-finding problem. To this end, we define cm bycm ≡ mqχ(λ )´ ∞−∞[w(y)]m−1φ(y)dy´ ∞−∞[w(y)]m dy, (2.30)and write the NLEP as (L0−λ )φ = cmwp, so that φ = cm(L0−λ )−1wp. Uponmultiplying both sides of this expression by wm−1, we integrate over the real lineand substitute the resulting expression back into (2.30). For eigenfunctions for31which´ ∞−∞wm−1φ dy 6= 0, we readily obtain that λ must be a root of A (λ ) = 0,whereA (λ )≡ C (λ )−F (λ ), C (λ )≡ 1χ(λ ),F (λ )≡ mq´ ∞−∞[w(y)]m−1(L0−λ )−1[w(y)]p dy´ ∞−∞[w(y)]m dy.(2.31)Since, it is readily shown that there are no unstable eigenvalues of the NLEP (2.28)for eigenfunctions for whichˆ ∞−∞wm−1φ dy = 0,the roots ofA (λ ) = 0 will provide all the unstable eigenvalues of the NLEP (2.28).For general Gierer-Meinhardt exponents, the spectral theory of the operatorL0 leads to some detailed properties of the term F (λ ) for various exponent sets(cf. [105]). In addition, to make further progress on the root-finding problem(2.31), we need some explicit results for the multiplier χ(λ ).For special sets of Gierer-Meinhardt exponents, known as the “explicitly solv-able cases” (cf. [70]), the termF (λ ) can be evaluated explicitly. We focus specif-ically on one such set (p,q,m,0) = (3,1,3,0) for which the key identity L0w2 =3w2 holds, where w =√2sechy from (2.8). Thus, after integrating by parts weobtainˆ ∞−∞w2(L0−λ )−1w3dy =´ ∞−∞(L0−λ )w2(L0−λ )−1w3dy3−λ=´ ∞−∞w2(L0−λ )(L0−λ )−1w3dy3−λ =´ ∞−∞w5dy3−λ .By making use of the identitiesˆ ∞−∞w5dy =3pi√2,ˆ ∞−∞w3dy =√2pi,we obtain that F (λ ) = 9/ [2(3−λ )], so that the root-finding problem (2.31) re-32duces to determining λ such thatA (λ )≡ 1χ(λ )− 9/23−λ = 0. (2.32)In addition to the explicitly solvable case (p,q,m,s) = (3,1,3,0), the root-finding problem (2.31) simplifies considerably for a general Gierer-Meinhardt ex-ponent set, when we focus on determining parameter thresholds for zero-eigenvaluecrossings (corresponding to asynchronous instabilities). Since L0w = w′′−w+pwp = (p−1)wp, it follows thatL −10 wp = 1p−1 w, from which we calculateF (0) = mq´ ∞−∞wm−1L −10 wp dy´ ∞−∞wm dy=mqp−1 .Therefore, a zero-eigenvalue crossing for a general Gierer-Meinhardt exponent setoccurs whenA (0) =1χ(0)− mqp−1 = 0. (2.33)2.3 Symmetric N-Spike Patterns: Equilibrium Solutionsand their StabilityFor the remainder of this chapter we will focus exclusively on symmetric N-spikesteady-states that are characterized by equidistant (in arc-length) spikes of equalheights. Due to the bulk-membrane coupling it is unclear whether such symmetricpatterns will exist for a general domain. Indeed it may be that a spike pattern withspikes of equal heights may require the equidistant requirement to be dropped.These more general considerations can perhaps be better approached by requiringthat the Green’s matrix G λ∂Ω admit the eigenvector e= (1, ...,1)T . A detailed studyof the geometries for which such an eigenvector can be found remains to be done.Avoiding these additional complications, we focus instead on two distinct casesfor which symmetric spike patterns, as we have defined them, can be constructed.The first case is the disk of radius R, denoted by Ω = BR(0), and the second casecorresponds to the well-mixed limit for which Db → ∞ in an arbitrary boundeddomain with C2 boundary. The membrane and bulk Green’s functions in these two33special cases can be found in the Appendices B.1.2 for the well-mixed case, andB.1.3 for the disk. In both cases the Green’s function is invariant under translations,satisfyingG∂Ω(σ +ϑ mod L,ζ +ϑ mod L) = G∂Ω(σ ,ζ ), ∀ σ ,ζ ∈ [0,L), ϑ ∈ R.By using this key property in (2.16a), we calculate the common spike height asve j = ve0 =[ωmN−1∑k=0G∂Ω(kLN,0)] 11−γm+s. (2.1)With a common spike height, the balance equations (2.16b) then reduce to[∂σG∂Ω(0+,0)+∂σG∂Ω(0−,0)]+2N−1∑k=1∂σG∂Ω(kLN,0)= 0, (2.2)which can be verified either explicitly or by using the symmetry of the Green’sfunction.For a symmetric N-spike steady-state the NLEP (2.28) can be simplified sig-nificantly. First the matrix E , defined in (2.25), simplifies toE = vˆγm−1e0 Gλ∂Ω.Therefore, from (2.27) it follows that χ(λ ) = vˆγm−1e0 µ(λ ), where µ(λ ) is an eigen-value of the Green’s matrix G λ∂Ω defined in (2.24). Furthermore, by using the bi-translation invariance and symmetry of Gλ∂Ω, we can defineHλ| j−i| ≡ Gλ∂Ω(|σi−σ j|,0) = Gλ∂Ω(|i− j|L/N,0), (2.3)which allows us to write the Green’s matrix asG λ∂Ω =Hλ0 Hλ1 Hλ2 · · · HλN−1HλN−1 Hλ0 Hλ1 · · · HλN−2.......... . ....Hλ1 Hλ2 Hλ3 · · · Hλ0 ,34which we recognize as a circulant matrix. As a result, the matrix spectrum of G λ∂Ωis readily available asµk(λ ) =N−1∑j=0Hλj ei 2pi jkN , ck(λ ) =(1,ei2pikN , · · · ,ei 2pi(N−2)kN ,ei 2pi(N−1)kN)T, (2.4)for all k = 0, . . . ,N−1.For each value of k = 0, . . . ,N− 1 we obtain a corresponding NLEP problemfrom (2.28). Since c0 = (1, . . . ,1)T we can interpret this “mode” as a synchronousperturbation. In contrast, the values k = 1, . . . ,N−1 for N ≥ 2 correspond to asyn-chronous perturbations, since the corresponding eigenvectors ck(λ ) are all orthog-onal to (1, . . . ,1)T . Any unstable asynchronous “mode” of this type is referred toas a competition instability, in the sense that the linear stability theory predicts thatthe heights of individual spikes may grow or decay, but that the overall sum of allthe spike heights remains fixed. For each value of k, the NLEP (2.28) becomesL0φ −mqχk(λ )wp´ ∞−∞[w(y)]m−1φ(y)dy´ ∞−∞[w(y)]m dy= λφ , (2.5a)whereχk(λ )≡ µk(λ )∑N−1j=1 G∂Ω( jL/N,0)=µk(λ )µ0(0). (2.5b)Thus, each NLEP leads to a distinct algebraic system of the form (2.31) corre-sponding to χk(λ ) for k = 0, ...,N. In particular, for the explicitly solvable case(2.32) becomesAk(λ ) =1χk(λ )− 9/23−λ .In addition, note that when K = 0 the bulk dependent term in (2.22) vanishes andGλ∂Ω reduces to the uncoupled periodic Green’s function (see Appendix B.1.1).In such a case the expression for χk(λ ) in (2.5) reduces to that of the classicaluncoupled case. When K > 0 further analysis of the NLEP (2.5) requires details ofthe Green’s function Gλ∂Ω, which are available in our two special cases.352.3.1 NLEP Multipliers for the Well-Mixed LimitIn the well-mixed limit, Db→∞, the membrane Green’s function, satisfying (2.22),is given by (see (B.5) of Appendix B.1)Gλ∂Ω(σ ,ζ ) = Γλ (|σ −ζ |)+ γλµ2sλ, γλ ≡K2/Aµ2sλ (µ2bλ +β )−Kβ, (2.6)where β ≡ KL/A. Here Γλ is the periodic Green’s function for the uncoupled(K = 0) problem, which is given explicitly by (B.2) of Appendix B.1 asΓλ (x) =12√Dvµsλcoth(µsλL2√Dv)cosh(µsλ√Dv|x|)− 12√Dvµsλsinh(µsλ√Dv|x|).After some algebra we use (2.4) to calculate the eigenvalues µk(λ ) of the Green’smatrix asµk(λ ) =N−1∑j=0Γλ ( jL/n)ei2pi jkN +δk0Nγλµ2sλ=12√Dvµsλcosh( µsλL2N√Dv)sinh( µsλL2N√Dv)sinh( µsλL2N√Dv+ ipikN)sinh( µsλL2N√Dv− ipikN) +δk0 Nγλµ2sλ ,where δk0 is the Kronecker symbol. In this way, we obtain from (2.5) that theNLEP multipliers are given byχ0(λ ) =12√Dvµsλcoth(µsλL2N√Dv)+ Nγλµ2sλ12√Dvµs0coth(µs0L2N√Dv)+ Nγ0µ2s0, (2.7a)χk(λ ) =12√Dvµsλcosh( µsλ L2N√Dv)sinh( µsλ L2N√Dv)sinh( µsλ L2N√Dv+ ipikN)sinh( µsλ L2N√Dv− ipikN)12√Dvµs0coth(µs0L2N√Dv)+ Nγ0µ2s0, (2.7b)for k = 1, ...,N−1. We observe from the χ0(λ ) term in (2.7), that any synchronousinstability will depend on the membrane diffusivity Dv only in the form N2Dv. Thisshows that a synchronous instability parameter threshold will be fully determined36by the one-spike case upon rescaling by 1/N2. We remark here that the numer-ator for χk(λ ) can be simplified by using the identity sinh(z+ ia)sinh(z− ia) =12 [cosh(2z)− cos(2a)] so that χk(λ ) is real valued whenever Imλ = 0.2.3.2 NLEP Multipliers for the DiskIn the disk we can calculate the membrane Green’s function as a Fourier series (see(B.7) of Appendix B.1)Gλ∂Ω(σ ,ζ ) =12piR∞∑n=−∞gλn ei nR (σ−ζ ), (2.8)where gλn is given explicitly bygλn =1Dv( nR)2+µ2sλ −K2aλn(2.9a)and whereaλn =1DbP′n(R)+K, Pn(r)≡I|n|(ωbλ r)I|n|(ωbλR), ωbλ ≡µbλ√Db. (2.9b)Here In(z) is the nth modified Bessel function of the first kind. From (2.4) theeigenvalues of the Green’s matrix becomeµk(λ ) =12piR∞∑n=1gλnN−1∑j=0ei2pi(k+n) jN .By using the identitiesN−1∑j=0ei2pi(k+n) jN =N n ∈ NZ− k,0 otherwise , and gλ−n = gλn ,the eigenvalues are given explicitly byµk(λ ) =N2piRgλk +N2piR∞∑n=1(gλnN+k +gλnN−k).37Therefore, since χk(λ ) = µk(λ )/µ0(0), the NLEP multipliers are given byχk(λ ) =gλk +∑∞n=1(gλnN+k +gλnN−k)g00+2∑∞n=1 g0nN, k = 0, . . . ,N−1. (2.10)2.3.3 Synchronous InstabilitiesFrom (2.33), and the special form of χk(λ ) given in (2.5), we deduce thatA0(0) = 1− mqp−1 < 0,where the strict inequality follows from the the usual assumption (2.2) on theGierer-Meinhardt exponents. As a result, synchronous instabilities do not occurthrough a zero-eigenvalue crossing, and can only arise through a Hopf bifurcation.To examine whether such a Hopf bifurcation for the synchronous mode can oc-cur, we now seek purely imaginary zeros of A0(λ ). Classically, in the uncoupledcase K = 0, such a threshold occurs along a Hopf bifurcation curve Dv = D?v(τs)(cf. [105]). We have an oscillatory instability if τs is sufficiently large, and no suchinstability when τs is small (cf. [105], [106]). Bulk-membrane coupling introducestwo additional parameters, τb and K, in addition to the quantities L and A for thewell-mixed case, or R and Db for the case of the disk. Thus, it is no longer clearhow the existence of a synchronous instability threshold Dv = D?v(τs) will be mod-ified by the additional parameters. Indeed, the analysis below reveals a variety ofnew phenomenon such as the existence of synchronous instabilities for τs = 0 andislands of stability for large values of τs. These are two behaviours that do notoccur for the classical uncoupled case K = 0.We begin by addressing the question of the existence of synchronous instabil-ity thresholds. The key assumption (supported below by numerical simulations)underlying this analysis is that synchronous instabilities persist as either the bulkand/or membrane diffusivities increase. While this assumption is heuristically rea-sonable (large diffusivities make it easier for neighbouring spikes to communicate)an open problem is to demonstrate it analytically. With this assumption it sufficesto seek parameter values of τb, τs, and K for which no Hopf bifurcations exist whenDv→ ∞ in the well-mixed limit Db→ ∞.38As a first step, we remark that in [105] it was shown that ReF (iλI) is mono-tone decreasing when λI > 0 for special choices of the Gierer-Meinhardt exponents(see also [106]). The monotonicity of this function for general Gierer-Meinhardtexponents is supported by numerical calculations. Thus we expect that ReF (iλI)decreases monotonically from ReF (0) = mqp−1 > 1 as λI > 0 increases. Further-more, numerical evidence suggests that ReC0(iλI) is monotone increasing in λI .Since C0(0) = 1 there must exist a unique root λI = λ ?I > 0 to ReA0(iλI) = 0bounded above by λFI , the unique solution to ReF (iλFI ) = 1, which depends solelyon the exponents (p,q,m,0). Therefore in the limit Dv→ ∞ the well-mixed NLEPmultiplier, as given in (2.7), becomesχ0(λ )∼ µ2s0(µ2b0+β )−Kβµ2sλ (µ2bλ +β )−Kβ(µ2bλ +βµ2b0+β).Seeking a purely imaginary root of A0(λ ) = 0 we focus first on the real part. WecalculateReA0(iλI) =1+β1+β +K(1+K− Kβ1+β11+( τbλI1+β)2)−ReF (iλI),and note that the root λI = λ ?I (τb,K) to ReA0(iλI) = 0 is independent of τs. Next,for the imaginary part we calculateImA0(iλ ?I ) =1+β1+β +K(τs+Kβ1+βτb1+β1+( τbλ ?I1+β)2)λ ?I − ImF (iλ ?I ).Fortunately, at each fixed value of τs the threshold K = K(τb) can be calculatedas the τs-level-set of a function depending only on K and τb. Indeed the conditionImA0(iλ ?I ) = 0 can equivalently be written asImA0(iλ ?I ) =1+β1+β +K(τs−M (τb,K))λ ?I = 0, (2.11)39where we have definedM (τb,K)≡(1+β +K1+β)ImF (iλ ?I )λ ?I− Kβ1+β τb1+β1+( τbλ ?I1+β)2 . (2.12)In the (p,q,m,s) = (3,1,3,0) explicitly solvable case we find that ImF (iλ ?I ) =13λ?I ReF (iλ ?I ), so that by solving ReA0(iλI) = 0 for ReF (iλ ?I ), (2.12) becomesM (τb,K) =1+K3− Kβ1+β τb1+β + 131+( τbλ ?I1+β)2 .By substituting this expression into (2.11), we deduce the existence of two distinctthreshold branches obtained by considering the limits K  1 and K  1. In thisway, we deriveτs−M (τb,K)∼ τs− 13 +1β0(τb− 13)+O(K−1) for K 1,τs−M (τb,K)∼ τs− 13 −13K+O(K2) for K 1,where β0 ≡ L/A. Notice that in ordering both of these asymptotic expansions wehave used that 0 < λ ?I ≤ λFI , where the upper bound is independent of K. In theK 1 regime we deduce that if τb = 13 −β0(τs− 13), then ImA0(iλ ?I ) = 0 forcesK → ∞, implying the existence of a threshold branch emerging from K = ∞ atthese parameter values. We remark here that in the K = ∞ limit we have V = von the boundary and therefore the contribution of V to the membrane equationis just the Dirichlet to Neumann map of v. Furthermore, since τb approaches 0when τs tends to 13( 1β0 + 1), we deduce that this branch will disappear for suffi-ciently large values of τs. In addition, in the K  1 regime we find that a newbranch given by K ≈ 3τs− 1 emerges when τs > 13 . The left panel of Figure 2.3shows the numerically-computed contours ofM (τb,K) for the explicitly solvablecase (p,q,m,s) = (3,1,3,0). The right panel of Figure 2.3 shows a qualitativelysimilar behaviour that occurs for the prototypical Gierer-Meinhardt parameter set(p,q,m,s) = (2,1,2,0).400 1 2 3 4 5 6 7 8b0.00.51.01.52.02.53.03.54.0K0.0000.3000.3500.3500.3800.3800.4000.4000.4500.450Contour Plot of ( B,K) for (p, q,m, s) = (3, 1, 3, 0)0 1 2 3 4 5 6 7 8b0.00.51.01.52.02.53.03.54.0K0.7000.8000.8000.9000.9000.9200.9200.9800.9801.0401.040Contour Plot of ( B,K) for (p, q,m, s) = (2, 1, 2, 0)Figure 2.3: Level sets of M (τb,K) for Gierer-Meinhardt exponents(p,q,m,s)= (3,1,3,0) (left) and (p,q,m,s)= (2,1,2,0) (right). In bothcases the level set value corresponds to a value of τs =M (τb,K). Notealso the contours tending to a vertical asymptote, and the emergence ofa horizontal asymptote as τs exceeds some threshold. Geometric param-eters are L = 2pi and A = pi .The preceding analysis does not directly predict in which regions synchronousinstabilities exist, as it only provides the boundaries of these regions. We now41Figure 2.4: Colour map of the synchronous instability threshold D?v in the Kversus τb parameter plane for the well-mixed explicitly solvable case forvarious values of τs with L = 2pi and A = pi . The dashed vertical linesindicate the asymptotic predictions for the large K threshold branch,while the dashed horizontal lines indicate the asymptotic predictionsfor the small K threshold branch. The unshaded regions correspond tothose parameter values for which synchronous instabilities are absent.outline a winding-number argument, related to that used in [106], that providesa hybrid analytical-numerical algorithm for calculating the synchronous instabil-ity threshold Dv = D?v(K,τb,τs). Furthermore, as we show below, this algorithmindicates that synchronous instabilities exist wheneverM (τb,K)< τs.Synchronous instabilities are identified with the zeros to (2.31) having a posi-tive real-part when χ(λ ) in (2.31) is replaced by χ0(λ ). By using a winding num-ber argument, the search for such zeros can be reduced from one over the entireright-half plane Re(λ ) > 0 to one along only the positive imaginary axis. Indeed,if we consider a counterclockwise contour composed of a segment of the imagi-nary axis, −ρ ≤ Imλ ≤ ρ , together with the semi-circle defined by |λ | = ρ and42−pi/2 < argλ < pi/2, then in the limit ρ → ∞ the change in the argument as thecontour is traversed is∆argA0(λ ) = 2pi(Z−1), (2.13)where Z is the number of zeros of A0 with positive real-part. Here we have usedthat χ0(λ ) 6= 0 when Re(λ )≥ 0, whileF (λ ) has exactly one simple (and real) polein the right-half plane corresponding to the only positive eigenvalue of the self-adjoint local operator L0 (cf. [107]). We immediately note that F (λ ) = O(λ−1)for |λ |  1, |argλ |< pi/2, whereas for |λ |  1 and |argλ |< pi/2C0(λ )∼ 2µ0(0)√τsDvλ 1/2, C0(λ )∼ µ0(0)N√DvτspiRλ 1/2, (2.14)for the well-mixed limit and the disk cases, respectively. Therefore, in both caseswe have A0(λ ) ∼ O(λ 1/2) for |λ |  1 with |argλ | < pi/2, so that the change inargument over the large semi-circle is pi/2. Furthermore, since the parameters inA0(λ ) are real-valued, the change in argument over the segment of the imaginaryaxis can be reduced to that over the positive imaginary axis. In this way, we deducethatZ =54+1pi∆argA0(iλI)∣∣λI∈(∞,0]. (2.15)We readily evaluate the limiting behaviour limλI→∞ argA0(iλI) = pi/4. Moreoversince χ0(0) = 1 we evaluate A0(0) = 1− mqp−1 < 0 by the assumption (2.2) onthe Gierer-Meinhardt exponents. Numerical evidence suggests that ReA0(iλI) in-creases monotonically with λI and there should therefore be a unique λ ?I for whichReA0(iλ ?I ) = 0. We conclude that there are two positive values for the change inargument, and hence the number of zeros ofA0(λ ) in Re(λ )> 0 is dictated by thesign of ImA0(iλ ?I ) as follows:Z = 2 if ImA0(iλ ?I )> 0, or Z = 0 if ImA0(iλ?I )< 0. (2.16)Note in particular that, in view of the expression (2.11) for ImA0(iλI), this crite-rion implies that synchronous instabilities will exist whenever M(τb,K)< τs in theprevious analysis. Within this region, the criterion (2.16) suggests a simple numer-ical algorithm for iteratively computing the threshold value of Dv = D?v(K,τb,τs).430 1 2 3 4 5K010203040506070D* vS=0.2, b = 1.50 1 2 3 4 5K02468101214D* vS=0.6, b = 1.0Db = 5.0Db = 10.0Db = 50.0Db = 100.0well-mixedFigure 2.5: Synchronous instability threshold D?v versus K for two pairs of(τs,τb) for a one-spike steady-state (N = 1) in the unit disk (R = 1).The quality of the well-mixed approximation rapidly improves as Db isincreased. The labels for Db in the right panel also apply to the left andmiddle panels.Specifically, with all parameters fixed, we first solve ReA0(iλI) = 0 for λ 0I . Then,we calculate ImA0(iλ 0I ) and increase (resp. decrease) Dv if ImA0(iλ 0I )< 0 (resp.ImA0(iλ 0I )> 0 until ImA0(iλ 0I ) = 0. This procedure is repeated until |A0(iλ 0I )| issufficiently small.Using the algorithm described above, the results in Figure 2.4 illustrate howthe synchronous instability threshold D?v depends on parameters τs, τb, and K forthe explicitly solvable case in the well-mixed limit. From these figures we observethat coupling can have both a stabilizing and a destabilizing effect with respect tosynchronous instabilities. Indeed, on the K = 0 axis we see, as expected from theclassical theory, that synchronous instabilities exist beyond some τs value. How-ever, well before this threshold of τs is even reached it is possible for synchronousinstabilities to exist when both τb and K are sufficiently large. In contrast, we alsosee from the panels in Fig. 2.4 with τs = 0.36, τs = 0.38, and τs = 0.4 that whenτb is sufficiently small, there are no synchronous instabilities when the couplingK is large enough. Perhaps the most perplexing feature of this bulk-membraneinteraction is the island of stability that arises around τs = 0.4 and appears to per-440 2 4 6 8 10K0.00.51.01.52.0DvL=6.283(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 2 4 6 8 10K0.02.55.07.510.012.515.017.5DvL=18.850(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)Figure 2.6: Asynchronous instability thresholds Dv versus the coupling K inthe well-mixed limit for different values of L, different (N,k) pairs, andfor domain areas A = 3.142 (solid), 1.571 (dashed), and 0.785 (dotted).sist, propagating to larger values of τb as τs increases (only shown up to τs = 0.6).Finally in Figure 2.5 we demonstrate how the synchronous instability threshold be-haves for finite bulk-diffusivity. A key observation from these plots is that that theinstability threshold increases with decreasing value of Db, which further supportsour earlier monotonicity assumption.2.3.4 Asynchronous InstabilitiesSince asynchronous instabilities emerge from a zero-eigenvalue crossing there aretwo significant simplifications. Firstly, the thresholds are determined by the non-linear algebraic problem Ak(0) = 0, for each mode k = 1, . . . ,N− 1, as given by(2.31) in which χ(λ ) is replaced by χk(λ ) as defined in (2.5). Secondly, by set-ting λ = 0, it follows that all τs and τb dependent terms in χk(λ ) vanish. There-fore, asynchronous instability thresholds are independent of these two parameters.The resulting nonlinear algebraic equations are readily solved with an appropriateroot finding algorithm (e.g. the brentq routine in the Python library SciPy). Fur-thermore, in the uncoupled case (K = 0) the threshold can be determined explic-itly (notice that when K = 0 the well-mixed and disk cases coincide). Indeed,defining z = L2N√Dv and y = pik/N, the algebraic problem Ak(0) = 0 becomes450 5 10 15 20K0.00.51.01.52.0Dv(N,k)=(2,1)0 5 10 15 20K0.000.250.500.751.00(N,k)=(3,1)0 5 10 15 20K0.00.20.40.60.8(N,k)=(4,1)0 5 10 15 20K0.00.20.4(N,k)=(4,2)Db = 5Db = 10Db = 20Db = 100well-mixedFigure 2.7: Asynchronous instability thresholds Dv versus the coupling K forthe unit disk with Gierer-Meinhardt exponents (3,1,3,0), and for dif-ferent Db. The dashed lines show the corresponding thresholds for thewell-mixed limit. The legend in the bottom right plot applies to eachplot.( mqp−1 − 1)sinh2(z) = sin2(y). From this relation it readily follows that the com-petition stability threshold for K = 0 isDv =[2NLlog(√p−1mq−p+1∣∣∣∣sin(pikN)∣∣∣∣+√p−1mq−p+1 sin2(pikN)+1)]−2. (2.17)Figure 2.6 illustrates the dependence of the asynchronous threshold on the ge-ometric parameters L and A for the well-mixed limit. In Figure 2.7 the effect offinite bulk diffusivity Db is explored for the unit disk. This figure also illustratesthat while the asynchronous threshold tends to zero as K→∞ for sufficiently largevalues of Db the same is not true for small values of Db. It is however worth re-membering that for large K, where the competition threshold value of Dv appearsto approach zero in these figures, the result is not uniformly valid since the NLEP46derivation required that Dv ε2.2.3.5 Numerical Support of the Asymptotic TheoryIn this subsection we verify some of the predictions of the steady-state and lin-ear stability theory by performing full numerical PDE simulations of the coupledbulk-membrane system (2.1). In particular we first give an outline of the numericalmethod used for solving (2.1). We then use full numerical simulations to quantita-tively support our predicted synchronous instability threshold. Finally we considera gallery of numerical simulations that qualitatively support the asymptotic theorydeveloped above.Outline of Numerical MethodsThe spatial discretization of (2.1) in the well-mixed limit is simplified by observingthat equation (2.1c) reduces to an ODE. In particular, to leading order V mustbe spatially homogeneous so that by integrating (2.1c) and using the divergencetheorem we obtain that V (t) must satisfy the ODEτbVt =−(β −1)V + βLˆ L0vdσ . (2.18)The remaining equations (2.1a) and (2.1b) can be discretized using a finite-differencemethod on a uniform discretization of the interval [0,L]. Using this same dis-cretization we can numerically evaluate the integral appearing in (2.18) using thetrapezoidal rule.When the bulk diffusivity Db is finite we use the finite-element method withlinear basis functions for the bulk equation (2.1c). Using the nodes on the bound-ary of the bulk triangulation we can use the finite difference method to discretizethe boundary equations (2.1a) and (2.1b). This aspect of the computation is simpli-fied by enforcing the bulk-triangulation to have boundary nodes that are uniformlydistributed with respect to the arc-length. In addition, we remark that the relevantintegral contribution of v(x, t) to the bulk finite-element discretization requires onlythe values of v(x, t) at the boundary nodes as can be seen by expanding v(x, t) interms of the restriction of the linear basis functions to the boundary.47For both the well-mixed case and the disk problem, the spatial discretizationultimately leads to a large system of ODEsdWdt= AW +F(W ). (2.19)Here the matrix A arises from the spatially discretized differential operators, whileF(W ) denotes the reaction kinetics and the bulk-membrane coupling terms.The choice of a time-stepping scheme for reaction diffusion systems is gener-ally non-trivial. Since the operator A is stiff, it is best handled using an implicittime-stepping method. On the other hand, the kinetics F(W ) are typically non-linear so explicit time-stepping is favourable. Using a purely implicit or explicittime-stepping algorithm therefore leads to substantial computation time, either byrequiring the use of a non-linear solver to handle the kinetics in the first case, or byrequiring a prohibitively small time-step to handle the stiff linear operator in thesecond case. This difficulty can be circumvented by using so-called mixed meth-ods, specifically the implicit-explicit methods described in [2]. We will use a sec-ond order semi-implicit backwards difference scheme (2-SBDF), which employsa second-order backwards difference to handle the diffusive term together with anexplicit time-stepping strategy for the nonlinear term (cf. [85]). This time-steppingstrategy is given by(3I−2∆tA)W n+1 = 4W n+4∆tF(W n)−W n−1−2∆tF(W n−1). (2.20)To initialize this second-order method we bootstrap with a first order semi-implicitbackwards difference scheme (1-SBDF) as follows:(I−∆tA)W n+1 =W n+∆tF(W n). (2.21)We will use the numerical method outlined above in the two proceeding sections.480 5 10 15 20 25 30 35 40K510152025DvN=1 synchronous threshold (well-mixed)asymptotic= 0.03=0.04=0.050 5 10 15 20 25 30 35 40K0.000.010.020.030.040.050.060.070.08rel. diff.Relative difference=0.03=0.04=0.05Figure 2.8: Comparison between numerical and asymptotic synchronous in-stability threshold for N = 1 with L= 2pi , A= pi , τs = 0.6, and τb = 0.01.Notice that, as expected, the agreement improves as ε decreases.Quantitative Numerical Validation: Numerically Computed SynchronousThresholdWe begin by describing a method for numerically calculating the synchronous in-stability threshold for a one (or more) spike pattern. Given an equilibrium solution(u0,v0,V0), for sufficiently small times the numerical solution will evolve approxi-mately as the linearizationu(σ , t)= u0(σ)+eλ tφ(σ), v(σ , t)= v0(σ)+eλ tψ(σ), V (σ , t)=V0(σ)+eλ tη(σ).For ε > 0 fixed and sufficiently small the steady-state will be very close to thatpredicted by the asymptotic theory. By initializing the numerical solver with oneof the steady-state solutions predicted by the asymptotic theory, and then trackingits time evolution, we will thus be able to approximate the value of Re(λ ). If wefix a location on the boundary σ? (e.g. one of the spike locations) and let t?1 <t?2 < ... denote the sequence of times at which u(σ?, t) attains a local maximum orminimum in t, then the sequence u?j = u(σ?, t?j ) ( j = 1, ..,) will approximate theenvelope of u(σ?, t). If this sequence is diverging from its average then Reλ ≥ 0,whereas if it is converging then Reλ < 0. Furthermore, by writing|u?n−u0(σ?)| ≈ etnRe(λ )|φ(σ?)|,49we can solve for Re(λ ) by taking two values t?n > t?m sufficiently far apart to getRe(λ )≈ log∣∣u?n−u0(σ?)∣∣− log∣∣u?m−u0(σ?)∣∣t?n − t?m.This motivates a simple method for estimating the synchronous instability thresh-old numerically. Starting with some point in parameter space (chosen close to thethreshold predicted by the asymptotic theory) we approximate Re(λ ) and then in-crease or decrease one of the parameters to drive Re(λ ) toward zero. Once Re(λ )is sufficiently close to zero we designate the resulting point in parameter space asa numerically-computed synchronous instability threshold point.In the well-mixed limit, we fix values of K and vary Dv using the numericalapproach described above until Re(λ ) is sufficiently small. The results in Figure2.8 compare the synchronous instability threshold for N = 1 in the well-mixedlimit as predicted by the asymptotic theory and by our full numerical approach forε = 0.3,0.4,0.5. We observe, as expected, that the asymptotic prediction improveswith decreasing values of ε , but that the agreement is non-uniform in the couplingparameter K.Qualitative Numerical Support: A Gallery of Numerical SimulationsWe conclude this section by first showcasing the dynamics of multiple spike pat-terns for several choices of the parameters K, Dv, τs, and τb in the well-mixed limit.We will focus exclusively on the explicitly solvable Gierer-Meinhardt exponentset (p,q,m,s) = (3,1,3,0) with ε = 0.05 and the geometric parameters L = 2piand A = pi . For the numerical computation we discretized the domain boundarywith 1200 uniformly distributed points (∆σ ≈ 0.00524) and used trapezoidal in-tegration for the bulk-inhibitor equation (2.18). Furthermore, we used 2-SBDFtime-stepping initialized by 1-SBDF with a time-step size of ∆t = 2.5(∆σ)2 ≈6.854× 10−4. In Figure 2.9 we plot the asymptotically predicted synchronousand asynchronous instability thresholds for two pairs of time-scale parameters:(τs,τb) = (0.2,2),(0.6,2). Each plot also contains several sample points whoseK and Dv values are given in Table 2.1 below. The corresponding full PDE nu-merical simulations, tracking the heights of the spikes versus time, at these sample500 2 4 6 8 10K012345678Dv123456s = 0.2, b = 20 1 2 3 4K012345678Dv1234567 8s = 0.6, b = 20.0 0.2 0.4K0.500.751.001.251.501.752.002.2547Figure 2.9: Synchronous (solid) and asynchronous (dashed) instabilitythresholds in the Dv versus K parameter plane in the well-mixed limitfor N = 1 (blue), N = 2 (orange), and N = 3 (green). At the top of eachof the three panels a different pair (τs,τb) is specified. See Table 2.1 forDv and K values at the numbered points in each panel. Figures 2.10 and2.11 show the corresponding spike dynamics from full PDE simulationsof (2.1) at the indicated points.points are shown in Figures 2.10 and 2.11. We observe that the initial instabilityonset in these figures is in agreement with that predicted by the linear stability the-ory. For example, when τs = 0.6 and τb = 2 an N = 3 spike pattern at point sixshould be stable with respect to an N = 3 synchronous instability but unstable withrespect to the N = 3 asynchronous instabilities. Indeed the initial instability onsetdepicted in the “point 6, N = 3” plot of Figure 2.11 showcases the non-oscillatorygrowth of two spikes and decay of one as expected. In addition the plots in Figures2.10 and 2.11 support two previously stated conjectures. First, pure Hopf bifurca-tions for N ≥ 2 should be supercritical (see “Point 4, N = 2” and “Point 7, N = 3”in Figure 2.11). Secondly, we observe that asynchronous instabilities lead to theeventual annihilation of some spikes and the growth of others. As a result, our PDE51Point K Dv1 8 72 4 63 4 24 1 35 1 1.256 1 0.5(a)Point K Dv1 3 72 1.5 53 0.75 2.54 0 1.755 1.5 1.256 0.75 1.257 0 0.98 1 0.5(b)Point K Dv1 0.5 182 2 103 2 3.54 1 0.55 0.025 1.8(c)Table 2.1: K and Dv values at the sampled points in the two panels of Fig. 2.9:(a) Left panel: (τs,τb) = (0.2,2), and (b) Right panel: (τs,τb) = (0.6,2).Table (d) shows the K and Dv values at the sampled points for the diskappearing in the left panel of Fig. 2.12.simulations suggest that these instabilities are subcritical.We now show that this agreement between predictions of our linear stabilitytheory and results from full PDE simulations continues to hold for the case of afinite bulk diffusivity. To illustrate this agreement, we consider the unit disk withDb = 10 for (τs,τb) = (0.6,0.1). For this parameter set, in the left panel of Figure2.12 we show the asymptotically predicted synchronous and asynchronous insta-bility thresholds in the Dv versus K parameter plane for N = 1 and N = 2. Thefaint grey dotted lines in this figure indicate the corresponding well-mixed thresh-olds. In the right panel of Figure 2.12 we plot the spike heights versus time, ascomputed numerically from (2.1), at the sample points indicated in the left panel.In each case, the numerically computed solution uses a 2% perturbation away fromthe asymptotically computed N-spike equilibrium. As in the well-mixed case,the full numerical simulations confirm the predictions of the linear stability the-ory. Furthermore, Figures 2.1 and 2.2 depict both the bulk-inhibitor and the twomembrane-bound species at certain times for an N = 2 spike pattern at points 2 and5 in the left panel of Figure 2.12, respectively. From this figure, we observe thatthe bulk-inhibitor field is largely constant except within a small near region near520 50 10001234point 1, N = 10 50 1000.00.51.01.52.0point 2, N = 10 50 1000123point 2, N = 20 50 1000.00.51.01.5point 3, N = 20 50 1000.00.20.40.60.81.0point 5, N = 20 50 1000.00.51.01.52.0point 3, N = 30 50 1000.00.51.01.5point 4, N = 30 50 1000.00.20.40.6point 6, N = 3Figure 2.10: Numerically computed spike heights (vertical axis) versus time(horizontal axis) from full PDE simulations of (2.1) for τs = 0.2 andτb = 2 at the points indicated in the left panel of Figure 2.9. Distinctspike heights are distinguished by line types (solid, dashed, and dot-ted).0 50 1000123point 1, N = 10 50 1000.00.51.01.52.0point 2, N = 10 50 1000.00.51.01.5point 3, N = 20 50 1000.00.51.01.52.0point 4, N = 20 50 1000.00.20.40.60.81.0point 5, N = 20 50 1000.00.51.01.5point 4, N = 30 50 1000.000.250.500.751.00point 5, N = 30 50 1000.00.20.40.60.81.0point 6, N = 30 50 1000.00.51.01.5point 7, N = 30 50 1000.00.20.40.6point 8, N = 3Figure 2.11: Numerically computed spike heights (vertical axis) versus time(horizontal axis) from full PDE simulations of (2.1) for τs = 0.6 andτb = 2 at the points indicated in the middle panel of Figure 2.9. Dis-tinct spike heights are distinguished by line types (solid, dashed, anddotted).530.0 0.5 1.0 1.5 2.0 2.5 3.0K0.02.55.07.510.012.515.017.520.0Dv12345s = 0.6, b = 0.10.00 0.05 0.10 0.15 0.20K1.41.61.82.05(a)0 50 1000246point 1, N = 10 50 1000.00.51.01.52.02.5point 2, N = 10 50 1000123point 2, N = 20 50 1000.00.51.01.5point 3, N = 20 50 1000.00.20.40.6point 4, N = 20 50 1000.00.51.01.5point 5, N = 2(b)Figure 2.12: Left panel (a): Synchronous (solid) and asynchronous (dashed)instability thresholds in the Dv versus K parameter plane for the unitdisk with Db = 10 and (τs,τb) = (0.6,0.1). N = 1 spike and N = 2spikes correspond to the (blue) and (orange) curves, respectively. Thefaint grey dotted lines are the corresponding well-mixed thresholds.Right panel (b): Numerically computed spike heights (vertical axis)versus time (horizontal axis) from full PDE simulations of (2.1) at thepoints indicated in the left panel for N = 1 and N = 2 spikes. Forvideos of the PDE simulations please see the supplementary materials.the spike locations.2.4 The Effect of Boundary Perturbations onAsynchronous InstabilitiesThe goal of this section is to calculate the leading order correction to the asyn-chronous instability thresholds for a perturbed disk. Specifically we consider thedomainΩδ ≡ {(r,θ) |0≤ r < R+δh(θ), 0≤ θ < 2pi},540 5 10 15K02468MN,kDb=50 (3,1,3,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 5 10 15K020406080MN,kDb=5 (3,1,3,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 5 10 15K05101520MN,kDb=50 (2,1,2,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 5 10 15K050100150200MN,kDb=5 (2,1,2,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 5 10 15K0.00.51.01.52.0DvDb=50 (3,1,3,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 5 10 15K0.00.51.01.52.0DvDb=5 (3,1,3,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 5 10 15K0123DvDb=50 (2,1,2,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)0 5 10 15K1234DvDb=5 (2,1,2,0)(N,k)=(2,1)(N,k)=(3,1)(N,k)=(4,1)(N,k)=(4,2)Figure 2.13: The effect of boundary perturbations on the asynchronous sta-bility of symmetric N-spike patterns for the unit disk. The toprow shows the multiplier MN,k, defined in (2.7), as a function of Kwhile the bottom row shows the leading order correction to the asyn-chronous instability threshold, with the dashed line indicating the un-perturbed threshold. Each column correspond to a choice of Db = 50or Db = 5 with Gierer-Meinhardt exponents of (p,q,m,s) = (3,1,3,0)or (p,q,m,s) = (2,1,2,0). In the second row the boundary perturba-tion has parameters ξ = 1 (indicating an outward bulge at the spikelocations), and δ = 0.01.where h(θ) is a smooth O(1) function with a Fourier series h(θ) = ∑∞n=−∞ hneinθ .Although our final results will be restricted to the specific formh(θ) = 2Rξ cos(Nθ) = RξeiNθ +Rξe−iNθ , (2.1)where ξ is a parameter, there is no additional difficulty in considering a generalFourier series in the analysis below. However, we remark that in using the gen-eral Fourier series given above we must impose appropriate symmetry conditionson h(θ) so that the symmetric N-spike pattern construction, and in particular theresulting NLEP (2.5), remain valid. Our main goal is to determine a two-term55asymptotic expansion in powers of δ for each asynchronous instability thresholdin the formDv ∼ D?vk0(Db,K,R)+D?vk1(Db,K,R)δ +O(δ 2),such that a zero-eigenvalue crossing is maintained to at least second order, i.e. forwhich λ = O(δ 2).Recall that the only component of the asynchronous NLEP (2.5) that dependson the problem geometry is the NLEP multiplier χk(λ ). To study the effect ofboundary perturbations, it therefore suffices to calculate the leading order correc-tions to the corresponding membrane Green’s function satisfying (2.22). Further-more, we note that since we are only interested in a first order expansion, whereasλ =O(δ 2), there is no loss in validity assuming that λ is an independent parameterthat we ultimately set to zero. Upon expanding Dv = Dv0(1+ Dv1Dv0 δ), a two-termexpansion for the perturbed membrane Green’s function is given by (see AppendixB.2)Gλ∂Ω(θ ,θ0)∼ Gλ∂Ω0(θ ,θ0)+Gλ∂Ω1(θ ,θ0)δ +O(δ 2),where Gλ∂Ω0 is the membrane Green’s function for the unperturbed disk calculatedpreviously in (B.7) and the leading-order correction isGλ∂Ω1(θ ,θ0) =−h(θ0)R Gλ∂Ω0(θ ,θ0)+ 12piR∞∑n=−∞∞∑k=−∞gˆλn,khn−kgλk gλn einθ−ikθ0− Dv12piR3∞∑n=−∞n2(gλn )2ein(θ−θ0).