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Oscillatory dynamics for PDE models coupling bulk diffusion and dynamically active compartments : theory,… Gou, Jia 2016

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Oscillatory Dynamics for PDE ModelsCoupling Bulk Diffusion andDynamically Active Compartments:Theory, Numerics and ApplicationsbyJia GouB.Sc., Beijing Normal University, China, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2016c© Jia Gou 2016AbstractWe formulate and investigate a relatively new modeling paradigm by whichspatially segregated dynamically active units communicate with each otherthrough a signaling molecule that diffuses in the bulk medium between ac-tive units. The modeling studies start with a simplified setting in a one-dimensional space, where two dynamically active compartments are locatedat boundaries of the domain and coupled through the feedback term to thelocal dynamics together with flux boundary conditions at the two ends. Forthe symmetric steady state solution, in-phase and anti-phase synchroniza-tions are found and Hopf bifurcation boundaries are studied using a windingnumber approach as well as parameter continuation methods of bifurcationtheory in the case of linear coupling. Numerical studies show the existenceof double Hopf points in the parameter space where center manifold andnormal form theory are used to reduce the dynamics into a system of am-plitude equations, which predicts the configurations of the Hopf bifurcationand stability of the two modes near the double Hopf point. The system witha periodic chain of cells is studied using Floquet theory. For the case of asingle active membrane bound component, rigorous spectral results for theonset of oscillatory dynamics are obtained and in the finite domain case, aweakly nonlinear theory is developed to predict the local branching behaviornear the Hopf bifurcation point. A previously developed model by Gomez etal.[23] is analyzed in detail, where the phase diagrams and the Hopf frequen-cies at onset are provided analytically with slow-fast type of local kinetics.A coupled cell-bulk system, with small signaling compartments, is also stud-ied in the case of a two-dimensional bounded domain using the method ofasymptotic expansions. In the very large diffusion limit we reduce the PDEcell-bulk system to a finite dimensional dynamical system, which is studiedboth analytically and numerically. When the diffusion rate is not very large,we show the effect of spatial distribution of cells and find the dependence ofthe quorum sensing threshold on influx rate.iiPrefaceThis thesis is an original work of the author, Jia Gou. This research projectwas originally initiated under the supervision of Dr. Yuexian Li and laterunder the supervision of Dr. Michael Ward all the way to the completion ofthe thesis.The work in chapter 2 has been submitted for publication along withmy supervisor Dr. Michael Ward, Dr. Yuexian Li and co-authors Dr. Pik-Yin Lai, Wei-Yin Chiang[24]. Dr. Yuexian Li was involved in the stageof model formulation and gave useful suggestions on design of biologicalmodels. Dr. Pik-Yin Lai provided helpful comments on numerical sim-ulations and manuscript edits. I conducted all computational work andderivations. Dr. Ward was the supervisory author on this project and wasinvolved throughout this project in project design, concept formation andmanuscript revision.A version of chapter 3 has been submitted for publication with Dr.Wayne Nagata and Dr. Yuexian Li[25]. I conducted all computationalwork. Dr. Nagata was the supervisory author on this project and wasinvolved throughout this project in conception and manuscript edits.A version of chapter 4 is published in SIAM Journal on Applied Dy-namical Systems along with Dr. Michael Ward, Dr. Wayne Nagata and Dr.Yuexian Li[26]. Dr. Nagata provided valuable suggestions in the developingof the weakly nonlinear theory. I conducted all computational work. Dr.Ward was the supervisory author on this project and was involved through-out this project in project design, concept formation and manuscript revi-sion.The materials in chapter 5 and chapter 6 have been submitted with Dr.Michael Ward[27, 28]. I conducted all computational work. Dr. Ward wasthe supervisory author on this project and was involved throughout thisproject in project design, concept formation and manuscript revision.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . 11.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Two Coupled Sel’kov Oscillators . . . . . . . . . . . . . . . . 72.1 Formulation of the Coupled Compartment-Bulk Model . . . 82.2 Linear Coupling Between the Compartments and the Bulk . 102.2.1 Linear Stability Analysis of the Steady State . . . . . 112.2.2 The Winding Number Analysis . . . . . . . . . . . . 172.3 A Periodic Chain of Active Units Coupled by Bulk Diffusion 212.3.1 The Steady-State Solution . . . . . . . . . . . . . . . 232.3.2 The Linear Stability Analysis . . . . . . . . . . . . . 242.3.3 Hopf Bifurcation Boundaries, Global Branches andNumerics . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.4 Large D Analysis for the Hopf Bifurcation Boundaries 292.4 Nonlinear Coupling Between Compartments and Bulk . . . . 342.4.1 Compartmental Dynamics Neglecting Bulk Diffusion 373 Nonlinear Analysis Near the Double Hopf Bifurcation Point 403.1 The Coupled Compartment-Bulk Diffusion Model . . . . . . 413.2 Linearized Stability . . . . . . . . . . . . . . . . . . . . . . . 42ivTable of Contents3.3 Double Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . 484 Generalized Model in One Dimensional Space . . . . . . . 584.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . 614.2 The Steady-State Solution and the Formulation of the LinearStability Problem . . . . . . . . . . . . . . . . . . . . . . . . 614.3 One-Component Membrane Dynamics . . . . . . . . . . . . . 654.3.1 Theoretical Results for a Hopf Bifurcation: The Infinite-Line Problem . . . . . . . . . . . . . . . . . . . . . . 664.3.2 A Finite Domain: Numerical Computations of theWinding Number . . . . . . . . . . . . . . . . . . . . 744.4 Examples of the Theory: One-Component Membrane Dy-namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.1 A Class of Feedback Models . . . . . . . . . . . . . . 764.4.2 A Phase Diagram for an Explicitly Solvable Model . . 794.4.3 A Model of Kinase Activity Regulation . . . . . . . . 834.4.4 Two Biologically-Inspired Models . . . . . . . . . . . 844.5 Two-Component Membrane Dynamics: Extension of the Ba-sic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.6 Weakly Nonlinear Theory for Synchronous Oscillations . . . 944.6.1 Numerical Validation of the Weakly Nonlinear TheoryWith the Explicitly Solvable Model . . . . . . . . . . 1064.6.2 Numerical Validation of the Weakly Nonlinear TheoryWith the Dictyostelium Model . . . . . . . . . . . . . 1105 A Model of Bulk-Diffusion Coupled to Active MembranesWith Slow-Fast Kinetics . . . . . . . . . . . . . . . . . . . . . 1135.1 Coupled Membrane-bulk Model With Activator-Inhibitor Dy-namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 The Steady-State Solution and the Formulation of the LinearStability Problem . . . . . . . . . . . . . . . . . . . . . . . . 1165.2.1 Formulation of the Linear Stability Problem . . . . . 1195.3 One Diffusive Species in the Bulk . . . . . . . . . . . . . . . 1225.3.1 Stability Analysis for the → 0 Limiting Problem . . 1255.3.2 Stability Analysis for the  = O(1) Problem . . . . . 1395.4 Two Diffusive Species in the Bulk . . . . . . . . . . . . . . . 1426 Models in a Two-Dimensional Domain . . . . . . . . . . . . 1496.1 Formulation of a 2-D Coupled Cell-Bulk System . . . . . . . 1516.2 Analysis of the Dimensionless 2-D Cell-Bulk System . . . . . 155vTable of Contents6.2.1 The Steady-State Solution for the m Cells System . . 1576.2.2 Formulation of the Linear Stability Problem . . . . . 1616.3 The Distinguished Limit of D = O(ν−1) 1 . . . . . . . . . 1656.4 Examples of the Theory: Finite Domain With D = O(ν−1) . 1706.4.1 Example 1: m Cells; One Local Component . . . . . 1706.4.2 Example 2: m Cells; Two Local Components . . . . . 1726.5 Finite Domain: Reduction to ODEs for D  O(ν−1) . . . . 1836.5.1 Large D Theory: Analysis of Reduced Dynamics . . . 1886.6 The Effect of the Spatial Configuration of the Small Cells:The D = O(1) Regime . . . . . . . . . . . . . . . . . . . . . 1956.6.1 Example: The Sel’kov Model . . . . . . . . . . . . . . 2006.7 Infinite Domain: Two Identical Cells . . . . . . . . . . . . . 2126.7.1 The Steady-State Solution . . . . . . . . . . . . . . . 2146.7.2 Linear Stability Analysis . . . . . . . . . . . . . . . . 2167 Conclusion and Future Work . . . . . . . . . . . . . . . . . . 2217.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225AppendicesA Formulation of the PDE-ODE System for a Periodic Chain 232B An Alternative PDE-ODE Formulation for a Periodic Chain 234C Calculation of Normal Form Coefficients . . . . . . . . . . . 236D Two Specific Biological Models . . . . . . . . . . . . . . . . . 241D.1 The Dictyostelium Model . . . . . . . . . . . . . . . . . . . . 241D.2 The GnRH Model . . . . . . . . . . . . . . . . . . . . . . . . 242viList of Figures2.1 Schematic plot of the geometry for the coupled model in 1-D 102.2 Phase diagram with D and β . . . . . . . . . . . . . . . . . . 142.3 Bifurcation diagram of V with respect to D and β with linearcoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Full numerical simulation of (2.1.1) . . . . . . . . . . . . . . . 162.5 Trace of the counterclockwise contour . . . . . . . . . . . . . 172.6 Plot of the characteristic function . . . . . . . . . . . . . . . . 202.7 Schematic diagram of a periodic chain of four cells . . . . . . 222.8 Phase diagram showing Hopf bifurcation boundaries for thecase of three cells in the D versus κ plane . . . . . . . . . . . 272.9 Global bifurcation diagram and plot of F(iλI) . . . . . . . . . 282.10 Full numerical results showing in-phase and anti-phase syn-chronous oscillations . . . . . . . . . . . . . . . . . . . . . . . 302.11 Plot of p1p2 − p3 versus κ for the Routh-Hurwitz criterion . . 322.12 Bifurcation diagram with respect to D and β with nonlinearcoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.13 Numerical simulation of the coupled PDE-ODE system (2.1.1)with nonlinear coupling . . . . . . . . . . . . . . . . . . . . . 372.14 Bifurcation diagram of the ODE system (2.4.6) versus β . . . 383.1 Parameteric portrait in the (µ1, µ2) and the (β,D) plane . . 533.2 Bifurcation diagram with parameters near the double-Hopfpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Full numerical simulation shows unstable torus bifurcationand two frequencies . . . . . . . . . . . . . . . . . . . . . . . . 574.1 Phase diagram in the (κ, γ) plane . . . . . . . . . . . . . . . 804.2 Two typical bifurcation diagrams for u versus γ with differentvalue of κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 Full numerical simulations of the PDE-ODE system for (4.4.6)for the finite-domain problem shows synchronized oscillationsof the two membranes . . . . . . . . . . . . . . . . . . . . . . 