(2.2)In this expression the coefficients gˆλn,k are given bygˆλn,k =Dv0R3 k(n+ k)+K2aλk(aˆλn,k +P′k(R)), (2.3)where gλk , aλk , and aˆλn,k are defined in (2.9), (B.6), and (B.14), respectively.Restricting our attention to perturbations of the form (2.1), and considering asymmetric N-spike pattern with spikes centred at θ j = 2pi( j−1)N for j = 1, ...,N, we56deduce from (2.2) thatGλ∂Ω1(θ ,θ j) =−2ξGλ∂Ω0(θ ,θ j)− Dv12piR3∞∑n=−∞n2(gλ0n)2ein(θ−θ j)+ξ2pi∞∑n=−∞{gˆλn,n+Ngλ0,n+N + gˆλn,n−Ngλ0,n−N}gλ0nein(θ−θ j).(2.4)Note that by symmetry the consistency and balance equations continue to hold fora symmetric N spike pattern. Furthermore the perturbed Green’s matrix remainscirculant, and therefore its eigenvalues can be read off asµk(λ )=N−1∑j=0Gλ∂Ω(2piN j,0)e2pii jkN ∼ µk0(λ )+δ{−2ξµk0(λ )+ξµk11(λ )+Dv1µk12(λ )}whereµk0(λ ) = N2piR∞∑n=−∞gλnN−k, (2.5a)µk11(λ ) = N2pi∞∑n=−∞{gˆλnN−k,(n+1)N−kgλ(n+1)N−k + gˆλnN−k,(n−1)N−kgλ(n−1)N−k}gλnN−k, (2.5b)µk12(λ ) =− N2piR3∞∑n=−∞(nN− k)2(gλnN−k)2. (2.5c)Finally, upon setting λ = 0 in the zero-eigenvalue crossing condition Ak(0) =[χk(0)]−1−mq/(p−1) for the asynchronous modes k = 1, . . . ,N−1 (see (2.33)),and noting χk(0) = µk(0)/µ0(0) from (2.5), we obtain thatµ00(0)+δ [−2ξµ00(0)+ξµ011(0)+Dv1µ012(0)]µk0(λ )+δ [−2ξµk0(λ )+ξµk11(λ )+Dv1µk12(λ )] −mqp−1 = 0, (2.6)for each k = 1, . . . ,N−1. The leading-order problem is satisfied by the previouslydetermined threshold Dv0 =D?vk0(K,Db,R). On the other hand, by expanding (2.6)in powers of δ , we obtain from equating O(δ ) terms in this expansion thatξ(µ011(0)− mqp−1µk11(0))+Dv1(µ012(0)− mqp−1µk12(0))= 0.57Upon solving for Dv1 = D?vk1(K,Db,R) in this expression, we conclude thatD?vk1 =−MN,kξ , where MN,k ≡µ011(0)− mqp−1µk11(0)µ012(0)− mqp−1µk12(0). (2.7)Therefore, the sign and magnitude of the multiplier MN,k determines how the asyn-chronous instability threshold changes when the boundary is perturbed by a singleFourier mode of the form (2.1).Figure 2.13 illustrates the effect of boundary perturbations of the form (2.1)by plotting the multiplier −MN,k in the top row, and the leading order correctedasynchronous threshold Dv ∼ D?vk0+D?vk1δ in the bottom row. Note that the (pos-itive) maximums of h(θ) correspond with the quasi-equilibrium spike locationsθ j for each j = 1, ...,N. From (2.7) we therefore conclude that positive values of−MN,k indicate an increase in stability when spike locations bulge out (ξ > 0), anda decrease in stability otherwise. The results of Figure 2.13 thus indicate that anoutward bulge at the location of each spike in a symmetric N-spike pattern leads toan improvement in stability of the pattern with respect to asynchronous instabili-ties. In addition, the magnitude of −MN,k shows that this stabilizing effect is mostpronounced at some finite value of K corresponding to a maximum of−MN,k. Fur-thermore, comparing the Db = 50 and Db = 5 plots we see that decreasing the bulkdiffusivity further accentuates the effect of boundary perturbations as is clear fromthe relative magnitude of −MN,k in these two cases. These numerical observationslead us to propose the following numerically supported proposition.Proposition 2.4.1. Consider a symmetric N-spike pattern for the Gierer-Meinhardtsystem (2.1) on the unit disk. Then a domain perturbation of the form (2.1), whichcreates an outward bulge at each spike location, will increase the asynchronousinstability threshold of the symmetric N-spike pattern.2.5 DiscussionWe have introduced a coupled bulk-membrane PDE model in which a scalar lin-ear 2-D bulk diffusion process is coupled through a linear Robin boundary con-dition to a two-component 1-D RD system with Gierer-Meinhardt (nonlinear) re-action kinetics defined on the domain boundary. For this coupled bulk-membrane58PDE model, in the singularly perturbed limit of a long-range inhibition and short-range activation for the membrane-bound species, we have studied the existenceand linear stability of localized steady-state multi-spike patterns defined on themembrane. Our primary goal was to study how the bulk diffusion process andthe bulk-membrane coupling modifies the well-known linear stability properties ofsteady-state spike patterns for the 1-D Gierer-Meinhardt model in the absence ofcoupling.By using a singular perturbation analysis on our coupled model (2.1) we firstderived a nonlinear algebraic system (2.16) characterizing the locations and heightsof steady-state multi-spike patterns on the membrane. Then we derived a newclass of NLEPs (nonlocal eigenvalue problems) characterizing the linear stabilityon O(1) time-scales of these steady-state patterns. In this NLEP, the multiplier ofthe nonlocal term is determined in terms of the model parameters together with anew coupled nonlocal Green’s function problem. More specifically, a novel featureof our steady-state and linear stability analysis is the appearance of a nonlocal 1-Dmembrane Green’s function Gλ∂Ω(σ ,ζ ) (see (2.22)), satisfyingDv∂ 2σGλ∂Ω(σ ,ζ )−(1+K+τsλ )Gλ∂Ω(σ ,ζ )+K2ˆ L0GλΩ(σ , σ˜)Gλ∂Ω(σ˜ ,ζ )dσ˜ =−δ (σ−ζ ),for 0 < σ ,ζ < L which is coupled to a 2-D bulk Green’s function GλΩ satisfying(see (2.19))Db∆GλΩ−(1+τbλ )GλΩ= 0, in Ω ; Db∂nGλΩ+KGλΩ= δ∂Ω(x−x0), on ∂Ω.Recall (2.1) for the description of all the model parameters including, the timeconstants τs and τb, the diffusivities Dv and Db, and the coupling constant K.To proceed with a more explicit linear stability theory we restricted our analysisto symmetric multi-spike patterns, which are characterized by equidistantly (in arc-length) separated spikes of equal height, for two analytically tractable cases. Thefirst case is when Ω is a disk of radius R, while the second case is when the bulk iswell mixed (i.e. Db 1). While our formulation is equally valid for more generalsettings there are two significant hurdles toward a more detailed stability analysis.First, the global coupling introduced by the nonlocal membrane Green’s functionmakes it unclear how to define symmetric multi-spike patterns. Although we re-59marked earlier that such a classification can be associated with the condition thatG λ∂Ω has the eigenvector e, apart from domains with certain rotational symmetries itis not clear how the geometry is related to this condition. Secondly, the numericalcomputation of the bulk Green’s function for more general domains remains a topicof ongoing research. For the two specific cases, we obtained analytical expressionsfor the relevant Green’s function, and consequently the NLEP multipliers, in theform of infinite series for the disk and explicit formulae for the well-mixed limit.Parameter thresholds for two distinct forms of linear instabilities, corresponding toeither synchronous or asynchronous perturbations of the heights of the steady-statespikes, were then computed from the NLEP. Our results indicate a non-monotonicdependence on the bulk-membrane coupling strength K for both modes of insta-bility, together with an intricate relationship between the time-scale and couplingparameters for the synchronous instabilities. Specifically, for the asynchronous in-stability modes the coupling has the effect of improving stability for smaller valuesof K by raising the instability threshold for Dv, but reducing the range of stabil-ity for larger values of K. This effect is amplified in the synchronous case wherefor certain choices of τs a small region in the K versus τb parameter space can befound for which no instabilities exist (see Figure 2.4). Finally, by using a FiniteElement / Finite Difference mixed IMEX scheme, we confirmed our linear stabilitythresholds with full numerical PDE simulations.We conclude the discussion by highlighting some open problems and directionsfor future research. Firstly, for our coupled model, additional work is required tocalculate and study the linear stability of asymmetric spike patterns. Secondly, wehave neglected the role of small O(ε2) eigenvalues corresponding to weak drift in-stabilities, which can be studied either through a more detailed asymptotic analysisor by deriving and analyzing a corresponding slow spike-dynamics ODE system.Thirdly, the numerical evidence provided by our PDE simulations suggests that,when N ≥ 2 in the absence of competition instabilities, the Hopf bifurcation is su-percritical, and leads to the emergence of a small amplitude time-periodic solutionnear the bifurcation point. The numerical evidence also suggests that competitioninstabilities are subcritical, and result in the annihilation of one or more spikes ina multi-spike pattern. It would be worthwhile to analytically establish these con-jectured branching behaviours from a weakly nonlinear analysis that is valid either60near a Hopf bifurcation point or near a zero-eigenvalue crossing. In particular,it would be interesting to perform a more detailed analysis of the observed time-periodic solution near the Hopf bifurcation threshold to determine whether it hassome regularity or is otherwise chaotic.Finally, there are several directions for extending our model and applying asimilar methodology. One direction would be to analyze similar problems in higherspace dimensions, such as a 3-D linear bulk diffusion process coupled to a nonlin-ear RD system on a 2-D surface. A common feature in the matched asymptoticscalculation for higher-dimensional problems is that the inhibitor and activator nolonger decouple in the inner problem. This leads to a nonlinear algebraic systemand globally coupled eigenvalue problem markedly different from those in one-dimensional problems. An analytical treatment of the effect of coupling on thesesystems has, as of yet, been unexplored. A further direction would be to considera two-component bulk diffusion process, with nonlinear bulk kinetics. For thismore complicated model it would be interesting to study the interplay between 1-D membrane-bound and 2-D bulk-bound localized patterns. Additionally it wouldbe instructive to asymptotically construct and analyze the localized patterns ob-served in the numerical study of Madzvamuse et. al. [57, 58] as well those of Rätzet. al. [81–83].61Chapter 3Localized Spot Patterns in aBulk-Membrane CoupledBrusselator ModelThe classical Brusselator model [79] is characterized by the reaction kineticsE −→U, B+U −→V +D, 2U +V −→ 3U, U −→ A, (3.1)for the activator U , inhibitor V , fuel E, and catalyst B. In most studies of the Brus-selator model the fuel and catalyst are typically assumed to be of constant concen-tration while the activator and inhibitor concentrations are assumed to depend onboth space and time. In particular, the spatiotemporal evolution of the activatorand inhibitor concentrations is determined by a two-component reaction diffusionsystem which is obtained by applying the Law of Mass Action and assuming thatboth chemical components have a finite diffusivity.In this chapter we analyze the structure and stability of localized spot patternsin a model which incorporates bulk-membrane coupling into a reaction-diffusion-system with Brusselator-like reaction kinetics. In particular we consider a modelin which the activator U and inhibitor V are bound to the membrane ∂Ω0 of a62three-dimensional domain Ω0 where they passively diffuse with reaction kineticsB+U −→V +D, 2U +V ka−→ 3U, (3.2)where B is a catalyst component and D a passive product. Bulk-membrane cou-pling is incorporated by assuming that the activator shuttles between membrane-and bulk-bound states through a Langmuir process of membrane attachment anddetachment [43]. Moreover we assume that the fuel necessary to sustain patternformation originates within the bulk and passively diffuses to the membrane. Theresulting bulk-membrane coupled reaction diffusion system for the membrane-bound inhibitor concentration V and membrane- and bulk-bound activator con-centrations U and W respectively is then given by the bulk-membrane coupledreaction-diffusion system∂TU =DU ∆∂Ω0U − (B+ kd +K1)U + kaU 2V +K2W , X ∈ ∂Ω0, (3.3a)∂TV =DV ∆∂Ω0V +BU − kaU 2V , X ∈ ∂Ω0, (3.3b)∂TW =DW ∆W +E (X ), X ∈Ω0, (3.3c)DW ∂NW =K1U −K2W , X ∈ ∂Ω0. (3.3d)where ∂T is the time derivative, ∂N is the derivative in the direction of the out-ward unit normal to ∂Ω0, ∆∂Ω0 is the Laplace-Beltrami operator on ∂Ω0 whichdescribes lateral diffusion on the membrane, DU , DV , and DW are the diffusiv-ities of U ,V , and W respectively, B is the conversion rate of U to V throughthe reaction B+U → V +D, K1 and K2 are the rates of activator membrane de-tachment and attachment respectively, ka is the rate of the autocatalytic reaction2U +V ka−→ 3U , kd is the activator membrane-degradation rate, and E (X ) is thebulk-bound activator fuel source which we assume is compactly supported in theinterior of Ω0.While the bulk-membrane coupled model (3.3) explicitly includes only two ofthe reactions from the Brusselator reaction-kinetics (3.1), the remaining two reac-tions are incorporated primarily through bulk-membrane coupling. In particularthe reaction U −→ A in (3.1) is replaced by a membrane-to-bulk detachment pro-cess characterized by the −K1U term in (3.3a) in addition to a generic membrane63degradation term characterized by the −kdU term. Similarly, the reaction E →Uin (3.1) required for sustaining pattern formation is replaced by the transport of a lo-calized fuel-term in the bulk to the membrane through passive diffusion. Note thatif K1 = 0 then the two-component system (3.3a)-(3.3b) is indistinguishable fromthe classically studied Brusselator system, albeit with a possibly spatially hetero-geneous fuel term given by K2W . However, the introduction of bulk-membranecoupling provides a systematic way of choosing the fuel term appearing in theclassical Brusselator reaction kinetics (3.1). In our model the fuel term E (X ) canencompass a wide variety of bulk-originating fuel terms that closely describe bio-logical processes being modelled. Such a modelling paradigm may be particularlyfruitful in the context of conifer growth models previously based on the Brusselatorreaction kinetics (see for example [35, 36] and the references therein). In additionbulk-membrane coupling introduces a nonlocal mechanism of chemical transporton the membrane characterized by a cycle of membrane detachment, bulk-bounddiffusion, and membrane reattachment. One of the key goals of this chapter isto explore the effects of this nonlocal mechanism on the structure, stability, anddynamics of localized patterns.To simplify our analysis and isolate key parameters we first perform a nondi-mensionalization of the model (3.3). Letting L be a characteristic length scalefor the domain Ω0 and its boundary ∂Ω0 we introduce non-dimensional spatialvariables x = L−1X so and define Ω = L−1Ω and ∂Ω = L−1∂Ω0. We further letDU ≡ L−2DU ,DV ≡ L−2DV , andDW ≡ L−2DW . In the limit of an asymptoticallysmall activator diffusivity DU = ε20  1 the resulting two-dimensional singularlyperturbed Brusselator system is known to support localized spot patterns when thefuel is O(ε0) [84, 91]. It follows that W = O(ε0) is needed to sustain patterns sowe assume that E (X ) = ε0E0E(x) where E(x) =O(1) in Ω and by introducing thenon-dimensional variablesU = ε−10 E0Lu, V = ε0BkaLE0v, W = ε0E0LK2w, T =tB+ kd +K1, (3.4)64we obtain the nondimensionalized bulk-membrane coupled system∂tu = ε2∆∂Ωu−u+ f u2v+ ε2w, x ∈ ∂Ω, t > 0 (3.5a)τv∂tv = Dv∆∂Ωv+ ε−2(u−u2v) x ∈ ∂Ω, t > 0, (3.5b)τw∂tw = Dw∆w+E(x), x ∈Ω, t > 0, (3.5c)Dw∂nw+w = ε−2Ku, x ∈ ∂Ω, t > 0, (3.5d)where ∆∂Ω is the Laplace-Beltrami operator on ∂Ω, ∂n denotes the derivative in thedirection of the outer unit normal, the time-constants and diffusivities are given byτv ≡ (B+ kd +K1)2 1kaE20 L2, Dv ≡ τvDVB+ kd +K1,τw ≡ (B+ kd +K1)LK2, Dw ≡ DW LK2,and the remaining parameter are given byε ≡ ε0√B+ kd +K1 1, K ≡ K1B+ kd +K1, f ≡ BB+ kd +K1. (3.6)Recall that the membrane-bound activator is reduced by three processes: con-version to V through the reaction B+U −→ V +D, membrane detachment at arate of K1, and degradation at a rate of kd . Therefore 0 ≤ K < 1 and 0 < f < 1correspond to the proportion of the rate of membrane-bound activator lost due tomembrane detachment and conversion to membrane-bound inhibitor respectively.An additional constraint on the two parameters f and K is obtained from (3.3) byusing the divergence theorem to calculateddTˆ∂Ω0(U +V )dS =−kdˆ∂ΩUdS+E0,from which it is clear that kd > 0 is needed to sustain stationary patterns and wededuce the constraint0 < f +K < 1. (3.7)In the absence of bulk-membrane coupling (e.g. with K = 0), several studieshave investigated the resulting singularly perturbed reaction diffusion system con-65sisting of (3.5a) and (3.5b). In particular when ∂Ω is the unit sphere in R3 thequasi-equilibrium structure, linear stability, and slow dynamics of multi-spot pat-terns in the presence of a spatially homogeneous and time-independent fuel havebeen analyzed using the method of matched asymptotic expansions in [84, 91].Assuming henceforth that Ω is the unit ball in R3 our aim is to extend the re-sults obtained in these previous studies to the bulk-membrane coupled model (3.5).The novel features of this extension are twofold. First, the bulk originating fuelterm will in general lead to a spatially heterogeneous fuel source for the membraneactivator equation. Presently, the effects of such a spatially heterogeneous fuelterm have been considered only in the context of the unit disk in R2 [96]. Sec-ondly, membrane-detachment and reattachment leads to a nonlocal mechanism ofmembrane-bound activator transport which has no yet been explored in the contextof localized patterns on two-dimensional surfaces.The remainder of this chapter is organized as follows. In §3.1 we use themethod of matched asymptotic expansions to construct quasi-equilibrium multi-spot patterns that are stationary on an O(1) time scale and in §3.2 we study theirlinear stability on an O(1) timescale. In §3.3 we derive a differential algebraicsystem of equations governing the slow evolution of multi-spot solutions on anO(ε−2) timescale. In §3.4 we explicitly construct and consider the stability anddynamics of one-, two-, and three-spot patterns. Finally, in §3.5 we provide a briefconclusion.3.1 Asymptotic Construction of N-SpotQuasi-EquilibriumThe method of matched asymptotic expansions has been extensively used to ana-lyze localized solutions to singularly perturbed reaction diffusion systems in one-,two-, and three-dimensional domains [41, 48, 97, 98]. It has likewise been used tostudy localized multi-spot patterns on the unit sphere [84, 91], and more recentlyon the torus [95]. Applying these techniques we now construct a quasi-equilibriumsolution consisting of N spots concentrating on ∂Ω atxi = (sinθi cosϕi,sinθi sinϕi,cosθi)T , for i = 1, ...,N, (3.8)66where 0< θi < pi and 0≤ϕi < 2pi (i= 1, ...,N) are the typical spherical coordinatesand for which we assume that the spots are well separated in the sense that |xi−x j|  ε for all i 6= j. This solution is to be understood only as a quasi-equilibriumbecause, unless additional constraints are imposed on x1, ...,xN , the spots will drifton an O(ε−2) timescale according to prescribed dynamics developed in §3.3. Withthese assumptions a local expansion of (3.5a) and (3.5b) near each xi (i = 1, ...,N)yields a system of core-problems which are coupled only through the prescriptionof far-field constants for the local inhibitor concentrations. An appropriate choiceof local coordinates greatly simplifies the resulting analysis. Motivated by the useof geodesic normal coordinates in [92] and more recently in [95] we first constructlocal normal coordinates near each spot.3.1.1 Geometric Preliminaries: Local Geodesic Normal Coordinateson the Unit SphereIn terms of spherical coordinates r > 0 and (θ ,ϕ) ∈ (0,pi)× [0,2pi) the Laplaceoperator and Laplace-Beltrami operator on the unit sphere are respectively givenby∆=1r2∂∂ rr2∂∂ r+1r2∆∂Ω, ∆∂Ω =1sinθ∂∂θsinθ∂∂θ+1sin2 θ∂ 2∂ϕ2. (3.9)Previous studies of multi-spot patterns on the unit sphere have used the local coor-dinates s2 = ε−1(θ −θi) and s2 = ε−1 sinθi(ϕ−ϕi) to parameterize points on ∂Ωwithin a O(ε) neighbourhood of each spot location xi [84, 91] (see also [13] for asimilar approach in the context of narrow escape problems). In terms of these localcoordinates it is then straightforward to calculate the two term expansion∆∂Ω = ε−2(∂ 2s1 +∂2s2)+ ε−1 cotθi(∂s1−2s1∂ 2s2)+O(1).The factor of cotθi in the O(ε−1) correction implies that the local behaviour ofspots depends on their location on the unit sphere which suggests this correction isan artifact of the choice of local coordinates. Indeed this term was recognized asa correction to the leading order tangent plane approximation by Trinh and Ward[91]. The accompanying analysis in the study of slow spot dynamics that arises due67Figure 3.1: Example of geodesic normal coordinates (ζ1,ζ2,ζ3) at xi ∈ ∂Ω.The blue (resp. orange) curves indicate geodesics obtained by varying−pi/2 < ζ1 < pi/2 (resp. −pi < ζ2 < pi) and fixing ζ3 = 0 and ζ2 = 0(resp. ζ1 = 0).to such an O(ε−1) correction can be bypassed through the use of local geodesicnormal coordinates in terms of which it can be shown that the O(ε−1) correctionin the local expansion of the Laplace-Beltrami operator vanishes identically [92].For the remainder of this section we explicitly construct such a coordinate systemfor the unit ball.We begin by defining new coordinates ζ =(ζ1,ζ2,ζ3)T ∈ (−pi/2,pi/2)×(−pi,pi)×[0,1] in Ω∪∂Ω such that ζ = 0 corresponds to xi ∈ ∂Ω, ξ3 > 0 corresponds to theinterior of Ω, and the curves obtained by setting ζ3 = 0 and fixing either ζ1 = 0 orζ2 = 0 are geodesics on ∂Ω. Specifically we first calculate the orthonormal vectorsxi =sinθi cosϕisinθi sinϕicosθi , ∂θxi =cosθi cosϕicosθi sinϕi−sinθi , xi×∂θxi =−sinϕicosϕi0 , (3.10a)and define ζ byxi(ζ ) = (1−ζ3)(cosζ1 cosζ2xi+ cosζ1 sinζ2∂θxi+ sinζ1xi×∂θxi). (3.10b)Thus ζ1 and ζ2 are the standard spherical coordinates in the reference frame withorthonormal basis {xi,∂θxi,xi× ∂θxi}, while ζ3 measures the distance from ∂Ω.68It is easy to verify that setting ζ = 0 corresponds to x = xi. Additionally thecoordinate curves obtained by setting ζ3 = 0 and fixing either ζ2 = 0 or ζ1 = 0 arerespectively given byxi(ζ1,0,0) = cosζ1xi+ sinζ1xi×∂θxi, xi(0,ζ2,0) = cosζ2xi+ sinζ1∂θxi,which correspond to intersections of ∂Ω with the planes spanned by {xi,xi×∂θxi}and {xi,∂θxi} respectively. In particular, these coordinates curves are geodesics on∂Ω so that in terms ofζ = εY = ε(Y1,Y2,Y3)T , Y = O(1) (3.11a)we obtain the local expansions (see Appendix A of [92])∆= ε−2∂ 2Y3 + ε−2∆(Y1,Y2)+O(ε−1), ∆∂Ω = ε−2∆(Y1,Y2)+O(1), (3.11b)where∆(Y1,Y2) ≡ ∂ 2Y1 +∂ 2Y2 . (3.11c)Moreover, by letting Y ′ ≡ (Y1,Y2)T , ρ ≡ |Y ′|=√Y 21 +Y22 , andJi ≡(xi×∂θxi∣∣∂θxi)=−sinϕi cosθi cosϕicosϕi cosθi sinϕi0 −sinθi ,we readily calculatexi(εY ) = xi+ ε(JiY ′−Y3xi)− ε2(ρ2xi2+Y3(Y2∂θxi+Y1xi×∂θxi))+O(ε3),(3.12)from which it follows that for all Y ′ = O(1)|xi(ε(Y ′,0))−ξ |2 = |xi−ξ |2+2εY ′TJ Ti (xi−ξ ) +O(ε2) (3.13a)69when |ξ − xi|=O(1), and|xi(ε(Y ′,0))−ξ |2 = ε2|Y ′−Z ′|2+O(ε4), (3.13b)when ξ = xi(ε(Z ′,0)) and Z ′ = O(1). In the second case we have usedJ Ti Ji =I2 (the 2×2 identity matrix) as well as J Ti xi = 0 to cancel the O(ε3) contribu-tion. On the other hand, writing any v ∈ R3 as v = v1xi + v2∂θxi + v3xi×∂θxi weobtainJiJTi v =Ji(v3v2)= v2∂θxi+ v3xi×∂θxi,from which we obtainJiJTi =I3− xixTi , (3.14)and deduce thatJiJ Ti is the projection onto the tangent plane of ∂Ω at xi.3.1.2 Matched Asymptotic Expansions and the Nonlinear AlgebraicSystemWe seek a quasi-equilibrium solution to (3.5) which is stationary on an O(1) timescale. First we construct the inner solution by letting Y be the local coordinates nearxi given by (3.11a). From (3.5c) we see that w(xi(εY )) = O(ε−1) and thereforethe leading order core problem for the membrane-bound activator and inhibitorare identical to those previously considered in [84] and [91]. In particular theleading order inner solution is completely determined by an unknown constantspot strength Si and is explicitly given byu(xi(εY ′,0)) =√DvUi0(ρ)+O(ε), v(xi(εY ′,0)) =1√DvVi0(ρ)+O(ε),(3.15)whereUi0(ρ)≡Uc(ρ;Si, f ), Vi0(ρ)≡Vc(ρ;Si, f ), (3.16)70and (Uc(ρ;S, f ),Vc(ρ;S, f )) is the radially symmetric solution to the core problem∂ 2ρUc+ρ−1∂ρUc−Uc+ fU2c Vc = 0, ρ > 0, (3.17a)∂ 2ρVc+ρ−1∂ρVc+Uc−U2c Vc = 0, ρ > 0, (3.17b)∂ρUc = ∂ρVc = 0, at ρ = 0, (3.17c)Uc→ 0, Vc ∼ S logρ+χ(S, f )+o(1), as ρ → ∞. (3.17d)The function χ(S, f ) indicates the far-field constant behaviour of the inner inhibitorsolution and can be numerically calculated by solving (3.17) on a truncated domain(see §2 of [84]). In Figure 3.2a we plot the numerically calculated χ(S, f ) versusS > 0 for select values of 0 < f < 1. By integrating (3.17b) over 0 < ρ < ∞ weobtain the useful identitySi =1− ffˆ ∞0Ui0(ρ)ρdρ =−ˆ ∞0(Ui0(ρ)−Ui0(ρ)2Vi0(ρ))ρdρ. (3.18)The leading order behaviour of the bulk-bound activator near xi is then readilycomputed by lettingw(xi(εY )) = ε−1Wi0(Y )+O(1), (3.19)so from (3.11b) we obtain∂ 2Y3Wi0+∆(Y1,Y2)Wi0 = 0, (Y′,Y3) ∈ R2× (0,∞),Dw∂Y3Wi0 = K√DvUi0(ρ), (Y ′,Y3) ∈ R2×{0}.This in turn can be explicitly solved using the Neumann Green’s function on theupper half-space which yieldsWi0(Y ) =K√Dv2piDwˆ ∞0ˆ ∞0Uc(√ξ 21 +ξ 22 )√(ξ1−Y1)2+(ξ2−Y2)2+Y 23dξ1dξ2+W i0, (3.20)710 2 4 6 8 10S0102030Far-field Constant (S, f)f0.20.40.60.8(a)0.0 0.2 0.4 0.6 0.8f010203040mInstability Threshold m(f)m234(b)0 5 10 15 20 25S0.00.51.01.5Dynamics Coefficient (S, f)f0.20.40.60.8(c)Figure 3.2: Numerically calculated (a) far-field constant χ(S, f ), (b) m-modeinstability threshold Σm( f ) for 2 ≤ m ≤ 4, and (c) slow dynamics mul-tiplier γ(S, f ). The solid circles in (c) indicate values of γ(S, f ) at thesplitting instability thresholds Σ2( f ). These functions depend only onthe local structure of each spot and are therefore independent of bulk-membrane coupling.where W i0 is an undetermined constant. Note that Wi0 is bounded for all Y3 ≥ 0,and is radially symmetric when Y3 = 0.When x ∈ ∂Ω and |x−xi|  ε for all i= 1, ...,N the exponential decay of eachUi0(ρ) as ρ → ∞, together with (3.13) implies that the membrane-bound activatoris, to leading order, given byu(x)∼ ε2w(x)+√DvN∑i=1Ui0(ε−1|x− xi|), (3.21)which in turn implies the distributional limitsu−u2vε2→ w(x)−2pi√DvN∑i=1Siδ∂Ω(x− xi), ε → 0+,uε2→ w(x)+ 2pi√Dv f1− fN∑i=1Siδ∂Ω(x− xi), ε → 0+.The leading order outer problems for the membrane-bound inhibitor and bulk-72bound activator are then respectively given byDv∆∂Ωv =−w(x)+2pi√DvN∑i=1Siδ∂Ω(x− xi), x ∈ ∂Ω, (3.22)andDw∆w =−E(x), x ∈Ω, (3.23a)Dw∂nw+(1−K)w = 2pi√DvK f1− fN∑i=1Siδ∂Ω(x− xi), x ∈ ∂Ω. (3.23b)By first integrating (3.22) over ∂Ω and then applying the divergence theorem to(3.23) we deduce the solvability conditionN∑i=1Si =12pi√Dv(1−K)ˆΩE(x)d3x+K f(1− f )(1−K)N∑i=1Si,which can be further rearranged into the formN∑i=1Si =12pi√Dv(1− K1− f)−1ˆΩE(x)d3x. (3.24)The dependence of the spot strengths on the total membrane-bound fuel is similarto that obtained for spot patterns to the Brusselator model in the unit disk withan inhomogeneous fuel source [96], whereas the coupling-dependence reflects afeedback mechanism introduced by the cycle of membrane detachment and reat-tachment. Moreover, using (3.6) we obtain that in terms of the original problem pa-rameters (1−K/(1− f ))−1 =K1/kd which indicates that the coupling-dependentmultiplier reflects the ratio between the rate of membrane-detachment to that ofdegradation.Next we solve for the membrane-bound inhibitor and bulk-bound activator inthe outer region where |x−xi| O(ε) for all i = 1, ...,N. First we note that (3.23)is solved byw(x)∼ 1DwwE(x)+2pi√DvDwK f1− fN∑i=1SiGmr(x,xi), (3.25)73where wE(x) is the unique solution to∆wE =−E(x), x ∈Ω; ∂nwE + 1−KDw wE = 0, x ∈ ∂Ω, (3.26)and Gmr(x,xi) is the membrane Robin Green’s function satisfying∆Gmr = 0, x ∈Ω; ∂nGmr + 1−KDw Gmr = δ∂Ω(x− xi), x ∈ ∂Ω,(3.27a)for which we note the explicit series expansionGmr(x,ξ ) =14pi∞∑l=0glPl(ξT x), gl ≡ 2l+1l+ 1−KDw, (3.27b)where Pl(x) is the Legendre polynomial of degree l. Assuming the solvabilitycondition (3.24) is satisfied, substituting (3.25) into (3.22) we explicitly calculatethe outer solution for the membrane-bound inhibitorv∼− 2pi√DvN∑i=1(Gs(x,xi)− 1DwK f1− fˆ∂ΩGs(x,ξ )Gmr(ξ ,xi)dAξ)Si+1√Dvv¯+1DwDvvE(x),(3.28)where v¯ is an undetermined constant, vE(x) is given byvE(x)≡ˆ∂ΩGs(x,ξ )wE(ξ )dAξ , (3.29)and Gs(x,ξ ) is the membrane Green’s function satisfying∆∂ΩGm(x,ξ ) =14pi−δ∂Ω(x−ξ ), x ∈ ∂Ω, (3.30a)ˆ∂ΩGs(x,ξ )dAx = 0, ξ ∈ ∂Ω, (3.30b)and given explicitly given by [84]Gs(x,ξ ) =− 12pi log |x−ξ |+R0, R0 ≡log4−14pi. (3.30c)74In addition we note that this membrane Green’s function has the series expansionGs(x,ξ ) =14pi∞∑l=12l+1l(l+1)Pl(ξT x), (3.31)so that by using the well known product formulaˆ∂ΩPl(xTi x)Pk(xT x j)dAx = δkl4pi2l+1Pl(xTi x j), (3.32)and expansion (3.27b) we obtainˆ∂ΩGs(x,ξ )Gmr(ξ ,xi)dAξ =14pi∞∑l=1gll(l+1)Pl(xTi x), x ∈ ∂Ω. (3.33)To determine the unknown spot strengths S1, · · · ,SN we match the behaviour ofthe inner solution (3.15) as ρ→∞ with the limiting behaviour of the outer solution(3.28) as |x−xi| → 0 for each i= 1, ...,N. Recalling the local expansion (3.13) andusing (3.30c) we calculateGs(xi(ε(Y ′,0)),ξ )∼ Gs(xi,ξ )− ε2piY′TJ Tix−ξ|x−ξ |2 +O(ε2), (3.34a)when |ξ − xi|  ε , andGs(xi(ε(Y ′,0)),ξ )∼− 12pi log |Y′−Z ′|+ 12piν+R0+O(ε2), (3.34b)when ξ = xi(ε(Z ′,0)) and Z ′ =O(1) and where we identifyν ≡− 1logε 1. (3.35)Evaluating (3.28) at x = xi(ε(Y ′,0)) we therefore obtain√Dvv(xi(ε(Y ′,0)))∼(logρ− 1ν)Si−2piN∑j=1Gi jS j +vE(xi)Dw√Dv+ v¯+ εY ′TJ Ti(∑j 6=iS jα i j +∇xvE(xi)Dw√Dv)+O(ε2).(3.36)75where for all i, j = 1, ...,N we defineGi j ≡− 1DwK f1− fˆ∂ΩGs(xi,ξ )Gmr(ξ ,x j)dAξ +R0, i = j,Gs(xi,x j), i 6= j, (3.37)andα i j ≡ xi− x j|xi− x j|2 −f1− fKDwˆ∂Ωxi−ξ|xi−ξ |2 Gmr(ξ ,x j)dAξ . (3.38)We also note that by using (3.29) and (3.30c) we explicitly calculate∇xvE(xi)≡− 12piˆ∂Ωxi−ξ|xi−ξ |2 wE(ξ )dAξ . (3.39)While theO(ε) correction in (3.36) does not play a role in either the quasi-equilibriumconstruction or the leading-order linear stability theory on an O(1) timescale, itdoes play a crucial role in deriving the slow dynamics taking place on an O(ε−2)timescale as detailed in §3.3 below.Comparing the behaviour of the outer solution (3.36) in the limit |x− xi| → 0with the limiting behaviour of the inner solution (3.15) as ρ→∞ yields, to leadingorder, the nonlinear systemν−1S+2piG S+χ (S, f ) = v¯e+1Dw√DvvE , (3.40)whereS ≡S1...SN , e ≡1...1 , vE ≡vE(x1)...vE(xN) , χ (S; f )≡χ(S1, f )...χ(SN , f ) , (3.41)and G is the N×N matrix with entries Gi j given by (3.37). The nonlinear system(3.40) must be solved subject to the solvability condition (3.24) which we canrewrite as eT S = NSc where the common spot strength Sc is defined bySc ≡ 12piN√Dv(1− K1− f)−1ˆΩE(x)d3x. (3.42)76Left-multiplying (3.40) by eT and using the solvability condition we calculatev¯ =1νSc+1NeT(2piG +χ (S, f ))− 1Dw√DvNeT vE , (3.43a)from which we find that the unknown spot strengths S must solve the nonlinearalgebraic system (NAS)S+2piν(IN−EN)G S+ν(IN−EN)χ (S, f ) = Sce+ νDw√Dv(IN−EN)vE(3.43b)where IN is the N×N identity matrix and EN ≡ N−1eeT . Summarizing, we havethe following proposition.Proposition 3.1.1. Let ε  1 and assume that x1, ...,xN ∈ ∂Ω are well separatedin the sense that |xi− x j|  ε for all i 6= j. Then, a quasi-equilibrium solutionto (3.5) consisting of N-spots concentrating at x1, ...,xN is asymptotically given by(3.15) and (3.19) when |x−xi|  ε for each i= 1, ...,N, and by (3.21), (3.28), and(3.25) when |x− xi|  ε for all i = 1, ...,N, where v¯ is given by (3.43a) and thespot strengths S1, ...,SN are found by solving the NAS (3.43b).Although the NAS (3.43b) must in general be solved numerically, it providesan asymptotic approximation that is accurate to all orders of ν = −1/ logε  1.Indeed, assuming that all problem parameters are O(1) with respect to ε  1, wecan easily calculate a two-term asymptotic expansion in powers of ν asS ∼ Sce+ν(IN−EN)(1Dw√DvvE −2piScG e)+O(ν2),for which we highlight that the logarithmic dependence of ν on ε may render thecorrection term ν2 impractically large. On the other hand, under certain condi-tions on the spot configuration x1, ...,xN and bulk-bound fuel source E(x) the NAS(3.43b) may be explicitly solved. In fact, since the range ofIN−EN is orthogonalto e we find that S = Sce is an exact solution of (3.43b) provided that(IN−EN)( 1Dw√DvvE −2piScG e)= 0.One such example occurs when the bulk-bound fuel has azimuthal symmetry about77an axis spanned by z 6= 0. In such a case vE = vE(zT x) and S = Sce exactly solves(3.43b) provided that x1, ...,xN are uniformly distributed on a ring making a com-mon angle with z. Indeed for such a configuration vE is proportional to e while Gis a circulant matrix and therefore has a constant row-sum. Finally, we remark thatdistinct asymptotic approximations to (3.43b) can be calculated when the remain-ing problem parameters are in different ε-dependent regimes [91].3.2 Linear Stability on an O(1) TimescaleIn this section we consider the linear stability on an O(1) timescale of the quasi-equilibrium solution (ue,ve,we) constructed in §3.1 above. We begin by first con-sidering perturbations of the formu(x) = ue(x)+ eλ tϕ(x), v(x) = ve(x)+ eλ tψ(x), w(x) = we(x)+ eλ tη(x),where we assume that λ = O(1) to reflect our restriction to instabilities arising onan O(1) timescale. Substituting into (3.5) and retaining only the linear terms in ϕ ,ψ , and η then yields the linearized eigenvalue problemε2∆∂Ωφ −φ +2 f ueveφ + f u2eψ+ ε2η = λφ , x ∈ ∂Ω, (3.44a)Dv∆∂Ωψ+ ε−2(φ −2ueveφ −u2eψ) = τvλψ x ∈ ∂Ω, (3.44b)Dw∆η = τwλη , x ∈Ω, (3.44c)Dw∂nη+η = ε−2Kφ , x ∈ ∂Ω, , (3.44d)Comparing this system with (3.5) we deduce that, as in §3.1, φ is localized at theN spot locations x1, ...,xN , while φ , ψ , and η retain the same relative scalingsin the inner-region near each spot location as in §3.1, mainly φ and ψ are O(1)and η is O(ε−1) near each x j ( j = 1, ...,N). In terms of polar coordinates Y ′ =(ρ cosΘ,ρ sinΘ) near the jth spot, we seek an inner expansion of the formφ(x j(εY ′,0)) =Φ j(ρ)eimΘ+O(ε), ψ(x j(εY ′,0)) = D−1v Ψ j(ρ)eimΘ+O(ε),(3.45)where m = 0,2,3... and we omit the neutrally stable translational m = 1 mode.Assuming that both τvλε2  1 and τwλε2  1 the leading order inner problem78near the jth spot is then given by∆ρM j− m2ρ2M j +Q(ρ;S j, f )M j = λE11M j, ρ > 0, (3.46a)where M j ≡ (Φ j(ρ),Ψ j(ρ))T andQ(ρ;S, f )≡(2 fUc(ρ;S, f )Vc(ρ;S, f )−1 fUc(ρ;S, f )2−2Uc(ρ;S, f )Vc(ρ;S, f )+1 −Uc(ρ;S, f )2), E11 =(1 00 0).At the origin we impose the boundary condition M ′j(0) = 0 while the limitingbehaviour as ρ → ∞ is found by first noting thatQ(ρ;S, f )→(−1 01 0)as ρ → ∞,for all S≥ 0 and henceΦ j→ 0, Ψ j ∼O(logρ) m = 0O(ρ−m) m≥ 1, ρ → ∞. (3.46b)Instabilities due to the m = 0 and m ≥ 2 modes arise through distinct mecha-nisms due to the logarithmic growth of the former and the algebraic decay of thelatter. Indeed, proceeding with the method of matched asymptotic expansions asin §3.1 we determine that the limiting behaviour of ψ as |x− x j| → 0 for eachj = 1, ...,N is O(1) and O(εm) when m = 0 and m≥ 2 respectively. Consequentlyglobal contributions are, to leading order in ε  1, absent in the calculation of in-stabilities to due m≥ 2 perturbations, whereas they may arise when m = 0. In theproceeding sections we discuss these two cases separately.3.2.1 The m≥ 2 Mode InstabilitiesDue to the algebraic decay of Ψ j(ρ) as ρ → ∞ for each j = 1, ...,N when m ≥ 2the inner problems are interact through the outer solution only weakly. Therefore,each spot may undergo an m ≥ 2 mode instability individually with the relevantinstability threshold being determined solely by the spot strength S j ( j = 1, ...,N).79In particular, it suffices to consider the spectrum of∆ρM− m2ρ2M +Q(ρ;S, f )M = λE11M , ρ > 0, (3.47a)M ′(0) = 0, M →(0M∞ρ−m)as ρ → ∞, (3.47b)as a function of 0 < f < 1, m ≥ 2, and S > 0. The eigenvalue problem (3.47) isidentical to that found in the study of multi-spot solutions for the Brusselator sys-tem on the unit sphere [84, 91] and the unit disk [96], and it also shares qualitativesimilarities with analogous problems derived for the two- and three-dimensionalSchnakenberg models [48, 98]. In particular, for a fixed value of m≥ 2 it is knownthat there exists a threshold Σm( f ) > 0 such that (3.47) admits an eigenvalue withpositive real part if and only if S > Σm( f ). Each threshold Σm( f ) must be calcu-lated numerically and this is easily accomplished by studying the spectrum of thematrix obtained by an appropriate discretization of (3.47) on a truncated domain(see §3.1 of [84] for details).The plots of Σ2( f ), Σ3( f ), and Σ4( f ) versus 0 < f < 1 shown in Figure 3.2bindicate that Σ2( f ) < Σ3( f ) < Σ4( f ) while further numerical evidence suggeststhat in fact Σ2( f ) < Σm( f ) for all m ≥ 2 so that Σ2( f ) is the appropriate instabil-ity threshold. It follows that a multi-spot pattern with spot strengths S1, ...,SN isunstable with respect to the m ≥ 2 modes if Si > Σ2( f ) for any i = 1, ..,N. More-over, numerical simulations of (3.5) have shown that m = 2 instabilities lead tononlinear splitting and self-replicating events [84]. This numerically observed be-haviour has more recently been analytically justified by a weakly nonlinear anal-ysis and derivation of normal form amplitude equations for the two-dimensionalSchnakenberg and Brusselator models for which it was demonstrated that m = 2mode instabilities are subcritical [113].803.2.2 The m = 0 Mode InstabilitiesFor each j = 1, ...,N we use the homogeneity of the eigenvalue problem (3.46a)and corresponding boundary conditions (3.46b) to letM j(ρ) = c jM(ρ;S j, f ), (3.48)where c j is an undetermined constant and M satisfies∆ρM +Q(ρ;S, f )M = λE11M , ρ > 0, (3.49a)M ′(0) = 0, M →(0logρ+Bλ (S, f )), as ρ → ∞. (3.49b)The constant far-field constant Bλ (S, f ) must in general be calculated numericallyfor λ 6= 0. On the other hand, differentiating the core problem (3.17) with respectto S we find that M = ∂S(Uc(ρ;S, f ),Vc(ρ;S, f ))T satisfies (3.49) with λ = 0 andin particular we obtain the important identityB0(S, f ) = χ ′(S, f ), χ ′(S, f )≡ ∂χ(S, f )∂S . (3.50)We can therefore use the numerically calculated function χ(S, f ) to approximatethe derivative χ ′(S, f ) to calculate zero eigenvalue crossing instability thresholds.For the remainder of this section we assume that λ = 0 and hence determine theeffect of bulk-membrane couping on zero-eigenvalue crossing instabilities.Integrating (3.46a) and using the divergence theorem together with (3.48) and(3.49) we calculateˆ ∞0[(1−2U j0(ρ)Vj0(ρ))Φ j(ρ)−U j0(ρ)2Ψ j(ρ)]ρdρ =−c j,ˆ ∞0Φ j(ρ)ρdρ =f1− f c j,81from which we then calculate the distributional limits(1−2ueve)φ −u2eψε2→ η(x)−2piN∑j=1c jδ∂Ω(x− x j), ε → 0+,φε2→ η(x)+ 2pi f1− fN∑j=1c jδ∂Ω(x− x j), ε → 0+,In the outer region for which |x− x j|  O(ε) for all j = 1, ...,N we then find that(3.44c) and (3.44d) becomeDw∆η = 0, x ∈Ω, (3.51a)Dw∂nη+(1−K)η = 2piK f1− fN∑j=1c jδ∂Ω(x− x j), x ∈ ∂Ω, (3.51b)for which we calculate the leading order outer solutionη(x)∼ 2pi f1− fKDwN∑j=1c jGmr(x,x j).Similarly, in the outer region (3.44b) becomesDv∆∂Ωψ = 2piN∑j=1c j(δ∂Ω(x− x j)−f1− fKDwGmr(x,x j)), x ∈ ∂Ω,for which integration over ∂Ω leads to the solvability conditionN∑j=1c j = 0, (3.52)where we have used both the identity´∂ΩGmr(x,ξ )dAx = (1−K)−1Dw as wellas the constraint (3.7). Assuming the solvability condition (3.52) holds, then wecalculate the following leading order asymptotic expansion in the outer regionψ(x)∼ 1Dvψ¯− 2piDvN∑j=1c j(Gm(x,x j)− f1− fKDwˆ∂ΩGm(x,ξ )Gmr(ξ ,x j)dAξ).82Comparing the limiting behaviour of the inner solutionΨi(ρ) as ρ→∞with that ofthe outer solution given above as |x− xi| → 0 for each i = 1, ...,N yields algebraicsystemν−1c+2piG c+B0c = ψ¯e.Left-multiplying by eT and using the solvability condition (3.52) then givesM0c = 0, M0 ≡ ν−1IN +2pi(IN−EN)G +(IN−EN)B0, (3.53a)whereB0 ≡ diag(χ ′(S1, f ), ...,χ ′(SN , f )). (3.53b)In particular, a zero-eigenvalue crossing instability threshold can be calculated byseeking parameter values for which M0 admits a zero eigenvalue or equivalentlydetM = 0. For appropriate choices of the multi-spot configuration x1, ...,xN sym-metry properties of the Green’s matrix G can be leveraged to further characterizethe modes of instabilities as described below.3.2.3 Instability Thresholds for Symmetric N-Spot PatternsWhen the spot configuration and bulk-bound fuel source are chosen such that theGreen’s matrix G is of constant row sum and the common spot strength solutionS = Sce satisfies the NAS (3.43b) exactly then we can derive an explicit criteria forzero-eigenvalue crossing instabilities in terms of the spectrum of G and the S 1behaviour of χ ′(S, f ). In particular, since the Green’s matrix G is symmetric theeigenpairs {µ j, p j}Nj=1 satisfyG p j = µ j p j, p1 = e, eT p j = 0 j = 2, ...,N.Additionally, sinceB0 = χ ′(Sc, f )IN we find that the spectrum ofM0 consists ofthe eigenpairs {A j, p j}Nj=1 whereA j ≡ν−1, j = 1,ν−1+2piµ j +χ ′(Sc, f ), 2≤ j ≤ N, (3.54)83Since A1 > 0 for all parameter values, the p1 mode does not lead to any zeroeigenvalue crossing instabilities. Therefore any zero eigenvalue instabilities thatarise must be the result of one of the p j modes for 2 ≤ j ≤ N and since eT p j = 0for all such modes the resulting instabilities are typically referred to as competitioninstabilities. Seeking parameter values such that A j = 0 for some j = 2, ...,N andnoting that µ j = O(1) we deduce that the ν−1 and χ ′(Sc, f ) terms must balance.From the small S 1 asymptotics (see Equation (4.20) in [84])χ(S, f )∼ d0S+d1S+O(S3),d0 =b(1− f )f 2, d1 =0.48931− f −0.4698, b≡ˆ ∞0w2ρdρ ≈ 4.934,(3.55)we further deduce Sc =O(ν1/2) is needed for A j = 0 to hold for some 2≤ j ≤ N.Previous studies of the Brusselator system on the unit sphere indicate that com-petition instabilities arise as S is decreased below a critical threshold [84] and wetherefore seek the largest value of Sc for whichA j = 0 for some j= 2, ...,N. In par-ticular, noting the S 1 asymptotics (3.55), the competition instability thresholdis determined by solving the algebraic equationA?(Scomp)≡ ν−1+2piµ?+χ ′(Scomp, f ) = 0, µ? ≡ minj=2,...,Nµ j, (3.56)for the critical spot strength Scomp. The resulting critical spot strength Scomp de-pends on both the spot configuration as well as the problem parameters f , K,and Dw. The resulting competition instability threshold is then found by lettingSc = Scomp and recalling the definition of Sc given by (3.42). Finally, since χ ′(S, f )is monotone increasing for S 1 we also remark that A?(S)≶ 0 if S≶ Scomp, andin particular, based on the past observations of [84, 91], we expect a common spotstrength pattern to be stable (resp. unstable) with respect to competition instabili-ties when A?(Sc)> 0 (resp. A?(Sc)< 0).3.3 Slow-Spot DynamicsIn the previous section we considered the stability of multi-spot quasi-equilibriumsolutions on an O(1) timescale by studying the O(1) eigenvalues of the eigen-84value problem (3.44). In the inner region we neglected the neutrally stable m = 1mode which results from local translational invariance and is closely related tothe slow dynamics of the spots on an O(ε−2) timescale. The particular timescaleof the slow-dynamics arises from a dominant balance in (3.5) and will becomeclearer in the proceeding derivation. Specifically we begin by introducing the slowtimescale σ = ε2t and assuming that xi = xi(σ) for each i= 1, ...,N. In the remain-der of this section we seek a higher order asymptotic expansion for a multi-spotquasi-equilibrium solution to (3.5) for which the leading order term is given by thequasi-equilibrium solution calculated in §3.1 while a solvability condition arisingfrom the higher order corrections yields a system of ordinary differential equations(ODEs) for the spot locations. The proceeding calculations follow closely those forthe sphere in the absence of bulk-membrane coupling in [91] as well as the morerecent computations on the torus in [95]. We remark that an analogous procedureis used for deriving the slow dynamics of localized multi-spot solutions in one-,two-, and three-dimensional domains [41, 96, 98].We begin by left-multiplying (3.12) byJ Ti and usingJTi Ji =I2 to calcu-latedY ′dσ=−1εT i+O(1),∂∂ t=−εT i ·∇Y ′+O(ε2), (3.57a)T i ≡J Tidxidσ, ∇Y ′ ≡ (∂Y1 ,∂Y2)T . (3.57b)We then consider the higher order inner expansionsu(xi(εY ′,0)) =√Dv(Ui0(ρ)+ εUi1(Y ′))+O(ε2), (3.58a)v(xi(εY ′,0)) =1√Dv(Vi0(ρ)+ εVi1(Y ′))+O(ε2), (3.58b)and recall the leading order inner region expansion of w(x) given by (3.19) near xifor each i = 1, ...,N. The leading order inner problem is satisfied identically whilethe first order correction problem is given by∆Y ′qi1(Y′)+Q(ρ;Si, f )qi1(Y ′) =− f i, Y ′ ∈ R, (3.59a)qi1(Y′)→ q∞i1(Y ′) as |Y ′| → ∞. (3.59b)85where qi1(y)≡ (Ui1(Y ′),Vi1(Y ′))T andf i ≡(T i ·∇Y ′Ui0(Y ′)+D−1/2v Wi0((Y ′,0))0), (3.59c)q∞i1(Y′)≡(0Y ′TJ Ti(∑ j 6=i S jα i j + 1Dw√Dv∇xvE(xi))) . (3.59d)for each i = 1, ...,N. The expression for f i is obtained by using (3.57a) as wellas the inner region expansion of w(x) given by (3.19). On the other hand theexpression for q∞i1(Y′) is obtained by matching with the O(ε) term in (3.36).To determine an appropriate solvability condition for (3.59a) we first note that∆Y ′ +Q(ρ;Si, f ) has a null-space of dimension at least two and is spanned in partby ∂Yk(Ui0(ρ),Vi0(ρ))T for k = 1,2. We assume that the corresponding adjointhomogeneous operator has a null-space of dimension exactly two. We then writesolutions to the homogeneous adjoint problem∆Y ′Ψ(Y′)+Q(ρ;Si, f )TΨ(Y ′) = 0, Y ′ ∈ R2,Ψ→(00)as |Y ′| → ∞,in terms of the polar coordinates introduced in §3.2 asΨc(ρ,Θ) =P(ρ)cosΘ, Ψs(ρ,Θ) =P(ρ)sinΘ.where P(ρ) = (P1(ρ),P2(ρ))T is the unique solution to∆ρP′(ρ)−ρ−2P(ρ)+QTi P(ρ) = 0, in ρ > 0, (3.60)P ∼ (ρ−1,ρ−1)T as ρ → ∞, (3.61)in which the normalized limiting behaviour as ρ → ∞ is obtained by noting thatQTi →(−1 10 0), as ρ → ∞.86Left-multiplying (3.59a) by ΨTc and integrating over a disk of radius R > 0 givesˆ 2pi0(PT∂qi1∂ρ−qTi1∂P∂ρ)∣∣∣∣ρ=Rcos(Θ)RdΘ=−ˆ R0ˆ 2pi0PT f i cos(Θ)ρdρdΘ.(3.62)Using (3.59b) and (3.61) we obtain the limiting behaviour(PT∂qi1∂ρ−qTi1∂P∂ρ)∣∣∣∣ρ=R∼ 2R(cosΘ,sinΘ)TJ Ti(∑j 6=iS jα i j +∇xvE(xi)Dw√Dv),as R→ ∞ and thereforelimR→∞ˆ 2pi0(PT∂qi1∂ρ−qTi1∂P∂ρ)∣∣∣∣ρ=Rcos(Θ)RdΘ= 2pieT1JTi(∑j 6=iS jα i j+∇xvE(xi)Dw√Dv),where e1 = (1,0)T . On the other hand the right side of (3.62) is evaluated by firstrecalling that Wi0((Y ′,0)) is radially symmetric so that its contribution vanisheswhereas∇Y ′Ui0 =U ′i0(ρ)(cosΘ,sinΘ)T . In particular we obtain the following limitfor the right hand side of (3.62)− limR→∞ˆ R0ˆ 2pi0PT f i cos(Θ)ρdρdΘ=−pieT1 T iˆ ∞0P1(ρ)U ′i0(ρ)ρdρ,from which we obtaineT1 T i =−2´ ∞0 P1(ρ)U′i0(ρ)ρdρeT1JTi(∑j 6=iS jα i j +∇xvE(xi)Dw√Dv).Proceeding similarly after left-multiplying (3.59a) by ΨTs we therefore obtainT i = γ(Si, f )J Ti(∑j 6=iS jα i j +1Dw√Dv∇xvE(xi)), (3.63a)whereγ(S, f )≡− 2´ ∞0 P1(ρ)U ′c(ρ;S, f )ρdρ. (3.63b)The multiplier γ(S, f ) is computed by numerically solving (3.60) for P1(ρ) usingstandard techniques. This multiplier depends only on the inner problem and is87identical to that found in studies of the Brusselator system on the unit sphere [91]and the unit disk [96]. For completeness we include a plot of γ(S, f ) for S > 0and select values of 0 < f < 1 from which we observe that γ(S, f ) > 0. Sincexi(σ) ∈ ∂Ω for all σ ≥ 0 we find(I3− xixTi)dxidσ=dxidσ,so that left-multiplying (3.63a) by Ji and using (3.14) we obtain the system ofODEsdxidσ= γ(Si, f )(I3− xixTi)(∑j 6=iS jα i j +1Dw√Dv∇xvE(xi)), (3.64)for each i = 1, ...,N.The effect of the bulk originating fuel source is analogous to that of a hetero-geneous fuel source for the Brusselator model on the unit disk [96]. Specifically,spots are drawn toward regions where the membrane-bound fuel is locally maxi-mized. On the other hand, the first term appearing in the definition of α i j given in(3.38) leads to mutual repulsion between spots whereas the second term introducesthe effects of bulk-membrane recirculation, though in its current form it is not clearwhether it leads to mutual repulsion or attraction between spots. To better under-stand its effect on the slow dynamics we state the following Lemma for which aproof is given in Appendix C.1.Lemma 3.3.1. Let z ∈Ω∪∂Ω\{0} and suppose that f (x,z) is defined on ∂Ω andhas the series expansionf (x,z) =14pi∞∑l=0flPl(zT x|z|), fl = fl(|z|), x ∈ ∂Ω. (3.65)Then, for any y ∈ ∂Ω that is not collinear with z we have the identityˆ∂Ωy− x|y− x|2 f (x,z)dAx =12f0(|z|)y+12∞∑l=1fl(|z|)l(l+1)P1l(zT y|z|)I3− yyT√1− (zT y/|z|)2z|z| ,(3.66)88where P1l (x) is the associated Legendre polynomial of first order and lth degree. Ifinstead y and z are collinear or fl = 0 for all l ≥ 1, then the second term on theright-hand-side of (3.66) vanishes identically.Using the series expansion (3.27b) and the above Lemma we calculate−(I3− xixTi )ˆ∂Ωxi−ξ|xi−ξ |2 Gmr(ξ ,x j)dAξ=12∞∑l=1gll(l+1)P1l (xTi x j)I3− xTi xi√1− (xTi x j)2(xi− x j).(3.67)for which in Appendix C.2 we show that∞∑l=1gll(l+1)P1l (z)< 0, for all −1 < z < 1. (3.68)This implies that bulk-membrane coupling induces a mutual attraction betweenspots. However, as the next proposition shows, the mutual attraction induced bybulk-membrane coupling is not enough to overcome the mutual repulsion betweenspots given by the first term of α i j.Proposition 3.3.1. Let ε → 0 and suppose that the N-spot pattern constructedin §3.1 is linearly stable on an O(1) timescale. Then, the spot locations varyon an O(ε−2) time-scale σ = ε2t according to the differential algebraic systemconsisting of the NAS (3.43b) and the system of ODEsdxidσ= γ(Si, f )(I3− xixTi)(12∑j 6=iS jC(xTi x j)(xi− x j)√1− (xTi x j)2+∇xvE(xi)Dw√Dv), (3.69a)for each i = 1, ...,N where vE(x) is given by (3.29) andC(z)≡√1+ z1− z +f1− fKDw∞∑l=1gll(l+1)P1l (z)> 0, −1 < z≤ 1, (3.69b)and C(−1) = 0.89The right-hand-side of (3.69a) is obtained by using (3.38) and (3.67) to get(I3− xixTi )α i j = 12C(xTi x j)(I3− xixTi )(xi− x j),and the sign of C(z)> 0 is derived in Appendix C.2.We conclude this section by deriving a system of ODEs for the spherical coor-dinates θi(σ) and ϕi(σ) for each i = 1, ...,N. First we calculateT i =J Ti(∂xi∂θidθidσ+∂xi∂ϕidϕidσ)=(sinθi dϕidσdθidσ),J Ti x j =(−sinθ j sin(ϕi−ϕ j)sinθ j cosθi cos(ϕi−ϕ j)− sinθi cosθ j.)Left-multiplying (3.69a) byJ Ti and using (3.14) we obtain for each i = 1, ...,N(sinθi dϕidσdθidσ)= γ(Si, f )(12∑j 6=iS jC(xTi x j)β i j√1− (xTi x j)2+J Ti ∇xvE(xi)Dw√Dv), (3.70a)whereβ i j =(sinθ j sin(ϕi−ϕ j)sinθi cosθ j− sinθ j cosθi cos(ϕi−ϕ j))(i, j = 1, ...,N). (3.70b)3.4 ExamplesTo illustrate the asymptotic theory developed in the previous sections we now con-sider examples of symmetric N-spot patterns for which we assume that the bulk-bound fuel source is of the formE(x) = E0δ (x− xsource), xsource = ηez, (3.71)where E0 > 0, 0≤ η < 1, and ez = (0,0,1)T . Such a fuel source may represent, forexample, a localized site of protein generation within the cell. In the following sec-tion we will analyze the effects of the source location parametrized by η ≥ 0. First90we note that the diffusion transported fuel is given by wE(x) = E0Gbr(x,xsource)where Gbr is the bulk Robin Green’s function satisfying∆Gbr =−δ (x− xsource), x ∈Ω, (3.72a)∂nGbr +1−KDwGbr = 0, x ∈ ∂Ω. (3.72b)When x ∈ ∂Ω the bulk Robin Green’s function has the series expansionGbr(x,xsource) =14pi∞∑l=0glη lPl(eTz x), x ∈ ∂Ω, (3.72c)and by using the series (3.31) and product formula (3.32) as well as Lemma 3.3.1we calculatevE(x) =E04pi∞∑l=1glη ll(l+1)Pl(eTz x), (3.73a)(I3− xxT)∇xvE(x) =E04piCE(eTz x)√1− (eTz x)2(I3− xxT)(x− ez), (3.73b)J Ti ∇xvE(xi) =E04piCE(cosθi)(01), (3.73c)whereCE(z)≡∞∑l=1glη ll(l+1)P1l (z). (3.73d)Note that CE(z) ≡ 0 for η = 0 and proceeding as in the derivation of (3.68) inAppendix C.2 we can similarly establish that CE(z) < 0 for all −1 < z < 1 andη > 0. Note in addition that CE(±1 = 0 since P1l (±1) = 0 (see (C.7)).Since vE(x) is constant along lines of constant latitude (i.e. depends only oneTz x) the common spot strength solution S = Sce exactly solves the NAS (3.43b)when the spots are uniformly distributed along a ring of fixed latitude, i.e.θi = θc, ϕi =2pi(i−1)N, i = 1, ...,N. (3.74)where θc is a common polar angle. The common spot strength is then explicitly91given bySc =E02piN√Dv(1− K1− f)−1. (3.75)Applying the linear stability results of §3.2 we deduce that the N-spot pattern islinearly stable on an O(1) timescale provided thatScomp <E02piN√Dv(1− K1− f)−1< Σ2( f ), (3.76)where Scomp is the numerically computed solution to (3.56) and Σ2( f ) the splittinginstability threshold calculated in §3.2.1. The effects of E0 and N on the compe-tition and splitting instability thresholds for Dv are seen to be identical to thosecalculated in [84], mainly increasing (resp. decreasing) E0 may lead to splitting(resp. competition) instabilities while increasing (resp. decreasing) N may lead tocompetition (resp. splitting) instabilities. On the other hand, increasing the bulk-membrane coupling parameter K increases the value of Sc and may therefore leadto splitting instabilities.Before proceeding with explicit examples of a one- and two-spot pattern, weuse (3.73) to rewrite (3.70) as(sinθi dϕidσdθidσ)= γ(Si, f )(FiGi). (3.77)whereFi(ϕ ,θ )≡ 12∑j 6=iS jC(xTi x j)√1− (xTi x j)2sinθ j sin(ϕi−ϕ j),Gi(ϕ ,θ )≡ 12∑j 6=iS jC(xTi x j)√1− (xTi x j)2(sinθi cosθ j− sinθ j cosθi cos(ϕi−ϕ j))+E0CE(cosθi)4piDw√Dvfor each i = 1, ...,N and where ϕ = (ϕ1, ...,ϕN)T and θ = (θ1, ...,θN)T .923.4.1 One-Spot PatternIn the case of a one-spot pattern with an arbitrary fuel source E(x) the slow dy-namics (3.69a) are explicitly given bydx1dσ=γ(S1, f )Dw√Dv(I3− x1xT1 )∇xvE(x1), (3.78)from which we deduce the equilibrium points x1 ∈ ∂Ω occur at both the criticalpoints of vE(x) and the points for which ∇xvE(x) is parallel to x. When the fuelsource is given by (3.71) the slow dynamics in spherical coordinates (3.77) areexplicitly given by (3.77) whereF1(ϕ1,θ1) = 0, G1(ϕ1,θ1) =E0CE(cosθ1)4piDw√Dv. (3.79)When η = 0 we calculate CE ≡ 0 and therefore any point x1 ∈ ∂Ω is an equilibriumwith respect to the slow dynamics. On the other hand when 0 < η < 1 we calculateCE(±1) = 0 and CE(z) < 0 for all −1 < z < 1. As a consequence x1 = ±ez areequilibrium points, but only x1 = ez is linearly stable. In addition x1 = ez is globallyattracting so that any spot concentrated at x1 ∈ ∂Ω tends to ez along a geodesic onan O(ε−2) timescale. Note that the remaining problem parameters E0, Dw, K, f ,and Dv only change the speed of the dynamics.3.4.2 Two-Spot PatternsNext we consider the case of a two-spot pattern. We begin by considering the caseη = 0 for which ∇xvE(x) = 0 on ∂Ω. Since the 2× 2 matrix G is symmetric weimmediately deduce that S = Sce exactly solves the NAS (3.43b) for all values ofx1,x2 ∈ ∂Ω. Substituting into (3.69a) we find that the slow dynamics are governedbydx1dσ=Scγ(Sc, f )2C(xT1 x2)√1− (xT1 x2)2(I3− x1xT1)(x1− x2),dx2dσ=Scγ(Sc, f )2C(xT2 x1)√1− (xT2 x1)2(I3− x2xT2)(x2− x1).93Letting xT1 x2 = cosβ we then calculatedβdσ=− 1sinβd(xT1 x2)dσ= Scγ(Sc, f )C(cosβ ). (3.80)Since C(z)> 0 for all−1 < z≤ 1 and C(−1) = 0 we deduce that β = pi is the onlyequilibrium and it is globally attracting. Therefore any antipodal configuration, inthe sense that x1 = −x2, is a stable equilibrium of the slow dynamics. In particu-lar, the slow dynamics when K > 0 are qualitatively identical to those previouslyinvestigated when K = 0 in [91].When η > 0 both spots are mutually repelled while simultaneously being at-tracted to ez. From (3.73) ad (3.69a) we immediately deduce that a North-South(NS) configuration for which x1 =±ez and x2 =∓ez is an equilibrium of the slowdynamics. Moreover, any equilibrium configuration in which a spot concentratesat ±ez must be a NS-configuration as can be deduced by noting that if x1 = ±ezthen ∇xvE(x1) = 0 and (3.69a) then implies x2 =∓ez. Note that the spot strengthsare not equal for the NS-configuration. In particular, if x1 = ez and x2 =−ez thenS1 > S2.Next we consider non NS-configurations by parameterizing each spot in termsof spherical coordinates ϕ = (ϕ1,ϕ2)T and θ = (θ1,θ2)T and assuming withoutloss of generality that 0 < θ1 ≤ θ2 < pi . The dynamic are then given by (3.77) withF1(ϕ ,θ ) =S22C(ξ )√1−ξ 2 f12, G1(ϕ ,θ )≡S22C(ξ )√1−ξ 2 g12+E0CE(cosθ1)4piDw√Dv,(3.81a)F2(ϕ ,θ ) =S12C(ξ )√1−ξ 2 f21, G2(ϕ ,θ )≡S12C(ξ )√1−ξ 2 g21+E0CE(cosθ2)4piDw√Dv(3.81b)wherefi j ≡ sinθ j sin(ϕi−ϕ j), gi j ≡ sinθi cosθ j− cosθi sinθ j cos(ϕi−ϕ j), (3.82)94Figure 3.3: Plots of θc/pi versus K and f for fixed values of Dw = 1 (top row)and Dw = 10 (bottom row) and η = 0.3,0.6,0.9 from left to right. Thesolid lines with in-line text are contours indicating fixed values of θc/pi .for each i, j ∈ {1,2} andξ ≡ xT1 x2 = sinθ1 sinθ2 cos(ϕ1−ϕ2)+ cosθ1 cosθ2. (3.83)Since two-spot configurations are at equilibrium when η = 0 if and only if they areantipodal, the upward drift toward ez implies that there are no non-NS antipodalsolutions when η > 0 and therefore ξ > −1. As an immediate consequence (1−ξ 2)−1/2C(ξ )> 0 so that F1 = F2 = 0 implies that f12 = f21 = 0 and therefore ϕ1−ϕ2 = 0 or |ϕ1−ϕ2| = pi . In the former case the assumption θ1 ≤ θ2 immediatelyimplies that G1 < 0 since x1 is propagated toward ez both by the repulsion fromx2 and attraction to ez due to ∇xvE(x1). Therefore any non NS-configuration musthave |ϕ1−ϕ2|= pi and without loss of generality we assume ϕ1 = 0 and ϕ2 = pi . Itfollows that gi j = sin(θ1+θ2)> 0 where the sign follows from noting that ξ >−1implies θ1 + θ2 < pi . As a consequence, non-NS equilibrium configurations are95determined by solvingG1(θ1,θ2) =S22C(cos(θ1+θ2))+E0CE(cosθ1)4piDw√Dv= 0, (3.84)G2(θ1,θ2) =S12C(cos(θ1+θ2))+E0CE(cosθ2)4piDw√Dv= 0, (3.85)for 0 < θ1 ≤ θ2 with θ1+θ2 < pi .As a special case, when θ1 = θ2 = θc we find that S1 = S2 = Sc and it sufficesto determine the common angle θc satisfyingC(cos2θc)+2Dw(1− K1− f)CE(cosθc) = 0, 0 < θc <pi2. (3.86)Since C(z)→ +∞ as z→ 1 and C(−1) = 0 whereas CE(z) remains bounded andstrictly negative on 0 < z < 1 the intermediate value theorem implies the existenceof a 0< θc < pi/2 satisfying (3.86). Numerical calculations further suggest that thecommon angle solution is unique and Gi(θ ,θ)≶ 0 for θ ≷ θc. In addition we notethat θc is a function only of θc = θc(K, f ,Dw,η). In Figure 3.3 we plot θc/pi versusK and f with K + f < 1 and for select values of Dw = 1,10 and η = 0.3,0.6,0.9from which we make the following observations. As 0 < K < 1− f increases,the mutual repulsion reflected by C(cos2θc) dominates the attraction toward ezreflected by CE(cosθc) and therefore θc increases, tending to the limit θc → pi/2as K→ 1− f . On the other hand, increasing η (resp. decreasing Dw) leads to anincrease of |∇xvE(x)| and therefore decreases θc.Next we analyze in more detail the stability of the common angle and NS so-lutions by considering the nullclines of (3.84). First note thatGi(θ ,θ)> 0, Gi(θ ,pi−θ)< 0, for all 0 < θ < θc,so that the intermediate value implies the existence of values θ ?i (θ) such thatGi(θ ,θ ?i (θ)) = 0 for all 0 < θ < θc (i = 1,2). Numerical calculations further sug-gest that both of these values are unique and monotone decreasing in θ . Clearlyθ ?i (θc) = θc for each i = 1,2 and we can further deduce that limθ→0+ θ ?1 (θ) = piand limθ→0+ θ ?2 (θ) < pi . Moreover, numerical calculations given below suggest96(a)(b) (c)Figure 3.4: (a) Plot of C′E(cosθc) at K = 0 versus 0 < η < 1 and Dw > 0with the solid orange line indicating values where C′E(cosθc) = 0 anddemarcating regions where C′E(cosθc) > 0 and C′E(cosθc) < 0. (b)-(c)Schematics showing the functions θ ?1 (θ) and θ ?2 (θ) in the absence andpresence of a tilted solution. In the latter case θ1 = θt and θ2 = θ ?1 (θt)are the polar angles of the tilted solution.that depending on the choice of problem parameters either θ ?2 (θ) < θ ?1 (θ) for all0< θ < θc, or else there exists a unique value 0< θt < θc such that θ ?2 (θt) = θ ?1 (θt)in which case we define θ ?t ≡ θi(θt) (i = 1,2) (see Figures 3.4b and 3.4c for aschematic demonstration of the emergence of 0 < θt < θc). In particular this im-plies the existence of a tilted two spot solution with θ1 = θt < θ ?t = θ2. Finally,since Gi(θ1,θ2)≶ 0 if θ2 ≷ θ ?i (θ1) we deduce that the NS configuration is linearlyunstable, the common angle solution is linearly stable only in the absence of a tiltedsolution, and the tilted solution, if it exists, is linearly stable.A criteria for the emergence of a tilted solution can be derived by two equiva-lent approaches: analyzing the stability of the common angle solution, or analyzingthe behaviour of θ ?1 (θ) and θ ?2 (θ) at θ = θc. We pursue the latter approach belowwhile the former is pursued in Appendix C.3. Specifically, the criteria for the emer-gence of a tilted solution is equivalent to the tangency conditiondθ ?1dθ∣∣∣∣θ=θc=dθ ?2dθ∣∣∣∣θ=θc. (3.87)97Indeed, in the absence of a tilted solution θ ?1 (θ) > θ ?2 (θ) for all 0 < θ < θc,whereas when a tilted solution emerges θ ?1 (θ) < θ ?2 (θ) for θt < θ < θc. By sym-metry it is easy to see that G2(θ ?1 (θ),θ) = 0 for all 0 < θ < θc so that θ ?2 (θ) =(θ ?1 )−1(θ). In particular dθ ?2 /dθ = [dθ ?1 /dθ ]−1 and the tangency condition (3.87)therefore becomes dθ ?1 /dθ |θ=θc = −1 where the sign is due to the, numericallyobserved, monotonicity of θ ?1 (θ). Implicitly differentiating G1(θ ,θ ?1 (θ) = 0 withrespect to θ and using (see Appendix C.3 for details)∂S2∂θ1∣∣∣∣(θ1,θ)=(θc,θc)=−∂S2∂θ2∣∣∣∣(θ1,θ)=(θc,θc)=− E0CE(cosθc)8piDw√DvA?(Sc),where A?(Sc) is given by (3.56), we deduce that the tangency condition (3.87) isequivalent toC(cos2θc)CE(cosθc)+2A?(Sc)C′E(cosθc)sinθc = 0. (3.88)In Appendix C.3 we show that the sign of the left-hand-side of (3.88), which wedenote by (3.88)LHS, determines the stability of the common angle solution (see(C.23b)). In particular, if (3.88)LHS < 0 (resp. (3.88)LHS > 0) then the commonangle solution is unstable (resp. stable). Assuming that Scomp < Sc < Σ2( f ) sothat the common angle solution is linearly stable with respect to competition andsplitting instabilities, we seek parameter values for which (3.88) is satisfied. FromA?(Sc) > 0 for Sc > Scomp and C(cos2θc)CE(cosθc) < 0 for all 0 < θc < pi/2 wededuce that C′E(cosθc) > 0 is a necessary condition for (3.88) to hold. Numericalevidence indicates that C′E(cosθ) has a unique zero 0< θe < pi/2 with C′E(cosθ)≶0 for θ ≷ θe, and furthermore this zero is monotone increasing in K though at aslower rate than the common angle θc. As a consequence if C′E(cosθc) ≤ 0 forK = 0 then C′E(cosθc) < 0 for all 0 < K < 1− f and the common angle solutionis linearly unstable. Noting that C′E(cosθc)|K=0 is a function only of η and Dw weobtain the plot of C′E(cosθc) shown in Figure 3.4a. Only within the indicated regionof η and Dw values where C′E(cosθc) > 0|K=0 can the common angle solution belinearly stable for appropriate choices of the remaining problem parameters when98K ≥ 0. In fact, when C′E(cosθc)> 0 we may solveC(cos2θc)CE(cosθc)+2A?