82viiList of Figures4.4 Full numerical simulations of the PDE-ODE system for (4.4.6)for the finite-domain problem shows phase-locking of the twomembranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5 Numerical simulation on the GnRH model and the windingnumber critieron . . . . . . . . . . . . . . . . . . . . . . . . . 854.6 Numerical simulation on the Dict model and winding numbercriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.7 Bifurcation diagram with respect to D of the Dict model . . 884.8 Heterogeneous cells: bifurcation diagram and numerical sim-ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.9 Winding number computation indicates the synchronous andasynchronous modes . . . . . . . . . . . . . . . . . . . . . . . 944.10 Numerical simulation of the heterogeneous membranes showsdifferent behaviors . . . . . . . . . . . . . . . . . . . . . . . . 954.11 Bifurcation diagrams with respect to D for different values of γ1074.12 Comparison of bifurcation diagrams near a subcritical Hopfbifurcation point . . . . . . . . . . . . . . . . . . . . . . . . . 1084.13 Comparison of bifurcation diagrams near a supercritical Hopfbifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.14 Delayed Hopf bifurcation behavior . . . . . . . . . . . . . . . 1104.15 Comparison of numerical and theoretical calculated bifurca-tion diagram for the Dictyostelium model . . . . . . . . . . . 1125.1 Plot of nullclines of the isolated membrane model . . . . . . . 1155.2 Stability boundary with different value of  . . . . . . . . . . 1195.3 Winding number calculation of the membrane-bulk couplingmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4 Phase diagram in the (l1/l0, L/l0) plane . . . . . . . . . . . . 1315.5 Winding number calculation for τ = 200 and τ = 1 . . . . . . 1325.6 Spectrum near the boundary of different region in phase dia-gram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.7 Bifurcation diagram with respect to kv and numerical simu-lation near asymmetric equilibrium solution . . . . . . . . . . 1365.8 Bifurcation diagram with respect to kv and plot of periods . . 1375.9 Numerical simulation shows in-phase and anti-phase synchro-nization of two membranes . . . . . . . . . . . . . . . . . . . 1385.10 Phase diagram in the (l1/l0, L/l0) plane and bifurcation dia-gram for fixed L . . . . . . . . . . . . . . . . . . . . . . . . . 1405.11 Full numerical simulation of (5.3.1) shows unequal amplitudeof oscillations on two membranes . . . . . . . . . . . . . . . . 141viiiList of Figures5.12 Phase diagram on (l1/l0, L/l0) plane and bifurcation diagramfor fixed L with two diffusive molecule . . . . . . . . . . . . . 1445.13 Numerical simulation of (5.1.1) shows in-phase oscillation . . 1455.14 Numerical simulation shows two periods oscillation and cor-respond bifurcation diagram . . . . . . . . . . . . . . . . . . . 1466.1 Schematic diagram showing the model setup in 2D . . . . . . 1556.2 Hopf bifurcation boundaries for the Sel’kov model in d1 versusd2 plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.3 Hopf bifurcation boundaries for the Sel’kov model in τ versusD0 plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1786.4 Comparison of the Hopf bifurcation boundaries inD = O(ν−1)and D  O(ν−1 regime . . . . . . . . . . . . . . . . . . . . . 1806.5 Hopf bifurcation boundaries for the FN system in the d1 ver-sus d2 plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.6 Hopf bifurcation boundaries for the FN system in the τ versusD0 plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.7 Comparison of D = O(ν−1) and D  O(ν−1) regime for theHopf boundaries for the Sel’kov model; Numerical simulationof the reduced system . . . . . . . . . . . . . . . . . . . . . . 1916.8 Plot of u1, u2 and U0 versus time for the reduced system; Plotof u1 versus u2 for the uncoupled system . . . . . . . . . . . . 1926.9 Bifurcation diagram of u1 with respect to d2 for the reducedsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.10 Comparison of D = O(ν−1) and D  O(ν−1) regime for theHopf boundaries for the FN system; Numerical simulation ofthe reduced system . . . . . . . . . . . . . . . . . . . . . . . . 1946.11 Bifurcation diagram of u1 versus d1 . . . . . . . . . . . . . . . 1956.12 Schematic diagram showing five cells on a 2D ring . . . . . . 1966.13 Hopf bifurcation boundaries in the τ versus D plane for m =2, r0 = 0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2036.14 Hopf bifurcation boundaries for the synchronous mode andthe asynchronous mode in the τ versus D plane . . . . . . . . 2056.15 Hopf bifurcation boundaries in the τ versus D plane for thesynchronous mode in the case of three cells . . . . . . . . . . 2066.16 Hopf bifurcation boundaries in the τ versus D plane for thesynchronous mode for five cells . . . . . . . . . . . . . . . . . 2076.17 Hopf bifurcation boundaries for the two distinct asynchronousmodes for two radius r0 values . . . . . . . . . . . . . . . . . 2086.18 Global bifurcation diagram of u1e versus τ for the Sel’kov model210ixList of Figures6.19 Quorum sensing threshold as a function of d1 . . . . . . . . . 2116.20 Schematic plot of the geometry of two cells in the infinite plane212xAcknowledgementsForemost, I would like to express my deepest gratitude to my supervisor,Dr. Michael Ward, whose expert guidance, patience and continuous supporthelped me in all of time of studies and the completion of this thesis. Hispassion for Mathematics always inspires me and will keep encouraging mein my future career.I am grateful to my co-supervisor Dr. Yuexian Li, who introduced meto the field of Mathematical Modeling in Biology, helped me understandconcepts in this interdisciplinary field and construct the model system usedin Chapter 2.My sincere thanks goes to Dr. Wayne Nagata for his various formsof support during my graduate study. I benefit a lot from many valuablecomments from Wayne. I wound like to thank the Dr. Rachel Kuske andDr. Anthony Peirce for their insightful comments and encouragements. Iam also thankful to Dr. Leah Keshet, Dr. Daniel Coombs and Dr. EricCytrynbaum for their support.Thanks to my friends Vincent Zhai, Jing Dong and Tara Cai. Withouttheir precious support this would be an impossible task.Last but not the least, I would like to thank my family especially myparents and my brother for their love and support throughout all of theseyears.xiDedicationTo my mother and fatherxiiChapter 1Introduction1.1 Background and MotivationIndividuals in a large network communicate with each other to engage andcoordinate their activities. This happens at almost all levels of the livingworld ranging from a colony of unicellular amoebae to highly sophisticatedsocial networks of people. For instance, the synchronous rhythmic flashingof fireflies is revealed to be an critical component of the mating process ofadult fireflies [4]. In neuronal systems, synchrony between different regionsof the brain, communicating through synaptic connections, is thought to bethe basis of many cognitive activities [75]. Among a variety of communica-tion methods, a common scenario is where the communication is carried outthrough diffusive chemicals. Examples of such kind of systems range fromthe signalling of the amoebae Dictyostelium discoideum through the releaseof cAMP into the medium [19] where it diffuses and act on each individ-ual, to some endocrine neurons that secrete a hormone to the extracellularmedium where it influences the secretion of this hormone from a pool ofsuch neurons [35, 40], and to girls sharing a dormitory room getting theirperiods synchronized [51] presumably through the secretion of a pheromone[66, 72] in the shared space. Further examples where this kind of signallingoccurs are related to quorum sensing behavior (cf. [12], [54], [55], [53], [13]).In many of these systems, the individual cells or localized units, can, underappropriate conditions, exhibit sustained temporal oscillations. In this way,signalling through a diffusive chemical often can switch on and/or off theoscillations and to synchronize the oscillations among all the individuals.Biological rhythms are ubiquitous in living organisms, especially in mam-mals including human being, with periods ranging from seconds to years.Examples include cardiac and respiratory rhythms, which are crucial for themaintenance of normal function of life, and the ultradian rhythms, whichrefers to a rhythm with period much shorter than the circadian rhythm,observed in the blood level of most hormones in mammals including humanbeing, which often plays a fundamental role in their physiological function.The creation of those periodic phenomena involve chemical reactions and11.1. Background and Motivationcooperations of individuals which might not be directly connected.The modeling paradigm we are interested in and will be studied in thisthesis includes spatially segregated dynamically active compartments, likecells or membranes, and coupling among those local units through diffusivesignals. This coupling can induce periodicities to the local compartments,which otherwise would not be present. The biological phenomenon that ini-tially inspired our study is the pulsatile variation in the concentration ofgonadotropin-releasing hormone (GnRH) in the portal blood that circulatesfrom the hypothalamus to the pituitary gland. This periodic signal of aboutone pulse per hour has been shown to be crucial in maintaining the nor-mal reproductive activities in mammals [73]. In order to generate pulsatileGnRH signals observed in the portal blood, synchronization in the secretoryactivities among the hundreds to thousands GnRH neurons is essential. Asynchronization mechanism was proposed in [47], whereby neurons are cou-pled through GnRH secreted into the extracellular space. Results from thismodel were shown to be consistent with in vivo experiments. However thekey limitation of this model of [47] is that it assumed that extracellularspace was continuously stirred so as to average out any spatial effects re-sulting from any chemical secretions. In the real experimental settings, whenthe diffusion rate of chemicals in the extracellular space is small, it is un-avoidable to introduce spatial ingredients into the model system. In thisway, one should consider the diffusion of GnRH in the bulk, which coupleslocalized secretory activity of individual neurons.In addition to the wide variety of cellular examples, the coupling of localcompartments to bulk diffusion arises in many applications, such as surfacescience, the effect of catalyst particles, etc. It was shown numerically in[23] that a two-component membrane-bulk dynamics on a 1-D spatial do-main can trigger synchronous oscillatory dynamics in the two membranes.Models of the multistage adsorption of viral particles trafficking across bi-ological membranes are studied in (cf. [10]). In the modeling of catalyticreactions occurring on solid surfaces, it was shown in [38] that oscillationsin the surface kinetics are triggered by the effect of spatial bulk diffusionin the gas phase near the catalytic