(Stilt)C′E(cosθc)sinθc = 0, (3.89)for the tilt instability threshold Stilt > Scomp. Since (3.88)LHS < 0 when Sc = Scompwe deduce that the common angle solution is unstable with respect to tilt instabili-ties for Sc < Stilt.We summarize the above discussion as follows. If C′E(cosθc) ≤ 0 then thecommon angle solution is linearly unstable with respect to a tilt instability over anO(ε−2) timescale. Otherwise, if C′E(cosθc) > 0 then the common angle solutionis linearly stable if and only if Stilt < Sc < Σ2( f ) where Σ2( f ) is the splitting in-stability threshold and Stilt is the tilt instability threshold satisfying (3.89). The NSconfiguration, for which x1 = ±ez and x2 = −x1 is always an equilibrium but itis linearly unstable. Finally, if the common angle solution is linearly stable withrespect to competition and splitting instabilities, i.e. Scomp < Sc < Σ2( f ), but unsta-ble with respect to tilt instabilities then it bifurcates to a tilted solution with polarangles satisfying θ1 < θc < θ2. The tilted solution is linearly stable with respect totilt instabilities and its stability with respect to competition and splitting instabili-ties depends on the proximity of Sc to Scomp and Σ2( f ). In particular recalling thatS = Sce+O(ν) (see §3.1) it suffices that Sc satisfy Scomp < Sc < Σ2( f ) and be anO(1) distance from these thresholds for the tilted solution to be linearly stable.To illustrate the above discussion, in Figures 3.5(a)-(c) we plot the competition(blue), splitting (orange), and tilt (green) instability thresholds in the form Dv/E20versus K for select values of Dw > 0 with η = 0.4, f = 0.6, and ν = 5× 10−3.Recalling (3.75) and the stability criteria Stilt < Sc < Σ2( f ), the common angle so-lution is linearly stable in the region bounded by the split (orange) and tilt (green)instability threshold curves provided that the latter exists. If the latter thresholddoes not exist then C′E(cosθc)≤ 0 for all K ≥ 0 and the common angle solution istherefore always unstable with respect to tilt instabilities. In Figures 3.5(d)-(f) weplot the common angle, θc, and tilted solution angles, θ1 and θ2, as 0≤ K < 1− fis varied for Dv/E20 = 0.1 with the remaining parameters equal to those used inFigures 3.5(a)-(c) respectively. Note that the tilted solution provides a connec-990.0 0.1 0.2 0.3 0.4K0.00.20.40.60.81.0Dv/E2 0Dw=1.0, = 0.4compsplittilt(a)0.0 0.1 0.2 0.3 0.4K0.00.20.40.60.81.0Dv/E2 0Dw=1.5, = 0.4compsplittilt(b)0.0 0.1 0.2 0.3 0.4K0.00.20.40.60.81.0Dv/E2 0Dw=2.0, = 0.4compsplit(c)0.0 0.1 0.2 0.3 0.4K0123Dw=1.0, = 0.4c12(d)0.0 0.1 0.2 0.3 0.4K0123Dw=1.5, = 0.4c12(e)0.0 0.1 0.2 0.3 0.4K0123Dw=2.0, = 0.4c12(f)Figure 3.5: (a)-(c) Plots of competition, splitting, and tilt instability thresh-olds as Dv/E20 versus K for select of Dw > 0 with η = 0.4, f = 0.6, andν = 5× 10−3. (d)-(f) Plots of the common angle θc, and tilted anglesθ1 < θc < θ2 versus K at Dv/E20 = 0.1 and with remaining parametersequal to those from (a)-(c) respectively. The solid (resp. dashed) lineindicates the stability (resp. instability) of the common angle solutionwith respect to tilt instabilities.tion between the common angle and NS two-spot configurations. Furthermore, asK → 1− f the tilted solution approaches the NS configuration, but it is destabi-lized by a splitting instability before reaching this new configuration. Additionally,as K increases the tilted solution rapidly approaches the NS configuration indicat-ing a preference of the system to align the two spots with the two local extrema(maximum at ez and minimum at −ez) of ∇xvE(x) on ∂Ω. Interestingly, the aboveanalysis indicates that while the existence of tilted solutions is closely tied to theheterogeneity of the fuel source in the absence of membrane-detachment, the lat-ter mechanism nevertheless plays an important role in the destabilization of the100common angle solution and resulting bifurcation to the stable tilted solution.3.5 DiscussionIn this chapter we have used the method of matched asymptotic expansions toderive hybrid asymptotic-numerical equations which determine the structure, sta-bility, and slow dynamics of strongly localized multi-spot solutions to a bulk-membrane-coupled Brusselator model posed on the unit sphere. Our particularchoice of bulk-membrane coupling was made to reflect a process in which the fuelnecessary to sustain pattern formation for the Brusselator model originates withinthe bulk and is transported to the membrane by passive diffusion. In addition, weintroduced a membrane-detachment mechanism controlled by the non-dimensionalparameter K which satisfies 0 < K < 1− f . Our analysis therefore focused on theeffects of the bulk-originating fuel source as well as the membrane detachmentmechanism on the structure, stability, and slow dynamics of the multi-spot pat-terns.To leading order in the small parameter ν =−1/ logε , we determined that thetotal bulk-bound fuel and the membrane-bound activator detachment rate have adirect effect on the strength of the membrane bound spot strengths, with one of thekey parameters being the common spot strength which is explicitly given bySc =12piN√Dv(1− K1− f)−1ˆΩE(x)d3x.The dependence on the total bulk-bound fuel source is analogous to results previ-ously obtained for the Brusselator model on the flat disk with a heterogeneous fuelsource [96]. The analysis of the structure of multi-spot solutions and their linearstability on anO(1) timescale with respect to competition and splitting instabilitiesis qualitatively similar to that in previous studies of the Brusselator model on theunit sphere without bulk-membrane coupling [84]. However two key differencesare the introduction of the membrane Robin Green’s function Gmr(x,ξ ) satisfying(3.27a) and playing a key role in modelling membrane-detachment, as well as therelated quantity ˆ∂ΩGs(x,ξ )Gmr(ξ ,xi)dAξ ,101which reflects a recirculation mechanism of membrane-detachment and reattach-ment.The effects of the recirculation mechanism and the bulk-originating fuel sourceare perhaps most prominent in the slow dynamics of multi-spot patterns occurringover an O(ε−2) timescale. Although our derivation of the slow dynamics closelyfollows that found in [91], our use of local geodesic normal coordinates streamlinesthis derivation. In particular geodesic normal coordinates lead to local expansionsof the Laplace-Beltrami operator that are free of artificial first order correctionterms [92, 95]. The resulting system of ODEs governing the slow spot dynam-ics consists of three terms of which the first reflects mutual repulsion betweenspots and is independent of bulk-membrane coupling. The second term reflectsmutual attraction and is a consequence of the recirculation mechanism. Howeverwe show that this term is weaker than the mutual repulsion due to the first term.The final term is a consequence of the bulk-originating fuel source and leads tothe attraction of spots toward local extrema of the resulting membrane-bound fuelterm. To more closely investigate the consequences of each of these three termswe considered an explicit example in which the bulk-bound fuel source is givenby a Dirac delta function concentrating at xsource = (0,0,η) for 0 ≤ η < 1. Wethen performed a detailed analysis of the dynamics of one- and two-spot config-urations. In particular we illustrated that a one-spot pattern is globally attractedto the only stable equilibrium located at x1 = ez. On the other hand our analysisof two-spot patterns revealed the existence of a common angle solution given byx1 =(sinθc,0,cosθc) and x2 =(−sinθc,0,cosθc), a North-South (NS) solution forwhich x1 =±ez and x2 =−x1, and a tilted solution for which x1 = (sinθt ,0,cosθt)and x2 = (−sinθ ?t ,0,cosθ ?t ) with θt < θc < θ ?t . By numerically calculating sta-bility thresholds for the common angle solution we the demonstrated an intricatebifurcation structure connecting the common angle solution to the tilted and NSconfigurations.102Chapter 4The Singularly PerturbedOne-DimensionalGierer-Meinhardt Model withNon-Zero Activator BoundaryFluxIn this chapter we consider the classically studied one-dimensional Gierer-Meinhardt(GM) modelut = Duuxx−u+u2v−1, vt = Dvvxx− v+u2, 0 < x < 1. (4.1a)In the singularly perturbed limit for which Du = ε2 1 this GM model is knownto exhibit multi-spike solutions. While boundary conditions have been identifiedas playing an important role in pattern formation [18, 59], relatively few studieshave investigated the role of boundary conditions on the structure and stabilityof multi-spike solutions to singularly perturbed reaction diffusion systems. In-stead most such studies have assumed either homogeneous Neumann or homoge-neous Dirichlet boundary conditions. Notable exceptions include the investigation103of homogeneous Robin boundary conditions for the activator in the GM model[4, 60] and inhomogeneous Robin boundary conditions for the inhibitor in the two-dimensional Brusselator model [96]. These two studies and their illustration ofthe effect of boundary conditions on the structure and stability of multi-spike pat-terns serve as the primary motivation for the present chapter in which we con-sider inhomogeneous Neumann boundary conditions for the activator in the one-dimensional singularly perturbed GM model (4.1a). Additionally, this chapter aimsto address some of the technical issues that arise in bulk-surface coupled reactiondiffusion systems for which inhomogeneous boundary conditions naturally arise[27, 54, 58, 83].We assume that that Du = ε2 while Dv = O(1) where ε  1 is an asymptoti-cally small parameter. The activator in an equilibrium solution will then concen-trate in intervals of O(ε) length and by integrating the inhibitor equation in (4.1a)it is easy to see that if v = O(1) then we must have u = O(ε−1/2) in each intervalon which it is concentrated. This motivates our choice of rescaling u = ε−1u˜ andv = ε−1v˜ which when substituted into (4.1a) and dropping the tildes givesut = ε2uxx−u+u2v−1, 0 < x < 1, (4.2a)τvt = Dvxx− v+ ε−1u2, 0 < x < 1, (4.2b)and for which we observe both u and v will be O(1) in each interval where u isconcentrated. In addition we impose inhomogeneous and homogeneous Neumannboundary conditions for the activator and inhibitor respectively which are given by− εux(0) = A, εux(1) = B, vx(0) = 0, vx(1) = 0, (4.2c)where we assume that A,B ≥ 0 and for which we note that the scaling for theactivator boundary conditions arises naturally from the scaling argument.In contrast to systems with homogeneous Neumann boundary conditions, wenote that (4.2) does not have a spatially homogeneous steady state when A > 0and/or B > 0 and therefore traditional Turing stability analysis methods no longerapply. In particular the inhomogeneous Neumann boundary conditions for the ac-tivator in (4.2c) necessitate that the activator forms a boundary layer near each1040.0 0.1 0.2 0.3 0.4 0.5x0.00.51.01.52.02.5One-Spike Solutions for A 0 and B=0A0.00.10.51.0(a)x0.0 0.2 0.4 0.6 0.8 1.0t0102030405060u,v0.00.10.20.30.40.5u(x, t)v(x, t)(b)Figure 4.1: (a) Examples of shifted one-spike solution concentrated at x = 0for various values of A≥ 0 and with ε = 0.05 and D = 5. (b) Evolutionof solution to GM problem with D= 0.6, ε = .005, τ = 0.1, and A=B=0.08. The initial condition is an unstable two-spike equilibrium whereboth spikes concentrate at the boundaries. A competition instabilitypredicted by our asymptotic results in Figure 4.5a is triggered and leadsto the solution settling at an asymmetric pattern.boundary when A > 0 and/or B = 0. As we demonstrate in §4.1 the appropriateboundary layer solution takes the form of a shifted spike (see Figure 4.1a) that isanalogous to the near-boundary spike solution of [4, 60] though it has differentstability properties which we investigate in §4.3. Furthermore, by considering ex-amples of one- and two-spike patterns in §4.4 we investigate the role of non-zeroboundary fluxes on the structure of symmetric and asymmetric patterns, as well astheir stability with respect to oscillatory (example 1), competition (examples 2-4),and drift (example 4) instabilities. In Figure 4.1b we plot the time-evolution of alinearly unstable equilibrium consisting of two boundary spikes of equal height.We see that the solution undergoes a competition instability, but rather than be-ing subcritical as is the case when A = B = 0 [104], the non-zero boundary fluxesforce the solution to settle to an asymmetric pattern. This illustrates that one of thekey features of introducing non-zero boundary fluxes is that it leads to a kind ofrobustness of asymmetric solutions similar to that observed in the presence of aninhomogeneous precursor gradient [49].The remainder of this chapter is organized as follows. In §4.1 we use the105method of matched asymptotic expansions to construct multi-spike equilibriumsolutions. The key idea of the construction is to leverage the localized of the spikesolution to reduce their construction to a problem of finding the spike heights andtheir locations. In §4.2 we derive a nonlocal eigenvalue problem which determinesthe linear stability of the multi-spike solutions on anO(1) time scale. Then, in §4.3we rigorously prove partial stability results for an equilibrium consisting of a singleboundary spike. This is done by analyzing a class of shifted nonlocal eigenvalueproblems analogous to those studied in [60]. In §4.4 we consider four examples forwhich we construct one- and two-spike equilibrium patterns and study their linearstability and dynamics. Finally in §4.5 we conclude with a summary of our resultsand highlight several open problems and suggestions for future research.4.1 Quasi-Equilibrium Multi-Spike Solutions and theirSlow DynamicsIn this section we use the method of matched asymptotic expansions to derivean algebraic system and an ordinary differential equation that determine the pro-file and slow dynamics of a multi-spike quasi-equilibrium solution to (4.2). Thederivation uses techniques that are now common in the study of localized patternsin one dimension. Our presentation will therefore be brief, highlighting only thenovel aspects introduced by the inhomogeneous Neumann boundary conditions forthe activator. We begin by supposing that there are two spikes concentrated at theboundaries xL = 0 and xR = 1 as well as N spikes concentrated in the interior at0 < x1 < ... < xN < 1. In addition we assume that the spikes are well separatedin the sense that |xi− x j| = O(1) as ε → 0+ for all i 6= j ∈ {L,1, ...,N,R}. Thislast assumption is key for effectively applying the method of matched asymptoticexpansions.We first construct an asymptotic approximation for the solution near x = 0 byletting x = εy where y = O(1) and expandingu∼ uL,0(y)+O(ε), v∼ vL,0(y)+ εvL,1(y)+O(ε2). (4.3)It is easy to see that vL = ξL where ξL is is an undetermined constant, and uL,0(y) =106ξLwc(y+ yL) where wc(y) is the unique homoclinic solution tow′′c −wc+w2c = 0, 0 < y < ∞, w′c(0) = 0, wc(y)→ 0 as y→ ∞,(4.4a)given explicitly bywc(y) =32sech2y2. (4.4b)Moreover, the undetermined shift parameter yL is chosen to satisfy the inhomoge-neous Neumann boundary conditionw′c(yL) =−A/ξL. (4.5)The unknown constants ξL and yL are found by matching with the outer solution.To determine the appropriate Neumann boundary conditions for the outer problemwe must first calculate v′L,1(y) as y→ ∞. This is done by integrating the O(ε)equationDv′′L,1 =−ξ 2L wc(y+ yL)2, 0 < y < ∞; v′L,1(0) = 0. (4.6)over 0 < y < ∞ to obtain the limitlimy→+∞v′L,1(y) =−ξ 2LDη(yL), (4.7)whereη(y0)≡ˆ ∞0wc(y+ y0)2dy =6e−2y0(3+ e−y0)(1+ e−y0)3(4.8)Note that η(0) = 3 and η → 0+ monotonically as z→ ∞. In a similar way weobtain the inner solution near x = 1 by letting x = 1− εy and finding thatu∼ ξRwc(y+ yR)+O(ε), v∼ ξR+ εvR,1(y)+O(ε2),where yR is determined by solvingw′c(yR) =−B/ξR, (4.9)107and for which we calculate the limitlimy→+∞v′R,1(y) =−ξ 2RDη(yR). (4.10)We now consider the inner solution at each interior spike location. By balanc-ing dominant terms in a higher order asymptotic expansion, it can be shown that theinterior spike locations var on an O(ε−2) timescale. Therefore we let xi = xi(ε2t)for each i = 1, ...,N and with x = xi(ε2t) + y we calculate the inner asymptoticexpansionsu∼ ξiwc(y)+O(ε), v∼ ξi+ εvi1(y)+O(ε2), i = 1, ...,N. (4.11)Furthermore, we must impose a solvability condition on the vi1 problem whichgives1ε2dxidt=− 1ξi(limy→+∞v′i1(y)+ limy→−∞v′i1(y)). (4.12)To determine the 2(N+2) undetermined constants ξi and yi where i∈{L,1, ...,N,R}we must now calculate the outer solution, defined for |x− xi| = O(1) for eachboundary and interior spike location, and match it with each of the inner solutions.Since wc decays to zero exponentially as y→±∞ we determine that the activator isasymptotically small in the outer region. On the other hand (4.7) and (4.10) implythe boundary conditionsvx(0)∼−ξ2LDη(yL), vx(1)∼ ξ2RDη(yR),while the exponential decay of wc implies that the following limits hold (in thesense of distributions)ε−1u2 −→ 6N∑j=1ξ 2j δ (x− x j), (ε → 0+),108Thus, to leading order in ε  1, the outer problem for the inhibitor is given byDvxx− v =−6N∑j=1ξ 2j δ (x− x j), 0 < x < 1, (4.13)Dvx(0) =−ξ 20η(yL), Dvx(1) = ξ 2N+1η(yR). (4.14)This boundary value problem can be solved explicitly by letting Gω be the Green’sfunction satisfyingGω,xx−ω2Gω =−δ (x−ξ ), 0 < x < 1;Gω,x(0,ξ ) = 0, Gω,x(1,ξ ) = 0, ω > 0(4.15a)and given explicitly byGω(x,ξ ) =1ω sinhωcoshωxcoshω(1−ξ ), 0 < x < ξ ,coshω(1− x)coshωξ , ξ < x < 1. (4.15b)Formally substituting ξ = 0 or ξ = 1 into the above expression givesGω(x,0) =coshω(1− x)ω sinhω, Gω(x,1) =coshωxω sinhω, (4.15c)which is readily seen to satisfyGω,xx−ω2Gω = 0, 0 < x < 1,with boundary conditionsGω,x(0,0) =−1, Gω,x(1,0) = 0, Gω,x(0,1) = 0, Gω,x(1,1) = 1.Letting ω0 ≡ D−1/2 we obtain the following leading order asymptotic expansion109for the quasi-equilibrium solution to (4.2)ue(x)∼ ξLwc( xε + yL)+ξRwc( 1−xε + yR)+N∑j=1ξ jwc( x−x jε), (4.16a)ve(x)∼ ω20(ξ 2Lη(yL)Gω0(x,0)+ξ2Rη(yR)Gω0(x,1)+6N∑j=1ξ 2j Gω0(x,x j)). (4.16b)Furthermore, by imposing the consistency condition ve(xi)= ξi for each i∈{L,1, ...,N,R}we obtain the system of N+2 nonlinear equationsB ≡ ξ −ω20Gω0N ξ 2 = 0, (4.17a)where Gω0 andN are the (N+2)× (N+2) matrices given by(Gω0)i j = Gω0(xi,x j) i, j = L,R,1, ...,N,N ≡ diag(η(yL),η(yR),6, ...,6),andξ ≡ (ξL,ξR,ξ1, ...,ξN)T , ξ 2 = (ξ 2L ,ξ 2R ,ξ 21 , ...,ξ 2N)T . (4.17b)Thus, for given spike configuration 0 < x1 < ... < xN < 1, the system (4.17a) to-gether with (4.5) and (4.9) can be solved for the unknown spike heights ξL,ξ1, ...,ξN ,ξRand boundary shifts yL and yR. Summarizing, we have the following proposition.Proposition 4.1.1. In the limit ε → 0+ and for t O(ε−2) an N+2 spike quasi-equilibrium solution to (4.2) consisting of two boundary spikes and N well sepa-rated interior spikes concentrated at specified locations 0 < x1 < ... < xN < 1 isgiven asymptotically by (4.16) where Gω0(x,ξ ) is given explicitly by (4.15b) andω0 = D−1/2. The boundary shifts, yL and yR, and spike heights, ξL,ξ1, ...,ξN ,ξR,are found by solving the system of N+4 equations (4.5), (4.9), and (4.17a).The asymptotic solution constructed in the above proposition will not generallybe an equilibrium of (4.2) due to the slow, O(ε−2), drift motion of the interiorspikes described by (4.12). However, this solution can be made into an equilibriumby choosing the interior spike locations x1, ...,xN appropriately.110Proposition 4.1.2. The interior spike locations of a multi-spike pattern consist-ing of two boundary spikes and N interior spikes vary on an O(ε−2) time scaleaccording to the differential equation1ε2dxidt=− 6ξiD〈∂xGω0(x,xi)〉x=xi− 12ξiD∑j 6=iξ 2j Gx(ξi,ξ j)− 2ξiD[ξ 2Lη(yR)Gx(xi,0)+ξ2Rη(yR)Gx(xi,1)],(4.18)for each i = 1, ...,N where〈f (x)〉x0= limx→x+0f (x)+ limx→x−0f (x), (4.19)which is to be solved together with (4.17a), (4.5), and (4.9) for the spike heightsξL,ξ1, ...,ξN ,ξR and shifts yL and yR. In particular, if the configuration x1, ...,xNis stationary with respect to the ODE (4.18), then to leading order the quasi-equilibrium solution of Proposition 4.1.1 is an equilibrium for all t ≥ 0.4.1.1 Equilibrium Multi-Spike Solutions by the Gluing MethodWe now use an alternative method for constructing asymmetric multi-spike equilib-rium solutions to (4.2). This method extends that of Ward and Wei [104] to accountfor inhomogeneous Neumann boundary conditions. The key idea is to construct asingle boundary spike solution in an interval of variable length and use this solu-tion to glue together a multi-spike solution. In particular, we begin considering theproblemε2uxx−u+ v−1u2 = 0, Dvxx− v+ ε−1u2 = 0, 0 < x < l, (4.20a)εux(0) =−A, εux(l) = 0, Dvx(0) = 0, Dvx(l) = 0, (4.20b)where l > 0 is fixed and for which we will use the method of matched asymptoticexpansions to construct a single spike solution concentrated at x = 0. Proceedingas in §4.1 we readily find that the equilibrium solution in the outer region (i.e. for111x =O(1)) is given byu(x; l,A)∼ ξ0wc(ε−1x+ y0), v(x; l,A)∼ ξ0 coshω0(l− x)coshω0l , (4.21)where ω0 ≡ D−1/2 while the shift parameter y0 and spike height ξ0 satisfyw′c(y0) =−Aξ0, (4.22)and for which, using (4.4b) and (4.8), we explicitly calculateξ0 =tanhω0lω0η(y0), y0 = log(1+3q+√9q2+10q+12), q≡ ω0Atanhω0l, (4.23)for which we remark that y0 ∼ 4q as q→ 0 and therefore ξ0 ∼ (3ω0)−1 tanhω0las A→ 0+. Finally, we note that y0 is monotone increasing in A and monotonedecreasing in D and l when A > 0 is fixed.A multi-spike pattern is constructed by first partitioning the unit interval 0 <x < 1 into N+2 subintervals defined byxL = 0, xi = lL+2i−1∑j=1l j + li (i = 1, ...,N), xR = 1,IL = [0, lL), Ii = [xi− li,xi+ li) (i = 1, ...,N), IR = [1− lR,1],where lL, l1, ..., lN , lR are chosen to satisfy the N+2 constraintslL+2l1+ · · ·2lN + lR = 1. (4.24a)v(lL; lL,A) = v(l1; l1,0) = · · ·= v(lN ; lN ,0) = v(lR; lR,B), . (4.24b)The first constraint guarantees that the intervals are mutually disjoint, while thesecond set of N + 1 constraints guarantees the continuity of the multi-spike equi-librium solutionue(x) =u(x; lL,A), x ∈ ILu(|x− xi|; li,0), x ∈ Iiu(1− x, lR,B), x ∈ IR, ve(x) =v(x; lL,A), x ∈ ILv(|x− xi|; li,0), x ∈ Iiv(1− x, lR,B), x ∈ IR. (4.24c)112We remark that the local symmetry of each interior spike implies that the inte-rior spikes are stationary with respect to the slow dynamics found in (4.12), andtherefore the multi-spike solution constructed above is an equilibrium of (4.2).4.2 Linear Stability of Multi-Spike PatternIn this section we derive a nonlocal eigenvalue problem (NLEP) that, to leadingorder in ε  1, determines the linear stability of the quasi-equilibrium solutiongiven in Proposition 4.1.1 on an O(1) timescale. Letting ue and ve be the quasi-equilibrium solution from Proposition 4.1.1, we consider the perturbations u =ue+ eλ tΦ and v = ve+ eλ tΨ with which (4.2) becomesε2Φxx−Φ+2ueveΦ−u2ev2eΨ= λΦ, 0 < x < 1, (4.25a)DΨxx−Ψ+2ε−1ueΦ= τλΨ, 0 < x < 1. (4.25b)This problem admits both large and small eigenvalues characterized by λ = O(1)and O(ε2) respectively. The small eigenvalues are closely related to the lineariza-tion of the slow-dynamics (4.18) and the resulting instabilities therefore take placeover a O(ε−2) timescale [111]. In contrast, the large eigenvalues lead to ampli-tude instabilities over a O(1) timescale. In this section we focus exclusively on thelarge eigenvalues and limit our discussion of the small eigenvalues to the specificexample given in §4.4.4 in which a two-spike solution consisting of one spike onthe boundary and one interior spike is considered.Using the method of matched asymptotic expansions as in §4.1 we readily findthat, to leading order in ε  1, the inhibitor perturbation Ψ satisfiesDΨxx− (1+ τλ )Ψ=−2N∑j=1ξ jˆ ∞−∞wc(y)φ j(y)dyδ (x− x j), 0 < x < 1, (4.26a)DΨx(0) =−2ξLˆ ∞0wc(y+ yL)φL(y)dy, (4.26b)DΨx(1) = 2ξRˆ ∞0wc(y+ yR)φR(y)dy, (4.26c)where φL and φR are the leading order inner expansions of the activator perturba-113tion Φ at the boundaries satisfyingLyiφi−wc(y+ yi)2Ψ(xi) = λφi, 0 < y < ∞,φ ′i (0) = 0, φi→ 0 as y→ ∞,(4.27a)for i = L,R respectively, while φ1, ...,φN are the leading order inner expansions ofΦ at each of the interior spike locations x1, ..,xN satisfyingL0φi−wc(y)2Ψ(xi) = λφi, −∞< y < ∞, φ → 0 as y→±∞, (4.27b)for each i = 1, ...,N respectively. The linear differential operatorLy0 parametrizedby y0 ≥ 0 appearing in each equation is explicitly given byLy0φ ≡ φ ′′−φ +2wc(y+ y0)φ . (4.28)Note that by decomposing each φi = φ eveni + φ oddi (i = 1, ...,N) where φ eveni andφ oddi are even and odd about y = 0 respectively, we find that either φ oddi = 0 or elseλ ≤ 0. In particular, the odd components of each φi (i = 1, ...,N) do not contributeto any instabilities and without loss of generality we may therefore assume thateach φi is even about y = 0. Hence it suffices to pose (4.27b) on the half line withthe same homogeneous Neumann boundary conditions used in (4.27a).Letting ωλ ≡√(1+ τλ )/D and recalling the definition of Gω in (4.15) wereadily find that the solution to (4.26) is explicitly given byΨ(x) = 2ω20N∑j=L,R,1ξˆ jGωλ (x,x j)ˆ ∞0wc(y+ y j)φ j(y)dy.where we lety1 = ...= yN = 0, ξˆi ≡ξi, i = L,R,2ξi, i = 1, ...,N, . (4.29)Evaluating Ψ(x) at each x = xi and substituting into (4.27) yields the system of114NLEPsLyiφi−2ω20 wc(y+ yi)2N∑j=L,R,1ξˆ jGωλ (xi,x j)ˆ ∞0wc(y+ y j)φ j(y)dy = λφi (4.30a)for y > 0 with boundary conditionsφ ′i (0) = 0, φi→ 0 as y→+∞. (4.30b)for each i = L,R,1, ...,N whereLyi is defined by (4.28).The NLEP system (4.30) has two key features that distinguish it from analo-gous NLEPs in singularly perturbed reaction diffusion systems [41, 104, 107, 111]and are explored in the rigorous stability results of §4.3 as well as in the spe-cific examples of §4.4. First, it considers both boundary-bound and interior-boundspikes. As explored in Examples 2 to 4 this has immediate consequences for boththe existence and stability of asymmetric patterns even in the zero-flux case whereA = B = 0. The second distinguishing feature of (4.30) is the introduction of theshift parameters yL ≥ 0 and yR ≥ 0. We remark that an analogous negative shiftparameter has been examined in the context of near-boundary spike solutions forhomogeneous Robin boundary conditions [4, 60]. However, as highlighted in thestability results of §4.3, the positive shift parameter plays a key role in the stabilityproperties of boundary-bound spikes. An important critical value of the shift pa-rameter is the unique value y0c > 0 such that w′′c (y0c) = 0 and which is explicitlygiven byy0c = log(2+√3). (4.31)In particular, it can be shown that if y0 ≶ y0c then Ly0 has an unstable and stablespectrum respectively (see Lemma 4.3.1 below). Moreover the operatorLy0c has aone-dimensional kernel spanned by w′c(y+ y0c).4.2.1 Reduction of NLEP to an Algebraic SystemIt is particularly useful to rewrite (4.30) as an algebraic system as follows. Assum-ing that λ is not an eigenvalue ofLyi for all i = L,R,1, ...,N we letφi = ci(Lyi−λ )−1wc(y+ yi)2, i ∈ {L,R,1, ...,N}, (4.32)115where the coefficients cL,cR,c1, ...,cN are undetermined. Note that in (4.32) thehomogeneous Neumann boundary condition φ ′i (0) = 0 is assumed. In addition,note that if λ = 0 then (4.32) is only valid if yL,yR 6= y0c.Substituting into (4.30) then yields the linear homogeneous system for c ≡(cL,cR,c1, ...,cN)TGωλDλc = (2ω20 )−1c, (4.33)where Gωλ is the (N+2)× (N+2) matrix with entries(Gωλ )i j = Gωλ (xi,x j), (i, j = L,R,1, ...,N), (4.34)while Dλ is the diagonal (N+2)× (N+2) matrix given by(Dλ )i j =1ω0η(yi)ξiFyL(λ ), i = j = L,R,6ξiF0(λ ), i = j = 1, ...,N,0, i 6= j,(4.35)whereFy0(λ )≡´ ∞0 wc(y+ y0)(Ly0−λ )−1wc(y+ y0)2dy´ ∞0 wc(y+ y0)2dy, (4.36)and for which (4.46) and (4.47) below imply that for all y0 6= y0cFy0(0) = 1+w′c(y0)wc(y0)22w′′c (y0)η(y0). (4.37)Comparing (4.33) and (4.30), it follows that λ is an eigenvalue of (4.30) if and onlyif (2ω20 )−1 is an eigenvalue of GωλDλ . In particular, when λ is not an eigenvalueof LyL , LyR , and L0 then it is an eigenvalue of the NLEP (4.30) if and only if itsatisfies the algebraic equationdet(IN+2−2ω20GωλDλ)= 0, (4.38)where IN+2 is the (N+2)× (N+2) identity matrix.We conclude by noting that if either yL = y0c and/or yR = y0c then the algebraicreduction fails when searching for a zero eigenvalue λ = 0 since Ly0c is not in-116vertible. However, in this case we can deduce an analogous system. In particularletting λ = 0 and assuming that yL = y0c and yR 6= y0c, we multiply the i= L NLEPin (4.30) by w′c(y+ y0c) and integrate over 0 < y < ∞ to getξLˆ ∞0wc(y+ yL)φLdy =−N∑j=R,1ξ˜ jGω0(0,x j)Gω0(0,0)ˆ ∞0wc(y+ y j)φ jdy. (4.39)Proceeding as above we then deduce that the NLEP (4.30) with λ = 0 is thenequivalent to the algebraic equationdet(IN+1−2ω20 ˜Gω0D˜) = 0, (4.40)where ˜Gω0 and D˜ are the (N+1)× (N+1) matrices with entries( ˜Gω0)i j = Gi j−1Gω0(0,0)Gω0(xi,0)Gω0(0,x j), (D˜)i j =Di j, (4.41)for i, j = R,1, ...,N. The same approach can likewise be used if yL = yR = y0c.4.2.2 Zero-Eigenvalues of the NLEP and the Consistency ConditionThe conditions under which λ = 0 is an eigenvalue of the NLEP (4.30) can be di-rectly linked to the system (4.17a) as highlighted in [111]. Specifically, assume thatx1, ...,xN are fixed (not necessarily at an equilibrium configuration of the slow dy-namics ODE (4.18)) and let ξL,ξR,ξ1, ...,ξN together with yL and yR solve (4.17a),(4.5), and (4.9) with the additional assumption that yL,yR 6= y0c. From the definitionof η in (4.8) and from (4.5) and (4.9) we calculate∂η(yi)∂ξi=wc(yi)2w′c(yi)ξiw′′c (yi), (4.42)for i = L,R. Taking the Jacobian of the quasi-equilibrium system (4.17a) and re-calling the definition of Dλ given in (4.35) we deduce that∇ξ B = I−2ω20Gω0D0. (4.43)117Together with the discussion of §4.2.1 we deduce that if yL,yR 6= 0 and each x1, ...,xNis independent of ξL,ξR,ξ1, ...,ξN , then λ = 0 is an eigenvalue of the NLEP (4.30)if and only if the Jacobian ∇ξ B is singular.4.3 Rigorous Stability and Instability Results for theShifted NLEPIn this section we rigorously prove instability and stability results for the shiftedNLEPLy0φ −µ´ ∞0 wφ´ ∞0 w2 w2 = λφ , 0 < y < ∞,φ ′(0) = 0; φ → 0 as y→ ∞,(4.44)where µ is a real constant and for a fixed value of y0 ≥ 0 we defineLy0φ ≡ φ ′′−φ +2wφ , w(y)≡ wc(y+ y0), (4.45)and where wc is the unique solution to (4.4a). When y0 = 0 the NLEP (4.44) isstable if µ > 1 and unstable if µ < 1 [107]. We begin by collecting a few factsabout the operatorLy0 and its spectrum. First, we calculateL −1y0 w2 = w− w′(0)w′′(0)w′, L −1y0 w = w+12yw′− 3w′(0)2w′′(0)w′(4.46)where the additional terms are chosen so that homogeneous Neumann boundaryconditions at y = 0 are satisfied and which we use to computeˆ ∞0wL −1y0 w2 =ˆ ∞0w2+w′(0)w(0)22w′′(0), (4.47a)ˆ ∞0wL −1y0 w =34ˆ ∞0w2+3w′(0)w(0)24w′′(0), (4.47b)ˆ ∞0w2L −1y0 w2 =ˆ ∞0w3+w′(0)w(0)33w′′(0), (4.47c)ˆ ∞0w3 =65ˆ ∞0w2+3w(0)w′(0)5. (4.47d)118In the next two lemmas, we describe some key properties of the eigenvalue problemLy0Φ= ΛΦ, 0 < y < ∞; Φ′(0) = 0; Φ→ 0, as y→+∞. (4.48)Lemma 4.3.1. Let y0 ≥ 0 an let Λ0 be the principal eigenvalue of (4.48). ThenΛ0 = 0 if y0 = y0c and Λ0 ≶ 0 if y0 ≷ y0c. Furthermore, the eigenfunction corre-sponding to the principal eigenvalue is of one sign.Proof. SinceLy0 is self-adjoint, the variational characterization−Λ0 = infΦ∈H2([0,∞))´ ∞0 |Φ′|2+ |Φ|2−2w|Φ|2´ ∞0 |Φ|2, (4.49)implies that the principal eigenfunctionΦ0 is of one sign. SinceΦ0(0) 6= 0 we may,without loss of generality, assume that Φ0(0) = 1 and Φ0 > 0. Now we multiply(4.48) by w′ and integrate by parts to getΛ0 =w′′(0)´ ∞0 w′Φ0dy, (4.50)where we remark that the denominator is negative since w′ ≤ 0 for all y ≥ 0. Theclaim follows by noting that w′′(0)= 0 when y0 = 0 and w′′(0)≷ 0 for y0≷ y0c.Lemma 4.3.2. Let Λ1 be the second eigenvalue ofLy0 . Then Λ1 < 0 for all y0 ≥ 0.Proof. First note that the second eigenfunction Φ1 must cross zero at least oncesince´ ∞0 Φ0Φ1dy = 0 and Φ0 is of one sign. Next we assume toward a contradic-tion that Λ1 ≥ 0. We begin by showing that Φ1 has exactly one zero in 0 < y < ∞.Assume that Φ1 has more than one zero and choose 0 < a < b < ∞ such thatΦ1(a) = Φ1(b) = 0 and Φ1 > 0 in a < y < b. Then Φ′1(a) > 0 and Φ′1(b) < 0 sowe obtain the contradiction0≥ Λ1ˆ baw′Φ1dy =ˆ baw′Ly0Φ1dy = w′(b)Φ′1(b)−w′(a)Φ′1(a)> 0, (4.51)where we have usedLy0w′ = 0 and w′ < 0 for all y > 0. Thus Φ1 has a unique zero0 < a < ∞ and we may assume that Φ1 ≶ 0 for y ≶ a. Setting b = ∞ in (4.51) weget a contradiction.119In Figure 4.2a we plot the principal and second eigenvalues of the operatorLy0which we calculated numerically (see Appendix D.2 for details on the numericalmethod).Lemma 4.3.1 implies that the NLEP will have different stability properties de-pending on whether y0 is greater than or smaller than y0c. We will henceforth referto 0 ≤ y0 < y0c and y0 > y0c as the small-shift and large-shift cases respectively.When y0 = 0 it is known that for µ > 0 sufficiently large, the NLEP (4.44) is sta-ble. In this sense the nonlocal term appearing in (4.44) can stabilize the spectrumof the linearized operatorLy0 . Since all the eigenvalues ofLy0 are negative in thelarge-shift case we expect the spectrum of the NLEP (4.44) to remain stable for allµ ≥ 0. Restricting our attention to real eigenvalues, we have the following stabilityresult for the large-shift case.Theorem 4.3.1. All real eigenvalues of the NLEP (4.44) are negative when y0 >y0c.To prove this, we first prove the following lemma.Lemma 4.3.3. Let y0 > y0c and suppose that φ satisfiesLy0φ −λφ ≥ 0, 0 < y < ∞; φ ′(0)≥ 0; φ → 0, as y→+∞, (4.52)where λ ≥ 0. Then φ < 0 for all y≥ 0.Proof. Assume toward a contradiction that φ > 0 in 0 ≤ a < y < b ≤ ∞. Withoutloss of generality we may assume that φ(a) = 0 if a > 0 and φ(0) > 0 if a = 0.Then, for any such 0≤ a < b≤ ∞ we haveφ(a)≥ 0, φ ′(a)≥ 0, φ(b) = 0, φ ′(b)≤ 0. (4.53)Let g(y) ≡ w′′(y)− βw′(y) where β ≡ maxy≥0 |w′′′(y)|w′′(a) is well-defined and positive.Then g > 0 for all y ≥ 0, g′(a) ≤ 0, and moreover (Ly0 − λ )g =Ly0w′′− λg =−(w′)2−λg < 0. Integrating by parts we obtain the contradiction0 <ˆ bag(Ly0−λ )φdy−ˆ baφ(Ly0−λ )gdy= g(b)φ ′(b)−g(a)φ ′(a)−g′(b)φ(b)+g′(a)φ(a)≤ 0.120Proof [Theorem 4.3.1. ] Suppose that λ ≥ 0 is an eigenvalue of (4.44) so that byLemma 4.3.1 the operatorLy0−λ is invertible and from (4.44) we calculateφ = µ´ ∞0 wφ´ ∞0 w2 (Ly0−λ )−1w2.But w2 > 0 so by Lemma 4.3.3 we obtain the contradiction1 = µ´ ∞0 w(Ly0−λ )−1w2´ ∞0 w2 < 0. (4.54)From Theorem 4.3.1 we immediately deduce that the NLEP does not admit azero eigenvalue for any µ ≥ 0 when y0 > y0c. On the other hand, when 0≤ y0 ≤ y0cwe suspect that the NLEP admits a zero eigenvalue for an appropriate choice ofµ ≥ 0. When y0 = y0c this is the case for µ = 0. When 0≤ y0 < y0c we set λ = 0in (4.44) and obtainLy0φ = µ´ ∞0 wφ´ ∞0 w2 w2. (4.55)Using (4.46) we calculate φ =L −1y0 w2 and substitute back into (4.55) to deducethat λ = 0 is an eigenvalue if and only ifµ = µc(y0)≡´ ∞0 w2´ ∞0 wL−1y0 w2=´ ∞0 w2´ ∞0 w2+ w′(0)w(0)22w′′(0). (4.56)Note that µc(y0) ≶ 0 if y0 ≷ y0c. In terms of the critical value µc we have thefollowing instability result for the small-shift case.Theorem 4.3.2. Let 0≤ y0 < y0c and 0≤ µ < µc where critical value µc is definedin (4.56). Then the NLEP (4.44) admits a positive real eigenvalue.Proof. Let Λ0 be the principal eigenvalue of Ly0 . First note that by Lemma 4.3.1and 4.3.2 the principal and second eigenvalues ofLy0 satisfy Λ1 < 0 < Λ0. More-over the corresponding eigenfunction Φ0 is of one sign and we may assume that121Φ0 > 0 and´ ∞0 Φ20 = 1. Observe that if λ0 6= Λ0 is a positive eigenvalue of theNLEP (4.44) thenφ = µ´ ∞0 wφ´ ∞0 w2 (Ly0−λ0)−1w2,and since´ ∞0 wφ 6= 0 the above equation is equivalent to h(λ0) = 0 whereh(λ )≡ˆ ∞0w(Ly0−λ )−1w2−´ ∞0 w2µ. (4.57)We now show that such a λ0 can always be found in 0 < λ0 < Λ0 for 0≤ µ < µc.First we calculate h(0) =´ ∞0 w2(µ−1c − µ−1) < 0. Next we we let ψ be theunique solution to(Ly0−λ )ψ = w2, 0 < y < ∞; ψ ′(0) = 0.Decomposing ψ = c0Φ0+ψ⊥ where´ ∞0 Φ0ψ⊥ = 0 we find that ψ⊥ satisfies(Ly0−λ )ψ⊥ = w2− c0(Λ0−λ )Φ0, 0 < y < ∞; (ψ⊥)′(0) = 0. (4.58)Multiplying by Φ0 and integrating by parts we obtain c0 = (Λ0− λ )−1´ ∞0 w2Φ0and thereforeh(λ ) =´ ∞0 w2Φ0´ ∞0 wΦ0Λ0−λ +ˆ ∞0wψ⊥−´ ∞0 w2µ. (4.59)On the other hand, if we multiply (4.58) by ψ⊥ and integrate then we obtain−ˆ ∞0|ψ⊥|2(λ +−´ ∞0 ψ⊥Ly0ψ⊥´ ∞0 |ψ⊥|2)=ˆ ∞0w2ψ⊥. (4.60)By Lemma 4.3.2 and the variational characterization of the second eigenvalue ofLy0 we obtain0 <−Λ1 = infΦ∈H2([0,∞))´ ∞0 ΦΦ0=0´ ∞0 |Φ′|2+ |Φ|2−2w|Φ|2|Φ|2 ≤−´ ∞0 ψ⊥Ly0ψ⊥´ ∞0 |ψ⊥|2.122Substituting into (4.60) we calculate ||ψ⊥||2L2([0,∞))≤ λ−1||w2||L2([0,∞))||ψ⊥||L2([0,∞))so that ||ψ⊥||L2([0,∞)) and hence also´ ∞0 wψ⊥ are bounded as λ → Λ0 > 0. There-fore, from (4.59) we deduce h(λ )→+∞ as λ →Λ−0 . By a continuity argument wededuce the existence of a λ0 ∈ (0,Λ1) such that h(λ0) = 0.We conclude this section by establishing sufficient conditions for the stabilityof the NLEP (4.44) in both the small- and large-shift cases. Suppose that φ =φR + iφI and λ = λR + iλI satisfies the NLEP. Separating real and imaginary partsin (4.44) then yields the systemLy0φR−µ´ ∞0 wφR´ ∞0 w2 w2 = λRφR−λIφI,Ly0φI−µ´ ∞0 wφI´ ∞0 w2 w2 = λRφI +λIφR.Multiplying the first and second equations by φR and φI respectively, integrating,and then adding them together givesλRˆ ∞0|φ |2 =−L1(φR,φR)−L1(φI,φI), (4.62)where we defineL1(Φ,Φ)≡ˆ ∞0|Φ′|2+Φ2−2wΦ2+µ´ ∞0 wΦ´ ∞0 w2Φ´ ∞0 w2 . (4.63)It is clear that if L1(Φ,Φ)> 0 for all Φ ∈ H2([0,∞)) then the NLEP (4.44) will belinearly stable. In the next theorem we determine sufficient conditions on µ ≥ 0and y0 ≥ 0 for which the NLEP is linearly stable.Theorem 4.3.3. If 0≤ y0 < y0c and µ1(y0)< µ < µ2(y0), or y0 > y0c and 0≤ µ <1230 1 2 3 4 5y01.00.50.00.51.0Eigenvalues of y001(a)0 2 4 6 8 10y0010203040Stability Thresholds12(b)0.0 0.5 1.0y00.0000.0050.0100.0150.0200.0251(y0) c(y0) for 0 y0 y0c(c)Figure 4.2: (a) Plot of the numerically computed principal and second eigen-values of the operator Ly0 . The dashed vertical line corresponds toy0 = y0c. (b) Plot of the stability thresholds µ1 and µ2 as functionsof y0. The dashed vertical and horizontal lines correspond to y0 = y0cand µ = 2 respectively. The NLEP (4.44) has been rigorously demon-strated to be stable in the region bounded by the curves µ1 and µ2. Notethat µ1 and µ2 are interchanged as y0 passes through y0c. (c) Plot ofµ1(y0)−µc(y0) for 0≤ y0 < y0c. The NLEP is unstable for µ < µc andstable for µ1 < µ < µ2 when 0 ≤ y0 < y0c. It is conjectured that theNLEP is stable for µ > µc.