surface. The models of the effect of thecoupling of diffusion to localized chemical reactions are given in [71] and[64]. Other applications include the analysis of Turing patterns arising fromcoupled bulk and surface diffusion (cf. [44]). In the study of cellular signaltransduction, the survey [33] emphasizes the need for developing detailedmodels of cell signaling that are not strictly ODE based, but that, instead,involve spatial diffusion processes coupled with bio-chemical reactions oc-curing within localized signaling compartments. A related class of models,21.2. Thesis Outlinereferred to here as quasi-static models, consist of linear bulk diffusion fieldsthat are coupled solely through nonlinear fluxes defined at specific spatiallattice sites. Such systems arise in the modeling of signal cascades in cel-lular signal transduction (cf. [45], [11]), and in the study of the effect ofcatalyst particles and defects on chemically active substrates (cf. [59], [56]).In [56] it was shown numerically that one such quasi-static model exhibitsan intricate spatial-temporal dynamics consisting of a period-doubling routeto chaotic dynamics.1.2 Thesis OutlineMotivated by the prior studies, the goal of this thesis is to formulate a rel-atively new modeling paradigm by which spatially segregated dynamicallyactive units, such as cells or localized signalling compartments, communi-cate with each other through a signalling molecule that diffuses in the bulkmedium between the active units. We will give a detailed analysis of the pos-sibility of the triggering of synchronous oscillations for the coupled system.The outline of this thesis is as follows.In chapter §2, we construct and analyze a coupled compartment-bulkdiffusive model with a one-dimensional domain. The coupling between eachcompartment and the bulk is due to both feedback terms to the compart-mental dynamics and flux boundary conditions at the interface between thecompartment and the bulk. The coupled model consists of dynamically ac-tive compartments located at the two ends x = 0 and x = 2L of a 1-Dbulk region of spatial extent 2L. The two compartments are assumed to beidentical chemical conditional oscillators, which is a term used to refer toa dynamical system that stays at a stable steady state when isolated fromothers, but is capable of generating sustained oscillations with some otherparameter values. The local dynamics is modeled by Sel’kov kinetics, whichis originally used to model glycolytic oscillations that occur in yeast andmuscle cells. Glycolysis is the metabolic pathway that breaks down glucoseto provide the energy for cellular metabolism. However, we emphasize thatthe particular choice of the local kinetics is not essential. The signallingmolecule between the two compartments is assumed to undergo both dif-fusion, with diffusivity D, and constant bulk degradation. For the result-ing PDE-ODE system, we construct a symmetric steady-state solution andanalyze the stability of this solution to either synchronous(in-phase mode,where the two compartments oscillate at identical frequencies with no phasedifference) or asynchronous(anti-phase mode, where the two compartments31.2. Thesis Outlineoscillate at identical frequencies with a phase difference of half a period)perturbations about the midline x = L. The conditions for the onset ofoscillatory dynamics, as obtained from a linearization of the steady-statesolution, are studied using a winding number approach. Global branches ofeither in-phase or anti-phase periodic solutions, and their associated stabil-ity properties, are determined with numerical bifurcation and continuationmethods. For the case of a linear coupling between the compartments andthe bulk, with coupling strength β, a phase diagram showing the Hopf bi-furcation boundaries in the parameter space D versus β is constructed thatshows the existence of a rather wide parameter regime where stable syn-chronized oscillations can occur between the two compartments. It alsoshows that there are parameter regions where bistability occurs, where bothin-phase and anti-phase synchronizations exist and both are stable. In ad-dition, the double Hopf (or Hopf-Hopf) points, parameter values where theHopf bifurcations of the in-phase and anti-phase modes coincide, are foundin the model with certain parameter values, and will be studied in detailin chapter §3. By using a Floquet-based approach, the analysis with linearcoupling is then extended to determine Hopf bifurcation thresholds for aperiodic chain of evenly-spaced dynamically active units. For one particularcase of nonlinear coupling between the compartments and the bulk, stablein-phase or anti-phase oscillations are also shown to occur in certain param-eter regimes, but as isolated solution branches that are disconnected fromthe steady state solution branch.In chapter §3 we consider the double Hopf bifurcation point that is foundthrough numerical study of the model system in chapter §2 with linear cou-pling. We use a center manifold approach and normal form theory to reducethe local dynamics of the model system to a system of two amplitude equa-tions, which determines the patterns of Hopf bifurcation and stability ofthe two modes near the double Hopf point. The normal form also showsthe existence of an unstable invariant torus in the dynamics of the modelsystem, and the location of the torus can be approximated from the normalform near the double Hopf point. Numerical simulations and continuation-bifurcation computations with the spatially discretized model are used toverify these predictions.We extend our study to a general class of coupled membrane-bulk dy-namics in the one-dimensional space in chapter §4. Firstly, we formulatea general model system that describes two dynamically active membranes,where n species are assumed to interact with each other, separated spatiallyby a distance 2L, that are coupled together through a linear bulk diffusionfield, with a fixed bulk decay rate. With this model setting, the algebraic41.2. Thesis Outlinesystem of equations that the steady state solutions should satisfy are derivedand the linear stability problem is also formulated. For this class of models,it is shown both analytically and numerically that bulk diffusion can triggera synchronous oscillatory instability in the temporal dynamics associatedwith the two active membranes. For the case of a single active componenton each membrane, and in the limit L→∞, rigorous spectral results for thelinearization around a steady-state solution, characterizing the possibility ofHopf bifurcations and temporal oscillations in the membranes, are obtained.For finite L, a weakly nonlinear theory, accounting for eigenvalue-dependentboundary conditions appearing in the linearization, is developed to predictthe local branching behavior near the Hopf bifurcation point. The analyt-ical theory, together with numerical bifurcation results and full numericalsimulations of the PDE-ODE system, are undertaken for various coupledmembrane-bulk systems, including two specific biologically relevant appli-cations. In addition, in the case of two heterogeneous membranes, numericalsimulations show the possibility of two sustained oscillations with distinctamplitudes on the two membranes, which serves as a modeling verification ofthe phenomenon observed in laboratory experiments where one cell exhibitsoscillatory dynamics and the other one is essentially quiescent.In chapter §5, we consider a coupled membrane-bulk PDE-ODE modelproposed by Gomez et al. [23]. A detailed analysis using a combination ofasymptotic analysis, linear stability theory, and numerical bifurcation soft-ware is given. The mathematical model consists of two dynamically activemembranes with Fitzhugh-Nagumo kinetics, which is often used to modelspike generation of excitable neurons, separated spatially by a distance L,that are coupled together through a diffusion field that occupies the bulkregion 0 < x < L. The flux of the diffusion field on the membranes at x = 0and x = L provides feedback to the local dynamics on the membranes. Inthe absence of membrane-bulk coupling the membrane kinetics has a stablefixed point. The effect of bulk diffusion is to trigger either synchronous andasynchronous oscillations in the two membranes. In the singular limit ofslow-fast membrane dynamics, and with only one diffusing species in thebulk, phase diagrams in parameter space showing where either synchronousor asynchronous oscillations occur, together with the corresponding Hopffrequencies at onset, are provided analytically. When the membrane ki-netics is not of slow-fast type, a numerical study of the stability problemtogether with numerical bifurcation software is used to to construct globalbifurcation diagrams of steady-states and the bifurcating periodic solutionbranches for the case of either one or two diffusing species in the bulk. Pre-dictions from the analytical and bifurcation theory are confirmed with full51.2. Thesis Outlinenumerical simulations of the PDE-ODE system.In chapter §6, we formulate and analyze a class of coupled cell-bulk ODE-PDE models in a two-dimensional domain, which is relevant to studyingquorum sensing behavior on thin substrates. In this model, spatially segre-gated dynamically active signaling cells of a common small radius  1 arecoupled through a passive bulk diffusion field. For this coupled system, themethod of matched asymptotic expansions is used to construct steady-statesolutions and to formulate a spectral problem that characterizes the linearstability properties of the steady-state solutions, with the aim of predictingwhether temporal oscillations can be triggered by the cell-bulk coupling.Phase diagrams in parameter space where such collective oscillations canoccur, as obtained from our linear stability analysis, are illustrated for twospecific choices of the intracellular kinetics. In the limit of very large bulkdiffusion, it is shown that solutions to the ODE-PDE cell-bulk system canbe approximated by a finite-dimensional dynamical system. This limitingsystem is studied both analytically, using a linear stability analysis, andglobally, using numerical bifurcation software. For one illustrative exam-ple of the theory it is shown that when the number of cells exceeds somecritical number, i.e. when a quorum is attained, the passive bulk diffusionfield can trigger oscillations that would otherwise not occur without thecoupling. Moreover, for two specific models for the intracellular dynamics,we show that there are rather wide regions in parameter space where thesetriggered oscillations are synchronous in nature. Unless the bulk diffusivityis asymptotically large, it is shown that a clustered spatial configuration ofcells inside the domain leads to larger regions in parameter space where syn-chronous collective oscillations between small cells can occur. Additionally,the linear stability analysis for these cell-bulk models is shown to be qualita-tively rather similar to the linear stability analysis of localized spot patternsfor activator-inhibitor reaction-diffusion systems in the limit of long-rangeinhibition and short-range activation.The chapter §7 is the conclusion chapter, where we summarize the mainresults and contributions of this thesis. Also we list several open problemsfor further explorations.6Chapter 2Two Coupled Sel’kovOscillatorsThe goal of this chapter is to formulate and investigate a simple cell-bulkcoupled model in a 1-D domain with two types of coupling. The symmet-ric steady state and its linear stability are studied both analytically andnumerically. The remainder of this chapter proceeds as follows.In §2.1 we formulate a 1-D model on the interval 0 < x < 2L, which con-sists of a PDE-ODE system that couples diffusion in the bulk 0 < x < 2L,with constant diffusivity D, to compartmental dynamics with Sel’kov kinet-ics on the boundaries x = 0 and x = 2L. The particular choice of Sel’kovkinetics is not essential, as the qualitative behavior of bulk-mediated oscilla-tory dynamics will also occur for other, more general, compartmental kinet-ics. In particular, the numerical study of [23] has revealed the possibility ofstable synchronous dynamics under Fitzhugh-Nagumo reaction-kinetics inthe compartments and an detailed analysis of the model in [23] is providedin §5.In §2.2 we consider the case where there is a linear coupling between thetwo compartments at x = 0 and x = 2L and the bulk, where β representsthe strength of this coupling. For this linearly coupled model, we constructa steady-state solution that is symmetric about the midline x = L. In §2.2.1we then derive a transcendental equation for the eigenvalue parameter λassociated with the linearization of the coupled compartment-bulk modelaround the symmetric steady-state solution. In our stability theory, wemust allow for perturbations that are either symmetric or anti-symmetricabout the midline, which leads to the possibility of either synchronous (in-phase) or asynchronous (out-of-phase) instabilities in the two compartments.To determine unstable eigenvalues of the linearization, in §2.2.2 we use thewinding number of complex analysis to determine the number of roots inRe(λ) > 0 to the transcendental equation for the eigenvalue. Branches ofperiodic solutions, either in-phase or anti-phase, that bifurcate from the sym-metric steady-state solution branch, together with their stability properties,are determined using the numerical bifurcation software package XPPAUT72.1. Formulation of the Coupled Compartment-Bulk Model[16] after first spatially discretizing the PDE-ODE system into a relativelylarge system of ODEs. In this way, a phase-diagram in the D versus βparameter space, characterizing the region where stable synchronous andasynchronous oscillations between the two compartments can occur is ob-tained. Our results show that there is a rather large parameter range whereeither stable synchronous or asynchronous oscillations occur. Full numericalcomputations of the PDE-ODE system of coupled compartmental-bulk dy-namics, undertaken using a method-of-lines approach, are used to validatethe theory.In §2.3, we extend the simple two-compartment case of §2.2 to allow fora periodic chain of evenly-spaced dynamically active units that are linearlycoupled to a bulk diffusion field. By using an approach based on Floquettheory, we analyze the linear stability problem to determine Hopf bifurcationthresholds associated with the various possible modes of oscillation. Com-parisons of predictions from the linear stability theory with full numericalsimulations are performed.In §2.4 we illustrate oscillatory compartmental dynamics for a specifictype of nonlinear coupling between the bulk and the two compartments.Although this nonlinearly coupled system possesses the same steady-stateas that of the uncoupled compartmental dynamics, we show using XPPAUT[16] that it can still generate compartment-bulk oscillations. In particular,our numerical computations show, in contrast to the case of a linear couplingbetween the compartments and the bulk, that the branches of synchronousand asynchronous periodic solutions are disconnected and do not bifurcateoff of the symmetric steady-state solution branch. Our global bifurcationdiagram also shows that there is a parameter range of bistability where eitherstable synchronous oscillations or stable asynchronous oscillations can co-exist with the stable symmetric steady-state solution branch. In §2.4.1 westudy an extended ODE compartmental dynamics model, closely related tothe nonlinear coupled compartment-bulk model, but where bulk diffusion isneglected.2.1 Formulation of the CoupledCompartment-Bulk ModelWe begin by formulating a simple model that describes the diffusion anddegradation of a signalling particle in a 1-D spatial domain. The concentra-tion/density of the particle is represented by C(x, t), defined on the bulkx ∈ [0, 2L] at time t. Two identical compartments are introduced at the two82.1. Formulation of the Coupled Compartment-Bulk Modelends of the interval. These compartments can either be regarded as two cellsor two dynamically active membranes, which can interact with the diffusivesignalling particle in the bulk. The dynamical process in each compartment,be it biochemical reactions inside a cell or other chemical reactions on themembrane, is described by a system of nonlinear ODEs. However, the dy-namical process in each compartment is modulated by the concentration ofthe diffusive particle near each boundary. Thus, the dynamics in the com-partment at the left end depends on C(0, t), while the one at the right endis modulated by C(2L, t). The release of signalling particles from the com-partments into the bulk is modeled as a flux boundary condition at each ofthe two compartments. In the bulk, we model the diffusion process as∂C∂t= DCxx − kC , 0 < x < 2L , t > 0,−DCx(0, t) = κ(V0(t)− C(0, t)) , DCx(2L, t) = κ(V1(t)− C(2L, t)) .(2.1.1a)Here D > 0 and k > 0 are the constant diffusion and degradation rates, re-spectively, while Vi(t) (i = 0, 1) are the concentrations of the particle in thetwo compartments. In our model, we assume the efflux of particles out ofeach compartment is proportional to the difference between the concentra-tion inside each compartment and that outside of it in the bulk. Therefore,the influence of each compartment on the diffusive particles is described bythe linear flux boundary condition of (2.1.1a).The dynamics governing the time evolution of the concentration Vi(t)and another variable Wi(t) inside each compartment is described by thefollowing system of nonlinear ODEs:dVidt= f(Vi,Wi) + βP (C(2Li, t), Vi(t)) ,dWidt= g(Vi,Wi) ; for i = 0, 1 .(2.1.1b)For simplicity, we assume that the compartment kinetics f(V,W ) and g(V,W ),as well as the coupling term βP (C, V (t)) to the bulk, are identical for thetwo compartments. We assume that this system, when isolated (i.e. whenβ = 0), and given favourable choices of parameter values, is capable of gen-erating sustained oscillations of limit cycle type. In addition, we furtherassume that, when isolated, the compartmental dynamics has a unique sta-ble steady-state. In Fig. 2.1 we give a schematic plot of the geometry for(2.1.1).To illustrate the new behavior that can be induced by compartment-bulk92.2. Linear Coupling Between the Compartments and the Bulkx=0 x=2LBulk region: Passive DiffusionLocalcompartmentFigure 2.1: Schematic plot of the geometry for (2.1.1) showing the bulkregion 0 < x < 2L, where passive diffusion occurs, and the two local com-partments at x = 0 and x = 2L. One of the local species can be exchangedbetween the compartment and the bulk.coupling, we will use Sel’kov model, for which the kinetics aref(V,W ) = αW +WV 2 − V , g(V,W ) = [µ− (αW +WV 2)],(2.1.1c)where 0 <  < 1 is a parameter. We remark that the qualitative conclusionsderived in the present study do not depend on the specific forms of the re-action kinetics, provided that limit cycle type oscillations in the dynamicscan occur. In later chapters, we will show modeling studies using other formof local kinetics. In our model, the influence of the concentration of parti-cles near each boundary on the compartment dynamics is described by thecoupling term βP (C, t), V ), where β represents the coupling strength. Twotypes of coupling will be considered. In §2.2 we consider a linear couplingterm, while in §2.4 we consider a specific form of nonlinear coupling.For this Sel’kov model, when each compartment is isolated, i.e. whenβ = 0, there is a unique steady state solution given by V0 = µ and W0 =µ/(α+ µ2), which is stable. In other words, the two compartments are“conditional oscillators” when decoupled from each other. Therefore, whenoscillations occur in the present study, they are caused by the couplingbetween the two compartments induced by the diffusive signalling particles.2.2 Linear Coupling Between the Compartmentsand the BulkWe first consider (2.1.1) with a linear coupling term P (C(2Li, t), V ) wherei = 0, 1. We specify thatP (C(2Li, t), V ) = C(2Li, t)− V (t) , i = 0, 1 . (2.2.1)102.2. Linear Coupling Between the Compartments and the BulkWith this choice, all interactions between the compartments and the diffu-sive particles are linear.We first determine a steady-state solution to (2.1.1), with (2.2.1), that issymmetric about the midline x = L. To construct this steady-state solutionwe solve (2.1.1) on 0 < x < L, while imposing a no-flux boundary conditionfor C at x = L. Since only the compartment at the left boundary x = 0 isconsider, we drop the subscripts for the compartmental variables V and W .We readily calculate that there is a unique symmetric steady-state solutionCe(x), Ve, and We, given byCe(x) = C0ecosh(ω(L− x))cosh(ωL), C0e =κµκ+Dω tanh(ωL)(1 + β), ω ≡√k/D ,Ve =µ1 + β+βC0e1 + β, We =µα+ V 2e.