µ1(y0) whereµ1(y0)≡2´ ∞0 w2´ ∞0 wL−1y0 w2+√´ ∞0 wL−1y0 w´ ∞0 w2L −1y0 w2, (4.64)µ2(y0)≡2´ ∞0 w2´ ∞0 wL−1y0 w2−√´ ∞0 wL−1y0 w´ ∞0 w2L −1y0 w2, (4.65)then Reλ < 0 for all eigenvalues of the NLEP (4.44).Proof. We first prove the result for 0≤ y0 < y0c. When µ = 2 and y0 = 0, Lemma5.1 (2) in [107] implies that L1(Φ,Φ) > 0 for all Φ ∈ H2((∞,∞)) and hence, byrestricting to even functions, also for all Φ ∈ H2([0,∞)). In particular, by the vari-ational characterization of the principal eigenvalue, this implies that the principal124eigenvalue of the self-adjoint operatorL1Φ≡Ly0Φ−µ2´ ∞0 wΦ´ ∞0 w2 w2− µ2´ ∞0 w2Φ´ ∞0 w2 w, (4.66)must be negative. We then perturb y0 ≥ 0 and µ until L1 has a zero eigenvalue andfor which we may solveΦ= c0L −1y0 w2+ c1L −1y0 w. (4.67)Substituting back into L1Φ= 0 we obtain the system(µ2´ ∞0 wL−1y0 w2´ ∞0 w2 −1)c0+µ2´ ∞0 wL−1y0 w´ ∞0 w2 c1 = 0,µ2´ ∞0 w2L −1y0 w2´ ∞0 w2 c0+(µ2´ ∞0 w2L −1y0 w´ ∞0 w2 −1)c1 = 0.‘Since´ ∞0 w2L −1y0 w =´ ∞0 wL−1y0 w2 a nontrivial solutions exists if and only if(µ2´ ∞0 wL−1y0 w2´ ∞0 w2 −1)2− µ24´ ∞0 wL−1y0 w´ ∞0 w2L −1y0 w2(´ ∞0 w2)2 = 0, (4.68)where explicit formulae for each integral can be found in (4.47). When µ = 2 andy0 = 0 the left hand side of (4.68) equals−9/10< 0 and therefore we have stabilityfor 0≤ y0 < y0c if(µ2´ ∞0 wL−1y0 w2´ ∞0 w2 −1)2− µ24´ ∞0 wL−1y0 w´ ∞0 w2L −1y0 w2(´ ∞0 w2)2 < 0, (4.69)which is easily seen to be equivalent to µ1 < µ < µ2.The thresholds µ1 and µ2 are singular as y0 → y0c and therefore the conti-nuity argument from above does not extend to y0 > y0c. However L1(Φ,Φ) > 0by Lemma 4.3.1 if y0 > y0c and µ = 0. Therefore we proceed with the samecontinuity argument as above, but starting from µ = 0. This yields the samecriteria, but since L −1y0 w2 < 0 by Lemma 4.3.3 the sufficient condition is nowµ2(y0)< 0≤ µ < µ1(y0).125Both of the stability thresholds µ1 and µ2 defined in (4.64) as well as the insta-bility threshold µc defined in (4.56) are easily computed using (4.47). In Figures4.2b and 4.2c we plot the stability thresholds and the difference µ1− µc respec-tively. In particular, from the plot in 4.2c we see that µ1 > µc. We conjecture,that as in the y0 = 0 case, the NLEP is stable for all µ > µc. In Appendix D.2 weprovide numerical support for this conjecture by plotting Reλ0 versus µ and y0 inFigure D.2a. In addition, we plot Λ0−Re(λ0) in Figure D.2b which suggest thatRe(λ0)≤ Λ0.4.4 ExamplesIn this section we illustrate the effect of introducing a nonzero boundary flux forthe activator by considering three distinct examples. Specifically, we first studythe stability of a single boundary spike concentrated at x = 0 when A ≥ 0 andB = 0. Using a winding number argument we illustrate that the stability of thesingle spike is improved by increasing the boundary flux A. Moreover, we illus-trate that if A exceed a threshold, then the spike is stable independently of theparameters τ ≥ 0 and D > 0. We then consider the structure and stability of a two-boundary-spike pattern when the boundary fluxes are equal, A= B≥ 0. One of thekey findings is that if A > 0 then the range of D > 0 values for which asymmet-ric patterns exist is extended. Additionally, by assuming that τ  1 we study thestability of both symmetric and asymmetric two-boundary-spike patterns to com-petition (zero eigenvalue crossing) instabilities. We demonstrate that one branch ofasymmetric patterns is always stable. Similarly, in our final example we consider atwo-boundary-spike pattern with a one-sided boundary flux A ≥ 0 and B = 0. Wedemonstrate the existence of several asymmetric patterns, with a certain branch ofthese patterns always being stable. For each example we include full numericalsimulations of the GM system (4.2) using the finite element software FlexPDE [1].1260.0 0.2 0.4 0.6 0.8A012345, IHopf Thresholds when Dhh(a)0 5 10 15D0246810h(D,A)A0.00.20.40.50.6(b)0 5 10 15D0.00.20.40.60.81.01.21.4hh(D,A)A0.00.20.40.50.6(c)Figure 4.3: Hopf bifurcation threshold and accompanying eigenvalue for asingle boundary-spike solution with one-sided boundary flux A ≥ 0 in(a) the shadow limit, D→ ∞, and (b and c) for finite D > 0 at selectvalues of 0≤ A < q0c. In (a) the dashed vertical line corresponds to thethreshold A = q0c beyond which no Hopf bifurcations occur.4.4.1 Example 1: One Boundary Spike at x = 0 with A > 0 and B = 0In this example we assume that B = 0 and investigate the role of a non-negativeflux, A ≥ 0, on the stability of a single boundary spike concentrated at x = 0. Wedenote the left boundary shift parameter by y0 = yL so that using (4.15b) and (4.23)we reduce (4.30) to (4.44) whereµ(λ ) = 2ω0 tanhω0ωλ tanhωλ. (4.70)Recalling that ωλ =√(1+ τλ )/D we first observe that 0 < µ(λ )≤ 2 for all real-valued λ ≥ 0 so that by Theorem 4.3.3 and Figure 4.2b the NLEP has no non-negative real eigenvalues. Next we determine whether the NLEP has any unstablecomplex-valued eigenvalues by using a winding number argument. Assuming thatλ is not in the spectrum of Ly0 we let φ = (Ly0 −λ )−1wc(y+ y0)2 so that as in§4.2.1 the NLEP reduces to the algebraic equationAy0(λ )≡1µ(λ )−Fy0(λ ) = 0, (4.71)127whereFy0(λ ) is defined in (4.36). In Figure D.1 we plotFy0(0) versus y0, as wellas the real and imaginary parts of Fy0(iλI) versus λI for select values of y0. Weintegrate in λ over a closed counter-clockwise contour consisting of the imaginaryaxis and a large semicircle in the right half-plane. SinceFy0(λ ) = O(|λ |−1) and µ(λ ) = O(λ−1/2) as |λ | → ∞, Reλ > 0, (4.72)the change in argument of Ay0(λ ) over the large semicircle ispi2 . Moreover, inReλ > 0, µ(λ ) 6= 0 whereas by Lemmas 4.3.1 and 4.3.2 we deduce that Fy0(λ )has one (resp. zero) pole(s) if y0 < y0c (resp. y0 ≥ y0c). Letting Z denote thenumber of zeros of Ay0(λ ) in Reλ > 0 it follows from the argument principle thatZ =1pi∆argAy0(iλI)∣∣0+∞+5/4, y0 < y0c,1/4, y0 ≥ y0c, (4.73)where the first term on the right hand side denotes the change in argument ofAy0(λ ) as λ follows the imaginary axis from λ =+i∞ to λ = 0. Note in additionthat we have usedAy0(λ¯ )=Ay0(λ ) to obtain ∆argAy0(iλI)∣∣−∞+∞= 2∆argAy0(iλI)∣∣0+∞.From (4.72) we immediately deduce that argAy0(+i∞) = pi/4. On the other hand,using (4.47), we evaluate Ay0(0) =12 −Fy0(0) ≶ 0 for y0 ≶ y0c (see also FigureD.1a). We will consider the cases y0 ≥ y0c and y0 < y0c separately below.If y0 ≥ y0c then ReFy0(iλI) < 0 for all λI > 0 (see Figure D.1b) so Ay0(iλI)never crosses the imaginary axis for all λI > 0. As a result ∆argAy0(iλI)|0∞=−pi/4and therefore Z = 0 if y0 > y0c. Since y0 is monotone decreasing in D when A > 0,we deduce that there is a threshold Dc(A) such that the single spike pattern is stablefor all τ ≥ 0 if D≥Dc(A). Substituting y0 = y0c and using (4.23) and (4.31) yieldsthe threshold parameterq0c =4+3√311≈ 0.83601, (4.74)with which Dc(A) is found by solving the transcendental equationtanhD−1/2c = q−10c AD−1/2c . (4.75)1280 20 40 60 80 100t0.00.51.01.52.02.53.03.54.0u(0,t)D=5, = 1A0.60.40.20.0(a)0 20 40 60 80 100t0.00.51.01.52.02.53.03.54.0u(0,t)D=5, = 2A0.60.40.20.0(b)0 20 40 60 80 100t0.00.51.01.52.02.53.03.54.0u(0,t)D=5, = 7A0.60.40.20.0(c)Figure 4.4: Plots of u(0, t) for a one boundary-spike solution with one-sidedboundary flux x = 0 (i.e. A ≥ 0 and B = 0) with ε = 0.005. Note thatincreasing the boundary flux A stabilizes the single boundary-spike so-lution for fixed values of D and τ .It is easy to see that if A ≥ q0c then Dc = ∞ is the only positive solution. On theother hand, if 0 < A < q0c then this equation has a unique positive solution that ismonotone increasing in A and satisfies Dc→ 0+ as A→ 0+ and Dc→∞ as A→ q−0c.In summary we deduce that the single spike pattern is stable for all D> 0 and τ ≥ 0if A ≥ q0c, or for all τ ≥ 0 if 0 < D ≤ Dc(A) and 0 < A < q0c. To determine thestability when 0≤ A < q0c and D > Dc(A) we must consider the case 0≤ y < y0c.Next we assume that 0≤ y0 < y0c. We begin by considering the shadow limit,defined by D→ ∞, for which µ(iλI)∼ 2(1+ iτλI)−1 and hence alsoReAy0(iλI)∼12−ReFy0(iλI), ImAy0(iλI)∼τλI2− ImFy0(iλI), D→∞.Since Fy0(0) > 1/2 and ReFy0(iλI)→ 0 as λI → ∞ (see Appendix D.1 and ac-companying Figure D.1) we deduce that there exists a solution to ReAy0(iλI) = 0.Moreover, in Figure D.1b we observe that when ReFy0(iλI) is positive it is alsomonotone decreasing in λI . Therefore there exists a unique eigenvalue λ∞h andtime constant τ∞h = 2ImFy0(iλI)/λ∞h such that Ay0(iλ∞h ) = 0. Furthermore, since129ImAy0(iλ∞h )≶ 0 if τ ≶ τ∞h we get∆argAy0(iλI)|0∞ =−5pi/4, τ < τ∞h ,3pi/4, τ > τ∞h .The single boundary spike solution therefore undergoes a Hopf bifurcation as τexceeds the Hopf bifurcation threshold τ∞h . Using the shadow limit threshold as aninitial guess, we numerically continue the Hopf bifurcation threshold for finite val-ues of D > 0 to obtain the Hopf bifurcation threshold τh(D,A) and accompanyingcritical eigenvalue λ = iλh(D,A) shown in Figure 4.3b and 4.3c respectively.The above analysis, together with the plots of τ∞h (A) and τh(D,A) in Figures4.3a and 4.3b respectively, indicate that the single boundary spike solution is sta-bilized as A > 0 is increased. Additionally, if A exceeds the threshold q0c givenin (4.74), then the single boundary spike is stable independently of the parametersτ ≥ 0 and D > 0. We illustrate the onset of oscillatory instabilities when D = 5 forτ = 1,2,7 and A = 0,0.2,0.4,0.6 by numerically computing the solution of (4.2)using FlexPDE 6 [1] and plotting u(0, t) in Figure 4.4. In particular we observethat the single spike pattern is stabilized by increasing the boundary flux A. Addi-tionally, our numerical simulations show good qualitative agreement with the Hopfbifurcation thresholds plotted in Figure 4.3b.4.4.2 Example 2: Two Boundary Spikes with A = B≥ 0In this example we investigate the role of equal boundary fluxes on the structureand stability of a two-boundary-spike pattern. Using the method of §4.1.1, a two-boundary-spike pattern is found by letting lL = l and lR = 1− l and solving (4.24b),which is explicitly given bytanhω0lη(yL)coshω0l− tanhω0(1− l)η(yR)coshω0(1− l) = 0, (4.76)130for 0 < l < 1 where η is given by (4.8) and yL and yR are given by (4.23). Note thatby (4.21) the algebraic equation (4.76) is equivalent toξLξR=coshω0lcoshω0(1− l) , (4.77)from which we deduce that l ≶ 1/2 implies ξL ≶ ξR. In particular l = 1/2 solves(4.76) for all A ≥ 0 and since in this case ξL = ξR we refer to it as the symmetricsolution. For the remainder of this example we will construct asymmetric two-spike patterns for which 0 < l < 1/2 (by symmetry the case l > 1/2 is identical)and then study the linear stability of both the symmetric and asymmetric patterns.Before constructing asymmetric two-boundary-spike patterns for A≥ 0 we firstrecall the following existence result from [104] in the case A = 0. Specifically, welet z = ω0lL and z˜ = ω0lR so that when A = 0 the system (4.24) (and hence also(4.76)) is equivalent toz+ z˜ = ω0, b(z) = b(z˜), b(z)≡ tanhzcoshz . (4.78)It follows from Result 2.3 (with k1 = k2 = 1, µ = 1, and r = 1) of [104] that (4.76)has a unique solution 0 < l < 1 if and only if0 < D < Dc1 ≡ [2log(1+√2)]−2 ≈ 0.322. (4.79)When A > 0 we solve (4.76) numerically and find that for given values of D andA > 0 it accepts zero, one, or two solutions in the range 0 < l < 1/2. Ratherthan solving (4.76) numerically for l as a function of A and D, we found it moreconvenient to solve for A = A(D, l). The results of our numerical calculations areshown in Figure 4.5a where we plot A = A(D, l) as well as the curve l = lmax(D)along which A(D, l) is maximized for a fixed value of D, and the curve l = lc1(D)along which A(D, lc1(D)) = A(D,1/2) for Dc1 < D < Dc2 ≈ 0.660 and lc1(D) = 0for 0 < D < Dc1. Consequently, (4.76) has zero solutions in 0 < l < 1/2 if A >Amax(D)≡A(D, lmax(D)), whereas it has two solutions, one with l < lc1(D) and theother with l > lc1(D), if Amax(D) > A > Ac1(D) ≡ A(D, lc1(D)) for 0 < D < Dc2.For all other values of D > 0 and A > 0 equation (4.76) has exactly one solution131(a) (b)Figure 4.5: Plots of (a) A = A(D, l) and (b) thresholds for the existence ofzero, one, or two asymmetric two-boundary-spike solutions in the pres-ence of equal boundary fluxes considered in Example 2.in 0 < l < 1/2. We summarize these existence thresholds in Figure 4.5b wherewe note in particular that A > 0 greatly extends the range of D values over whichasymmetric two-boundary-spike patterns exist.We now consider the linear stability of both the symmetric and asymmetrictwo-boundary-spike patterns constructed above. Note that since xL = 0 and xR = 1are fixed there are no slow drift dynamics and the linear stability of the two-spikepatterns is completely determined by the O(1) eigenvalues calculated from theNLEP (4.30). Moreover, we assume that τ = 0 so that no oscillatory instabilitiesarise (see Example 1) and for which we can exclusively focus on zero eigenvaluecrossing, or competition, instabilities. We proceed by first using the rigorous re-sults of §4.3 to determine the linear stability of the symmetric two-spike patternsconstructed above, and we will then use the algebraic reduction outlined in §4.2.1to determine the stability of the remaining asymmetric two-spike patterns.Using (4.15b) and (4.21), the NLEP (4.30) for the symmetric two-spike patternconstructed above is explicitly given byLy0φ −2ω0 tanh(ω02)´ ∞0 wc(y+ y0)Gω0φ dy´ ∞0 wc(y+ y0)2dywc(y+ y0)2 = λφ , 0 < y < ∞,φ ′(0) = 0,132whereφ =(φ1φ2), Gω0 =1ω0 sinhω0(coshω0 11 coshω0). (4.80)The Green’s matrix is symmetric and of constant row sum and therefore has eigen-vectors (1,±1)T . Substituting φ = (φ ,±φ)T into the NLEP therefore yields twouncoupled scalar NLEPs of the form (4.44) with µ = µ± whereµ+ ≡ 2, µ− ≡ 2tanh2(ω02). (4.81)From Theorem 4.3.3 and accompanying Figure 4.2b we immediately deduce thatthe φ+ mode is linearly stable. On the other hand, by Theorem 4.3.2 the φ− modeis unstable if µ− < µc, where µc is defined by (4.56). We therefore calculate thecompetition instability threshold by numerically solving 2tanh2 ω02 = µc(y0) wherey0 = yL = yR is the shift parameter given by (4.23) with l = 1/2. Our numeri-cal calculations indicate that the resulting instability threshold coincides with thevaluesA(D,1/2) =Ac1(D), Dc1 < D < Dc2,Amax(D), D > Dc2, (4.82)calculated above. In particular, the symmetric two-spike pattern is linearly unstablefor all A < A(D,1/2) when D > Dc1. Furthermore, since µc(y0) < 0 for y0 > y0cwe determine from (4.24b) that there are no competition instabilities ifA > ω−10 q0c tanh(ω0/2), (4.83)where q0c is the threshold identified in Example 1 and is explicitly given by (4.74).Note that, analogous to the results in Example 1, in the shadow limit (D→∞) thereare no competition instabilities for the symmetric pattern if A > q0c/2 (see Figure4.5b). As in §4.3 we conjecture and have numerically supported that the symmetrictwo-spike pattern is linearly stable for µ > µc and hence for all A > A(D,1/2).Finally, as is clear from Figure 4.5b, increasing A > 0 expands the range of Dvalues over which the symmetric two-boundary-spike pattern is linearly stable.For the asymmetric two-boundary-spike solutions constructed above the NLEP(4.30) is not diagonalizable since wc(y+ yL) 6≡ wc(y+ yR). We therefore can’t133directly apply the rigorous results of §4.3. To determine the competition instabilitywe instead use the algebraic reduction outlined in §4.2.1 and seek parameter valuessuch thatdet(I2−2ω20Gω0D0) = 0 (4.84)where I2 is the 2×2 identity matrix, Gω0 is the 2×2 Green’s matrix given in (4.80),andD0 =1ω0(tanhω0lFyL(0) 00 tanhω0(1− l)FyR(0)). (4.85)Substituting the function A = A(D, l) calculated above into (4.84) we can solvefor l as a function of D using standard numerical methods (specifically we used acombination of Scipy’s brentq and fsolve function in Python 3.6.8). Our compu-tations indicate that the resulting competition instability threshold coincides withthe curves lmax(D) for D > 0 and l = 1/2 for Dc1 < D < Dc2. In fact, we can showthat this is the case explicitly by first differentiating the quasi-equilibrium equationB = 0 with respect to l to get∇ξB(∂ξ∂ l+∂ξ∂A∂A∂ l)+∂B∂A∂A∂ l= 0. (4.86)Along the curve lmax(D) for D > 0 the function A(D, l) is maximized whereas,by symmetry, it is minimized along l = 1/2 for Dc1 < D < Dc2. In both cases∂A/∂ l = 0 along these curves so that (4.86) becomes ∇ξ B∂ξ /∂ l = 0. Differen-tiating (4.77) with respect to l implies that ∂ξ /∂ l 6= 0 and therefore we deducethat ∇ξ B is singular along l = lmax(D) and l = 1/2. By the discussion of §4.2.1 itfollows that along these curves the algebraic equation (4.84) is satisfied and theytherefore correspond to competition instability thresholds. Note in particular thatthe competition instability threshold along l = 1/2 corresponds to the competitioninstability threshold for the symmetric two-spike pattern. As an immediate conse-quence it follows that Amax→ q0c/2 as D→ ∞.To determine in which of the regions demarcated by the competition instabilitythresholds the asymmetric two-boundary-spike patterns are linearly stable and un-stable, we calculate the stability of the asymmetric patterns along the A = 0 curve.As outlined in Appendix D.3, the asymmetric two-boundary spike patterns when1340 10 20 30 40 50t0.000.050.100.150.200.250.30u(0,t) and u(1,t)D=0.30, A=0.02l0.5000.2200.005(a)0 20 40 60 80 100t0.00.10.20.30.4u(0,t) and u(1,t)D=0.60, A=0.08l0.5000.043(b)0 20 40 60 80 100t0.00.10.20.30.40.5u(0,t) and u(1,t)D=0.90, A=0.18l0.5000.217(c)Figure 4.6: Results of numerical simulation of (4.2) using FlexPDE 6 [1] withε = 0.005, τ = 0.1, and select values of D and A. In each plot thesolid (resp. dashed) lines correspond tot he spike height at x = 0 (resp.x = 1). Both the asymptotically constructed symmetric (l = 1/2) andasymmetric 0 < l < 1/2 solutions were used as initial conditions. SeeFigure 4.5a for position of parameter values relative to existence andstability thresholds.A = 0 are always linearly unstable, and therefore we deduce that the asymmetrictwo-boundary-spike patterns in the region bounded by l = 1/2 and l = lmax(D)for 0 < D < Dc2 are linearly unstable. On the other hand, numerically calculat-ing the dominant eigenvalue of the NLEP (4.30) (see Appendix D.2 for descrip-tion of numerical method) for select parameter values with l < lmax(D) we findthat such asymmetric two-boundary spike patterns are linearly stable. In partic-ular, we note that in the region where there are two asymmetric patterns (i.e. forAc1(D)< A < Amax(D)), the pattern with l > lmax(D) is linearly unstable while thatwith l < lmax(D) is linearly stable. Moreover, the single asymmetric two-boundary-spike pattern that exists for A < Amax(D) and D > Dc1 is linearly stable.Finally we support our asymptotic predictions by numerically solving (4.2)using FlexPDE 6 [1] with parameters ε = 0.005 and τ = 0.1 for select values of Dand A. Letting ue and ve be the any of the symmetric or asymmetric two-boundary-135spike patterns constructed above, we useu(x,0) = (1+0.025cos(20x))ue(x),v(x,0) = (1+0.025cos(20x))ve(x),(4.87)as initial conditions and simulate (4.2) sufficiently long that the solution settles.The results of our numerical simulations indicate good agreement with the asymp-totically calculated linear stability thresholds for both symmetric and asymmetrictwo-boundary-spike patterns. We include in Figure 4.6 the results of our numericalcalculations for select values of D, l, and A indicated by black markers in Figure4.5a. In particular, in Figure 4.6a we plot the spike heights at xL = 0 (solid) andxR = 1 (dashed) for D= 0.30 and A= 0.02 with initial conditions given by the two-boundary-spike pattern constructed with l = 0.5 which is symmetric and predictedto be stable, as well as l = 0.220 and l = 0.005 which are both asymmetric but pre-dicted to be linearly unstable and stable respectively. It is clear from the resultingplots that our asymptotic predictions hold in this numerical simulations. Addition-ally, we observe that the unstable asymmetric two-spike pattern tends toward thelinearly stable pattern. We observed this trend for all our numerical simulationsin which lmax < l < 1/2 though predicting this long-time behaviour analytically isbeyond the scope of this chapter. Similarly we numerically simulate the dynamicsof a symmetric and asymmetric two-boundary-spike pattern when D = 0.60 andA = 0.08 (Figure 4.6b) and when D = 0.90 and A = 0.18 (Figure 4.6c). In bothcases the symmetric and asymmetric patterns are predicted to be linearly unstableand stable respectively, which agrees with the outcomes observed in our numericalsimulations.4.4.3 Example 3: Two Boundary Spikes with a One Sided Flux(A≥ 0 and B = 0)In this example we investigate the effect of a one sided boundary flux (A ≥ 0 andB= 0) on the structure and linear stability of a two-boundary-spike pattern. Letting136(a) (b)0.0 0.2 0.4 0.6 0.8 1.0D0.000.050.100.150.200.250.300.350.40AA(D) ThresholdsAtopmax=A(D, ltopmax)Abotmax=A(D, lbotmax)Abotmax 0.16148(c)Figure 4.7: Plot of A = A(D, l) for Example 3 obtained by solving (4.88)when (a) 0 < l < 1/2 and (b) 1/2 < l < 1. The solid curves l = ltopmax(D)and lbotmax(D) indicate the values of l at which A(D, l) is maximized aswell as the competition instability threshold in the l > 1/2 and l < 1/2regions respectively. The corresponding existence thresholds of A ver-sus D are plotted against D in (c).lL = l and lR = 1− l, the gluing equation (4.24b) of §4.1.1 becomestanhω0lη(yL)coshω0l− tanhω0(1− l)3coshω0(1− l) = 0, (4.88)which is to be solved for 0 < l < 1 where η(yL) and yL = y0( ω0Atanhω0 ) are given by(4.8) and (4.23) respectively. Note that since η(yL)< 3 for all A > 0 it follows thatl = 1/2 is a solution of (4.88) if and only if A = 0. In particular ξL 6= ξR for allA > 0 and by the asymmetry of the boundary fluxes the cases l ≶ 1/2, for whichξL ≶ ξR, must be considered separately. On the other hand, when A = 0 we applythe same results from [104] summarized in Example 2.Proceeding as in Example 2 we numerically solve (4.88) for A = A(D, l) > 0when l < 1/2 and l > 1/2. In addition we compute l = lbotmax(D) and l = ltopmax(D)defined as the curves along which A(D, l) is maximized in the regions l < 1/2 andl > 1/2 respectively. In Figures 4.7a and 4.7b we plot A = A(D, l) together withltopmax and lbotmax in the regions 1/2 < l < 1 and 0 < l < 1/2 respectively. In eachregion the maximum value of A given by Atopmax(D)≡ A(D, ltopmax(D)) and Abotmax(D)≡A(D, lbotmax(D) and plotted in Figure 4.7c gives an existence threshold for the bound-137ary flux beyond which no two-boundary-spike with ξL > ξR and ξL < ξR existsrespectively. In particular, a two-boundary-spike pattern with ξL > ξR only existsif A < Atopmax(D) and D satisfies (4.79), whereas a two-boundary-spike pattern withξL < ξR exists for all D > 0 provided that A < Abotmax(D). Furthermore, by lettingD→∞ in (4.88) we numerically calculate lbotmax→ 0.13772 and Abotmax(D)→ 0.16148as D→ ∞ and this horizontal asymptote is indicated in Figure 4.7c.Next we consider the linear stability of the two-boundary-spike patterns con-structed above when τ = 0. In particular we restrict our attention to competitioninstabilities which arise through a zero eigenvalue crossing. Proceeding as in Ex-ample 2 we first deduce that both l = ltopmax(D) and l = lbotmax(D) yield a competitioninstability threshold. Furthermore, we verify that these are the only competition in-stability thresholds by numerically computing the algebraic equation (4.38) whereGω0 and D0 are given by (4.80) and (4.85) respectively. Since all asymmetric two-boundary-spike patterns when A= 0 are unstable with respect to competition insta-bilities (see Example 2 and Appendix D.3), we immediately deduce that all asym-metric two-boundary spike patterns with ξL > ξR and ξL < ξR are linearly unstablewhen l > ltopmax(D) and l > lbotmax(D) respectively, and are linearly stable otherwise.In particular, the non-zero boundary flux A> 0 both extends the range of parametervalues for which asymmetric patterns exists and are linearly stable.To support our asymptotic predictions we numerically calculate solutions to(4.2) when ε = 0.005 and τ = 0.1 for select values of D and A<max{Atopmax(D),Abotmax(D)}.For each pair (D,A) we let 0 < l < 1 be any of the values for which A(D, l) = Aand then let (ue(x),ve(x)) be the corresponding equilibrium pattern constructedabove. Using (4.87) as an initial condition we then solve (4.2) numerically usingFlexPDE 6 [1]. The results of our numerical simulations are illustrated in Figures4.8 and 4.9 when l is chosen to be in 1/2 < l < 1 and 0 < l < 1/2 respectively.Specifically, in Figure 4.8a (resp. 4.9a) we indicate with a blue or orange markerrespectively whether the solution settles (after simulating for 0 < t < 200) to theasymptotically predicted stable equilibrium with l < ltopmax(D) (resp. l < lbotmax(D)) orto a one-boundary spike solution in which the spike at x= 1 collapses respectively.In Figures 4.8b and 4.8c (resp. 4.9b and 4.9c) we show the spike heights as func-1380.1 0.2 0.3D0.50.60.70.80.91.0lSimulation Outcomes ( L> R)l= ltopmax(D)A=0.023A=0.038A=0.264A=0.370(a)0 20 40 60 80 100t0.00.10.20.30.40.5u(0,t), u(1,t)D= 0.054A(l)0.370 (0.852)0.370 (0.892)0.264 (0.748)0.264 (0.952)(b)0 20 40 60 80 100t0.000.050.100.150.200.25u(0,t), u(1,t)D= 0.204A(l)0.038 (0.705)0.038 (0.755)0.023 (0.610)0.023 (0.821)(c)Figure 4.8: Numerical simulations for Example 2 when ξL > ξR. (a) Out-come of numerical simulation of (4.2) starting from the asymmetrictwo-boundary-spike pattern constructed using the indicated values ofD, l, and A. Blue and orange markers indicate the two-boundary spikepattern settled to the stable two-spike pattern (i.e. with l < ltopmax(D))or collapsed to a single spike pattern respectively. Black dots indicatevalues of D, A, and l for which the spike heights are plotted over timein Figures (b) and (c). The left and right dashed vertical lines indicateD = 0.054 and D = 0.204 respectively. In (b) and (c) we plot spikeheights at x = 0 (solid) and x = 1 (dashed) at given values of D and Aand with initial condition specified by indicated value of l.tions of time at select values of A and D using an unstable and stable value of l in1/2 < l < 1 (resp. 0 < l < 1/2) to construct the initial condition (see captions formore details). The results of these numerical simulations are in good agreementwith our asymptotic predictions. However, we comment that in the numerical out-comes shown in Figure 4.8a some of the asymmetric patterns which are predictedto be stable collapse. We expect that this error due to a combination of small errorsfrom the asymptotic theory, numerical errors from the time integration of (4.2), aswell as the close proximity to the fold point A = Atopmax(D) for these values of l.1390.5 1.0 1.5 2.0D0.00.10.20.30.40.5lSimulation Outcome ( L< R)A0.0190.0310.0680.112(a)0 20 40 60 80 100t0.000.050.100.150.200.250.300.35u(0,t), u(1,t)D= 0.592A(l)0.068 (0.069)0.068 (0.130)0.019 (0.002)0.019 (0.414)(b)0 20 40 60 80 100t0.00.10.20.30.4u(0,t), u(1,t)D= 1.508A(l)0.112 (0.092)0.112 (0.172)0.031 (0.003)0.031 (0.432)(c)Figure 4.9: Numerical simulations for Example 2 when ξL < ξR. (a) Out-come of numerical simulation of (4.2) starting from the asymmetrictwo-boundary-spike pattern constructed using the indicated values ofD, l, and A. Blue and orange markers indicate the two-boundary spikepattern settled to the stable two-spike pattern (i.e. with l < lbotmax(D))or collapsed to a single spike pattern respectively. Black dots indicatevalues of D, A, and l for which the spike heights are plotted over timein Figures (b) and (c). The left and right dashed vertical lines indicateD = 0.592 and D = 1.508 respectively. In (b) and (c) we plot spikeheights at x = 0 (solid) and x = 1 (dashed) at given values of D and Aand with initial condition specified by indicated value of l.4.4.4 Example 4: One Boundary and Interior Spike with One-SidedFeed (A≥ 0, B = 0)In this final example we extend the results of Example 3 to the case where thereis one boundary spike at xL = 0 and one interior spike at 0 < x1 < 1. The asymp-totic construction of the resulting two spike patterns, as well as the analysis of theirlinear stability on an O(1) timescale proceeds as in the previous example. How-ever, since x1 is an equilibrium of the slow dynamics equation (4.18), we must nowalso determine the stability of the two-spike pattern on an O(ε−2) timescale byanalyzing the linearization of (4.18). We remark that an alternative approach todetermine the stability of multi-spike patterns on an O(ε−2) timescale is to calcu-late the O(ε2) eigenvalues of the linearization (4.25) though we do not pursue thisapproach further (see for example [41] for the analysis of O(ε2) eigenvalues).140Using the method of §4.1.1, with lL = l and l1 = (1− l)/2 equation (4.24b)becomestanhω0lη(yL)coshω0l− tanhω01−l23coshω0 1−l2= 0, (4.89)which is to be solved for 0 < l < 1 where η(yL) and yL = y0(ω0A/ tanhω0l) aregiven by (4.8) and (4.23) respectively. As in the previous examples we observe thatξL = ξ1 if and only if l = 1/3 which is a solution to (4.89) if and only if A = 0.In particular, together with the discussion in Example 3 we deduce that there areno symmetric two-spike patterns when there is a one-sided positive boundary fluxA > 0. As in the previous examples we also note that ξL ≶ ξ1 if l ≶ 1/3. Nextwe note that since l1 = (1− l)/2 the relevant asymmetric equilibrium results forA= 0 from [104] summarized in Example 2 must be modified. In particular, lettingz = ω0lL and z˜ = ω0l1, equations (4.24a) and (4.24b) when A = 0 becomez+2z˜ = ω0, b(z) = b(z˜), (4.90)where b(z) is given in (4.78). Then Result 2.3 of [104] (with k1 = 1 and k2 = 2)implies that a unique asymmetric two-spike solution with z≤ z˜ exists if and only ifD < Dm ≡ [3log(1+√2)]−2 ≈ 0.143, (4.91)whereas Result 2.4 of [104] (with k1 = 2 and k2 = 1) implies that there are eitherexactly one or two asymmetric two-spike solutions with z > z˜ if and only ifD < Dm or Dm < D < Dm1 ≡ [2sinh−1(1/2)+ sinh−1(2)]−2 ≈ 0.17274, (4.92)respectively. Proceeding as in Examples 2 and 3 we can then numerically calculateA = A(D, l) from (4.89) in the appropriate regions with 0 < l < 1/3 and 1/3 <l < 1. In Figures 4.10a and 4.10b we plot A = A(D, l) together with the curvesl = ltopmax(D) and l = lbotmax(D) along which A(D, l) is maximized. The resultingexistence thresholds Atopmax(D) ≡ A(D, ltopmax(D)) and Abotmax(D) ≡ A(D, lbotmax(D)) areplotted in Figure 4.10c. In particular a two spike pattern with a two-spike patternconsisting of one boundary and one interior spike with ξL > ξ1 only exists forD < Dm1 when A < Atopmax(D), whereas such a two-spike pattern with ξL < ξ1 exists141(a) (b)0.2 0.4 0.6 0.8 1.0D0.000.020.040.060.080.100.120.14AA(D) ThresholdsAtopmaxAtopcompAbotmaxAbotcomp, 1Abotcomp, 2(c)Figure 4.10: Plot of A = A(D, l) obtained by solving (4.88) when (a) 0 < l <1/3 and (b) 1/3 < l < 1. The curves l = ltopmax(D) and lbotmax(D) indi-cate the values of l at which A(D, l) is maximized while the curvesltopcomp(D), lbotcomp,1(D), and lbotcomp,2(D) indicate the competition instabil-ity thresholds. The corresponding existence thresholds, Atopmax(D) andAbotmax(D) are plotted in (c).for all D > 0 provided that A < Abotmax(D). Finally, by taking the limit D→ ∞ in(4.89) we numerically obtain the limiting values lbotmax(D)→ 0.0857 and Abotmax(D)→0.087174 as D→ ∞.Next we consider the linear stability of the two-spike patterns constructedabove on anO(1) timescale. As in Examples 2 and 3 we focus exclusively on com-petition instabilities by assuming that τ = 0 and seeking a zero eigenvalue crossingof the NLEP (4.30). In contrast to Examples 2 and 3 above the relevant competitioninstability threshold does not necessarily coincide with the curves l = ltopmax(D) andl = lbotmax. Indeed, fixing D > 0 and differentiating the quasi-equilibrium equationB = 0 with respect to l gives∇ξB(∂ξ∂ l+∂ξ∂A∂A∂ l)+∂B∂A∂A∂ l+∂B∂x1dx1dl= 0, (4.93)which along either l = ltopmax(D) or l = lbotmax(D) reduces to∇ξ B∂ξ∂ l=−12∂B∂x1.142Since ∂B/∂x1 6= 0 it follows that ∇ξ B is not necessarily singular along ltopmax(D)or lbotmax(D) and in particular these curves need not coincide with the competitioninstability thresholds. Note however that ∂B/∂x1→ 0 as D→ ∞ and therefore thecompetition instability threshold will coincide with lbotmax(D) but only in the limitD→ ∞. Thus, to calculate the appropriate competition instability thresholds weuse the algebraic reduction of §4.2.1 and numerically solve (4.38) when λ = 0 andwhere the matrices Gω0 and Dω0 are respectively given byGω0 =1ω0 sinhω0(coshω0 coshω0(1− x1)coshω0(1− x1) coshω0x1 coshω0(1− x1)), (4.94a)andD0 =1ω0(tanh(ω0l)FyL(0) 00 2tanh(ω0 1−l2)F0(0)). (4.94b)The resulting competition instability threshold ltopcomp(D) when ξL > ξ1 as well aslbotcomp,i(D) (i = 1,2) when ξL < ξ1 are indicated in Figures 4.10a and 4.10b respec-tively. Additionally, in Figure 4.10c we have plotted Atopcomp(D) ≡ A(D, ltopcomp(D))and Atopcomp,i(D) ≡ A(D, ltopcomp,i(D)) (i = 1,2). Finally, the stability result along theA = 0 curve calculated in Appendix D.3 implies that the two-spike pattern withξL > ξ1 is linearly unstable on an O(1) timescale when l > ltopcomp(D), and similarlywhen ξL < ξ1 the two-spike pattern is linearly unstable in the region bounded bythe curves l = lbotcomp,1(D) and l = lbotcomp,2(D).Next we consider the linear stability of the two-spike patterns on an O(ε−2)timescale. We explicitly calculate the right-hand-side of (4.18) by first calculating〈∂xGω0(x,x1)〉x1=sinhω0(2x1−1)sinhω0, ∂xGω0(x,0)∣∣x1=−sinhω0(1− x1)sinhω0,and rearranging the quasi-equilibrium equation (4.17a) asω20ξ 2Lη(yL)ξ1=1−6ω20ξ1Gω0(x1,x1)Gω0(x1,0).143so that (4.18) becomes1ε2dx1dt=−6ω20 f (x1), f (x1,ξ1) = ξ1−tanhω0(1− x1)3ω0, (4.95)where ξ1 together with ξL and yL are functions of x1 found by solving (4.17a) and(4.5). Note that the asymmetric two-spike equilibrium solutions constructed aboveusing the method of §4.1.1 immediately satisfy f (x1) = 0. The linear stabilityof these asymmetric two-spike patterns on an O(ε−2) timescale is determined bythe sign of f ′(x1); it is stable if f ′(x1) > 0 and unstable otherwise. We explicitlycalculated fdx1=∂ξ1∂x1+13sech2ω0(1− x1), (4.96)where ∂ξ1/∂x1 is calculated by first differentiating the quasi-equilibrium equationB = 0 with respect to x1∇ξ B∂ξ∂x1=− ∂B∂x1, (4.97)and then solving for ∂ξ /∂x1 which we can do since we are assuming the two-spikepattern is stable on an O(1) timescale and the matrix ∇ξ B is therefore invertible.Numerically evaluating f ′(x1) we find that the drift instability thresholds for whichf ′(x1) = 0 coincide with the curves ltopmax(D) and lbotmax(D). In fact, we can showthat this is the case analytically by first evaluating (4.93) along either ltopmax(D) orlbotmax(D) to get∇ξ B∂ξ∂ l=−12∂B∂x1. (4.98)Since the competition instability thresholds do not coincide with the curves ltopmax(D)and lbotmax(D), the matrix ∇ξ B is invertible along these curves and comparing (4.97)with (4.98) we obtain∂ξ1∂x1= 2∂ξ∂ l=−13sech2ω01− l2=−13sech2ω0(1− x1). (4.99)In particular f ′(x1) = 0 along the curves l = ltopmax(D) and lbotmax(D). Numericallyevaluating f ′(x1) at select values of l above and below these thresholds we deter-mine that the two-spike patterns constructed above with ξL > ξ1 or ξL < ξ1 are144linearly stable on an O(ε−2) timescale if and only if l < ltopmax(D) or l < lbotmax(D)respectively.As in the previous examples we performed full numerical simulations of (4.2)with FlexPDE 6 [1] to support our asymptotic predictions. Our numerical sim-ulations were found to strongly agree with the predicted stability thresholds. Inparticular, we observed the following dynamics. For values of l that are stablewith respect to both competition and drift instabilities, that is when l < ltopmax (resp.l < lbotmax) for l > 1/3 (resp. l < 1/2) the two-spike pattern was observed to be sta-ble. In the remaining regions (both stable and unstable with respect to competitioninstabilities) we observed that the interior spike either collapses and the bound-ary spike collapses to the one-boundary-spike solution, or else the interior spikechanges height to the height of the stable pattern and then slowly drifts toward thelocation of the interior spike in the stable two-spike solution. As in Example 3 wenoticed sensitivity to the competition instability threshold which we believe to beprimarily due to the flatness of A in this region4.5 DiscussionWe have extended the asymptotic theory developed for the singularly perturbedone-dimensional GM model to include the possibility of inhomogeneous Neumannboundary conditions for the activator. Additionally, we have rigorously establishedpartial stability and instability results for a class of shifted NLEPs. While theshifted NLEPs we considered are closely related to those in [60] we highlight thatthe difference in sign of the shift parameter leads substantial differences in the sta-bility properties of the NLEP. Finally we considered four examples to illustrate theasymptotic and rigorous theory as well as to explore the behaviour of the GM sys-tem with non-zero flux boundary conditions. For a one-boundary spike solution wefound that the non-zero Neumann boundary condition improves the stability withrespect to oscillatory instabilities arising through a Hopf bifurcation. Moreover,by considering a two-boundary-spike pattern with equal inhomogeneous boundaryfluxes we illustrated that the non-zero boundary flux improves the stability of sym-metric two-spike patterns and also extends the region of D > 0 values for whichasymmetric patterns exist provided that A = B > 0 is not larger than a computed145threshold. Similar results were obtained when considering a one-sided boundaryflux for which we considered patterns where both spikes concentrate on the bound-ary and where one concentrates on the boundary and the other in the interior. Ineach of our two-spike pattern examples we observed that there are two asymmetricpatterns, where one is always linearly unstable and the other is always linearly sta-ble. In a sense, the stable asymmetric pattern can be considered a boundary layersolution that is a direct consequence of the inhomogeneous Neumann boundarycondition. In particular its existence is mandated by the inhomogeneous boundarycondition which makes a direct comparison with asymmetric spike patterns in theabsence of boundary flux conditions difficult. However, our results illustrate thatinhomogeneities at the boundaries predispose the GM to forming patterns concen-trating at the boundaries in both symmetric and asymmetric configurations. Webelieve the distinction between interior and boundary-layer like localized patternwill play a key role in understanding more complicated mathematical models suchas those incorporating bulk-surface coupling (see Figure 3 in [58] for an exampleof a boundary-layer type pattern in a bulk-surface model).There are several key open problems and directions for future research. First,our rigorous results for the shifted NLEP do not provide tight bounds for regionsof stability and instability. Specifically, in the small shift-case we have determinedthat the NLEP is unstable if µ < µc(y0), and stable if µ1(y0)< µ < µ2(y0) wherewe have highlighted that µc(y0) < µ1(y0). As indicated in §4.3 we conjecturethat in fact the shifted NLEP is stable for all µ > µc and in Appendix D.2 weprovide numerical support for this conjecture. Proving this conjecture is our firstopen problem. In addition, to calculate the stability of asymmetric patterns forwhich the shift parameters are different we could not directly use the rigorousresults established in §4.3 since the NLEP (4.30) could not be diagonalized. Thedevelopment of a rigorous stability theory for NLEP systems of this form is anadditional direction for future research.One of the key insights from our investigation of a two-boundary spike config-uration is that the presence of equal or one sided boundary fluxes for the activatorgreatly extends the range of diffusivity values for which asymmetric patterns ex-ist and are linearly stable. This expanded region of existence and stability paral-lels that found when spatially inhomogeneous precursors are included in the GM146model. However, it can be argued that introducing inhomogeneous flux conditionsprovides a simpler alternative for generating asymmetric patterns. This warrantsfurther research into the role of inhomogeneous boundary conditions for the acti-vator in both activator-inhibitor and activator-substrate models in one-, two-, andthree-dimensional domains.147Chapter 5Localized Patterns in the 3D GMModelIn this chapter we analyze the existence, linear stability, and slow dynamics of lo-calized N-spot patterns for the singularly perturbed dimensionless Gierer-Meinhardt(GM) reaction-diffusion (RD) model (cf. [23])vt = ε2∆v− v+v2u, τut = D∆u−u+ ε−2v2 , x ∈Ω ;∂nv = ∂nu = 0 , x ∈ ∂Ω ,(5.1)where Ω ⊂ R3 is a bounded domain, ε  1, and v and u denote the activator andinhibitor fields, respectively. While the shadow limit in which D→ ∞ has beenextensively studied (cf. [108], [112], [105]), there have relatively few studies oflocalized RD patterns in 3-D with a finite inhibitor diffusivity D (see [11], [22],[53], [98] and some references therein). For 3-D spot patterns, the existence, sta-bility, and slow-dynamics of multi-spot quasi-equilibrium solutions for the singu-larly perturbed Schnakenberg RD model was analyzed using asymptotic methodsin [98]. Although our current study is heavily influenced by [98], our results forthe GM model offer some new insights into the structure of localized spot solu-tions for RD systems in three-dimensions. In particular, one of our key findings isthe existence of two regimes, the D = O(1) and D = O(ε−1) regimes, for whichlocalized patterns can be constructed in the GM-model, in contrast to the single148D = O(ε−1) regime where such patterns occur for the Schnakenberg model. Fur-thermore, our analysis traces this distinction back to the specific far-field behaviourof the appropriate core problem, characterizing the local behaviour of a spot, forthe GM-model. By numerically solving the core problem, we formulate a conjec-ture regarding the far-field limiting behaviour of the solution to the core problem.With the numerically established properties of the core problem, strong localizedperturbation theory (cf. [103]) is used to construct N-spot quasi-equilibrium solu-tions to (5.1), to study their linear stability, and to determine their slow-dynamics.We now give a more detailed outline of this chapter.In the limit ε → 0, in §5.1 we construct N-spot quasi-equilibrium solutionsto (5.1). To do so, we first formulate an appropriate core problem for a localizedspot, from which we numerically compute certain key properties of its far fieldbehaviour. Using the method of matched asymptotic expansions, we then estab-lish two distinguished regimes for the inhibitor diffusivity D, the D = O(1) andD = O(ε−1) regimes, for which N-spot quasi-equilibrium solutions exist. By for-mulating and analyzing a nonlinear algebraic system, we then demonstrate thatonly symmetric patterns can be constructed in the D =O(1) regime, whereas bothsymmetric and asymmetric patterns can be constructed in the D =O(ε−1) regime.In §5.2 we study the linear stability on an O(1) time scale of the N-spot quasi-equilibrium solutions constructed in §5.1. More specifically, we use the method ofmatched asymptotic expansions to reduce a linearized eigenvalue problem to a sin-gle globally coupled eigenvalue problem. We determine that the symmetric quasi-equilibrium patterns analyzed in §5.1 are always linearly stable in the D = O(1)regime but that they may undergo both oscillatory and competition instabilities inthe D = O(ε−1) regime. Furthermore, we demonstrate that the asymmetric pat-terns studied in §5.1 for the D =O(ε−1) regime are always unstable. Our stabilitypredictions are then illustrated in §5.4 where the finite element software FlexPDE6[1] is used to perform full numerical simulations of (5.1) for select parameter val-ues.In §5.5 we consider the weak interaction limit, defined by D = O(ε2), wherelocalized spots interact weakly through exponentially small terms. In this regime,(5.1) can be reduced to a modified core problem from which we numerically cal-culate quasi-equilibrium solutions and determine their linear stability properties.149Unlike in the D = O(1) and D = O(ε−1) regimes, we establish that spot solutionsin the D=O(ε2) regime can undergo peanut-splitting instabilities. By performingfull numerical simulations using FlexPDE6 [1], we demonstrate that these insta-bilities lead to a cascade of spot self-replication events in 3-D. Although spikeself-replication for the 1-D GM model have been studied previously in the weakinteraction regime D=O(ε2) (cf. [19], [46], [72]), spot self-replication for the 3-DGM model has not previously been reported.In §5.6 we briefly consider the generalized GM system characterized by dif-ferent exponent sets for the nonlinear kinetics. We numerically verify that thefar-field behaviour associated with the new core problem for the generalized GMsystem has the same qualitative properties as for the classical GM model (5.1) Thisdirectly implies that many of the qualitative results derived for (5.1) in §5.1–5.3still hold in this more general setting. Finally, in §5.7 we summarize our findingsand highlight some key open problems for future research.5.1 Asymptotic Construction of an N-SpotQuasi-Equilibrium SolutionIn this section we asymptotically construct an N-spot quasi-equilibrium solutionwhere the activator is concentrated at N specified points that are well-separated inthe sense that x1, . . . ,xN ∈Ω, |xi−x j|=O(1) for i 6= j, and dist(xi,∂Ω) =O(1) fori = 1, . . . ,N. In particular, we first outline the relevant core problem and describesome of its properties using asymptotic and numerical calculations. Then, themethod of matched asymptotic expansions is used to derive a nonlinear algebraicsystem whose solution determines the quasi-equilibrium pattern. A key feature ofthis nonlinear system, in contrast to that derived in [98] for the 3-D Schnakenbergmodel, is is that it supports different solutions depending on whether D = O(1)or D = O(ε−1). More specifically, we will show that the D = O(1) regime ad-mits only N-spot quasi-equilibrium solutions that are symmetric to leading order,whereas the D = O(ε−1) regime admits both symmetric and asymmetric N-spotquasi-equilibrium solutions.1500.00 0.05 0.10 0.15 0.20 0.25S0.0000.0050.0100.0150.0200.0250.030(S)(a)0.0 2.5 5.0 7.5 10.0 12.5 15.00.00.10.20.3V( , S)S0.010.050.1S(b)0.0 2.5 5.0 7.5 10.0 12.5 15.00.0000.0250.0500.0750.1000.1250.150U( , S)S0.010.050.1S(c)Figure 5.1: Plots of numerical solutions of the core problem (5.2): (a) µ(S)versus S, as well as the (b) activator V and (c) inhibitor U , at a fewselect values of S. The value S = S? ≈ 0.23865 corresponds to the rootof µ(S) = 0.5.1.1 The Core ProblemA key step in the application of the method of matched asymptotic expansions toconstruct localized spot patterns is the study of the core problem∆ρV −V +U−1V 2 = 0 , ∆ρU =−V 2 , ρ > 0 , (5.2a)∂ρV (0) = ∂ρU(0) = 0; V −→ 0 and U ∼ µ(S)+S/ρ , ρ → ∞ , (5.2b)where ∆ρ ≡ ρ−2∂ρ[ρ2∂ρ]. For a given value of the spot strength S > 0, the system(5.2) is solved for V =V (ρ;S), U =U(ρ;S), and µ = µ(S). Specifying the valueof S > 0 is equivalent to specifying the L2(R3) norm of V , as can be verified byapplying the divergence theorem to the second equation in (5.2a) over an infinitelylarge ball, which yields the identity S =´ ∞0 ρ2 [V (ρ)]2 dρ . Specifying the valueof the spot strength therefore yields a unique solution to (5.2) and in the contextof constructing multi-spot solutions by using the method of matched asymptoticexpansions the spot strengths completely determine the local profile of each spot.When S 1 we deduce from this identity that V = O(√S). By applying thedivergence theorem to the first equation in (5.2a) we get U = O(√S), while from(5.2b) we conclude that µ = O(√S). It is then straightforward to compute the151leading order asymptoticsV (ρ;S)∼√Sbwc(ρ) , U(ρ;S)∼√Sb, µ(S)∼√Sb, for S 1 , (5.3)where b≡ ´ ∞0 ρ2 [wc(ρ)]2 dρ ≈ 10.423 and wc > 0 is the unique nontrivial solutionto∆ρwc−wc+w2c = 0 , ρ > 0; ∂ρwc(0) = 0 , wc→ 0 as ρ→∞ . (5.4)We remark that (5.4) has been well studied, with existence being proved using aconstrained variational method, while its symmetry and decay properties are es-tablished by a maximum principle (see for example Appendix 13.2 of [112]). Thelimit case S 1 is related to the shadow limit obtained by taking D→∞, for whichnumerous rigorous and asymptotic results have previously been obtained (cf. [108],[112], [105]).Although the existence of solutions to (5.2) have not been rigorously estab-lished, we can use the small S asymptotics given in (5.3) as an initial guess tonumerically path-follow solutions to (5.2) as S is increased. The results of our nu-merical computations are shown in Figure 5.1 where we have plotted µ(S), V (ρ;S),and U(ρ;S) for select values of S > 0. A key feature of the plot of µ(S) is that ithas a zero crossing at S = 0 and S = S? ≈ 0.23865, while it attains a unique max-imum on the interval 0 ≤ S ≤ S? at S = Scrit ≈ 0.04993. Moreover, our numericalcalculations indicate that µ ′′(S)< 0 on 0 < S≤ S?. The majority of our subsequentanalysis hinges on these numerically determined properties of µ(S). We leave thetask of rigorously proving the existence of solutions to (5.2) and establishing thenumerically verified properties of µ(S) as an open problem, which we summarizein the following conjecture:Conjecture 5.1.1. There exists a unique value of S? > 0 such that (5.2) admitsa ground state solution with the properties that V,U > 0 in ρ > 0 and for whichµ(S?) = 0. Moreover, µ(S) satisfies µ(S)> 0 and µ ′′(S)< 0 for all 0 < S < S?.1525.1.2 Derivation of the Nonlinear Algebraic System (NAS)We now proceed with the method of matched asymptotic expansions to constructquasi-equilibrium solutions to (5.1). First we seek an inner solution by introducinglocal coordinates y = ε−1(x− xi) near the ith spot and letting v ∼ DVi(y) and u ∼DUi(y) so that the local steady-state problem for (5.1) becomes∆yVi−Vi+U−1i V 2i = 0 , ∆yUi− ε2D−1Ui+V 2i = 0 , y ∈ R3 . (5.5)In terms of the solution to the core problem (5.2) we determine thatVi ∼V (ρ,Siε)+O(D−1ε2) , Ui ∼U(ρ,Siε)+O(D−1ε2), (5.6)where ρ ≡ |y| = ε−1|x− xi| and Siε is an unknown constant that depends weaklyon ε . We remark that the derivation of the next order term requires that x1, . . . ,xNbe allowed to vary on a slow time scale. This higher order analysis is done in §5.3where we derive a system of ODE’s for the spot locations.To determine S1ε , . . . ,SNε we now derive a nonlinear algebraic system (NAS)by matching inner and outer solutions for the inhibitor field. As a first step, wecalculate in the sense of distributions that ε−3v2 −→ 4piD2∑Nj=1 S jε δ (x− x j) +O(ε2) as ε → 0+. Therefore, in the outer region the inhibitor satisfies∆u−D−1u =−4piεDN∑j=1S jεδ (x− x j)+O(ε3) , x ∈Ω ;∂nu = 0 , x ∈ ∂Ω .(5.7)To solve (5.7), we let G(x;ξ ) denote the reduced-wave Green’s function satisfying∆G−D−1G =−δ (x−ξ ) , x ∈Ω ; ∂nG = 0 , x ∈ ∂Ω ,G(x;ξ )∼ 14pi|x−ξ | +R(ξ )+∇xR(x;ξ ) · (x−ξ ) , as x→ ξ ,(5.8)where R(ξ ) is the regular part of G. The solution to (5.7) can be written asu∼ 4piεDN∑j=1S jεG(x;x j)+O(ε3) . (5.9)153Before we begin matching inner and outer expansions to determine S1ε , . . . ,SNεwe first motivate two distinguished limits for the relative size of D with respect toε . To do so, we note that when D 1 the Green’s function satisfying (5.8) has theregular asymptotic expansionG(x,ξ )∼ D|Ω|−1+G0(x,ξ )+O(D−1) , (5.10)where G0(x,ξ ) is the Neumann Green’s function satisfying∆G0 =1|Ω| −δ (x−ξ ) , x ∈Ω ; ∂nG0 = 0 , x ∈ ∂Ω ; (5.11a)ˆΩG0 dx = 0 , (5.11b)with asymptoticsG0(x,ξ )∼ 14pi|x−ξ | +R0(ξ )+∇xR0(x;ξ ) · (x−ξ ) , as x→ ξ , (5.11c)and where R0(ξ ) is the regular part of G0. In summary, for the two ranges of D wehaveG(x,ξ )∼ 14pi|x−ξ | +R(ξ )+o(1) , D = O(1) ,D|Ω|−1+R0(ξ )+o(1) , D 1 , as |x−ξ | → 0 , (5.12)where R(ξ ) is the regular part of G(x,ξ ). By matching the ρ → ∞ behaviour ofUi(ρ) given by (5.6) with the behaviour of u given by (5.9) as |x− xi| → 0, weobtain in the two regimes of D thatµ(Siε) = 4piεSiεR(xi)+∑ j 6=i S jεG(xi,x j) , D = O(1) ,SiεR0(xi)+∑ j 6=i S jεG0(xi,x j)+D|Ω|−1∑Nj=1 S jε , D 1 . (5.13)From the D  1 case we see that D = O(ε−1) is a distinguished regime forwhich the right-hand side has an O(1) contribution. Defining the vectors Sε ≡(S1ε , . . . ,SNε)T , µ(Sε)≡ (µ(S1ε), . . . ,µ(SNε))T , and e ≡ (1, . . . ,1)T , as well as the154matrices EN , G , and G0 byEN ≡ 1N eeT , (G )i j =R(xi) , i = jG(xi,x j) , i 6= j , (G0)i j =R0(xi) , i = jG0(xi,x j) , i 6= j , (5.14)we obtain from (5.13) that the unknowns S1ε , . . . ,SNε must satisfy the NASµ(Sε) = 4piεGSε , (5.15a)for D =O(1) andµ(Sε) = κENSε +4piεG0Sε , κ ≡ 4piND0|Ω| , (5.15b)for D = ε−1D0.5.1.3 Symmetric and Asymmetric N-Spot Quasi-EquilibriumWe now determine solutions to the NAS (5.15) in both the D = O(1) and the D =O(ε−1) regimes. In particular, we show that it is possible to construct symmetricN-spot solutions to (5.1) by finding a solution to the NAS (5.15) with Sε = Scεein both the D = O(1) and D = O(ε−1) regimes. Moreover, when D = O(ε−1) wewill show that it is possible to construct asymmetric quasi-equilibrium solutions to(5.1) characterized by spots each having one of two strengths.When D = O(1) the NAS (5.15a) implies that to leading order µ(Siε) = 0 forall i = 1, . . . ,N. From the properties of µ(S) outlined in §5.1.1 and in particu-lar the plot of µ(S) in Figure 5.1a, we deduce that Siε ∼ S? for all i = 1, . . . ,N.Thus, to leading order, N-spot quasi-equilibrium solutions in the D =O(1) regimehave spots with a common height, which we refer to as a symmetric pattern. Bycalculating the next order term using (5.15a) we readily obtain the two term resultSε ∼ S?e+ 4piεS?µ ′(S?)Ge . (5.16)We conclude that the configuration x1, . . . ,xN of spots only affects the spot strengthsat O(ε) through the Green’s matrix G . Note that if e is an eigenvector of G witheigenvalue g0 then the solution to (5.15a) is Siε = Scεe where Scε satisfies the scalar155equation µ(Scε) = 4piεg0Scε .Next, we consider solutions to the NAS (5.15b) in the D = ε−1D0 regime.Seeking a solution Sε ∼ S0+ εS1+ · · · we obtain the leading order problemµ(S0) = κENS0. (5.17)Note that the concavity of µ(S) (see Figure 5.1a) implies the existence of twovalues 0< Sl < Sr < S? such that µ(Sl) = µ(Sr). Thus, in addition to the symmetricsolutions already encountered in the D=O(1) regime, we also have the possibilityof asymmetric solutions, where the spots can have two different heights. We firstconsider symmetric solutions, where to leading order S0 = Sce in which Sc satisfiesµ(Sc) = κSc . (5.18)The plot of µ(S) in Figure 5.1a, together with the S  1 asymptotics given in(5.3), imply that a solution to (5.18) can be found in the interval 0 < Sc ≤ S? for allκ > 0. In Figure 5.3a we illustrate graphically that the common spot strength Sc isobtained by the intersection of µ(S) with the line κS. We refer to Figure 5.4 forplots of the symmetric solution strengths as a function of κ . In addition, we readilycalculate thatSc ∼ S?(1+κµ ′(S?))+O(κ2) , for κ  1 ,Sc ∼ 1bκ2 +O(κ−3) , for κ  1 ,(5.19)which provides a connection between the D = O(1) and D→ ∞ (shadow limit)regimes, respectively. From (5.15b), the next order correction S1 satisfies µ ′(Sc)S1−κENS1 = 4piScG0e. Upon left-multiplying this expression by eT we can determineeTS1. Then, by recalling the definition of EN ≡ N−1eeT we can calculate S1. Sum-marizing, a two term asymptotic expansion for the symmetric solution to (5.15b)isSε ∼ Sce+ 4piεµ ′(Sc)(ScIN +µ(Sc)µ ′(Sc)−κ EN)G0e , (5.20)provided that µ ′(Sc) 6= 0 (i.e. Sc 6= Scrit). Note that µ ′(Sc)−κ = 0 is impossible1560.05 0.10 0.15 0.20Sr0.000.010.020.030.040.05Sl(Sr)(a)0.05 0.10 0.15 0.20Sr1.00.80.60.40.20.0S ′l(Sr)(b)0.05 0.10 0.15 0.20S0.000.050.100.150.200.25f(S; )00.20.50.81(c)Figure 5.2: Plots of (a) Sl(Sr) and (b) S′l(Sr) for the construction of asymmet-ric N-spot patterns. (c) Plots of f (S,θ) for select values of θ ≡ n/N.For 0 < θ < 0.5 the function f (S,θ) attains an interior minimum inScrit < S < S?.by the following simple argument. First, for this equality to hold we require that0 < S < Scrit since otherwise µ ′(Sc) < 0. Moreover, we can solve (5.18) for κ toget µ ′(Sc)− κ = S−1c g(Sc) where g(S) ≡ Sµ ′(S)− µ(S). However, we calculateg′(S) = Sµ ′′(S) < 0 and moreover, using the small S asymptotics found in (5.3)we determine that g(S) ∼ −√S/(4b) < 0 as S→ 0+. Therefore, g(Sc) < 0 forall 0 < Sc < Scrit so that µ ′(Sc) < κ holds. Finally, as for the D = O(1) case, ifG0e = g00e then the common source values extends to higher order and we haveSε = Scεe where Scε is the unique solution to the scalar problemµ(Scε) = (κ+4piεg00)Scε . (5.21)Next, we construct of asymmetric N-spot configurations. The plot of µ(S) in-dicates that for any value of Sr ∈ (Scrit,S?] there exists a unique value Sl = Sl(Sr) ∈[0,Scrit) satisfying µ(Sl) = µ(Sr). A plot of Sl(Sr) is shown in Figure 5.2a. ClearlySl(Scrit) = Scrit and Sl(S?) = 0. We suppose that to leading order the N-spot config-uration has n large spots of strength Sr and N−n small spots of strengths Sl . Morespecifically, we seek a solution of the formSε ∼ (Sr, . . . ,Sr,Sl(Sr), . . . ,Sl(Sr))T , (5.22)157so that (5.17) reduces to the single scalar nonlinear equationµ(Sr) = κ f (Sr;n/N) , f (S;θ)≡ θS+(1−θ)Sl(S) , (5.23)for Scrit < Sr < S?. Since µ(Scrit)−κ f (Scrit;n/N) = µ(Scrit)−κScrit and µ(S?)−κ f (S?;n/N) = −κnS?/N < 0, we obtain by the intermediate value theorem thatthere exists at least one solution to (5.23) for any 0 < n≤ N when0 < κ < κc1 ≡ µ(Scrit)/Scrit ≈ 0.64619 .Next, we calculatef ′(S;θ) = (1−θ)(θ1−θ +S′l(S)),where S′l(S) is computed numerically (see Figure 5.2b). We observe that −1 ≤S′l(Sr) ≤ 0 with S′l(Scrit) = −1 and S′l(S?) = 0. In particular, f (S;n/N) is mono-tone increasing if θ/(1−θ) = n/(N−n)> 1, while it attains a local minimum in(Scrit,S?) if n/(N−n)< 1. A plot of f (S;θ) is shown in Figure 5.2c. In either case,we deduce that the solution to (5.23) when 0 < κ < κc1 is unique (see Figure 5.3a).On the other hand, when n/(N−n)< 1 we anticipate an additional range of valuesκc1 < κ < κc2 for which (5.23) has two distinct solutions Scrit < S˜r < Sr < S?. In-deed, this threshold can be found by demanding that µ(S) and κ f (S;n/N) intersecttangentially. In this way, we find that the threshold κc2 can be written asκc2 = κc2(n/N)≡ µ(S?r )f (S?r ;n/N), (5.24a)where S?r is the unique solution tof (S?r ;n/N)µ′(S?r ) = f′(S?r ;n/N)µ(S?r ) . (5.24b)In Figure 5.3c we plot κc2−κc1 as a functions of n/N where we observe thatκc2 > κc1 with κc2−κc1→ 0+ and κc2−κc1→ ∞ as n/N→ 0.5− and n/N→ 0+respectively. Furthermore, in Figure 5.3b we graphically illustrate how multiplesolutions to (5.23) arise as θ = n/N and κ are varied. We remark that the condi-1580.00 0.05 0.10 0.15 0.20 0.25S0.000.010.020.030.04(S) and S(S)S(a)0.05 0.10 0.15 0.20S0.000.010.020.030.040.05(S) and f(S; )(S)0.75 f(S,0.2)0.75 f(S,0.6)0.2 f(S,0.2)0.2 f(S,0.6)(b)0.0 0.1 0.2 0.3 0.4 0.5n/N0.00.51.01.52.0c2(n/N) c1(c)Figure 5.3: (a) Illustration of solutions to (5.18) as the intersection betweenµ(S) and κS. There is a unique solution if κ < κc1 ≡ µ(Scrit)/Scrit. (b)Illustration of solutions to (5.23) as the intersection between µ(S) andκ f (S,θ) where θ = n/N denotes the fraction of large spots in an asym-metric pattern. Note that when θ = 0.2 < 0.5 and κ > κc1 ≈ 0.64619there exist two solutions. (c) Plot of κc2−κc1 versus n/N. Observe thatκc2−κc1 increases as the fraction of large spots decreases.tion n/(N− n) < 1 implies that n < N/2, so that there are more small than largespots. The appearance of two distinct asymmetric patterns in this regime has adirect analogy to results obtained for the 1-D and 2-D GM model in [104] and[109], respectively. The resulting bifurcation diagrams are shown in Figure 5.4 forn/N = 0.2,0.4,0.6. We summarize our results for quasi-equilibrium solutions inthe following proposition.Proposition 5.1.1. (Quasi-Equilibrium Solutions): Let ε → 0 and x1, . . . ,xN ∈Ω be well-separated. Then, the 3-D GM model (5.1) admits an N-spot quasi-equilibrium solution with inner asymptoticsv∼ DVi(ε−1|x− xi|) , u∼ DUi(ε−1|x− xi|) , (5.25)as x→ xi for each i = 1, . . . ,N where Vi and Ui are given by (5.6). When |x− xi|=O(1), the activator is exponentially small while the inhibitor is given by (5.9). Thespot strengths Siε for i = 1, . . . ,N completely determine the asymptotic solutionand there are two distinguished limits. When D = O(1) the spot strengths satisfy1590.00 0.25 0.50 0.75 1.00 1.250.000.050.100.150.20Spot Strengths for n/N = 0.20ScSrSlSrSl(a)0.00 0.25 0.50 0.75 1.00 1.250.000.050.100.150.20Spot Strengths for n/N = 0.40ScSrSlSrSl(b)0.00 0.25 0.50 0.75 1.00 1.250.000.050.100.150.20Spot Strengths for n/N = 0.60ScSrSl(c)Figure 5.4: Bifurcation diagram illustrating the dependence on κ of the com-mon spot strength Sc as well as the asymmetric spot strengths Sr andSl or S˜r and S˜l . In (a) and (b) we have n/N < 0.5 so that there aremore small spots than large spots in an asymmetric pattern. As a re-sult, we observe that there can be two types of asymmetric patterns withstrengths Sr and Sl or S˜r and S˜l . In (c) the number of large spots exceedsthat of small spots and only one type of asymmetric pattern is possible.the NAS (5.15a), which has the leading order asymptotics (5.16). In particular,Siε ∼ S? so all N-spot patterns are symmetric to leading order. When D = ε−1D0the spot strengths satisfy the NAS (5.15b). A symmetric solution with asymptotics(5.20) where Sc satisfies (5.18) always exists. Moreover, if0 <4piND0|Ω| < κc1 ≈ 0.64619 ,then an asymmetric pattern with n large spots of strength Sr ∈ (Scrit,S?) and N−nsmall spots of strength Sl ∈ (0,Scrit) can be found by solving (5.23) for Sr andcalculating Sl from µ(Sl) = µ(Sr). If, in addition we have n/(N − n) < 1, then(5.23) admits two solutions on the range0.64619≈ κc1 < 4piND0|Ω| < κc2(n/N) ,where κc2(n/N) is found by solving the system (5.24).As we have already remarked, in the D = D0/ε regime, if D0  1 then the160symmetric N-spot solution (5.20) coincides with the symmetric solution for theD=O(1) regime given by (5.16). The asymmetric solutions predicted for the D=D0/ε regime persist as D0 decreases and it is, therefore, natural to ask what thesesolutions correspond to in the D = O(1) regime. From the small S asymptotics(5.3) we note that the NAS (5.15a) does admit an asymmetric solution, albeit onein which the spot strengths of the small spots are ofO(ε2). Specifically, for a giveninteger n in 1 < n≤ N we can construct a solution whereSε ∼ (S?, . . . ,S?,ε2Sn+1,0, . . . ,ε2SN,0)T . (5.26)By using the small S asymptotic expansion for µ(S) given in (5.3), we obtain from(5.15a) thatSi,0 = b(4piS?n∑j=1G(xi,x j))2, i = n+1, . . . ,N . (5.27)We observe that in order to support N− n spots of strength O(ε2), we require atleast one spot of strength O(1). Setting D = D0/ε , we use the large D asymptoticsfor G(x,ξ ) in (5.10) to reduce (5.27) toSi,0 ∼ bε−2(4piD0nS?|Ω|)2, i = n+1, . . . ,N . (5.28)Alternatively, by taking κ  1 in the NAS (5.15b) for the D = D0/ε regime, weconclude that Sr ∼ S? and Sl ∼ b(κnS?/N)2. Since κn/N = 4piD0n/|Ω|, as ob-tained from (5.15b), we confirm that the asymmetric patterns in the D = D0/εregime lead to an asymmetric pattern consisting of spots of strength O(1) andO(ε2) in the D =O(1) regime.5.2 Linear StabilityLet (vqe,uqe) be an N-spot quasi-equilibrium solution as constructed in §5.1. Wewill analyze instabilities for quasi-equilibrium solutions that occur on O(1) time-161scales. To do so, we substitutev = vqe+ eλ tφ , u = uqe+ eλ tψ , (5.29)into (5.1) and, upon linearizing, we obtain the eigenvalue problemε2∆φ −φ + 2vqeuqeφ − v2qeu2qeψ = λφ ,D∆ψ−ψ+2ε−2vqeφ = τλψ ,(5.30)where ∂nφ = ∂nψ = 0 on ∂Ω. In the inner region near the jth spot, we introducea local expansion in terms of the associated Legendre polynomials Pml (cosθ) ofdegree l = 0,2,3, . . . , and order m = 0,1, . . . , lφ ∼ c jDPml (cosθ)eimϕΦ j(ρ) , ψ ∼ c jDPml (cosθ)eimϕΨ j(ρ) , (5.31)where ρ = ε−1|x− x j|, and (θ ,ϕ) ∈ (0,pi)× [0,2pi). Suppressing subscripts forthe moment, and assuming that ε2τλ/D 1, we obtain the leading order innerproblem∆ρΦ− l(l+1)ρ2 Φ−Φ+2VUΦ− V2U2Ψ= λΦ , ρ > 0 ,∆ρΨ− l(l+1)ρ2 Ψ+2VΦ= 0 , ρ > 0 ,(5.32a)with the boundary conditions Φ′(0) = Ψ′(0) = 0, and Φ→ 0 as ρ → ∞. Here(V,U) satisfy the core problem (5.2). The behaviour of Ψ as ρ → ∞ depends onthe parameter l. More specifically, we have thatΨ∼B(λ ,S)+ρ−1 , for l = 0 ,ρ−(1/2+γl) , for l > 0 , as ρ → ∞ , (5.32b)where γl ≡√14 + l(l+1) and B(λ ,S) is a constant. Here we have normalized Ψby fixing to unity the multiplicative factor in the decay rate in (5.32b). Next, we162introduce the Green’s function Gl(ρ, ρ˜) solving∆ρGl− l(l+1)ρ2 Gl =−ρ−2δ (ρ− ρ˜) , ρ, ρ˜ > 0 , (5.33a)and explicitly given byGl(ρ, ρ˜) =12γl√ρρ˜(ρ/ρ˜)γl , 0 < ρ < ρ˜ ,(ρ˜/ρ)γl , ρ > ρ˜ , (5.33b)when l > 0. For l = 0 the same expression applies, but an arbitrary constant maybe added. For convenience we fix this constant to be zero. In terms of this Green’sfunction we can solve for Ψ explicitly in (5.32a) asΨ= 2ˆ ∞0Gl(ρ, ρ˜)V (ρ˜)Φ(ρ˜)ρ˜2 dρ˜+B(λ ,S) , for l = 0 ,0 , for l > 0 . (5.34)Upon substituting this expression into (5.32a) we obtain the nonlocal spectral prob-lemsM0Φ= λΦ+B(λ ,S)V 2U2, for l = 0; MlΦ= λΦ , for l > 0 . (5.35a)Here the integro-differential operatorMl is defined for every l ≥ 0 byMlΦ≡ ∆ρΦ− l(l+1)ρ2 Φ−Φ+2VUΦ− 2V2U2ˆ ∞0Gl(ρ, ρ˜)V (ρ˜)Φ(ρ˜)ρ˜2 dρ˜ . (5.35b)A key difference between the l = 0 and l > 0 linear stability problems is theappearance of an unknown constant B(λ ,S) in the l = 0 equation. This unknownconstant is determined by matching the far-field behaviour of the inner inhibitorexpansion with the outer solution. In this sense, we expect that B(λ ,S) will encap-sulate global contributions from all spots, so that instabilities for the mode l = 0are due to the interactions between spots. In contrast, the absence of an unknownconstant for instabilities for the l > 0 modes indicates that these instabilities arelocalized, and that the weak effect of any interactions between spots occurs only163through higher order terms. In this way, instabilities for modes with l > 0 are de-termined solely by the spectrum of the operator Ml . In Figure 5.5a we plot thenumerically-computed dominant eigenvalue ofMl for l = 0,2,3 as well as the subdominant eigenvalue for l = 0 for 0 < S < S?. This spectrum is calculated fromthe discretization of Ml obtained by truncating the infinite domain to 0 < ρ < L,with L 1, and using a finite difference approximation for spatial derivatives com-bined with a trapezoidal rule discretization of the integral terms. The l = 1 modealways admits a zero eigenvalue, as this simply reflects the translation invarianceof the inner problem. Indeed, these instabilities will be briefly considered in §5.3where we consider the slow dynamics of quasi-equilibrium spot patterns. FromFigure 5.5a we observe that the dominant eigenvalues of Ml for l = 2,3 satisfyRe(λ ) < 0 (numerically we observe the same for larger values of l). Therefore,since the modes l > 1 are always linearly stable, for the 3-D GM model there willbe no peanut-splitting or spot self-replication instabilities such as observed for the3-D Schnakenberg model in [98]. In the next subsection we will focus on analyzinginstabilities associated with l = 0 mode, which involves a global coupling betweenlocalized spots.5.2.1 Competition and Synchronous Instabilities for the l = 0 ModeFrom (5.35a) we observe that λ is in the spectrum ofM0 if and only if B(λ ,S) = 0.Assuming that B(λ ,S) 6= 0 we can then solve for Φ in (5.35a) asΦ= B(λ ,S)(M0−λ )−1(V 2/U2) . (5.36)Upon substituting (5.36) into the expression (5.34) forΨ when l = 0, we let ρ→∞and use G0(ρ, ρ˜)∼ 1/ρ as ρ→ ∞, as obtained from (5.33), to deduce the far-fieldbehaviourΨ∼ B+ 2Bρˆ ∞0V (M0−λ )−1(V 2/U2)ρ2dρ , as ρ → ∞ . (5.37)1640.00 0.05 0.10 0.15 0.20 0.25S1.00.50.00.51.0Real Part of Dominant Eigenvalues of ll023(a) (b)Figure 5.5: (a) Spectrum of the operator Ml defined in (5.35b). The dashedblue line indicates the eigenvalue with second largest real part for l = 0.Notice that the dominant eigenvalue of M0 is zero when S = Scrit ≈0.04993, corresponding to the maximum of µ(S) (see Figure 5.1a). (b)Plot of B(λ ,S). The dashed line black indicates the largest positiveeigenvalue of M0(S) and also corresponds to the contour B(λ ,S) = 0.We observe that B(λ ,S) is both continuous and negative for S > Scrit ≈0.04993.We compare this expression with the normalized decay condition on Ψ in (5.32b)for l = 0 to conclude thatB(λ ,S) =12´ ∞0 V (M0−λ )−1(V 2/U2)ρ2 dρ. (5.38)We now solve the outer problem and through a matching condition derive analgebraic equation for the eigenvalue λ . Since the interaction of spots will beimportant for analyzing instabilities for the l = 0 mode, we re-introduce the sub-script j to label the spot. First, since ∂ρΨ j ∼ −ρ−2 as ρ → ∞, as obtained from(5.32b) for l = 0, an application of the divergence theorem to ∆ρΨ j = −2VjΦ jyields that´ ∞0 VjΦ jρ2 dρ = 1/2. Next, by using vqe ∼ DVj(ρ) and φ ∼ c jDΦ j(ρ)for |x− x j| = O(ε) as obtained from (5.25) and (5.31), respectively, we calculatein the sense of distributions for ε → 0 that2ε−2vqeφ → 8piεD2N∑j=1c j(ˆ ∞0VjΦ jρ2 dρ)δ (x− x j) = 4piεD2N∑j=1c jδ (x− x j) .165Therefore, by using this distributional limit in the equation for ψ in (5.30), theouter problem for ψ is∆ψ− (1+ τλ )Dψ =−4piεDN∑j=1c jδ (x− x j) , x ∈Ω ; ∂nψ = 0 , x ∈ ∂Ω . (5.39)The solution to (5.39) is represented asψ = 4piεDN∑j=1c jGλ (x,x j) , (5.40)where Gλ (x,ξ ) is the eigenvalue-dependent Green’s function satisfying∆Gλ − (1+ τλ )DGλ =−δ (x−ξ ) , x ∈Ω ; ∂nGλ = 0 , x ∈ ∂Ω ,Gλ (x,ξ )∼ 14pi|x−ξ | +Rλ (ξ )+o(1) , as x→ ξ .