(2.2.2)We observe that the steady-state solution in the compartment for the cou-pled system differs from that of the uncoupled problem, and reduces toVe = V0 ≡ µ and We = W0 ≡ µ/(α+ µ2) in the absence of coupling.2.2.1 Linear Stability Analysis of the Steady StateTo analyze the linear stability of the symmetric steady-state solution, weintroduce the perturbationC(x, t) = Ce(x) + eλtη(x) , V (t) = Ve + eλtϕ , W (t) = We + eλtφ ,(2.2.3)into (2.1.1). Upon linearizing the resulting system, we obtain the followingeigenvalue problem for the eigenvalue parameter λ:λη = Dηxx − kη , 0 < x < L ; −Dηx(0) = κ(ϕ− η0) , (2.2.4a)λϕ = feV ϕ+ feWφ+ β (PeCη0 + PeV ϕ) , λφ = geV ϕ+ geWφ . (2.2.4b)Here we have defined η0 ≡ η(0), feV ≡ fV (Ve,We), feW ≡ fW (Ve,We), P eC ≡PC(C0e , Ve), etc.The formulation of the linear stability problem is complete after imposinga boundary condition for η(x) on the midline x = L. We will consider twodistinct choices. The choice η(L) = 0 corresponds to an asynchronous, oranti-phase, perturbation, while the condition ηx(L) = 0 corresponds to anin-phase synchronization of the two compartments. We will consider bothpossibilities in our analysis below.112.2. Linear Coupling Between the Compartments and the BulkFor either choice of the boundary condition, we can readily solve (2.2.4)to derive that λ must be a root of the transcendental equation F(λ) = 0,where F(λ) is defined byF(λ) ≡ 1p±(λ)− geW − λdet(Je − λI) , Je ≡ feV , feWgeV geW . (2.2.5a)Here Je is the Jacobian matrix of the uncoupled compartmental dynamicsevaluated at the steady-state (2.2.2) for the coupled system. In (2.2.5a),p±(λ) are defined byp+(λ) ≡ βDΩλ tanh(ΩλL)κ+DΩλ tanh(ΩλL), p−(λ) ≡ βDΩλ coth(ΩλL)κ+DΩλ coth(ΩλL),Ωλ ≡√k + λD,(2.2.5b)where p+ corresponds to synchronous (in-phase) perturbations, while p−corresponds to asynchronous (anti-phase) perturbations. In (2.2.5b), wemust specify the principal branch of the square root to ensure that η(x) isanalytic in Re(λ) > 0.To classify any instabilities that can occur with compartment-bulk cou-pling we need to determine the number of roots of (2.2.5a) and their distri-bution in the right-half of the complex λ-plane (i.e. Re(λ) > 0). We willapproach this problem in two ways. One method is to numerically implementa winding number approach, as done below in §2.2.2. The second method,which we dicuss here, is to use the bifurcation software XPPAUT[16]. Firstwe spatially discretize (2.1.1) into a relatively large system of ODEs, andthen we use XPPAUT to path-follow solution branches that bifurcate offthe steady-state solution (2.2.2). In this way, in Fig. 2.3 we show two typi-cal bifurcation diagrams with respect to the diffusivity D and the couplingstrength β, for fixed values of the other parameters as shown in the fig-ure caption. As seen from these plots, there are Hopf bifurcation points atwhich the steady-state solution loses its stability to either synchronous orasynchronous oscillatory instabilities in the two compartments. Moreover,in some regions of the (β,D) parameter space only either the synchronousor asynchronous mode is present. In the left panel of Fig. 2.3, where we plotthe bifurcation diagram for V versus D when β = 0.8, we observe that thesynchronous and asynchronous periodic solution branches change stabilityat D ≈ 0.25 and D ≈ 0.55, respectively. These bifurcation points correspond122.2. Linear Coupling Between the Compartments and the Bulkto Torus bifurcations. By tuning the parameter β, these bifurcation pointscan occur at a common value of D, and correspond to the intersection ofthe black and magenta curves in Fig. 2.2. For this co-dimension-2 case, suchdouble Hopf bifurcations were analyzed in detail using normal form theoryin §3. For β = 0.8, we further observe from the left panel of Fig. 2.3 thatboth the synchronous and asynchronous oscillations are stable on the range0.25 < D < 0.55. A similar bifurcation diagram, but with fixed D = 0.4and β a parameter, is shown in the right panel of Fig. 2.3.By varying the values of D and β, we can obtain a series of bifurcationdiagrams, representing slices through the (β,D) phase space. By amalga-mating these slices, we generate the phase diagram in the (β,D) parameterplane as shown in Fig. 2.2. We remark that the diffusivity D effectivelyrepresents the length scale of this system. When D is small, effectively thedistance between the two cells is large. However, when D is large, effectivelyone can consider that the two cells are close together. Therefore, changingD is equivalent to changing the distance between the two cells. Variationsin the coupling strength β determine the importance of the feedback in thecompartment-bulk interactions.The phase diagram in Fig. 2.2 shows the region of stability of the steady-state solution, and regions where either synchronous or asynchronous oscil-lations, or both, can occur as the diffusivity D and the coupling strengthβ are varied. From this plot, we observe that when D is relatively small,then as the coupling strength β is increased it is the anti-phase mode thatbecomes unstable first. This phenomenon is also plausible biologically, sincewhen D and β are both small the communication between the two cells isnot very efficient, and so it is hard to synchronize their dynamics with acommon phase.From Fig. 2.2, we also observe that whenD is relatively large, only the in-phase synchronized oscillation can occur. In the region of Fig. 2.2 boundedby the blue solid curve, the steady-state solution is unstable to the in-phasemode, but it is only above the black solid curve where a stable synchronizedoscillation between the two compartments can occur. Similarly, inside thered dashed curve, the steady-state solution is unstable to the anti-phasemode, but it is only under the magenta dashed curve where the asynchronousmode is stable. Therefore, in the region of Fig. 2.2 bounded by the blackand magenta curves, stable synchronous and stable asynchronous periodicoscillations can co-exist. The determination which mode would result fromnumerical computations of the initial value problem (2.1.1) should dependon the initial conditions at time t = 0.To confirm predictions obtained from the bifurcation analysis, full time-132.2. Linear Coupling Between the Compartments and the Bulk0.4 0.6 0.8 1 1.200.511.5βD  Anti−phaseIn−phaseFigure 2.2: Phase diagram in the D versus β parameter plane, for the Sel’kovmodel (2.1.1) with linear coupling (2.2.1) for both the synchronous (in-phase) and asynchronous (anti-phase) modes. The other fixed parametersin (2.1.1) are µ = 2, α = 0.9,  = 0.15, κ = 1, k = 1, and L = 1. Theparameter regime where compartment oscillations occur is within the bluesolid curve (in-phase synchronization) and the red dashed curve (anti-phasesynchronization). Above the black solid line, the in-phase periodic solutionis stable, while below the dashed magenta curve the anti-phase periodicsolution is stable. The horizontal and vertical slices at D = 0.4 and β = 0.8,respectively, through the phase diagram are discussed in Fig. 2.3.142.2. Linear Coupling Between the Compartments and the Bulk0 0.5 11.52DV0.4 0.8 1.211.52βVFigure 2.3: Bifurcation diagram of V corresponding to the vertical and hor-izontal slices through the phase diagram of Fig. 2.2, as computed usingXPPAUT [16]. Left panel: V versus D for β = 0.8 (vertical slice). Rightpanel: V versus β for D = 0.4 (horizontal slice). In these panels the solidand dashed lines denote linearly stable and unstable branches of steady-statesolutions, respectively. The two closed loops correspond to branches of syn-chronous and asynchronous periodic solutions. In the left panel, the branchthat bifurcates from the steady-state near D = 1 is the synchronous branchand in the right panel, the outer loop is the asynchronous branch. Thesolid/open circles on these loops denote linearly stable/unstable periodicsolutions, respectively.152.2. Linear Coupling Between the Compartments and the BulkFigure 2.4: Full numerical solutions of the ODE-PDE system (2.1.1) demon-strating either in-phase or anti-phase oscillations of the two compartments.Time increases from bottom to top and the horizontal axis indicate the bulkregion where L = 1. Left panel: synchronous oscillations for D = 1 andβ = 0.7 (black dot in Fig 2.2). Right panel: asynchronous oscillations forD = 0.4 and β = 0.5 (magenta open circle in Fig. 2.2.) The other parametervalues are the same as in the caption of Fig.2.2.162.2. Linear Coupling Between the Compartments and the Bulkdependent numerical solutions of the coupled ODE-PDE system (2.1.1) werecomputed using a method of lines approach based on a second-order spatialdiscretization of the bulk diffusion operator. In our computation, we pickedtwo points in the phase diagram in Fig. 2.2 indicated in the figure by theblack solid dot and the magenta open circle. For these parameter sets, fullnumerical solutions of the ODE-PDE system (2.1.1) are shown in Fig. 2.4starting with the initial value C(x, 0) = 0.2, and with randomly generatedinitial values for Vi and Wi for i = 0, 1 at t = 0. The plots in Fig. 2.4 fort large confirm the theoretical predictions of the phase diagram by showingsynchronous in-phase oscillations for D = 1 and β = 0.7 (left panel), andasynchronous anti-phase oscillations for D = 0.4 and β = 0.5 (right panel).2.2.2 The Winding Number AnalysisIn this section, we show how to use the winding number criterion of complexanalysis to determine the number of roots of F(λ) = 0 in Re(λ) > 0, whereF(λ) is defined in (2.2.5). The analysis below is similar to that used in[57] to analyze the stability of localized pulse solutions to reaction-diffusionsystems.To determine the number N of roots of F(λ) = 0 in Re(λ) > 0 of thespectral plane, we calculate the winding number of F(λ) over the contourconsisting of the imaginary axis −iR ≤ Imλ ≤ iR, decomposed as ΓI+ = iλIand ΓI− = −iλI where 0 < λI < R, together with the semi-circle |λ| = R,with | arg λ| ≤ pi/2, which we denote by ΓR, as shown in Fig. 2.5.ΓRΓ+Γ-Figure 2.5: Counterclockwise contour consisting of the imaginary axis−iR ≤Imλ ≤ iR, denoted by Γ−∪Γ+, and the semicircle ΓR, given by |λ| = R > 0,for |argλ| ≤ pi/2.Assuming that there are no roots of F(λ) = 0 on the imaginary axis, we172.2. Linear Coupling Between the Compartments and the Bulkuse the argument principle of complex analysis to determine N asN =12pi(limR→∞[argF ]ΓR + 2 limR→∞[argF ]ΓI+)+ P , (2.2.6)where P is the number of poles of F(λ) in Re(λ) > 0. Here [argF ]Γ denotesthe change in the argument of F(λ) over the contour Γ oriented in thecounterclockwise direction. In deriving (2.2.6), we have used F(λ) = F(λ¯)to obtain the relation limR→∞[argF ]ΓI− = limR→∞[argF ]ΓI+ .To determine P , we first observe from (2.2.5) that the choice of theprincipal branch of the square root for Ωλ ensures that 1/p±(λ) is analyticin Re(λ) > 0. Therefore, P is determined by the number of zeros of thequadratic function det(Je − λI) = λ2 − tr(Je)λ + det(Je) in Re(λ) > 0.By using the specific forms of the nonlinearities f(V,W ) and g(V,W ) in(2.1.1c), we readily calculate det(Je) = (α + V2e ) > 0. Therefore, in termsof the trace of Je, which we have denoted by tr(Je), we have that P = 2 iftr(Je) > 0 and P = 0 if tr(Je) < 0.Next, we determine the change in the argument of F(λ) over ΓR asR → +∞. Since det(Je − λI) is a quadratic function of λ and 1/p±(λ) ∼β−1 +O(Ω−1/2λ ) as |λ| → +∞ in Re(λ) > 0, we estimate from (2.2.5a) and(2.2.5b) that, for either the synchronous or asynchronous modes,F(λ) ∼ 1β+κDΩλβ+O(1λ), as |λ| = R→ +∞ , (2.2.7)where | arg λ| ≤ pi/2. Hence, we have limR→∞[argF ]ΓR = 0, so that (2.2.6)becomesN =1pilimR→∞[argF ]ΓI+ + P , (2.2.8)where P = 2 if tr(Je) > 0 and P = 0 if tr(Je) < 0.In this way, the problem of determining N is reduced to the simplerproblem of calculating [argF ]ΓI+ where ΓI+ is traversed in the downwardsdirection. On ΓI+ , we let λ = iλI for 0 < λI < ∞, and decompose F(iλI)in (2.2.5a) into real and imaginary parts as F(iλI) = FR(λI) + iFI(λI). AsλI decreases from +∞ to 0, we use (2.2.5a) to determine how many timesF(iλI) wraps around the origin in the (FR,FI) plane. By using (2.2.5b) tocalculate the asymptotics of p± as λI → +∞, we conclude that FR → 1/β >0 and FI → 0 as λI → +∞. This shows that argF(iλI) → 0 as λI → ∞.182.2. Linear Coupling Between the Compartments and the BulkIn contrast, as λI → 0, we further calculate from (2.2.5a) and (2.2.5b) thatF(0) = 1p±(0)− geWdet(Je), where1p±(0)=1β +κβDω tanh(ωL) > 0 ,1β +κβDω coth(ωL) > 0 ,(2.2.9)and ω ≡√k/D. Then, from the specific form of g(V,W ) in (2.1.1c), we getthat geW = −(α + V 2e ) < 0. Upon recalling that det(Je) = (α + V 2e ) > 0,we conclude from (2.2.9) that F(0) = [p±(0)]−1 + 1 > 0.This indicates that as we traverse ΓI+ , the path of F(iλI) both startsand ends on the positive real axis of the (FR,FI) plane. It follows that thechange in the argument of F(λ) on ΓI+ can only be an integer number of2pi, so that[argF ]∣∣∣ΓI+= 2mpi, m = 0,±1,±2, . . . .Consequently, we have from (2.2.8) thatN = 2m+ P , P ={2 , when tr(Je) > 0 ,0 , when tr(Je) < 0 .(2.2.10)Although we cannot, in general, determine m analytically, it is readilycalculated numerically from (2.2.5a). To illustrate the numerical compu-tation of the winding number, we consider (2.1.1) with the linear coupling(2.2.1) for the parameter value D = 1 and β = 0.7, corresponding to themarked black dot in Fig. 2.2. The other parameter values for (2.1.1) aregiven in the caption of Fig. 2.2. For this parameter set we calculate thattr(Je) > 0 so that P = 2 from (2.2.10). In the right panel of Fig. 2.6,we plot the path of F(iλI) in the (FR,FI) parameter plane for both thein-phase synchronous mode (solid curve) and the anti-phase asynchronousmode (dashed curve). For the asynchronous mode we observe that as λIdecreases from a very large initial value, F(λ) wraps around the origin oncein clockwise direction, so that [argF ]|ΓI+ = −2pi. Therefore, since m = −1,we get N = 0 from (2.2.10). In contrast, for the synchronous mode weobserve from Fig. 2.6 that [argF ]|ΓI+ = 0, so that m = 0 and N = 2 from(2.2.10). These winding number computations show that, at this parameterset, the steady-state solution is unstable only to synchronous perturbations.To determine the location of the two unstable eigenvalues for the syn-chronous mode when D = 1 and β = 0.7 we look for roots of F(λ) on the192.2. Linear Coupling Between the Compartments and the Bulk0 0.2 0.4 0.6 0.8 1−0.8−0.400.40.81.2λ   G+(λ) G−(λ) H(λ)0 1 2 3 4−1012ImFImF  Sym AsyFigure 2.6: Left panel: G+(λ), G−(λ), and H(λ), as defined in (2.2.11), areplotted on λ > 0 real for D = 1 and β = 0.7, with the other parameters as inthe caption of Fig. 2.2. There is no intersection between G±(λ) and H(λ),which shows that F(λ) has no real roots λ for either the synchronous andasynchronous modes. Right panel: FI(λI) = Im(F(iλI)) is plotted versusFR(λI) = Re(F(iλI)) for both the synchronous and asynchronous modesas λI is decreased from 1000 to 0. The open circle represents the startingpoint at λI = 1000. For the asynchronous mode (dashed curve), we havem = −1 in (2.2.10) since the trajectory wraps around the origin once in theclockwise direction. For the synchronous mode (solid curve), the plot showsthat m = 0 in (2.2.10).202.3. A Periodic Chain of Active Units Coupled by Bulk Diffusionpositive real axis λ > 0. To do so, it is convenient to rewrite F asF(λ) = H(λ)−G±(λ)p±(λ) det(Je − λI) , (2.2.11a)whereH(λ) ≡ det(Je − λI) , G±(λ) ≡ p±(λ)(geW − λ) . (2.2.11b)In the left panel of Fig. 2.6 we plot H(λ) and G±(λ) on λ > 0 real for D = 1and β = 0.7. This plot shows that there are no intersections between H(λ)and G±(λ). Since, consequently, there is no real positive root to F(λ) = 0,we conclude that the initial instability associated with the in-phase-modeis a synchronous oscillatory instability of the compartmental dynamics. Abifurcation diagram (not shown) similar to that in Fig. 2.3 predicts thatthis initial instability leads to a large-scale stable synchronous oscillation.The full numerical results of the ODE-PDE system (2.1.1) shown in the leftpanel of Fig. 2.4, as computed using a method of lines approach, confirmsthis prediction of a stable synchronous oscillation in the two compartments.We remark that this strategy of computing the winding number, andthen using (2.2.10) to determine N , was used for mapping out the regions inthe phase diagram of Fig. 2.2 characterizing the linear stability propertiesof the steady-state solution to either in-phase or anti-phase perturbations.2.3 A Periodic Chain of Active Units Coupled byBulk DiffusionIn this section we extend the analysis in §2.2 to the case where m identicalcompartments, or cells, are evenly-spaced, with spacing 2L, on a 1-D ring.These cells are then coupled by a bulk-diffusion field. A schematic diagramof this periodic arrangement of active cells is shown in the left panel ofFig. 2.7. Equivalently, we consider a 1-D domain on the interval [−L, (2m−1)L], with cells located at 2jL for j = 0, . . . ,m − 1, with the imposition ofperiodic boundary conditions for the bulk diffusion field at the endpoints.A schematic plot of four such cells is shown in the right panel of Fig. 2.7.With the same notation used in §2.1, we model the system with m iden-tical cells on a 1-D structure as follows. Firstly, the bulk diffusion process212.3. A Periodic Chain of Active Units Coupled by Bulk Diffusion-L 0 7L2L 4L 6L-L L0(A) (B)(C)Figure 2.7: Left panel: Schematic diagram of four identical cells on a ringstructure. The green solid dots represent cells. Top right panel: Schematicdiagram of four identical cells on the domain [−L, 7L] with periodic bound-ary conditions at the two ends. Bottom right panel: schematic of one cellon [−L,L].is modeled byCt = DCxx − kC , t > 0 , x ∈ (−L, (2m− 1)L) ,with x 6= 2jL , j = 0, . . . ,m− 1 ,C(−L, t) = C(2mL− L, t) , Cx(−L, t) = Cx(2mL− L, t) .(2.3.1a)Inside each cell, we suppose that there are n locally interacting chemicalsspecies. As shown in Appendix A, the local dynamics in each cell, with thelinear coupling to the bulk diffusion field, is governed bydujdt= F (uj)+e1[κ2(C(2jL+, t) + C(2jL−, t))− κu1j], j = 0, . . . ,m−1 ,(2.3.1b)where uj = (u1j , u2j , . . . , unj)T denotes the n species inside the j-th cell,e1 ≡ (1, 0, . . . , 0)T , with u1j being the first chemical species inside the jthcell. Moreover, F is the common local reaction kinetics, since the cells areassumed to be identical. Here C(2jL−, t) and C(2jL+, t) represent the bulkconcentration field at the left and right boundary of the j-th cell. As shownin Appendix A, the boundary conditions for the bulk concentration C at thecell boundaries, where j = 0, . . . ,m− 1, areDCx(2jL+, t) = κ(C(2jL+, t)− u1j(t)),DCx(2jL−, t) = κ(u1j(t)− C(2jL−, t)),(2.3.1c)where κ > 0 is the common cell permeability parameter.We remark that in our formulation, we have only assumed that C(x, t) ispiecewise continuous on the ring, and so in general C(2jL+, t) 6= C(2jL−, t).222.3. A Periodic Chain of Active Units Coupled by Bulk DiffusionAn alternative, but simpler formulation, would be to impose that C is con-tinuous on the ring, and that there is a jump in the flux DCx across eachcell. Although we do not pursue this simpler problem here, the linear sta-bility analysis associated with this problem is discussed briefly in AppendixB.2.3.1 The Steady-State SolutionWe first calculate the symmetric steady-state solution of (2.3.1). For thissteady-state, the bulk concentration is symmetric with respect to the midlineof every two cells, and the local cell variables are the same for each cell.Although there might be other asymmetric steady-state solutions for the fullsystem (2.3.1) of m coupled cells, we focus only on the symmetric steady-state solution and its linear stability properties.To construct the symmetric steady-state, we need only consider the do-main [−L,L], as shown in Fig. 2.7, with a cell located at x = 0 and withperiodic boundary conditions at x = ±L. We denote this steady-state solu-tion by Ce(x) and the corresponding local steady-state cell variables as ue.Then the symmetric steady-state solution for C in the full system (2.3.1)is constructed by a simple period extension of this basic solution. Hence,focusing on the interval [−L,L], the steady-state solution Ce(x) satisfiesCexx =kDCe , x ∈ (−L, 0) ∪ (0, L) ; Ce(−L) = Ce(L) , Cex(−L) = Cex(L) ,DCex(0+) = κ(Ce(0+)− ue1), DCex(0−) = κ(ue1 − Ce(0−)).(2.3.2)The steady-state solution for the compartmental variable ue satisfiesF (ue) + e1[κ2(Ce(0+) + Ce(0−))− κue1]= 0 . (2.3.3)On each subinterval, we can calculate the steady state solution Ce(x) sepa-rately asCe(x) ={A cosh((x− L)ω) , 0 < x < L ,A cosh((x+ L)ω) , −L < x < 0 , (2.3.4a)where A and ω are given byω ≡√kD, A =κue1κ cosh(Lω) +Dω sinh(Lω). (2.3.4b)232.3. A Periodic Chain of Active Units Coupled by Bulk DiffusionAs expected this steady-state is continuous across the cells.For the special case where the local cell variable u has two componentsu = (V,W )T with local reaction term F = (f, g)T , where f and g are theSel’kov kinetics given in (2.1.1c), we can use (2.3.3) and (2.3.4) to explicitlyidentify a unique steady-state V e and W e asV e =µ(κ+Dω tanh(Lω))κ+ (1 + κ)Dω tanh(Lω), W e =µα+ (V e)2. (2.3.4c)2.3.2 The Linear Stability AnalysisNext, we study the linear stability of the steady-state solution (2.3.4) for thecase of Sel’kov kinetics. By introducing the perturbation (2.2.3) into (2.3.1)and linearizing, we obtain the eigenvalue problemη′′ =(k + λ)Dη , x ∈ (−L, (2m− 1)L) , with x 6= 2jL , j = 0, . . . ,m− 1 ,Dη′(2jL+) = κ(η(2jL+)− ϕ) ,Dη′(2jL−) = κ(ϕ− η(2jL−)) , j = 0, . . . ,m− 1 ,(2.3.5a)subject to the periodic boundary conditionsη(−L) = η(2mL− L) , η′(−L) = η′(2mL− L) . (2.3.5b)Upon linearizing the reaction kinetics we have thatλϕ = feV ϕ+feWφ−κϕ+κ2(η(0+) + η(0−)), λφ = geV ϕ+geWφ , (2.3.5c)where feV , feW , geV , and geW are evaluated at the steady-state (2.3.4c).Instead of considering (2.3.5a) with periodic boundary condition (2.3.5b),we make use of Floquet theory and consider (2.3.5a) on the fundamental in-terval [−L,L] with the Floquet boundary conditionsη(L) = zη(−L) , η′(L) = zη′(−L) . (2.3.6)The solution can then be extended to the interval [L, 3L] by defining η(x) ≡zη(x−2L) for x ∈ [L, 3L] and using translation invariance. Since the m cellsare identical, it is clear that η(x) satisfies (2.3.5a). By iterating this process,we construct the solution of (2.3.5a) on the whole domain [−L, (2m− 1)L]242.3. A Periodic Chain of Active Units Coupled by Bulk Diffusionprovided that η(2mL − L) = zmη(−L). Therefore, we obtain that z mustbe one of the m-th roots of unityz ≡ e2piil/m , where l = 0, . . . ,m− 1 . (2.3.7)In this way we have recovered the periodic solution to (2.3.5a) on [−L, (2m−1)L].Next, we solve (2.3.5a) on [−L,L] subject to the Floquet boundary con-ditions (2.3.6). The solution to (2.3.5a) and (2.3.6) isη(x) ={ [zA cosh((x− L)Ωλ) + zB sinh((x− L)Ωλ)]ϕ , 0 < x < L ,[A cosh((x+ L)Ωλ) +B sinh((x+ L)Ωλ)]ϕ , −L < x < 0 ,(2.3.8)where A, B, and Ωλ are defined byA ≡ κ(z + 1)/z2DΩλ sinh(LΩλ) + 2κ cosh(LΩλ),B ≡ κ(z − 1)/z2DΩλ cosh(LΩλ) + 2κ sinh(LΩλ), Ωλ ≡√k + λD,(2.3.9)and where we choose the principal branch of Ωλ if λ is complex. From thissolution we then calculateη(0+) + η(0−) = A(z + 1) cosh(ΩλL) +B(1− z) sinh(ΩλL) . (2.3.10)Upon substituting these expressions into (2.3.5c), we obtain a homogeneouslinear system for ϕ and φ given by(feV + ∆λ)ϕ+ feWφ = λϕ , geV ϕ+ geWφ− λφ = 0 ,∆λ ≡κ2[A(1 + z) cosh(ΩλL) +B(1− z) sinh(ΩλL)]− κ . (2.3.11)By writing (2.3.11) in matrix form, and then using (2.3.9) together with(2.3.7) for z, we readily derive, after some algebra, that the the discreteeigenvalues λ satisfy the transcendental equation F(λ) = 0, whereF(λ) ≡ 1∆λ+geW − λdet(Je − λI) , (2.3.12a)and where for each possible mode l of instability, with l = 0, . . . ,m− 1, wehave∆λ ≡κ2ΩλD[Re(zl)− cosh (2ΩλL)]− κΩ2λD2 sinh (2ΩλL)(Ω2λD2 + κ2)sinh (2ΩλL) + 2κDΩλ cosh (2ΩλL),Re(zl) = cos(2pilm).