(5.41)By matching the limit as x→ xi of ψ in (5.40) with the far-field behaviour ψ ∼DciB(λ ,Si) of the inner solution, as obtained from (5.37) and (5.31), we obtain thematching conditionB(λ ,Si)ci = 4piε(ciRλ (xi)+N∑j 6=ic jGλ (xi,x j)). (5.42)As similar to the construction of quasi-equilibrium solutions in §5.1, there are twodistinguished limits D = O(1) and D = D0/ε to consider. The stability propertiesare shown to be significantly different in these two regimes.In the D=O(1) regime, we recall that Si∼ S? for i= 1, . . . ,N where µ(S?)= 0.From (5.42), we conclude to leading order that B(λ ,S?) = 0, so that λ must bean eigenvalue of M0 when S = S?. However, from Figure 5.5a we find that alleigenvalues ofM0 when S= S? satisfy Re(λ )< 0. As such, from our leading ordercalculation we conclude that N-spot quasi-equilibrium solutions in the D = O(1)regime are all linearly stable.For the remainder of this section we focus exclusively on the D=D0/ε regime.Assuming that ε|1+τλ |/D0 1 we calculate Gλ (x,ξ )∼ ε−1D0/ [(1+ τλ )|Ω|]+G0(x,ξ ), where G0 is the Neumann Green’s function satisfying (5.11). We substi-166tute this limiting behaviour into (5.42) and, after rewriting the the resulting homo-geneous linear system for c ≡ (c1, . . . ,cN)T in matrix form, we obtainBc =κ1+ τλENc+4piεG0c , (5.43a)whereB ≡ diag(B(λ ,S1), . . . ,B(λ ,SN)) , EN ≡ N−1eeT . (5.43b)Here G0 is the Neumann Green’s matrix and κ ≡ 4piND0/|Ω| (see (5.15b)). Next,we separate the proceeding analysis into the two cases: symmetric quasi-equilibriumpatterns and asymmetric quasi-equilibrium solutions.Stability of Symmetric Patterns in the D = D0/ε RegimeWe suppose that the quasi-equilibrium solution is symmetric so that to leadingorder S1 = . . . = SN = Sc where Sc is found by solving the nonlinear algebraicequation (5.18). Then, from (5.43), the leading order stability problem isB(λ ,Sc)c =κ1+ τλENc . (5.44)We first consider competition instabilities for N ≥ 2 characterized by cTe = 0so that ENc = 0. Since B(λ ,Sc) = 0 from (5.44), it follows that λ must be aneigenvalue of M0, defined in (5.35b), at S = Sc. From Figure 5.5a we deducethat the pattern is unstable for S below some threshold where the dominant eigen-value ofM0 equals zero. In fact, this threshold is easily determined to correspondto Sc = Scrit, where µ ′(Scrit) = 0, since by differentiating the core problem (5.2)with respect to S and comparing the resulting system with (5.32) when l = 0, weconclude that B(0,Sc) = µ ′(Sc). The dotted curve in Figure 5.5b shows that thezero level curve B(λ ,Sc) = 0 is such that λ > 0 for Sc < Scrit. As such, we con-clude from (5.18) that symmetric N-spot quasi-equilibrium solutions are unstableto competition instabilities when κ > κc1 ≡ µ(Scrit)/Scrit.For special spot configurations {x1, . . . ,xN} where e is an eigenvector of G0 wecan easily calculate a higher order correction to this instability threshold. Since G0is symmetric, there are N−1 mutually orthogonal eigenvectors q2, . . . ,qN such thatG0qk = gkqk with qTk e = 0. Setting c =qk in (5.43), and using B(0,S)∼ εµ ′′(Scrit)δ167for S = Scrit+ εδ , we can determine the perturbed stability threshold where λ = 0associated with each eigenvector qk. By taking the minimum of such values, andby recalling the refined approximation (5.21), we obtain that N-spot symmetricquasi-equilibrium solutions are all unstable on the rangeScε < Scrit+4piεµ ′′(Scrit)mink=2,...,Ngk . (5.45)Next we consider the case c = e for which we find from (5.43) that, to leadingorder, λ satisfiesB(λ ,Sc)− κ1+ τλ = 0 . (5.46)First, we note that λ = 0 is not a solution of (5.46) since, by using B(0,S) = µ ′(S),this would require that µ ′(Sc) = κ , which the short argument following (5.20)demonstrates is impossible. Therefore, the c = e mode does not admit a zero-eigenvalue crossing and any instability that arises must occur through a Hopf bi-furcation. We will seek a leading order threshold τ = τh(κ) beyond which a Hopfbifurcation is triggered. To motivate the existence of such a threshold we considerfirst the κ→∞ limit for which the asymptotics (5.19) implies that Sc = 1/(bκ2)1 so that from the small S expansion (5.3) of the core solution we calculate from(5.35b) that M0Φ ∼ ∆ρΦ−Φ+ 2wcΦ+O(κ−1). Then, by substituting this ex-pression, together with the small S asymptotics (5.3) where Sc ∼ 1/bκ2 1, into(5.38) we can determine B(λ ,Sc) when κ  1. Then, by using the resulting ex-pression for B in (5.46), we obtain the following well-known nonlocal eigenvalueproblem (NLEP) corresponding to the shadow limit κ = 4piND0/|Ω| → ∞:1+ τλ − 2´ ∞0 wc(∆ρ −1+2wc−λ )−1w2cρ2 dρ´ ∞0 w2cρ2 dρ= 0 . (5.47)From Table 1 in [105], this NLEP has a Hopf bifurcation at τ = τ∞h ≈ 0.373 withcorresponding critical eigenvalue λ = iλ∞h with λ∞h ≈ 2.174. To determine τh(κ)for κ = O(1), we set λ = iλh in (5.46) and separate the resulting expression intoreal and imaginary parts to obtainτh =− Im(B(iλh,Sc))λhRe(B(iλh,Sc)) ,|B(iλh,Sc)|2Re(B(iλh,Sc))−κ = 0 , (5.48)168where Sc depends on κ from (5.18). Starting with κ = 50 we solve the secondequation for λh using Newton’s method with λh = λ∞h as an initial guess. We thenuse the first equation to calculate τh. Decreasing κ and using the previous solutionas an initial guess we obtain the curves τh(κ) and λh(κ) as shown in Figure 5.6.We conclude this section by noting that as seen in Figures 5.6a and 5.6c theleading order Hopf bifurcation threshold diverges as κ→ κ+c1, where κc1 = µ(Scrit)/Scrit.This is a direct consequence of the assumption that ε|1+ τλ |/D0 1 which failsto hold as τ gets increasingly large. Indeed, by using the series expansion in (3.12)–(3.14) of [89] for the reduced wave Green’s function in the sphere, we can solve(5.42) directly using Newton’s method for an N = 1 spot configuration centred atthe origin of the unit ball. Fixing ε = 0.001, this yields the higher order asymptoticapproximation for the Hopf bifurcation threshold indicated by the dashed lines inFigure 5.6. This shows that to higher order the bifurcation threshold is large butfinite in the region κ ≤ κc1. Moreover, it hints at an ε dependent rescaling of τin the region κ ≤ κc1 for which a counterpart to (5.44) may be derived. While wedo not undertake this rescaling in this chapter we remark that for 2-D spot patternsthis rescaling led to the discovery in [99] of an anomalous scaling law for the Hopfbifurcation threshold.Stability of Asymmetric Patterns in the D = D0/ε RegimeWhen the N-spot pattern consists of n large spots of strength S1 = . . . = Sn = Srand N− n small spots of strength Sn+1 = . . . = SN = Sl , the leading order linearstability is characterized by the blocked matrix system(B(λ ,Sr)In 00 B(λ ,Sl)IN−n)c =κ1+ τλENc , (5.49)where Im denotes the m×m identity matrix. In particular, an asymmetric quasi-equilibrium solution is linearly unstable if this system admits any nontrivial modes,c, for which λ has a positive real part. We will show that asymmetric patterns arealways unstable by explicitly constructing unstable modes.1691 2 3 4 5012345Hopf Bifurcation Thresholdleading orderhigher order(N=1, =0.01)(a)1 2 3 4 50.00.51.01.52.0IHopf Threshold Eigenvalueleading orderhigher order(N=1, =0.01)(b)1 2 3 4 5100101102103104Hopf Bifurcation Thresholdleading orderhigher order(N=1, =0.01)(c)Figure 5.6: Leading order (a) Hopf bifurcation threshold τh(κ) and (b) crit-ical eigenvalue λ = iλh for a symmetric N-spot pattern as calculatedby solving (5.48) numerically. The leading order theory assumesε|1+ τλ |/D0  1 and is independent of the spot locations. We cal-culate the higher order Hopf bifurcation threshold for an N = 1 spotpattern centred at the origin of the unit ball with ε = 0.01 by solving(5.42) directly (note κ = 3D0). In (c) we see that although the leadingorder Hopf bifurcation threshold diverges as κ → κc1, going to higherorder demonstrates that a large but finite threshold persists.First, we assume that 1≤ n < N−1 and we choose c to be a mode satisfyingc1 = · · ·= cn = 0 , cn+1+ · · ·+ cN = 0 . (5.50)Note that this mode describes competition among the N−n small spots of strengthSl . For such a mode, (5.49) reduces to the single equation B(λ ,Sl) = 0, whichimplies that λ must be an eigenvalue of M0 at S = Sl . However, since Sl < Scrit,we deduce from Figure 5.5a that there exists a real and positive λ forM0 at S= Sl .As such, any mode c satisfying (5.50) is linearly unstable.We must consider the n = N−1 case separately since (5.50) fails to yield non-trivial modes. Instead of considering competition between the small spots, weinstead consider competition between large and small spots collectively. We as-sume that n ≥ N− n, for which n = N− 1 is a special case, and we try to exhibitan unstable mode c of the formc1 = . . .= cn = cr , cn+1 = . . .= cN = cl . (5.51)170Then, (5.49) reduces to the system of two equations(B(λ ,Sr)− κ1+τλ nN)cr− κ1+τλ (N−n)N cl = 0 ,− κ1+τλ nN cr +(B(λ ,Sl)− κ1+τλ (N−n)N)cl = 0 ,which admits a nontrivial solution if and only if the determinant of this 2×2 systemvanishes. Therefore, to show that this mode is unstable it suffices to prove that thezero-determinant condition, written asF(λ )≡ B(λ ,Sl)B(λ ,Sr)− κ1+ τλ(nNB(λ ,Sl)+(N−n)NB(λ ,Sr))= 0 , (5.52)has a solution λ > 0. To establish this, we first differentiate µ(Sr) = µ(Sl) withrespect to Sr to obtain the identity µ ′(Sl)S′l(Sr) = µ′(Sr). Combining this resultwith B(0,S) = µ ′(S) we calculate thatF(0) = µ ′(Sl)[µ ′(Sr)−κ (N−n)N(n(N−n) +dSldSr)]. (5.53)Using µ ′(Sl) > 0 and µ ′(Sr) < 0 together with S′l(Sr) > −1 (see Figure 5.2b) andthe assumption n/(N− n) ≥ 1, we immediately deduce that F(0) < 0. Next, welet λ0 > 0 be the dominant eigenvalue ofM0 when S = Sl (see Figure 5.5a) so thatB(λ0,Sl) = 0. Then, from (5.52) we obtainF(λ0) =− κ1+ τλ0(N−n)NB(λ0,Sr) . (5.54)However, sinceM0 at S = Sr > Scrit has no positive eigenvalues (see Figure 5.5a),we deduce that B(λ ,Sr) is of one sign for λ ≥ 0 and, furthermore, it must benegative since B(0,Sr) = µ ′(Sr)< 0 (see Figure 5.5b for a plot of B showing bothits continuity and negativity for all λ > 0 when S > Scrit). Therefore, we haveF(λ0) > 0 and so, combined with (5.53), by the intermediate value theorem itfollows that F(λ ) = 0 has a positive solution. We summarize our leading orderlinear stability results in the following proposition:Proposition 5.2.1. (Linear Stability): Let ε  1 and assume that t  O(ε−3).When D = O(1), the N-spot symmetric pattern from Proposition 5.1.1 is linearly171stable. If D = ε−1D0 then the symmetric N-spot pattern from Proposition 5.1.1 islinearly stable with respect to zero-eigenvalue crossing instabilities if κ < κc1 ≡µ(Scrit)/Scrit ≈ 0.64619 and is unstable otherwise. Moreover, it is stable with re-spect to oscillatory instabilities on the range κ > κc1 if τ < τh(κ) where τh(κ) isplotted in Figure 5.6a. Finally, every asymmetric N-spot pattern in the D = ε−1D0regime is always linearly unstable.5.3 Slow Spot DynamicsA wide variety of singularly perturbed RD systems are known to exhibit slow dy-namics of multi-spot solutions in 2-D domains (cf. [48], [12], [91], [103]). In thissection we derive a system of ODE’s which characterize the motion of the spotlocations x1, . . . ,xN for the 3-D GM model on a slow time scale. Since the onlyN-spot patterns that may be stable on an O(1) time scale are (to leading order)symmetric we find that the ODE system reduces to a gradient flow. We remark thatboth the derivation and final ODE system are closely related to those in [98] for the3-D Schnakenberg model.The derivation of slow spot dynamics hinges on establishing a solvability con-dition for higher order terms in the asymptotic expansion in the inner region neareach spot. As a result, we begin by collecting higher order expansions of the lim-iting behaviour as |x− xi| → 0 of the Green’s functions G(x,x j) and G0(x,x j) thatsatisfy (5.8) and (5.11), respectively. In particular, we calculate thatG(xi+ εy,x j)∼G(xi,x j)+ εy ·∇1G(xi,x j) , i 6= j ,14piερ +R(xi)+ εy ·∇1R(xi;xi) , i = j ,(5.55a)as |x− xi| → 0 where ρ = |y| and ∇1R(xi;xi) ≡ ∇xR(x;x1)|x=x1 . Likewise, for theNeumann Green’s function, we haveG0(xi+ εy,x j)∼ D0ε|Ω| +G0(xi,x j)+ εy ·∇1G0(xi,x j) , i 6= j ,14piερ +R0(xi)+ εy ·∇1R0(xi;xi) , i = j ,(5.55b)as |x− xi| → 0 where ∇1 again denotes the gradient with respect to the first argu-ment. We next extend the asymptotic construction of quasi-equilibrium patterns172in §5.1 by allowing the spot locations to vary on a slow time scale. In particu-lar, a dominant balance in the asymptotic expansion requires that xi = xi(σ) whereσ = ε3t. For x near xi we introduce the two term inner expansionv∼ DVi ∼ D(Viε(ρ)+ ε2Vi2(y)+ · · ·) ,u∼ DUi ∼ D(Uiε(ρ)+ ε2Ui2(y)+ · · ·),(5.56)where we note the leading order terms are Viε(ρ)≡V (ρ,Siε) and Uiε(ρ)≡U(ρ,Siε).By using the chain rule we calculate ∂tVi =−ε2x′i(σ) ·∇yVi and ∂tUi =−ε2x′i(σ) ·∇yUi. In this way, upon substituting (5.56) into (5.1) we collect the O(ε2) terms toobtain that Vi2 and Ui2 satisfyLiεW i2 ≡ ∆yW i2+QiεW i2 =− f iε , y ∈ R2 , (5.57a)whereW i2 ≡(Vi2Ui2), f iε ≡(ρ−1V ′iε(ρ)x′i(σ) · y−D−1Uiε),Qiε ≡(−1+2U−1iε Viε −U−2iε V 2iε2Viε 0).(5.57b)It remains to determine the appropriate limiting behaviour as ρ → ∞. From thefirst row of Qiε , we conclude that Vi2→ 0 exponentially as ρ → ∞. However, thelimiting behaviour of Ui2 must be established by matching with the outer solution.To perform this matching, we first use the distributional limitε−2v2 −→ 4piεD2N∑j=1S jεδ (x− x j)+2ε3D2N∑j=1(ˆR3VjεVj2 dy)δ (x− x j) ,as ε → 0 where the localization at each x1, . . . ,xN eliminates all cross terms. Wethen update (5.9) to include the O(ε3) correction term. This leads to the refinedapproximation for the outer solutionu∼ 4piεDN∑j=1S jεG(x;x j)+2ε3DN∑j=1(ˆR3VjεVj2 dy)G(x;x j) . (5.58)173We observe that the leading order matching condition is immediately satisfied inboth the D=O(1) and the D=D0/ε regimes. To establish the higher order match-ing condition we distinguish between the D = O(1) and D = ε−1D0 regimes anduse the higher order expansions of the Green’s functions as given by (5.55a) and(5.55b). In this way, in the D = O(1) regime we obtain the far-field behaviour as|y| → ∞ given byUi2 ∼ 12piρˆR3ViεVi2 dy+ y ·biε , (5.59a)wherebiε4pi≡ Siε∇1R(xi;xi)+∑j 6=iS jε∇1G(xi,x j) . (5.59b)Similarly, in the D = D0/ε regime we obtain the following far-field matching con-dition as |y| → ∞:Ui2 ∼ 12piρˆR3ViεVi2 dy+2D0|Ω|N∑j=1ˆR3VjεVj2 dy+ y ·b0iε , (5.60a)whereb0iε4pi≡ Siε∇1R0(xi;xi)+∑j 6=iS jε∇1G0(xi,x j) . (5.60b)In both cases, our calculations below will show that only biε and b0iε affect theslow spot dynamics.To characterize slow spot dynamics we calculate x′i(σ) by formulating an ap-propriate solvability condition. We observe for each k = 1,2,3 that the functions∂ykW iε where W iε ≡ (Viε ,Uiε)T satisfy the homogeneous problem Liε∂ykW iε = 0.Therefore, the null-space of the adjoint operator L ?iε is at least three-dimensional.Assuming it is exactly three dimensional we consider the three linearly indepen-dent solutions Ψik ≡ ykPi(ρ)/ρ to the homogeneous adjoint problem, where eachPi(ρ) = (Pi1(ρ),Pi2(ρ)T solves∆ρPi− 2ρ2 Pi+QTiεPi = 0 , ρ > 0; P′i(0) =(00), (5.61)174and for which we noteQTiε −→(−1 00 0)as ρ → ∞ . (5.62)Owing to this limiting far-field behaviour of the matrix QTiε , we immediately de-duce that Pi2 = O(ρ−2) and that Pi1 decays exponentially to zero as ρ → ∞. En-forcing, for convenience, the point normalization condition Pi2 ∼ ρ−2 as ρ → ∞,we find that (5.61) admits a unique solution. We use each Ψik to impose a solv-ability condition by multiplying (5.57a) by ΨTik and integrating over the ball, Bρ0 ,centred at the origin and of radius ρ0 with ρ0 1. Then, by using the divergencetheorem, we calculatelimρ0→∞ˆBρ0(ΨTikLiW i2−W i2L ?i Ψik)dy= limρ0→∞ˆ∂Bρ0(ΨTik∂ρW i2−W Ti2∂ρΨik)∣∣∣∣ρ=ρ0ρ20 dΘ ,(5.63)where Θ denotes the solid angle for the unit sphere.To proceed, we use the following simple identities given in terms of the Kro-necker symbol δkl:ˆBρ0yk f (ρ)dy = 0 ,ˆBρ0ykyl f (ρ)dy = δkl4pi3ˆ ρ00ρ4 f (ρ)dρ , (5.64)for l,k = 1,2,3. Since L ?i Ψik = 0, we can use (5.57a) and (5.64) to calculate theleft-hand side of (5.63) aslimρ0→∞ˆBρ0ΨTikLiW i2dy = limρ0→∞(−3∑l=1x′il(σ)ˆBρ0ykylPi1(ρ)V ′iε(ρ)ρ2dy+1DˆBρ0ykPi2(ρ)Uiε(ρ)ρdy)=−4pi3x′ik(σ)ˆ ∞0Pi1(ρ)V ′iε(ρ)ρ2 dρ .(5.65)Next, in calculating the right-hand side of (5.63) by using the far-field behaviour175(5.59) and (5.60), we observe that only biε and b0iε terms play a role in the limit.In particular, in the D = O(1) regime we calculate in terms of the components ofbiεl of the vector biε , as given in (5.59), thatlimρ0→∞ˆ∂Bρ0ΨTik∂ρW i2∣∣ρ=ρ0ρ20 dΘ= limρ0→∞3∑l=1biεlˆ∂Bρ0ykylρ20dΘ=4pi3biεk ,limρ0→∞ˆ∂Bρ0W Ti2∂ρΨik∣∣ρ=ρ0ρ20 dΘ=−2 limρ0→∞3∑l=1biεlˆ∂Bρ0ykylρ20dΘ=−8pi3biεk .(5.66)From (5.63), (5.65), and (5.66), we conclude for the D = O(1) regime thatx′ik(σ) =−3γ(Siε)biεk , γ(Siε)≡ˆ ∞0Pi1(ρ)V ′i (ρ,Siε)ρ2 dρ , (5.67)which holds for each component k = 1,2,3 and each spot i = 1, . . . ,N. Fromsymmetry considerations we see that the constant contribution to the far-field be-haviour, as given by the first term in (5.59), is eliminated when integrated over theboundary. In an identical way, we can determine x′ik for the D = D0/ε regime. Insummary, in terms of the gradients of the Green’s functions and γiε ≡ γ(Siε), asdefined in (5.67), we obtain the following vector-valued ODE systems for the twodistinguished ranges of D:dxidσ=−12piγiε(Siε∇1R(xi;xi)+∑ j 6=i S jε∇1G(xi,x j)), for D = O(1) ,(Siε∇1R0(xi;xi)+∑ j 6=i S jε∇1G0(xi,x j)), for D = D0/ε .(5.68)Since only the symmetric N-spot configurations can be stable on an O(1) timescale (see Proposition 5.2.1), it suffices to consider the ODE systems in (5.68)when Siε = S?+O(ε) in the D = O(1) regime and when Siε = Sc +O(ε), whereSc solves (5.18), in the D = ε−1D0 regime. In particular, we find that to leadingorder, where the O(ε) corrections to the spot strengths are neglected, the ODEsystems in (5.68) can be reduced to the gradient flow dynamicsdxidσ=− 6piSγ(S)∇xiH (x1, . . . ,xN) , γ(S) =ˆ ∞0P1(ρ)V1(ρ,S)ρ2 dρ , (5.69a)where S = S? or S = Sc depending on whether D = O(1) or D = ε−1D0, respectively.In (5.69) the discrete energy H , which depends on the instantaneous spot locations, is1760.00 0.05 0.10 0.15 0.20S0.000.020.040.060.08(S)Figure 5.7: Plot of the numerically-computed multiplier γ(S) as defined inthe slow gradient flow dynamics (5.69).defined byH (x1, . . . ,xN)≡∑Ni=1 R(xi)+2∑Ni=1∑ j>i G(xi,x j) , for D = O(1) ,∑Ni=1 R0(xi)+2∑Ni=1∑ j>i G0(xi,x j) , for D = ε−1D0 . (5.69b)In accounting for the factor of two between (5.69) and (5.68), we used the reci-procity relations for the Green’s functions. In this leading order ODE system,the integral γ(S) is the same for each spot, since P1(ρ) is computed numericallyfrom the homogeneous adjoint problem (5.61) using the core solution V1(ρ,S)and U1(ρ,S) to calculate the matrix QTiε in (5.61). In Figure 5.7 we plot thenumerically-computed γ(S), where we note that γ(S) > 0. Since γ(S) > 0, localminima ofH are linearly stable equilibrium solutions to (5.69).We remark that this gradient flow system (5.69) differs from that derived in[98] for the 3-D Schnakenberg model only through the constant γ(S). Since thisparameter affects only the time-scale of the slow dynamics we deduce that theequilibrium configurations and stability properties for the ODE dynamics will beidentical to those of the Schnakenberg model. As such, we do not analyze (5.69)further and instead refer to [98] for more detailed numerical investigations. Finallywe note that the methods employed here and in [98] should be applicable to other3-D RD systems yielding similar limiting ODE systems for slow spot dynamics.The similarity between slow dynamics for a variety of RD systems in 2-D has beenpreviously observed and a general asymptotic framework has been pursued in [91]for the dynamics on the sphere.1770.5 1.0 1.5 2.0D00.000.250.500.751.001.251.501.752.00h(D0)Hopf Bifurcation Threshold for N=1(a)0 10 20 30024v(0,t)D0=0.60=0.69=1.150 10 20 300.02.55.07.5D0=1.00=0.49=0.810 10 20 30t2.55.07.5v(0,t)D0=1.40=0.42=0.700 10 20 30t0510D0=1.80=0.38=0.64(b)Figure 5.8: (a) Leading order Hopf bifurcation threshold for a one-spot pat-tern. (b) Plots of the spot height v(0, t) from numerically solving (5.1)using FlexPDE6 [1] in the unit ball with ε = 0.05 at the indicated τ andD0 values.5.4 Numerical ExamplesIn this section we use FlexPDE6 [1] to numerically solve (5.1) when Ω is the unitball. In particular, we illustrate the emergence of oscillatory and competition insta-bilities, as predicted in §5.2 for symmetric spot patterns in the D = D0/ε regimes.We begin by considering a single spot centred at the origin in the unit ball,for the D = ε−1D0 regime. Since no competition instabilities occur for a singlespot solution, we focus exclusively on the onset of oscillatory instabilities as τ isincreased. In Figure 5.8a we plot the Hopf bifurcation threshold obtained fromour linear stability theory, and indicate several sample points below and abovethe threshold. Using FlexPDE6 [1], we performed full numerical simulations of(5.1) in the unit ball with ε = 0.05 and parameters D0 and τ corresponding to thelabelled points in Figure 5.8a. The resulting activator height at the origin, v(0, t),computed from FlexPDE6 is shown in Figure 5.8b for these indicated parametervalues. We observe that there is good agreement with the onset of Hopf bifurcationsas predicted by our linear stability theory.1780 100 200 300 400 500t0.00.10.20.30.40.50.6v(x 1,t),v(x 2,t)Activator Spike HeightsD00.090.112(a) (b)Figure 5.9: (a) Plots of the spot heights (solid and dashed lines) in a two-spotsymmetric pattern at the indicated values of D0. Results were obtainedby using FlexPDE6 [1] to solve (5.1) in the unit ball with ε = 0.05 andτ = 0.2. (b) plot of three-dimensional contours of v(x, t) for D0 = 0.112,with contours chosen at v = 0.1,0.2,0.4.Next, we illustrate the onset of a competition instability by considering a sym-metric two-spot configurations with spots centred at (±0.51565,0,0) in the unitball and with τ = 0.2 (chosen small enough to avoid Hopf bifurcations) and ε =0.05. The critical value of κc1≈ 0.64619 then implies that the leading order compe-tition instability threshold for the unit ball with |Ω|= 4pi/3 is D0≈ 0.64619/(3N)=0.108. We performed full numerical simulations of (5.1) using FlexPDE6 [1] withvalues of D0 = 0.09 and D0 = 0.112. The results of our numerical simulations areshown in Figure 5.9, where we observe that a competition instability occurs forD0 = 0.112, as predicted by the linear stability theory. Moreover, in agreementwith previous studies of competition instabilities (cf. [98], [12]), we observe that acompetition instability triggers a nonlinear event leading to the annihilation of onespot.5.5 The Weak Interaction Limit D = O(ε2)In §5.2 we have shown in both the D =O(1) and D =O(ε−1) regimes that N-spotquasi-equilibrium solutions are not susceptible to locally non-radially symmetricinstabilities. Here we consider the weak-interaction regime D = D0ε2, where we17910 15 20 25 30 35D002468101214V(0) versus D0(a)0 2 4 6 8 10 12 14V01.21.00.80.60.40.20.00.20.4Dominant Eigenvalue 0(l)l0234(b)Figure 5.10: (a) Bifurcation diagram for solutions to the core problem (5.70)in the D = ε2D0 regime. (b) Dominant eigenvalue of the linearizationof the core problem for each mode l = 0,2,3,4, as computed numeri-cally from (5.74).numerically determine that locally non-radially symmetric instabilities of a local-ized spot are possible. First, we let ξ ∈ Ω satisfy dist(ξ ,∂Ω) O(ε2) and weintroduce the local coordinates x = ξ + εy and the inner variables v∼ ε2V (ρ) andu ∼ ε2U(ρ). With this scaling, and with D = D0ε2, the steady-state problem for(5.1) becomes∆ρV −V +U−1V 2 = 0 , D0∆ρU−U +V 2 = 0 , ρ = |y|> 0 . (5.70)For this core problem, we impose the boundary conditions Vρ(0) = Uρ(0) = 0and (V,U)→ 0 exponentially as ρ → ∞. Unlike the D = O(1) and D = O(ε−1)regimes, u and v are both exponentially small in the outer region. Therefore, for anywell-separated configuration x1, . . . ,xN , the inner problems near each spot centreare essentially identical and independent. In Figure 5.10a we plot V (0) versus D0obtained by numerically solving (5.70). From this figure, we observe that for allD0' 14.825, corresponding to a saddle-node point, the core problem (5.70) admitstwo distinct radially-symmetric solutions.Since both the activator V and inhibitor U decay exponentially there are onlyexponentially weak interactions between individual spots. As a result, it suffices toconsider only the linear stability of the core problem (5.70). Upon linearizing (5.1)180about the core solution we obtain the eigenvalue problem∆ρΦ− l(l+1)ρ2 Φ−Φ+2VUΦ− V2U2Ψ= λΦ ,D0∆ρΨ− l(l+1)ρ2 Ψ−Ψ+2VΦ= 0 ,(5.71)for each l ≥ 0 and for which we impose that Φ′(0) = Ψ′(0) = 0 and (Φ,Ψ)→ 0exponentially as ρ → ∞. We reduce (5.71) to a single nonlocal equation by notingthat the Green’s function Gl(ρ,ρ0) satisfyingD0∆ρGl− l(l+1)ρ2 Gl−Gl =−δ (ρ−ρ0)ρ2, (5.72)is given explicitly byGl(ρ,ρ0) =1D0√ρ0ρIl+1/2(ρ/√D0)Kl+1/2(ρ0/√D0) , ρ < ρ0 ,Il+1/2(ρ0/√D0)Kl+1/2(ρ/√D0) , ρ > ρ0 ,(5.73)where In(·) and Kn(·) are the nth order modified Bessel Functions of the first andsecond kind, respectively. As a result, by proceeding as in §5.2 we reduce (5.71)to the nonlocal spectral problemMlΦ= λΦ whereMlΦ≡ ∆ρΦ− l(l+1)ρ2 Φ−Φ+2VUΦ− 2V2U2ˆ ∞0Gl(ρ, ρ˜)V (ρ˜)Φ(ρ˜)ρ˜2 dρ˜ .(5.74)In Figure 5.10b we plot the real part of the largest numerically-computed eigen-value of Ml as a function of V (0) for l = 0,2,3,4. From this figure, we observethat the entire lower solution branch in the V (0) versus D0 bifurcation diagram inFigure 5.10a is unstable. However, in contrast to the D = O(1) and D = O(ε−1)regimes, we observe from the orange curve in Figure 5.10b for the l = 2 mode thatwhen D= ε2D0 there is a range of D0 values for which a peanut-splitting instabilityis the only unstable mode.In previous studies of singularly perturbed RD systems supporting peanut-181Figure 5.11: Snapshots of FlexPDE6 [1] simulation of (5.1) in the unit ballwith ε = 0.05, D = 16ε2, and τ = 1 and with initial condition givenby a single spot solution in the weak interaction limit calculated from(5.70) with V (0) = 5. The snapshots show contour plots of the acti-vator v(x, t) at different times where for each spot the outermost, mid-dle, and innermost contours correspond to values of 0.006, 0.009, and0.012 respectively. Note that the asymptotic theory predicts a maxi-mum peak height of v∼ ε2V (0)≈ 0.0125.splitting instabilities it has typically been observed that such linear instabilitiestrigger nonlinear spot self-replication events (cf. [98], [48], [91], and [12]). Re-cently, in [113] it has been shown using a hybrid analytical-numerical approachthat peanut-splitting instabilities are subcritical for the 2-D Schnakenberg, Gray-Scott, and Brusselator models, although the corresponding issue in a 3-D settingis still an open problem. Our numerical computations below suggest that peanut-splitting instabilities for the 3-D GM model in the D = ε2D0 regime are also sub-critical. Moreover, due to the exponentially small interaction between spots, wealso hypothesize that a peanut-splitting instability triggers a cascade of spot self-replication events that will eventually pack the domain with identical spots. Toexplore this proposed behaviour we use FlexPDE6 [1] to numerically solve (5.1) inthe unit ball with parameters τ = 1, ε = 0.05 and D0 = 16ε2, where the initial con-dition is a single spot pattern given asymptotically by the solution to (5.70) withV (0) = 5. From the bifurcation and stability plots of Figure 5.10 our parametervalues and initial conditions are in the range where a peanut-splitting instabilityoccurs. In Figure 5.11 we plot contours of the solution v(x, t) at various times. We182observe that the peanut-splitting instability triggered between t = 20 and t = 60leads to a self-replication process resulting in two identical spots at t = 110. Thepeanut-splitting instability is triggered for each of these two spots and this processrepeats, leading to a packing of the domain with N = 8 identical spots.5.6 General Gierer-Meinhardt ExponentsNext, we briefly consider the generalized GM modelvt = ε2∆v− v+u−qvp , τut = D∆u−u+ ε−2u−svm , x ∈Ω ;∂nv = ∂nu = 0 , x ∈ ∂Ω ,(5.75)where the GM exponents (p,q,m,s) satisfy the usual conditions p > 1, q > 0,m > 1, s≥ 0, and ζ ≡ mq/(p−1)− (s+1)> 0 (cf. [105]). Although this generalexponent set leads to some quantitative differences as compared to the prototypi-cal set (p,q,m,s) = (2,1,2,0) considered in this chapter, many of the qualitativeproperties resulting from the properties of µ(S) in Conjecture 5.1.1, such as theexistence of symmetric quasi-equilibrium spot patterns in the D = O(1) regime,remain unchanged.Suppose that (5.75) has an N-spot quasi-equilibrium solution with well-separatedspots. Near the ith spot we introduce the inner expansion v∼DαVi(y), u∼DβUi(y),and y = ε−1(x− xi), where∆Vi−Vi+D(p−1)α−qβU−qi V pi = 0 , y ∈ R3∆Ui− ε2D−1Ui =−Dmα−(s+1)β−1U−si V mi , y ∈ R3 .Choosing α and β such that (p−1)α−qβ = 0 and mα− (s+1)β = 1 we obtainα = ν/ζ , β = 1/ζ , ν = q/(p−1) ,with which the inner expansion takes the form v∼Dν/ζV (ρ;Siε) and u∼D1/ζU(ρ;Siε),where V (ρ;S) and U(ρ;S) are radially-symmetric solutions to the D-independent183core problem∆ρV −V +U−qV p = 0 , ∆ρU =−U−sV m , ρ > 0 , (5.76a)∂ρV (0) = ∂ρU(0) = 0 , V → 0 and U ∼ µ(S)+S/ρ , ρ → ∞ . (5.76b)By using the divergence theorem, we obtain the identity S =´ ∞0 U−sV mρ2 dρ > 0.By solving the core problem (5.76) numerically, we now illustrate that thefunction µ(S) retains several of the key qualitative properties of the exponentset (p,q,m,s) = (2,1,2,0) observed in §5.1.1, which were central to the analy-sis in §5.1 and §5.2. To path-follow solutions, we proceed as in §5.1.1 by firstapproximating solutions to (5.76) for S  1. For S  1, we use the identityS=´ ∞0 U−sV mρ2 dρ > 0 to motivate a small S scaling law, and from this we readilycalculate thatV (ρ;S)∼(Sb) νζ+1wc(ρ) , U(ρ;S)∼(Sb) 1ζ+1, µ(S)∼(Sb) 1ζ+1, (5.77)whereb≡ˆ ∞0wmc ρ2dρ ,and wc > 0 is the radially-symmetric solution of∆ρwc−wc+wpc = 0 , ρ > 0; ∂ρwc(0) = 0, wc→ 0, ρ → ∞ . (5.78)With this approximate solution for S 1, we proceed as in §5.1.1 to calculateµ(S) in (5.76) for different GM exponent sets by path-following in S. In Figure5.12b we plot µ(S) when (p,q,m,s) = (p,1, p,0) with p = 2,3,4, while a similarplot is shown in Figure 5.12a for other typical exponent sets in [105]. For each setconsidered, we find that µ(S) satisfies the properties in Conjecture 5.1.1. Finally,to obtain the NAS for the spot strengths we proceed as in §5.1.2 to obtain thatthe outer solution for the inhibitor field is given by simply replacing D with D1/ζin (5.9). Then, by using the matching condition u∼ D1/ζ (µ(S jε)+S jεε/|x− x j|)as x→ x j, for each j = 1, . . . ,N, we conclude that the NAS (5.15) still holds fora general GM exponent set provided that µ(S) is now defined by the generalized1840.0 0.1 0.2 0.3 0.4 0.5S0.000.020.040.060.080.10(S) for GM Exponents (p, q,m, s)(p,q,m,s)(2,1,3,0)(3,2,3,1)(3,2,2,0)(2,1,2,0)(a)0.00 0.05 0.10 0.15 0.20 0.25S0.0000.0050.0100.0150.0200.0250.030(S) for GM Exponents (p, 1, p, 0)p234(b)Figure 5.12: Left panel: Plot of µ(S), computed from the generalized GMcore problem (5.76), for the indicated exponent sets (p,q,m,s). Rightpanel: µ(S) for exponent sets (p,1, p,0) with p = 2,3,4. For each set,there is a unique S = S? for which µ(S?) = 0. The properties of µ(S)in Conjecture 5.1.1 for the prototypical set (2,1,2,0) still hold.core problem (5.76).5.7 DiscussionWe have used the method of matched asymptotic expansions to construct and studythe linear stability of N-spot quasi-equilibrium solutions to the 3-D GM model (5.1)in the limit of an asymptotically small activator diffusivity ε  1. Our key con-tribution has been the identification of two distinguished regimes for the inhibitordiffusivity, the D = O(1) and D = O(ε−1) regimes, for which we constructed N-spot quasi-equilibrium patterns, analyzed their linear stability, and derived an ODEsystem governing their slow spot dynamics. We determined that in the D = O(1)regime all N-spot patterns are, to leading order in ε , symmetric and linearly sta-ble on an O(1) time scale. On the other hand, in the D = O(ε−1) regime wefound the existence of both symmetric and asymmetric N-spot patterns. How-ever, we demonstrated that all asymmetric patterns are unstable on an O(1) timescale, while for the symmetric patterns we calculated synchronous and competi-tion instability thresholds. These GM results are related to those in [98] for the1850.0 0.1 0.2 0.3S0.0000.0050.0100.0150.0200.0250.030(S) for the GMS Model150100500(a)0 10 20 30 40S68101214161820(S) for the S/GS Model(b)0 10 20 30S01020304050(S) for the Brusselator Modelf0.20.40.60.8(c)Figure 5.13: Plots of the far-field constant behaviour for the (a) Gierer-Meinhardt with saturation, (b) Schnakenberg or Gray-Scott, and (c)Brusselator models. See Table 5.1 for the explicit form of the kineticsF(v,u) and G(v,u) for each model. A zero-crossing of µ(S) at someS > 0 occurs only for the GMS model.3-D singularly perturbed Schnakenberg model, with one of the key new featuresbeing the emergence of two distinguished limits, and in particular the existence oflocalized solutions in the D=O(1) regime for the GM model. For D=O(1), con-centration behaviour for the Schnakenberg model as ε → 0 is no longer at discretepoints typical of spot patterns, but instead appears to occur on higher co-dimensionstructures such as thin sheets and tubes in 3-D (cf. [94]). For the GM model, weillustrated the onset of both oscillatory and competition instabilities by numericallysolving the full GM PDE system using the finite element software FlexPDE6 [1].We have also considered the weak-interaction regime D = O(ε2), where we useda hybrid analytical-numerical approach to calculate steady-state solutions and de-termine their linear stability properties. In this small D regime we found that spotpatterns are susceptible to peanut-splitting instabilities. Finally, using FlexPDE6we illustrated how the weak-interaction between spots together with the peanut-splitting instability leads to a cascade of spot self-replication events.We conclude by highlighting directions for future work and open problems.First, although we have provided numerical evidence for the properties of µ(S)highlighted in Conjecture 5.1.1, a rigorous proof remains to be found. In partic-186RD Model F(V,U) G(V,U) Decay behaviourGierer-Meinhardt w/ Saturation (GMS) −V + V 2U(1+κU2) V 2 U ∼ µ(S)+S/ρSchnakenberg or Gray-Scott (S/GS) −V +V 2U −V 2U U ∼ µ(S)−S/ρBrusselator (B) −V + fV 2U V −V 2U U ∼ µ(S)−S/ρTable 5.1: Core problems and inhibitor decay behaviour for some RD sys-tems. In each case the activator decays exponentially.ular, we believe that it would be significant contribution to rigorously prove theexistence and uniqueness of the ground state solution to the core problem (5.2),which we numerically calculated when S = S?. A broader and more ambitious fu-ture direction is to characterize the reaction kinetics F(V,U) and G(V,U) for whichthe core problem∆ρV +F(V,U) = 0, ∆ρU +G(V,U) = 0, in ρ > 0 , (5.79)admits a radially-symmetric ground state solution for which V → 0 exponentiallyand U = O(1) as ρ → ∞. The existence of such a ground state plays a key rolein determining the regimes of D for which localized solutions can be constructed.For example, in the study of the 3-D singularly perturbed Schnakenberg modelit was found that the core problem does not admit such a solution and as a re-sult localized spot solutions could not be constructed in the D = O(1) regime(cf. [98]). To further motivate such an investigation of (5.79) we extend our numer-ical method from §5.1.1 to calculate and plot in Figure 5.13 the far-field constantµ(S) for the core problems associated with the GM model with saturation (GMS),the Schnakenberg/Gray-Scott (S/GS) model, and the Brusselator (B) model (seeTable 5.1 for more details). Note that for the GMS model we can find values ofS? such that µ(S?) = 0, but such a zero-crossing does not appear to occur for the(S/GS) and (B) models. As a consequence, for these three specific RD systems,localized spot patterns in the D = O(1) regime should only occur for the GMSmodel. Additionally, understanding how properties of µ(S), such as convexity andpositiveness, are inherited from the reaction kinetics would be a significant contri-187bution. In this direction, it would be interesting to try extend the rigorous numericsmethodology of [3] to try to establish Conjecture 5.1.1.188Bibliography[1] Flexpde 6. 