(2.3.12b)252.3. A Periodic Chain of Active Units Coupled by Bulk DiffusionHere Je is the Jacobian of the reaction kinetics, as defined in (2.2.5a), eval-uated at the steady-state (2.3.4).Our goal below is to determine Hopf bifurcation thresholds for whichF(±iλI) = 0 in (2.3.12a), for some λI > 0. Such pure imaginary eigenval-ues depend on Re(z) through ∆λ, as defined in (2.3.12b). To examine thepossible modes of instability, we observe that if zl is one of the m-th rootsof unity, thenzl = z¯m−l , l = 1, . . . , bm2c , (2.3.13)where the floor function bxc is defined as the largest integer not greater thanx. Therefore, if m is odd, there are (m+ 1)/2 different values of Re(z), andthus (m+ 1)/2 different possible modes of linear instability. Alternatively,if m is even, there are m2 + 1 different possible modes of linear instability.The eigenvalue of multiplicity one corresponds to z = 1 (and also z = −1if m is even). The remaining eigenvalues always have multiplicity two. Inother words, the eigenvalue corresponding to zl is also an eigenvalue forz = zm−l. Therefore, if we find a Hopf bifurcation point for z 6= ±1, thenthere are always two possible spatial modes of oscillation for that specificpair of purely imaginary eigenvalues. Finally, to determine the predictedspatial pattern of any Hopf bifurcation point λ = iλI , we observe that atthe midpoint between the cells the perturbation Re(eλtη[(2j − 1)L)] to thebulk diffusion field C(x, t) isRe(eiλIη[(2j − 1)L]) = cos(λIt+ 2piljm), j = 0, . . . ,m . (2.3.14)2.3.3 Hopf Bifurcation Boundaries, Global Branches andNumericsNext, we use (2.3.12) to compute the Hopf bifurcation boundaries for thedifferent possible modes of instability in the D versus κ parameter plane. Weremark that the choice l = 0 in (2.3.12) corresponds to in-phase synchronousperturbations across the cells, whereas the bm2 c other eigenvalues correspondto the various anti-phase modes across the m cells. For m = 3, and for oneparticular parameter set for the Sel’kov model (2.1.1c), in Fig. 2.8 we plotthe Hopf bifurcation thresholds in the D versus κ plane.Next, for the m = 3 cell problem with κ = 1, we use XPPAUT [16] tocompute the global bifurcation diagram, as a function of D, for the in-phasesynchronous periodic solution branch, which bifurcates from the symmetricsteady-state solution in (2.3.4) at the two distinct values of D shown in the262.3. A Periodic Chain of Active Units Coupled by Bulk Diffusion0 1 2 300.20.40.60.81κD0 1 2 3020406080100κDFigure 2.8: Left: Phase diagram showing Hopf bifurcation boundaries forthe case of three (m = 3) cells in the D versus κ plane for k = 1, L = 1, andwhere the Sel’kov parameters in (2.1.1c) are  = 0.15, µ = 2 and α = 0.9.The black curves corresponds to l = 0 and the red curves corresponds tol = 1, 2. The black and red curves almost coincide on the lower boundary.In the region bounded by the two black and two red curves the symmetricsteady-state is linearly unstable to the l = 0 and l = 1, 2 modes, respectively.Right: Same as the left panel, but with a larger range of D for the verticalaxis. For these parameter values we observe that the region of instability isunbounded in the D versus κ plane.272.3. A Periodic Chain of Active Units Coupled by Bulk Diffusion0 0.2 0.4 0.6 0.811.522.5DV−4 −3 −2 −1 0 1−2−1.5−1−0.500.511.5ReFImFFigure 2.9: Left: Global bifurcation diagram with m = 3 cells on the domain[−1, 5] for κ = 1, with the other parameters as in the caption of Fig. 2.8.The solid and dashed lines denote linearly stable and unstable branches ofsteady-state solutions, respectively. The closed loop is the global branch ofin-phase synchronous periodic solutions. The upper Hopf bifurcation valueD ≈ 0.54299 is for the l = 0 in-phase mode. The solid/open circles on thisloop denotes a linearly stable/unstable periodic solution, respectively. Thered dot at D ≈ 0.48482 corresponds to the Hopf bifurcation point for thedegenerate l = 1, 2 mode. Right panel: Plot of F(iλI) as λI decreases from1000 to 0 with D = 0.5. The blue curve corresponds to l = 0, and themagenta curve is for l = 1, 2. The inner panel shows the curves near theorigin. The trace and determinant of Je are trJe = 0.4879 and det Je =0.4474, so that P = 2 in (2.3.15). We obtain N = 2 unstable eigenvalues forl = 0, and N = 0 for l = 1, 2 from (2.3.15).282.3. A Periodic Chain of Active Units Coupled by Bulk Diffusionleft panel of Fig. 2.8. The computations, done by first discretizing (2.3.1),are displayed in the left panel of Fig. 2.9. From this figure we observe thatfor larger values of D the in-phase synchronous periodic solution branch islinearly stable, but it then destabilizes as D is decreased towards the lowerHopf bifurcation threshold.To verify the linear stability properties of the steady-state solution forthe l = 0 and l = 1, 2 modes off of the Hopf bifurcation boundaries, we canuse a similar winding number criterion for F(λ), defined in (2.3.12), as wasdeveloped in §2.2.2. With the same notation as in §2.2.2, the number N ofunstable eigenvalues of the linearization of the symmetric steady-state forthe periodic cell problem isN =1pi[argF ]∣∣∣ΓI++ P , P ={2 , when tr(Je) > 0 ,0 , when tr(Je) < 0 .(2.3.15)For κ = 1 and D = 0.5, a numerical computation of the winding numbershown in the right panel of Fig. 2.9 yields [argF ]∣∣∣ΓI+= 0 for l = 0 and[argF ]∣∣∣ΓI+= −2pi for l = 1, 2. Therefore, N = 2 for l = 0 and N = 0 forl = 1, 2. These results agree with those predicted from the phase diagramin the left panel of Fig. 2.8, since it is only the in-phase l = 0 mode that iswithin the region of instability.Finally, to confirm predictions obtained from the linear stability analysisand the global bifurcation diagram, full time-dependent numerical solutionsof the coupled PDE-ODE system (2.3.1) were computed for two values of Dwhen κ = 1 by using a method of lines approach based on a second-orderspatial discretization for the bulk diffusion. In the upper row in Fig. 2.10for D = 0.5 we observe, as expected, a stable in-phase synchronous periodicsolution. In the lower row of Fig. 2.10 where D = 0.2, the full numericalsimulations show a stable asynchronous oscillation where the dynamics inthe cells are phase-shifted. The phase-shifting observed in the lower rowof Fig. 2.10 is consistent with the l = 2 mode (with m = 3) in the result(2.3.14) from the linear stability analysis, in that the bulk diffusion field atthe midpoint of the cells and the cell dynamics V1, V2, V3 have the formcos(λIt), cos(λIt+ 4pi/3), and cos(λIt+ 8pi/3).2.3.4 Large D Analysis for the Hopf Bifurcation BoundariesIn this subsection, we examine analytically some qualitative aspects of theregion in the D versus κ phase diagram shown in Fig. 2.8 where the sym-292.3. A Periodic Chain of Active Units Coupled by Bulk DiffusionFigure 2.10: Full numerical results computed from (2.3.1) with D = 0.5(upper row) and D = 0.2 (lower row). Other parameters are the same as inFig. 2.8 with κ = 1. The initial conditions for D = 0.5 are V0 = [0.5, 1.5, 0.5],W0 = [1, 1, 1], and C0(x) = 1. For D = 0.2 the initial conditions areV0 = [0.5, 1.5, 0.5], W0 = [1, 1, 1], and C0(x) = 1 if x > 0, C0(x) = sin(x) + 1if x < 0. The V1, V2 and V3 curves are in blue, green and red respectively.For D = 0.5 there are stable in-phase synchronous oscillations, whereas forD = 0.2 stable phase-shifted synchronous oscillations occur. The phase shiftamong V1, V2 and V3, is consistent with the mode l = 2 in the linear stabilityanalysis (2.3.14). The right panel in each row is a contour plot of C(x, t).302.3. A Periodic Chain of Active Units Coupled by Bulk Diffusionmetric steady-state is linearly unstable. In particular, we will study thelarge D behavior of the Hopf bifurcation boundaries in this plane. Fromthis analysis we will also formulate a simple criterion that can be used topredict whether the lobe of instability in the D versus κ plane is boundedin D for other domain lengths L and bulk degradation parameter k. For thechoice L = k = 1 the instability regions were unbounded as D → ∞ (seethe right panel of Fig. 2.8).Firstly, we determine the limiting behavior of F(λ) in (2.3.12) asD →∞.Upon using ΩλD sinh (2ΩλL) ∼ 2Ω2λLD = 2(k + λ)L and cosh (2ΩλL) ∼ 1,we obtain from (2.3.12b) thatlimD→∞∆λ = ∆λ,∞ ≡ κ2(Re(zl)− 1)− 2κL(k + λ)2L(k + λ) + 2κ. (2.3.16)Therefore, F(λ) in (2.3.12a) has the following limiting form as D →∞:limD→∞F(λ) ≡ F∞(λ) ≡ 1∆λ,∞+geW − λdet(Je − λI) . (2.3.17)In addition, for D → ∞, we can also find an approximate expression forthe steady state V e from (2.3.4c), which is needed to calculate the terms in(2.3.17). By using Dω tanh(ωL) ∼ Dω2L ∼ kL, we obtain from (2.3.4c),that for D →∞,limD→∞V e = V e∞ ≡µ(κ+ Lk)κ+ (1 + κ)kL, limD→∞W e = W e∞ ≡µα+ (V e∞)2 .(2.3.18)We observe from (2.3.16), (2.3.17) and (2.3.18), that upon setting F∞(λ) =0, and rearranging the resulting expression, we obtain a cubic equation in λof the formλ3 + λ2p1 + λp2 + p3 = 0 , (2.3.19a)where we have identify p1, p2 and p3 byp1 ≡ a2L+ κ− tr(Je) ,p2 ≡ det(Je)− κgeW −(a tr(Je) + b)2L,p3 ≡(a det(Je) + bgeW)2L,(2.3.19b)and where we have defined a and b bya ≡ 2(κ+ kL) , b ≡ κ2(Re(zl)− 1)− 2Lκk . (2.3.19c)312.3. A Periodic Chain of Active Units Coupled by Bulk DiffusionNext, we simplify (2.3.19b) for the Sel’kov kinetics (2.1.1c), for whichdet(Je) = (α+ (V e∞)2)= −geW > 0 , tr(Je) = 2V e∞W e∞ − 1− det(Je) .(2.3.20)By substituting (2.3.20) into (2.3.19b), we readily calculate thatp1 ≡ κ(1 +1L)+ k + 1 + det(Je)− 2µVe∞(α+ (V e∞)2) ,p2 ≡(1 + κ(1 +1L)+ k)det(Je) +ξ2L− 2(κ+ kL)µVe∞L(α+ (V e∞)2) ,p3 ≡ ξ2Ldet(Je) , ξ ≡ a− b = 2kL (1 + κ) + 2κ+ κ2(1− Re(zl))> 0 .