2019. URL http://www.pdesolutions.com. → pagesxvii, xx, xxi, 19, 126, 130, 135, 138, 145, 149, 150, 178, 179, 182, 186[2] U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton. Implicit-explicit methodsfor time-dependent partial differential equations. SIAM J. Numer. Anal., 32(3):797–823, 1995. ISSN 0036-1429. doi:10.1137/0732037. 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Weakly nonlinear analysis of peanut-shapeddeformations for localized spots of singularly perturbed reaction-diffusionsystems. arXiv preprint arXiv:2002.01453, 2020. → pages 15, 80, 182203Appendix ABulk-Membrane-CoupledReaction-Diffusion-Systems are aLeading Order ApproximationIn this appendix we demonstrate the sense in which bulk-membrane coupled reaction-diffusion systems of the form (1.8) are to be understood as a leading order approx-imation in the limit of a thin membrane.A.1 Geometric PreliminariesWe first derive a three-term approximation to the Laplacian near a smooth andcompact (N−1)-dimensional manifold S⊂RN for N = 2,3. Let X(s1, ...,sN−1)∈ Sbe a parametrization of S where (s1, ...,sN−1)∈U ⊂RN−1 and X :U→ S is smooth.When N = 2 we will assume that the curve S is parameterized by arc-length so thatthe tangent vector |dX/ds1|= 1. The unit normal to S at X is defined byν(s) = κ−1d2Xds21(N = 2), ν(s1,s2) =∂1X×∂2X|∂1X×∂2X | (N = 3), (A.1)204where κ is the curvature of the curve S when N = 2, and where we use the notation∂i ≡ ∂/∂ si (i = 1,2). Next we define the tensors g, b, and c with entiresgi j ≡ ∂iX ·∂ jX , bi j ≡ ∂i∂ jX ·ν =−∂iX ·∂ jν , ci j = ∂iν ·∂ jν (A.2)where the second equality in the definition of bi j follows from differentiating theorthogonality relation ∂iX · ν = 0. Note that g corresponds to the metric tensor,while b corresponds to the second fundamental form when N = 3. Explicitly wecalculateg = 1, b = κ, (A.3a)for N = 2, andg =(∂1X ·∂1X ∂1X ·∂2X∂2X ·∂1X ∂2X ·∂2X), b =−(∂1X ·∂1ν ∂1X ·∂2ν∂2X ·∂1ν ∂2X ·∂2ν)(A.3b)for N = 3. Note that both g and b are symmetric.Next we derive a useful formula for c in terms of b and g. Throughout the nextcalculations we use the Einstein summation convention in which we sum over allrepeated indices. Since |ν | = 1 it follows that ∂iν · ν = 0 and we can thereforewrite ∂iν in terms of the tangent vectors ∂iX as ∂ jν = a jk∂kX for some unknowncoefficients ai j. Taking the inner product with ∂iX and recalling the definition ofgi j and bi j in (A.2) we obtain −bi j = a jkgki and therefore ai j =−bikgk j, where gi jare the entries of g−1. Calculating ci j = aika jqgkq = bilglkb jrgrqgqk = bilglkbk j wededuce thatc = κ2 (N = 2), c = bg−1b (N = 3). (A.4)We now consider the parametrization of a region near S in RN given byX˜(s1, ...,sN) = X(s1, ...,sN−1)+ sNν(s1, ...,sN), (A.5)where sN is assumed to be sufficiently small so that X˜(s1, ...,sN) is well definedand in particular 1+ κs2 > 0 when N = 2 and (I + sNg−1b) is positive definitewhen N = 3 where I is the 2×2 identity matrix Using (A.2) and (A.4) we readilydetermine that the metric tensor g˜ with entries g˜i j = ∂iX˜ ·∂ jX˜ is block diagonal and205explicitly given byg˜=((1+ s2κ)2 00 1)(N = 2), g˜=(g(I+ sNg−1b)2 00 1)(N = 3). (A.6)Since g˜ is block diagonal, the Laplacian in this parametrization is given by∆φ =1√det g˜N∑i, j=1∂∂ si(√det g˜g˜i j∂φ∂ s j)= ∂ 2Nφ +∂N det g˜2det g˜∂Nφ +1√det g˜N−1∑i, j=1∂i(√det g˜g˜i j∂ jφ).(A.7)When N = 2 we use (A.6) to easily calculate the exact formula∆φ =∂ 2φ∂ s22+κ1+κs2∂φ∂ s2+11+κs2∂∂ s1(11+κs2∂φ∂ s1). (A.8)For N = 3 an exact formula is more involved and we instead focus on retaining thefirst three terms in an expansion of the Laplacian when sN  1. First we calculatedet g˜ = detgdet2(I+ sNg−1b). (A.9)Next we use Jacobi’s formula for the derivative of a determinant to calculate∂ det(I+ sNg−1b)∂ sN∣∣∣∣sN=0= tr(g−1b) = 2H,∂ 2 det(I+ sNg−1b)∂ s2N∣∣∣∣sN=0= tr2(g−1b)− tr(g−1b)2 = 4H2− (κ21 +κ22 ).where we have used that the principal curvatures, κ1 and κ2 are the eigenvalues ofg−1b and the mean curvature H isH =κ1+κ22. (A.10)206Thereforedet g˜ = detg(1+2HsN + 12(4H2−κ21 −κ22 )s2N +O(s3N))2= detg(1+4HsN +(8H2−κ21 −κ22 )s2N +O(s3N)),and∂N det g˜2det g˜=2H +(8H2−κ21 −κ22 )sN +O(s2N)1+4HsN +(8H2−κ21 −κ22 )s2N +O(s3N)= 2H− (κ21 +κ22 )sN +O(s2N).Retaining leading order terms in sN and letting ∆S be the Laplace-Beltrami operatoron S we obtain∆φ =∂ 2φ∂ s2N+2H∂φ∂ sN− (κ21 +κ22 )sN∂φ∂ sN+∆Sφ +O(sN). (A.11)Summarizing, we let sN = δη where δ  1 and η = O(1) to obtain∆φ = δ−2∂ 2φ∂η2+δ−1κ∂φ∂η−κ2η ∂φ∂η+∂ 2φ∂ s2+O(δ ), (A.12)∆φ = δ−2∂ 2φ∂η2+2δ−1H∂φ∂η− (κ21 +κ22 )η∂φ∂η+∆Sφ +O(δ ). (A.13)for N = 2 and N = 3 respectively, and where we have used s to denote the arc-lengthalong S in N = 2.A.2 Derivation of Bulk-Membrane-CoupledReaction-Diffusion SystemWe begin by assuming that the cell bulk is given by a bounded domain Ω ⊂ RN(N = 2,3) with smooth boundary ∂Ω. Next we let ν(x) be the outward unit normalto ∂Ω at s ∈ ∂Ω and let the cell membrane be given byΩδ ≡ {x+δην(x) |x ∈ ∂Ω, 0 < η < 1}, (A.14)207where δ > 0 is the membrane thickness and is assumed to be sufficiently small sothat Ωδ is well defined. Furthermore, we separate ∂Ωδ into two disjoint (N−1)-dimensional surfaces ∂Ωδ = ∂Ωiδ ∪ ∂Ωeδ where ∂Ωiδ = ∂Ω denotes the interfacebetween the bulk and membrane and corresponds to setting η = 0 in (A.14) while∂Ωeδ denotes the interface between the membrane and the extracellular space andcorresponds to setting η = 1 in (A.14).Next we suppose that there are n and m chemical species in the bulk and in themembrane with concentrations respectively given byU = (U1, ...,Un)T , u = (u1, ...,um)T .In both the bulk and the membrane we further assume that these chemical speciesundergo isotropic diffusion and reaction kinetics so that the spatio-temporal evolu-tion of their concentrations is governed by the system of reaction-diffusion equa-tionsτB∂tU = DB∆U +F (U ), x ∈Ω, (A.15a)DB∂nU = γqδ (u,U ), x ∈ ∂Ω, (A.15b)andτM∂tu = DM∆u+ f δ (u), x ∈Ωδ , (A.16a)DM∂nu =−qδ (u,U ), x ∈ ∂Ωiδ , (A.16b)DM∂nu = 0, x ∈ ∂Ωeδ . (A.16c)where DB = diag(DB1, ...,DBn) and DM = diag(DM1, ...,DMm) are the diffusion co-efficients of each species, τB = diag(τB1, ...,τBn) and τM = diag(τM1, ...,τMm) arethe time constants of each species, F and f δ describe the bulk- and membrane-bound reaction kinetics, and qδ describes the interchange between bulk- and membrane-bound species across the bulk-membrane interface ∂Ω. The constant γ in (A.15b)is included to reflect possible asymmetries in the interchange between bulk- andmembrane-bound species. While we may anticipate that the bulk-bound reactionkinetics F (·) are independent of the membrane-thickness, the same cannot be said208of the membrane-bound kinetics f δ (·) and boundary interchange qδ (·, ·). Indeed,integrating (A.16a) over Ωδ and using the divergence theorem givesτMddtˆΩδu =ˆ∂Ωiδqδ (u,U )+ˆΩδf δ (u).Since vol(Ωδ ) = O(δ ) and area(∂Ω) = O(1), the three terms are balanced forδ  1 provided that they satisfy O(δ )O(u) = O(qδ ) = O(δ )O( f δ ). Anticipatingthat U = O(1), (A.15) and (A.16) imply thatqδ = O(1), u = O(δ−1), f δ = O(δ−1). (A.17)We now derive a leading order approximation to the bulk-membrane coupledreaction-diffusion system (A.15) and (A.16) in the δ  1 limit. We let X(s) ∈ ∂Ωparametrize ∂Ω where s ∈ B ⊂ RN−1. In particular we assume that s is the arc-length along ∂Ω when N = 2 and s = (s1,s2) when N = 3. Next we choose thesign of the curvature κ(s) of X(s) when N = 2 and the orientation of the local basis(X1,X2) when N = 3 such that the unit normal at X(s) ∈ ∂Ω given byν(s) = κ(s)−1d2Xds21(N = 2), ν(s1,s2) =∂1X×∂2X|∂1X×∂2X | (N = 3), (A.18)points away from the interior of Ω. Parameterizing Ωδ in terms of the boundaryfitted coordinates (s,η) ∈U× (0,1) asX(s)+δην(s) ∈Ωδ , (A.19)the results from Appendix A.1 we imply the following approximations for theLaplacian in Ωδ∆= δ−2∂ 2η +δ−1κ∂η −κ2η∂η +∂ 2s +O(δ ), (N = 2) (A.20)∆= δ−2∂ 2η +2δ−1H∂Ω∂η − (κ21 +κ22 )η∂η +∆∂Ω+O(δ ), (N = 3) (A.21)where κi (i = 1,2) are the principal curvatures of ∂Ω, H = (κ1 + κ2)/2 is themean curvature, and ∆∂Ω is the Laplace-Beltrami operator on ∂Ω. Throughout this209section we will use ∆∂Ω= ∂ 2s when N = 2. Based on the scaling (A.17) we supposethat f δ (u) = δ−1 f (δu) and qδ (u,U ) = q(δu,U ) and letu = δ−1u0+u1+δu2+O(δ 2), U =U 0+O(δ ), (A.22)where each ui = O(1) and U 0 = O(1). Note that whereas u denotes a volumeconcentration, u0 denotes a surface concentration. Substituting both ∂n =−δ−1∂ηand (A.20) into the membrane-bound system (A.16) and collecting powers of δ ,we find that the leading order, O(δ−3), equation is given byDM∂ηηu0 = 0,(s,η) ∈ ∂Ω× (0,1), DM∂ηu0 = 0, η = 0,1which implies u0 = u0(s). Similarly, from the O(δ−2) equation we find that thefirst order correction is independent of η , u1 = u1(s). The O(δ−1) problem is thengiven by∂tu0 = DM∂ηηu2+DM∆∂Ωu0+ f (u0), (s,η) ∈ ∂Ω× (0,1),DM∂ηu2 = q(u0,U 0), η = 0, DM∂ηu2 = 0, η = 1.Integrating over 0 < η < 1 and using both the boundary conditions of u2 and theη-independence of u0 we obtain∂tu0 = DM∆∂Ωu0+ f (u0)−q(u0,U 0), in ∂Ω.Summarizing, in the limit of a thin membrane, δ  1, the bulk-membranecoupled reaction-diffusion system (A.15) and (A.16) is approximated byτ∂tU 0 = DB∆U 0+F (U 0), in Ω, (A.23a)DB∂nU 0 = γq(u0,U 0), on ∂Ω, (A.23b)∂tu0 = DM∆∂Ωu0+ f (u0)−q(u0,U 0), in ∂Ω, (A.23c)whereU 0 =O(1) and u0 =O(1) are the leading order terms in the expansion (A.22)and where we assume that f (u0) = O(1) and q(u0,U 0) = O(1). Note that we haveassumed no interaction between the extracellular space and the cell membrane. It210is clear from the above derivation that including such an interaction would addan additional term to (A.23c). Additionally we have assumed that the membranethickness is constant and equal to δ > 0. It is a straightforward extension to includea non-constant thickness by letting 0 < η < δh(s) for s ∈ ∂Ω.211Appendix BAppendix for Chapter 2B.1 Green’s Functions in the Well-Mixed Limit and forthe DiskIn this appendix we collect all the relevant Green’s functions and indicate someof their key properties. We focus specifically on the uncoupled (K = 0) Green’sfunction, the well-mixed Green’s function (Db→∞), and the disk Green’s function(Ω= BR(0)). For the first two cases explicit formulae can be derived, while for thefinal case we must rely on a Fourier series expansion representation.B.1.1 Uncoupled Membrane Green’s FunctionWhen the bulk and membrane are uncoupled there is no direct dependence on thebulk Green’s function. Indeed the only relevant geometric dependent parameterbecomes the perimeter of the domain L = |∂Ω|. Thus, Ω may be an arbitrarybounded and simply connected subset of R2 with C2 boundary. We define theuncoupled Green’s function Γλ as the solution toDv∂ 2σΓ−µ2Γ=−δ (σ −ζ ), 0 < σ < L, Γ is L-periodic. (B.1)212The solution to (B.1) is readily calculated asΓ(σ ,ζ ) =coth(µL2√Dv)cosh(µ√Dv|σ −ζ |)2√Dvµ−sinh(µ√Dv|σ −ζ |)2√Dvµ.(B.2)B.1.2 Bulk and Membrane Green’s functions in the Well-MixedLimitWe now derive the leading order expression for the membrane Green’s function,defined by (2.22), when Db→ ∞. To leading order GλΩ, defined by (2.19), is con-stant and from the divergence theorem we findGλΩ(σ , σ˜)∼ GλΩ0 =1KL+µ2bλA=β/Kµ2bλ +β1L, where β ≡ K LA. (B.3)Here L≡ |∂Ω| and A≡ |Ω|. The leading order problem for the membrane Green’sfunction in (2.22) is thenDv∂ 2σGλ∂Ω−µ2sλGλ∂Ω+K2GλΩ0ˆ L0Gλ∂Ω(σ˜ ;ζ )dσ˜ =−δ (σ −ζ ). (B.4)Upon integrating this equation and using the periodic boundary conditions we getˆ L0Gλ∂Ω(σ˜ ;ζ )dσ˜ =1µ2sλ −K2LGλΩ0=(1µ2sλ (µ2bλ +β )−Kβ)1AGλΩ0,where GλΩ0 is defined in (B.3). Therefore, from (B.4), we find that Gλ∂Ω satisfiesDv∂ 2σGλ∂Ω−µ2sλGλ∂Ω =−δ (σ −ζ )−K2/Aµ2sλ (µ2bλ +β )−Kβ.This problem is readily solved in terms of the uncoupled Green’s function of (B.2)by definingΓλ (σ ,ζ ) := Γ(σ ,ζ )∣∣µ=µsλ,213and then using the decompositionGλ∂Ω(σ ,ζ ) = Γλ (σ ,ζ )+γλµ2sλ, γλ ≡K2/Aµ2sλ (µ2bλ +β )−Kβ. (B.5)B.1.3 Bulk and Membrane Green’s functions in the DiskHere we consider the bulk Green’s function defined by (2.19). By using separationof variables (in polar coordinates), and applying the boundary condition in (2.19),we can write this Green’s function as a Fourier seriesGλΩ(r,σ , σ˜) =12piR∞∑n=−∞aλn Pn(r)einR (σ−σ˜),Pn(r)≡I|n|(ωbλ r)I|n|(ωbλR), aλn ≡1DbP′n(R)+K, ωbλ ≡µbλ√Db.(B.6)We remark that the singularity lies on the boundary and for this reason the radialdependence is given only in terms of the modified Bessel functions of the first kindIn(z). Similarly, we can represent the membrane Green’s function in (2.22) for thedisk in terms of the Fourier seriesGλ∂Ω(σ ,σ0) =12piR∞∑n=−∞gλn einR (σ−σ0), gλn ≡1Dv n2R2 +µ2sλ −K2aλn. (B.7)B.1.4 A Useful Summation Formula for the Disk Green’s FunctionsWe make note here of a useful summation formula for numerically evaluatingthe Green’s function eigenvalues for the disk. By integrating the function (ζ 2 +z2)−1 cot( piN (ζ − k))over the contour enclosing [−R,R]× [−R,R], and then takingthe limit R→ ∞, we obtainS(z;N,k) :=∞∑n=−∞1(nN+ k)2+ z2=pi2Nz[coth(piN(z+ ik))+coth(piN(z− ik))]. (B.8)214B.2 Derivation of Membrane Green’s Function for thePerturbed DiskIn this appendix we provide the details for calculating the leading-order correctionto the perturbed disk Green’s function given in (2.2). Recall that the bulk Green’sfunction solvesDb∆GλΩ−µ2bλGλΩ = 0, in Ωδ ,Db∂nGλΩ+KGλΩ = δ∂Ωδ (x− x˜), on ∂Ωδ .(B.9)On the boundary r = R+δh(θ) of the perturbed disk we calculate in terms of polarcoordinates thatnˆ(θ) =[1+( δh′(θ)R+δh(θ))2]− 12 (eˆr− δh′(θ)R+δh(θ) eˆθ), ∇= eˆr∂r + 1r eˆθ∂θ ,δ∂Ωδ (x− x˜) =[1+( δh′(θ)R+δh(θ))2]− 12 δ (θ − θ˜)R+δh(θ),which yields the following asymptotic behaviour as δ → 0:nˆ(θ)∼ eˆr−δ h′(θ)Reˆθ +O(δ 2),δ∂Ωδ (x− x˜)∼1Rδ (θ − θ˜)−δ h(θ)R2δ (θ − θ˜)+O(δ 2).Next, for δ  1, we seek a solution of the formGλΩ(r,θ , θ˜ ∼ GλΩ0(r,θ , θ˜)+GλΩ1(r,θ , θ˜)δ +O(δ 2).Upon substituting these expansions into (B.9), and collecting powers of δ , we ob-tain the following zeroth-order and first-order problems:Db∆GλΩ0−µ2bλGλΩ0 = 0, in Ω0,B0GλΩ0 =δ (θ − θ˜)R, on ∂Ω0,215andDb∆GλΩ1−µ2bλGλΩ1 = 0, in Ω0,B0GλΩ1 =−h(θ)Rδ (θ − θ˜)R−B1GλΩ0, on ∂Ω0,where the boundary operatorsB0 andB1 are defined byB0 ≡ Db∂r +K B1 ≡ Db(h(θ)∂ 2r −h′(θ)R2∂θ)+Kh(θ)∂r.The zeroth-order solution is the unperturbed disk bulk Green’s function given in(B.6). For the problem for the leading order correction, we use linearity to decom-pose its solution in the formGλΩ1(r,θ , θ˜) =−h(θ˜)RGλΩ0(r,θ , θ˜)+ G˜λΩ1(r,θ , θ˜),G˜λΩ1(r,θ , θ˜) =12piR∞∑n=−∞a˜λ1n(θ˜)Pn(r)einθ ,(B.10)for some coefficients a˜λ1n to be found. To determine an expression for these coeffi-cients, we first multiply the boundary conditionB0G˜λΩ1 =−B1GλΩ0 by e−inθ , andthen integrate from 0 to 2pi . This gives1R(DbP′n(R)+K)a˜λ1n(θ˜) =−ˆ 2pi0e−inθB1GλΩ0 dθ . (B.11)Then, by using the differential equation satisfied by GλΩ0 we calculate the right-hand side of this expression asˆ 2pi0e−inθB1GλΩ0(R,θ , θ˜)dθ =Dbˆ 2pi0h(θ)GλΩ0rr(R,θ , θ˜)e−inθ dθ− DbR2ˆ 2pi0h′(θ)GλΩ0θ (R,θ , θ˜)e−inθ dθ+Kˆ 2pi0h(θ)GλΩ0r(R,θ , θ˜)e−inθ dθ .(B.12)Next, we assume that the boundary perturbation h(θ) is sufficiently smooth so that216each of the following hold:h(θ) =∞∑n=−∞hneinθ , h′(θ) = i∞∑n=−∞nhneinθ , h′′(θ) =−∞∑n=−∞n2hneinθ . (B.13)This allows us to calculate the individual terms on the right-hand side of (B.12) asˆ 2pi0h(θ)GλΩ0rr(R,θ , θ˜)e−inθdθ =1R∞∑k=−∞P′′k (R)aλk hn−ke−ikθ˜ ,ˆ 2pi0h′(θ)GλΩ0θ (R,θ , θ˜)e−inθdθ =− 1R∞∑k=−∞k(n− k)aλk hn−ke−ikθ˜ ,ˆ 2pi0h(θ)GλΩ0r(R,θ , θ˜)e−inθdθ =1R∞∑k=−∞P′k(R)aλk hn−ke−ikθ˜ ,where aλk are the Fourier coefficients of the leading-order Green’s function, as de-fined in (B.6). By substituting these relations into (B.12), and then using (B.11),we determine the coefficients asa˜λ1n(θ˜) =∞∑k=−∞aˆλn,kaλk hn−ke−ikθ˜ , aˆλn,k ≡−DbP′′k (R)+KP′k(R)+DbR2 k(n− k)DbP′n(R)+K. (B.14)In (B.14), to calculate various derivatives of Pn(R), as defined in (B.6), we makerepeated use of the identityI′n(z) =nzIn(z)+ In+1(z),to readily derive thatP′n(R) =|n|R+ωbλI|n+1|(ωbλR)I|n|(ωbλR),P′′n (R) =|n|(|n|−1)R2+2|n|+1RωbλI|n+1|(ωbλR)I|n|(ωbλR)+ω2bλI|n+2|(ωbλR)I|n|(ωbλR).This completes the derivation of the leading-order correction for the bulk Green’sfunction, defined in (B.10).Next, we derive a two-term approximation for the membrane Green’s function217problem on the perturbed disk. This Green’s function satisfiesDv∂ 2σGλ∂Ω(σ ,σ0)−µ2sλGλ∂Ω(σ ,σ0)+K2ˆ |∂Ωδ |0GλΩ(σ , σ˜)Gλ∂Ω(σ˜ ,σ0)dσ˜ =−δ (σ −σ0),(B.15)for 0≤ σ < |∂Ωδ |. Repeated use of the chain rule to the arc-length formulaσ(θ) =ˆ θ0(R+δh(ϑ))√1+(δh′(ϑ)R+δh(ϑ))2dϑ ,gives∂ 2σ =1(R+δh(θ))2+(δh′(θ))2∂ 2θ −δh′(θ)R+δh(θ)+δh′′(θ)[(R+δh(θ))2+(δh′(θ))2]2∂θ .Multiplying the membrane equation through by (R+δh(θ))2+(δh′(θ))2, writingDv = Dv0(1+ Dv1Dv0 δ), and then dividing through by R2(1+ Dv1Dv0 δ), we obtain theperturbed problemDv0R2 ∂2θGλ∂Ω(θ ,θ0)− Dv0R2 δh′(θ)R+δ [h(θ)+h′′(θ)](R+δh(θ))2+(δh′(θ))2 ∂θGλ∂Ω(θ ,θ0)−µ2sλR2(R+δh(θ))2+(δh′(θ))21+Dv1Dv0δGλ∂Ω(θ ,θ0)+ K2R2(R+δh(θ))2+(δh′(θ))21+Dv1Dv0δˆ 2pi0(GλΩ0(R,θ , θ˜)+δGλΩ1(R,θ , θ˜)+δh(θ)GλΩ0r(R,θ , θ˜))×Gλ∂Ω(θ˜ ,θ0)√(R+δh(θ˜))2+(δh′(θ˜))2 dθ˜=− 1R2√(R+δh(θ))2+(δh′(θ))21+Dv1Dv0δδ (θ −θ0).To determine a two-term asymptotic solution to this problem, we expand the mem-brane Green’s function asGλ∂Ω(θ ,θ0)∼ Gλ∂Ω0(θ ,θ0)+δGλ∂Ω1(θ ,θ0)+O(δ 2).Upon substituting this expansion into the perturbed problem, and collecting powers218of δ , we obtain the following zeroth-order and first-order problems:M0Gλ∂Ω0(θ ,θ0) =− 1Rδ (θ −θ0),M0Gλ∂Ω1(θ ,θ0) =−(h(θ)R − Dv1Dv0) 1Rδ (θ −θ0)−M1Gλ∂Ω0(θ ,θ0).Here we have defined the unperturbed membrane operatorM0 byM0ψ(θ ,θ0)≡ Dv0R2 ∂ 2θψ(θ ,θ0)−µ2sλψ(θ ,θ0)+K2ˆ 2pi0GλΩ0(R,θ , θ˜)ψ(θ˜ ,θ0)Rdθ˜ ,and its leading-order correctionM1 byM1ψ(θ ,θ0)≡− Dv0R3 h′(θ)∂θψ(θ ,θ0)−µ2sλ(2h(θ)R − Dv1Dv0)ψ(θ ,θ0)+K2(2h(θ)R − Dv1Dv0)ˆ 2pi0GλΩ0(R,θ , θ˜)ψ(θ˜ ,θ0)Rdθ˜+K2ˆ 2pi0GλΩ1(R,θ , θ˜)ψ(θ˜ ,θ0)Rdθ˜+K2h(θ)ˆ 2pi0GλΩ0r(R,θ , θ˜)ψ(θ˜ ,θ0)Rdθ˜+K2ˆ 2pi0GλΩ0(R,θ , θ˜)ψ(θ˜ ,θ0)h(θ˜)dθ˜ .(B.16)The zeroth-order solution is that of the unperturbed disk and is given by (B.7).By linearity, we then seek the solution for the leading order correction in the formGλ∂Ω1(θ ,θ0) =(h(θ0)R − Dv1Dv0)Gλ∂Ω0(θ ,θ0)+ G˜λ∂Ω1(θ ,θ0), (B.17)where G˜λ∂Ω1(θ ,φ) now satisfiesM0G˜λ∂Ω1(θ ,θ0) =−M1Gλ∂Ω0(θ ,θ0).We will represent the solution G˜λ∂Ω1 in terms of a Fourier series asG˜λ∂Ω1(θ ,θ0) =12piR∞∑n=−∞g˜λ1n(θ0)einθ , (B.18)for some coefficients g˜λ1n(θ0) to be found. Similar to the calculation provided above219for the perturbed bulk Green’s function, we obtain thatg˜λ1n(θ0) = Rgλ0nˆ 2pi0e−inθM1Gλ∂Ω0(θ ,θ0)dθ . (B.19)By using (B.16) we calculate the right-hand side of this expression asˆ 2pi0e−inθM1Gλ∂Ω0(θ ,θ0)dθ =−Dv0R3 J1n(θ0)−2µ2sλR J2n(θ0)+µ2sλDv1Dv0J3n(θ0) (B.20)+2K2J4n(θ0)− K2RDv1Dv0J5n(θ0)+K2RJ6n(θ0)+K2RJ7n(θ0), (B.21)where the various integrals J1n, . . . ,J7n are defined byJ1n(θ0) =ˆ 2pi0h′(θ)Gλ∂Ω0θ (θ ,θ0)e−inθdθ ,J2n(θ0) =ˆ 2pi0h(θ)G∂Ω0(θ ,θ0)e−inθdθ ,J3n(θ0) =ˆ 2pi0Gλ∂Ω0(θ ,θ0)e−inθdθ ,J4n(θ0) =ˆ 2pi0ˆ 2pi0h(θ)GλΩ0(R,θ , θ˜)Gλ∂Ω0(θ˜ ,θ0)e−inθdθ˜dθ ,J5n(θ0) =ˆ 2pi0ˆ 2pi0GλΩ0(R,θ , θ˜)Gλ∂Ω0(θ˜ ,θ0)e−inθdθ˜dθ ,J6n(θ0) =ˆ 2pi0ˆ 2pi0G˜λΩ1(R,θ , θ˜)Gλ∂Ω0(θ˜ ,θ0)e−inθdθ˜dθ ,J7n(θ0) =ˆ 2pi0ˆ 2pi0h(θ)GλΩ0r(R,θ , θ˜)Gλ∂Ω0(θ˜ ,θ0)e−inθdθ˜dθ .By using the Fourier series representations for the leading-order bulk and mem-brane Green’s functions given in (B.6) and (B.7), respectively, together with (B.13)220for h(θ), we calculate explicitly thatJ1n(θ0) =− 1R∞∑k=−∞k(n− k)hn−kgλk e−ikθ0 ,J2n(θ0) =1R∞∑k=−∞hn−kgλk e−ikθ0 ,J3n(θ0) =1Rgλn e−inθ0 ,J4n(θ0) =1R2∞∑k=−∞hn−kaλk gλk e−ikθ0 ,J5n(θ0) =1R2aλn gλn e−inθ0 ,J6n(θ0) =1R2∞∑k=−∞hn−kaˆλn,kaλk gλk e−ikθ0 ,J7n(θ0) =1R2∞∑k=−∞hn−kP′k(R)aλk gλk e−ikθ0 .Upon substituting these expressions into (B.20), and then recalling (B.19), we con-clude thatg˜λ1n(θ0) =gλn∞∑k=−∞{Dv0R3 k(n− k)−2µ2sλR +2K2R aλk +K2aˆλn,kaλk +K2P′k(R)aλk}hn−kgλk e−ikθ0+ Dv1Dv0 gλn(µ2sλ −2piK2Raλn)gλn e−inθ0 ,where the coefficients aλk are defined in (B.6). We can use the definition of thecoefficients gλn , as given in (B.7), to write µ2sλ −K2aλn = 1gλn −Dv0R2 n2. In this way,we getg˜λ1n(θ0) =∞∑k=−∞gˆλn,khn−kgλk e−ikθ0gλn +(Dv1Dv0− 2h(θ0)R)gλn e−inθ0− Dv1R2 n2(gλn )2e−inθ0 ,wheregˆλn,k =Dv0R3 k(n+ k)+K2aλk(aˆλn,k +P′k(R)).Finally, from (B.17) and (B.18), we conclude that the first order correction for the221membrane Green’s function isGλ∂Ω1(θ ,θ0) =−h(θ0)R Gλ∂Ω0(θ ,θ0)+ 12piR∞∑n=−∞∞∑k=−∞gˆλn,khn−kgλk gλn einθ−ikθ0− Dv12piR3∞∑n=−∞n2(gλn )2ein(θ−θ0).222Appendix CAppendix for Chapter 3C.1 Derivation of Lemma 3.3.1Denote by I the left hand side of (3.66) and letRy be any rotation matrix such thatRyy = ez ≡ (0,0,1)T . Since f (x,z) = f (Ryx,Ryz) and |y− x|= |Ryy−Ryx| weimmediately getRyI =ˆ∂Ωez−Ryx|Ryy−Ryx|2 f (Ryx,Ryz)dAx =ˆ∂Ωez− x|ez− x|2 f (x,Ryz)dAx.Since |ez− x|2 = 2(1− eTz x) for all x ∈ ∂Ω the z-component of the left-hand sideabove is given byeTz (RyI) =12ˆ∂Ωf (x,Ryz)dAx =12f0(|z|) (C.1)To compute the x and y components of the left-hand-side we use spherical coordi-nates and find thateTx (RyI) = ReJ, eTy (RyI) =−ImJ, (C.2a)whereJ =−12ˆ pi0ˆ 2pi0e−iϕ sinθ1− cosθ f (x,Ryz)sinθdθdϕ (C.2b)223and ex = (1,0,0)T , ey = (0,1,0)T , and we have used the notationx = (sinθ cosϕ,sinθ sinϕ,cosθ)T , Ryz = |z|(sin θ˜ cos ϕ˜,sin θ˜ sin ϕ˜,cos θ˜)T .The summation formulaPl(xTi x j) =l∑m=−l(l−m)!(l+m)!Pml (cosθi)Pml (cosθ j)eim(ϕi−ϕ j), (C.3)then implies thatJ =− 18pi∞∑l=0l∑m=−l(l−m)!(l+m)!flPml (cos θ˜)e−imϕ˜×ˆ pi0ˆ 2pi0sin2 θ1− cosθ Pml (cosθ)ei(m−1)ϕdθdϕ,of which only the m = 1 term is nonzero. The identityˆ 1−1√1+ x1− xP1l (x)dx =−2,obtained using integration by parts as well as P1l (x) =−√1− x2P′l (x) and Pl(1) = 1then gives the seriesJ =12∞∑l=1(l−m)!(l+m)!flP1l (cos θ˜)e−iϕ˜ . (C.4)Combining (C.1), (C.2), and (C.4), together withcos θ˜ = eTzRyz|z| = (RTy ez)T z|z| = yT z|z| .andRTycos ϕ˜sin ϕ˜0= 1sin θ˜RTy(Ryz|z| − cos θ˜ez)=I3− yyT√1− (yT z/|z|)2z|z| ,224then gives (3.66).C.2 Sign of Dynamic TermsIn this appendix we derive the inequalities (3.68) and (3.69b). To prove (3.68) wefirst calculateddz(√1− z2∞∑l=1gll(l+1)P1l (z))=∞∑l=1glPl(z) = 4pi(gmr(z)− 14pi g0), (C.5)wheregmr(z)≡ Gmr((√1− z2,0,z)T ,ez) (C.6)and where the first equality was obtained usingP1l (z) =−√1− z2P′l (z), (C.7)andddz[(1− z2)dPldz]+ l(l+1)Pl(z) = 0, (C.8)while the second equality was obtained using (3.27b). Integrating (C.5) from −1to z and using P1l (−1) = 0 for all l ≥ 1 we obtain∞∑l=1gll(l+1)P1l (z) =4pi√1− z2ˆ z−1(gmr(ξ )− 12ˆ 1−1gmr(ζ )dζ)dξ ,from which the inequality (3.68) follows by noting that gmr(z) is positive andmonotone increasing in −1 < z < 1. The sign of C(z) is similarly found by calcu-latingddz(√1− z2C(z))= 1− f1− fKDwg0+4pi f1− fKDwgmr(z)> 0,where the inequality follows by noting that gmr > 0 together with g0 =Dw/(1−K)and the constraint (3.7) on f and K. The inequality (3.69b) then follows by notingthat√1− z2C(z) = 0 at z=−1. Note in addition that C(−1) = 0 since P1l (−1) = 0for all l ≥ 1.225C.3 Linear Stability of the Common Angle SolutionIn this appendix we consider the linear stability with respect to the slow dynamicsof the common angle solution in Section 3.4.2 when the fuel source is given by(3.71). When N = 2 the 2× 2 Green’s matrix G is symmetric and of constantrow-sum. It therefore admits the eigenvectors p± = (1,±1)T with correspondingeigenvalues µ± respectively. Additionally the Green’s matrix G depends on thespot locations x1 and x2 only through the quantityξ ≡ xT1 x2 = sinθ1 sinθ2 cos(ϕ1−ϕ2)+ cosθ1 cosθ2, (C.9)for which we note that when ϕ2−ϕ1 = pi and θ1 = θ2 = θc∂ξ∂ϕ1=− ∂ξ∂ϕ2= 0,∂ξ∂θ1=∂ξ∂θ2=−sin2θc. (C.10)SubstitutingS =S1+S22p1+S1−S22p2, (C.11)into the NAS (3.43b) and left-multiplying by pT1 and pT2 givesS1+S2−2Sc = 0, (C.12)(ν−1+2piµ−)(S1−S2)+χ(S1, f )−χ(S2, f )− vE(x1)− vE(x2)Dw√Dv= 0. (C.13)Differentiating the first equation with respect to any parameter z ∈ {ϕ1,ϕ2,θ1,θ2}we find that∂S2∂ z=−∂S1∂ z.On the other hand, differentiating the second equation with respect to z and assum-ing thatA2(S1,S2)≡ ν−1+2piµ−+ χ′(S1, f )+χ ′(S2, f )26= 0, (C.14)we obtain∂S1∂ z=∂vE (x1)∂ z − ∂vE (x2)∂ z2Dw√DvA2(S1,S2)−pi S1−S2A2(S1,S2)∂µ−∂ z. (C.15)226Equation (3.73a) implies that vE(x1) and vE(x2) are functions only of cosθ1 andcosθ2 respectively. Therefore when ϕ2−ϕ1 = pi , θ1 = θ2 = θc, and S1 = S2 = Scwe obtain∂Si∂ϕ j= 0, (C.16a)for all i, j ∈ {1,2}, and∂S1∂θ1=−∂S2∂θ1=12Dw√DvA?(Sc)∂vE(x1)∂θ1∂S2∂θ2=−∂S1∂θ2=12Dw√DvA?(Sc)∂vE(x2)∂θ2,whereA?(Sc) is given by (3.56) and for which we note µ? = µ− when N = 2. Notethat from (3.73a), (C.7), and (3.73d) we calculate∂vE(x)∂θ=E04pi∞∑l=1glη ll(l+1)(−sinθP′l (cosθ)) =E04piCE(cosθ), (C.17)which vanishes at θ = 0,pi and is strictly negative otherwise. In particular wededuce that∂S1∂θ1=−∂S2∂θ1=∂S2∂θ2=−∂S1∂θ2=E0CE(cosθc)8piDw√DvA?(Sc). (C.18)Using (C.16a) and (C.18) together with (3.81) we calculate the Jacobian matri-ces of the dynamics (3.77) evaluated at the common angle solution∂ (F1,F2)∂ (θ1,θ2)=(0 00 0),∂ (G1,G2)∂ (ϕ1,ϕ2)G =(0 00 0), (C.19)as well as∂ (F1,F2)∂ (ϕ1,ϕ1)=ScC(cos(2θc))sinθc2sin(2θc)(−1 11 −1), (C.20)227and∂ (G1,G2)∂ (θ1,θ2)=(1− K1− f )ScCE(cosθc)C(cos2θc)4DwA?(Sc)(−1 11 −1)− ScC′(cos2θc)sin2θc2(1 11 1)−(1− K1− f )ScC′E(cosθc)sinθcDw(1 00 1).(C.21)In particular the Jacobian ∂ (F ,G)/∂ (ϕ ,θ ) is block diagonal and it therefore suf-fices to consider the stability with respect to ϕ and θ separately. Sincedet(∂ (F1,F2)∂ (ϕ1,ϕ1))= 0, tr(∂ (F1,F2)∂ (ϕ1,ϕ1))=−ScC(cos(2θc))sinθcsin(2θc)< 0, (C.22)we deduce that ∂ (F1,F2)/∂ (ϕ1,ϕ1) has one neutral zero eigenvalue correspond-ing to rotational invariance and one negative eigenvalue. In particular the sta-bility of the common angle solution is determined solely by the eigenvalues of∂ (G1,G2)/∂ (θ1,θ1)which we calculate explicitly by noting that (1,1)T and (1,−1)Tare its eigenvectors with corresponding eigenvaluesd+ =− E04pi√Dv(C′(cos2θc)sin2θc1− K1− f+C′E(cosθc)sinθcDw), (C.23a)d− =− E04piDw√Dv(C(cos2θc)CE(cosθc)2A?(Sc)+C′E(cosθc)sinθc), (C.23b)where we have used (3.75) for Sc. Note that d+ corresponds to perturbations whereθ1 and θ2 both increase or decrease synchronously. As a result the numerically sup-ported observation that Gi(θ ,θ)≶ 0 when θ ≷ θc for i = 1,2 implies that d+ < 0.Therefore the common angle solution can only be destabilized by increasing onepolar angle and decreasing the other. Specifically, the common angle solution islinearly unstable with respect to a tilt instability when d− > 0 and is stable oth-erwise. The corresponding tilt instability threshold is obtained by setting d− = 0which we note is equivalent to (3.88).228Appendix DAppendix for Chapter 4D.1 Large λI Asymptotics ofFy0(iλI)In this appendix we determine some key properties of Fy0(λ ) defined in (4.36).Recalling (4.37), in Figure D.1a we plot Fy0(0) versus y0 ≥ 0. Next we calculatethe limiting behaviour of Fy0(iλI) as λI → ∞. First, we let (Ly0 − iλI)−1wc(y+y0)2 =ΦR+ iΦI where ΦR and ΦI solveLy0ΦR+λIΦI = wc(y+ y0)2, Ly0ΦI−λIΦR = 0, (D.1)with the boundary conditions Φ′R(0) = Φ′I(0) and ΦR,ΦL → 0 as y→ ∞. TakingλI  1 and assuming that y = O(1) we obtainΦI(y)∼ 1λI wc(y+y0)2, ΦR(y)∼ 1λILy0ΦI =1λ 2I(2w′c(y+y0)2+wc(y+y0)2).If y0 > 0 then Φ′R(0) = 0 and Φ′I(0) = 0 are not satisfied and we must thereforeconsider the boundary layer at y = 0. Setting z = λ 1/2I y we consider the innerexpansion ΦR ∼ Φ˜R(z) and ΦI ∼ Φ˜I(z) where Φ˜I satisfiesd4Φ˜Idz4+ Φ˜I =1λIwc(y0)2 z > 0;dΦ˜Idz=d3Φ˜Idz3= 0, z = 0,2290 1 2 3y0432101234y0(0) versus y0(a)0 10 20 30I0.0100.0050.0000.0050.0100.0150.020Re y0(i I)y000.81.32.0(b)0 2 4 6 8 10I0.00.20.40.60.81.0Im y0(i I)y000.81.32.0(c)Figure D.1: (a) Plot ofFy0(0) versus the shift parameter y0. (b) and (c) Realand imaginary parts ofFy0(iλI) for select values of y0 ≥ 0. The dashedlines indicate the λI  1 asymptotics.and must be matched to the outer, y = O(1), solution through the far-field be-haviourΦ˜I ∼ 1λI wc(y0)2,d2Φ˜Idz2∼ 1λ 2I(2w′c(y+ y0)2+wc(y+ y0)2), z→ ∞.It is clear that the leading order solution is Φ˜I(z) ∼ λ−1I wc(y0)2. The constantbehaviour of ΦI at the boundary layer therefore does not contribute to the leadingorder behaviour of the integralˆ ∞0wc(y+ y0)ΦI(y)dy∼ λ−1Iˆ ∞0wc(y+ y0)3dy, λI  1.Moreover, multiplying the right equation in (D.1) by wc(y+ y0) and integratingwe calculateˆ ∞0wc(y+ y0)ΦR(y)dy =1λI(w′c(y0)ΦI(0)+ˆ ∞0ΦILy0wc(y+ y0)dy)∼ 1λ 2I(w′c(y0)wc(y0)2+ˆ ∞0wc(y+ y0)4dy),230for λI 1 where we have usedLy0wc(y+y0) =wc(y+y0)2. In summary, we havethe large λI asymptoticsFy0(iλI)∼1λ 2Iw′c(y0)wc(y0)2+´ ∞0 wc(y+ y0)4dy´ ∞0 wc(y+ y0)2dy+iλI´ ∞0 wc(y+ y0)3dy´ ∞0 wc(y+ y0)2dy, λI 1.(D.2)Note that the real part changes from positive to negative as y0 exceeds y0 ≈ 1.0487.In Figures D.1b and D.1c we plot the real and imaginary parts ofFy0(iλI) respec-tively for select values of y0. In addition, we have included the large λI asymptoticswhich indicate close agreement for moderately large values of λI .D.2 Numerical Support for Stability ConjectureIn this appendix we provide numerical support for the conjecture that the shiftedNLEP (4.44) has a stable spectrum when µ > µc(y0) by numerically calculating thedominant eigenvalue of the NLEP for 0≤ y0 ≤ 1.5 and 0≤ µ ≤ 10. The numericalcalculation of the spectrum was performed by truncating the domain 0 < y < ∞to 0 < y < 20 and discretizing it with 600 uniformly distributed points. Then, weused a finite difference approximation for the second derivatives and a trapezoidalrule discretization for the integral term to approximate the NLEP (4.44) with adiscrete matrix eigenvalue problem. We then numerically calculated the dominanteigenvalue of matrix by using the eig function in the Python scipy.linalg libraryfor our numerical computation of the dominant eigenvalue. In Figure D.2a weplot Reλ0 versus y0 and µ . We observe the real part of the dominant eigenvalueis negative when µ exceeds the threshold µc(y0). Additionally, in Figure D.2bwe plot Λ0−Re(λ0) for the same range of y0 and µ values. We observe that thisdifference is non-negative which suggest that Reλ0 ≤ Λ0.D.3 Stability of Asymmetric Two-Boundary SpikePattern when A = 0Previous results on the stability of asymmetric two spike equilibrium solutions of(4.2) when A = 0 have focused exclusively on interior multi-spike solutions [104].To compare the A = B = 0 theory with our results obtained in Examples 2-4 we231(a) (b)Figure D.2: (a) Plot of the real part of the dominant eigenvalue of the shiftedNLEP (4.44) versus shift parameter y0 and multiplier µ . The dotted redline corresponds to the critical threshold µc defined in (4.56) and thesolid dark line is the zero-contour of Reλ0. (b) Plot of the differencebetween dominant eigenvalues ofLy0 and the NLEP (4.44).include here a summary of the stability of an asymmetric two-spike solution whereone spike concentrates at x = 0 and the other concentrates at either x = 1 or in theinterior 0 < x < 1. In both cases the NLEP (4.30) with A = 0 can be written asL0φ −2wc(y)2´ ∞0 wc(y)Eφ dy´ ∞0 wc(y)2dy= λφ , (D.3)where for two boundary spikes we letE = Ebb ≡(cothω0 tanhω0l cschω0 tanhω0(1− l)cschω0 tanhω0l cothω0 tanhω0(1− l)), (D.4)and in the case of one boundary and one interior spike we letE = Ebi ≡(cothω0 tanhω0l 2cschω0 sinhω0 1−l2cschω0 tanhω0l coshω0 1−l2 2cschω0 coshω01+l2 sinhω01−l2).(D.5)It is then straightforward to verify that σ = 1 is an eigenvalue of both matrices Ebb2320.0 0.2 0.4 0.6 0.8 1.0l0.00.20.40.60.81.01.2NLEP Multiplier for Two-Spike Pattern2det bb2det bi(a)0.00 0.05 0.10 0.15 0.20 0.25 0.30D0.00.20.40.60.81.0lAsymmetric Two-Spike Patternsboundary-boundaryboundary-interior(b)Figure D.3: (a) NLEP multipliers for a boundary-boundary and boundary-interior configuration. (b) Plot of l versus D determining asymmetricboundary-boundary and boundary-interior spike patterns. Solid (resp.)dashed lines indicate the two-spike pattern is linearly stable (resp. un-stable) with respect to competition instabilities.and Ebi. The remaining eigenvalue in each case is then given by the determinant.By diagonalizing E the NLEP (D.3) can therefore be written as (4.44) with µ = 2as well as with µ = 2detEbb and µ = 2detEbi for the boundary-boundary andboundary-interior cases respectively. Since the A = B = 0 stability theory impliesthat the NLEP (4.44) is stable if and only if µ > 1 [107] we immediately deducethat the µ = 2 modes are stable in both the boundary-boundary and boundary-interior cases. To determine the stability of the remaining modes we explicitlycalculatedetEbb = tanhω0l tanhω0(1− l), detEbi = 2tanhω0l sinhω0 1−l2 sinhω01+l2sinhω0.(D.6)Finally, for the boundary-boundary and boundary-interior cases we solve (4.76)and (4.89) for D = D(l) respectively and the resulting values of 2detEbb and2detEbi for 0 < l < 1 are shown in Figure D.3a. In particular the asymmetricpattern with two boundary spikes is always linearly unstable, while the patternwith one boundary and one interior spike has a region of stability (with respect to233the O(1) eigenvalues). In Figure D.3b we plot l = l(D) (cf. Figures 4.7 and 4.10)for both two-spike configurations, indicating where the pattern is stable (solid line)and unstable (dashed line).234

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