(2.3.21)0 0.2 0.4 0.6 0.8 1−0.500.511.52κp 1p 2−p 30 0.2 0.4 0.6 0.8 1−0.500.511.52κp 1p 2−p 3Figure 2.11: Plot of p1p2−p3 versus κ for l = 0 (black) and l = 1 (red) for aring of m = 3 cells. The blue horizontal line is the threshold p1p2 = p3. Anyintersections of the black (red) curve with the blue line yields the bifurcationpoints for κ for l = 0 (l = 1). Left panel: the parameter values as givenin Fig. 2.8. Between the two bifurcation points, the black (red) curve liesbelow the threshold p1p2 = p3, and so by the Routh-Hurwitz criterion thereare unstable eigenvalues. Numerically we verify p1 > 0. Right panel: sameparameters except that now the bulk decay is smaller at k = 0.3. There arenow no Hopf bifurcation values of κ in the D →∞ regime.For the Sel’kov model with  = 0.15, µ = 2, and α = 0.9, we now use the322.3. A Periodic Chain of Active Units Coupled by Bulk Diffusioncubic (2.3.19a) with coefficients (2.3.21) to calculate the the limiting Hopfbifurcation values of κ, valid as D → ∞, when k = 1 and L = 1. By theRouth-Hurwitz criterion, a necessary and sufficient condition for all of theroots of (2.3.19a) to satisfy Re(λ) < 0 is that the following three inequalitieshold:p1 > 0 , p3 > 0 , p1p2 > p3 . (2.3.22)From (2.3.21), we have p3 > 0 for any parameter set. Moreover, the Hopfbifurcation boundary satisfiesp1p2 = p3 . (2.3.23)provided that p1 > 0 and p3 > 0.Our numerical computations, from enforcing (2.3.23) for m = 3, predictthat there is a Hopf bifurcation for D  1 whenl = 0 , κ ≈ 0.1313 and 0.6564 ,l = 1 , κ ≈ 0.1407 and 0.3633 . (2.3.24)This is shown in Fig. 2.11. In contrast, from the phase diagram of D versusκ, as seen in the right panel of Fig. 2.8, we obtain for D = 100 that theHopf bifurcation values for κ arel = 0 , κ ≈ 0.1314 and 0.6579 ,l = 1 , κ ≈ 0.1405 and 0.3665 , (2.3.25)which are remarkably close to the values calculated in (2.3.24) from theD →∞ theory.Finally, to obtain a bounded lobe of instability in the D versus κ plane,rather than the unbounded region as D → ∞ shown in Fig 2.8, all that isneeded is to seek conditions on the domain length L and bulk parameterk such that the Routh-Hurwitz stability condition (2.3.22) holds for all κ.This can be achieved by decreasing either L or k. We remark that if wedecrease either the domain length L or bulk decay parameter k, then theblack and red curves in Fig. 2.11 move up, and so there no longer any Hopfbifurcation points for the D → ∞ regime. An example of this is shown inthe right panel of Fig. 2.11 for the same parameters as in the left panel ofFig 2.11 except that now k = 0.3. In this case, the instability lobe in thephase diagram of D versus κ would be bounded in D for both the l = 0 andl = 1, 2 modes.332.4. Nonlinear Coupling Between Compartments and Bulk2.4 Nonlinear Coupling Between Compartmentsand BulkIn the previous section, we considered the case where there is a linear cou-pling between the compartment and the bulk. Such a linear coupling termshifts the steady-state of the original ODE system from Ve = µ to a newvalue that depends on the coupling strength β.In this section, we will study a nonlinear coupling between the compart-ments and the bulk that possesses the same stable steady-state as that ofthe uncoupled ODE system in the compartment, but that still has the effectof generating compartment-bulk oscillations. To illustrate such a possibility,we consider the coupling term P (C(2Li, t), V ), for i = 0, 1 with the formP (C(2Li, t), V ) = βh(C(2Li, t))q(V (t)) ,h(C(2Li, t)) =C(2Li, t)(C(2Li, t)− c0)Kc + C(2Li, t)2, q(V ) =V (V − µ)Lv + V 2, i = 0, 1 ,(2.4.1)where Lv > 0, Kc > 0, and where we have defined c0 byc0 = γµ , γ ≡ κκ+Dω tanh(ωL), ω ≡√k/D . (2.4.2)As in §2.2 we will determine the symmetric steady-state solution to(2.1.1) with (2.4.1) and analyze its linear stability. Upon solving the time-independent problem for (2.1.1) on the domain [0, L], with no-flux boundarycondition for C at x = L, we readily obtain thatCe(x) = C0ecosh(ω(L− x))cosh(ωL), C0e =κVsκ+Dω tanh(ωL),We =µα+ V 2e,(2.4.3)where Ve satisfies the following fifth order polynomial:(Ve − µ)Q(Ve) = 0 , Q(Ve) ≡ (V 4e − βV 3e + V 2e (Kcγ2+ Lv + βµ) +KcLvγ2) .(2.4.4)Here γ, ω are defined in (2.4.2). We observe that with the nonlinear couplingfunction (2.4.1), the steady state solution of the uncoupled ODE compart-mental dynamics is still a steady-state of the coupled ODE system. Specif-ically, we have the uncoupled steady-stateVe = µ , We =µα+ µ2, C0e = c0 , (2.4.5)342.4. Nonlinear Coupling Between Compartments and Bulkwhere corresponds to setting β = 0 in (2.2.2). In addition, there can be atmost four other steady-state solutions, corresponding to the roots of Q(Ve) =0 in (2.4.4). However, since Q(µ) > 0, none of these additional steady-statesolution branches bifurcate from the uncoupled steady-state branch (2.4.5).To examine the stability of the steady state (2.4.5), we introduce thesame perturbation as in (2.2.3). Upon linearizing (2.1.1), we obtain, aftersome algebra, that the associated eigenvalue λ satisfiesλ2 − λ(geW + feV + β(P eV + δP eC)) + (geW feV − geV feW + βgeW (P eV + δP eC)) = 0,whereδ ≡ κκ+DΩλ tanh(ΩλL)(in-phase) ,δ ≡ κκ+DΩλ coth(ΩλL)(anti-phase) .We observe that with the special choice (2.4.1) of nonlinear couplingP (C(0, t), V ), we have P eC = 0 and PeV = 0, so that the characteristic equa-tion for λ becomesλ2 − λ(geW + feV ) + (geW feV − geV feW ) = 0 ,which is the same as that for the uncoupled problem. Since we assumedthat the uncoupled problem has stable dynamics, we have Re(λ) < 0. Thus,our linear stability analysis predicts that the steady-state (2.4.5) can neverbe destabilized by the nonlinear coupling (2.4.1).To determine whether, nevertheless, there can be any compartment-bulkoscillations, we used XPPAUT[16] to compute global bifurcation diagramsafter first spatially discretizing (2.1.1) with the coupling (2.4.1). In Fig. 2.12we show two typical bifurcation diagrams of the compartmental variable V .In the left panel of Fig. 2.12 we plot V versus the coupling strength β forthe fixed diffusivity D = 0.1 showing the stable steady-state solution andthe branch of synchronous periodic oscillations. There is also a branch ofasynchronous periodic solutions (not shown), that essentially overlaps thesynchronous branch. This overlap occurs since for D = 0.1, the bulk diffu-sion field decays rather quickly away from x = 0 and x = 2L, which leads to arather weak coupling between the two compartments. The key feature fromthe left panel of Fig. 2.12 is that there is some parameter regime in β, withD = 0.1, where stable synchronous time-periodic solutions co-exist with thestable steady-state solution (2.4.5). This phenomenon cannot be revealedfrom a local linear stability analysis along the solution branch (2.4.5). Forβ = 4, in the right panel of Fig. 2.12 we plot a bifurcation diagram of V ver-sus D showing the stable steady-states together with disconnected branches352.4. Nonlinear Coupling Between Compartments and Bulkof synchronous and asynchronous periodic solutions. Both the asynchronousand synchronous branches have a saddle-node bifurcation point at D ≈ 0.57and D ≈ 0.67, respectively. The synchronous branches are always unsta-ble. Stable asynchronous time-periodic solutions co-exist with the stablesteady-state solution (2.4.5) when D < 0.57.To confirm predictions from the bifurcation diagram, we computed fullnumerical solutions of the PDE-ODE system (2.1.1) with nonlinear coupling(2.4.1) for D = 0.5 and β = 4, with the other parameter values as givenin the caption of Fig. 2.13. From Fig. 2.12, we observe for these parametervalues that the asynchronous mode is stable. The full numerical resultsshown in Fig. 2.13 confirm this prediction of stable asynchronous oscillatorydynamics.3 4 5 6 7 802.55βV0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80123DVFigure 2.12: Bifurcation diagram of the local varible V with respect to thecoupling strength β and diffusivity D for the parameter set L = 1, k = 2,κ = 3,  = 0.15, µ = 0.9, α = 0.55, Kc = 1 and Lv = 0.8. The solid/dashedline represents stable/unstable steady state solution of V , open/solid circleindicates unstable/stable periodic solution, respectively. The steady-state(2.4.5) is the solid horizontal line. Left panel: V versus β for D = 0.1.The periodic solution branches shown correspond only to the synchronousoscillations. Stable synchronous oscillations and stable steady-states willcoexist only for some range of β. Right panel: V versus D for β = 4.The periodic solution branch that is unstable, with a saddle-node point atD ≈ 0.67, is the synchronous branch. The other periodic solution branch,with a saddle-node point at D ≈ 0.57 represents asynchronous oscillations.This plot shows that synchronous oscillations are unstable for β = 4, butthat stable asynchronous oscillations and stable steady-state solutions willco-exist in some range of D when β = 4.362.4. Nonlinear Coupling Between Compartments and BulkFigure 2.13: Numerical simulation of the coupled PDE-ODE system (2.1.1)with nonlinear coupling (2.4.1) for D = 0.5, β = 4, and L = 1. The otherparameter values are the same as in the caption of Fig. 2.12. The initialconditions are C(x, 0) = 2, v1 = 1, w1 = 0.3, v2 = 0.2, and w2 = 0.3. Stableasynchronous oscillations for C(x, t) are observed.2.4.1 Compartmental Dynamics Neglecting Bulk DiffusionAs shown above, a local stability analysis around the steady-state (2.4.5)does not provide any insight into the occurrence of oscillatory behavior of thecoupled ODE-PDE system (2.1.1) with coupling (2.4.1). In this subsection,we consider an ODE model in the compartment, where we have neglectedthe bulk diffusion process, and simply set P (C, V ) = q(V ) in (2.4.1). Theresulting ODE model is written asdVdt= f0(V,W )− V + βq(V ) , dWdt= (µ− f0(V,W )) ,f0(V,W ) ≡ αW +WV 2 , q(V ) ≡ V (V − µ)Lv + V 2.(2.4.6)A typical bifurcation diagram of this ODE system is shown in Fig. 2.14.From this numerically calculated bifurcation diagram we observe that thereare three types of critical points; three Hopf bifurcation (HB) points, thesaddle node (SN) point, and the transcritcal point where two steady-statebranches intersect (IS).To determine the location of these points we first determine the steady-states of (2.4.6), which satisfy(Ve − µ)(V 2e − βVe + Lv) = 0 . (2.4.7)372.4. Nonlinear Coupling Between Compartments and Bulk0 1 2 3 4 500.511.52βVFigure 2.14: Bifurcation diagram of the ODE system (2.4.6) versus β for theparameter set  = 0.15, µ = 0.4, α = 0.55, and Lv = 0.8. The solid/dashedline represents stable/unstable steady state solution of V .Therefore, Ve = µ is a steady-state, and there are two additional steady-statesolutions given byV ±e =β2±√(β2)2− Lv , (2.4.8)when Lv < β2/4. At the SN point, we have V +e = V−e , which givesβSN = 2√Lv , and V±e =√Lv . (2.4.9)For the parameter values in Fig. 2.14 we get βSN ≈ 1.789.At the IS point, since one of V ±e must equal µ, we obtain thatβ2±√(β2)2− Lv = µ , (2.4.10)which yields β = µ + Lv/µ. Since, µ − β/2 = (µ2 − Lv)(2µ), we concludethat V −e = µ when Lv > µ2, and V +e = µ when Lv < µ2. The parameter setin the caption of Fig. 2.14 corresponds to this first possibility. The IS pointoccurs at βIS =(0.42 + 0.8)/0.4 = 2.4.To determine the HB points, we calculate the trace and the determinantof the Jacobian matrix Je associated with (2.4.6) astr(Je) = fe0V − 1 + βq′(Ve)− fe0W , det(Je) = −fe0W(β q′(Ve)− 1).The Hopf bifurcation occurs when tr(Je) = 0 and det(Je) > 0, which givesβq′(Ve) = 1 + fe0W − fe0V , (2.4.11a)382.4. Nonlinear Coupling Between Compartments and Bulkprovided thatdet(Je) = −fe0W (fe0W − fe0V ) = [2Veµ− (α+ V 2e)2]> 0 . (2.4.11b)To determine the HB point off of the Ve = µ = 0.4 steady-state branch inFig. 2.14, we set Ve = µ in (2.4.11b) to calculate that det(Je) ≈ 0.0367 > 0.By using (2.4.6) for q(V ) to calculate q′(µ), we obtain from (2.4.11a) thatthe HB point βHB isβHB =(Lv + µ2)µ(α+ µ2)[α− µ2 + (α+ µ2)2]. (2.4.12)For the parameter set of Fig. 2.14 this yields βHB ≈ 1.574. The other twoHB points in Fig. 2.14, corresponding to bifurcations from the V ±e steady-states, are also readily calculated from (2.4.11). We find that the Hopfbifurcation on the v+ branch is at βHB ≈ 1.8641 with det(Je) ≈ 0.0554 > 0,while the Hopf bifurcation on the v− branch occurs at βHB ≈ 2.9884 withdet(Je) ≈ 0.0265 > 0.We conclude that the bifurcation diagram of the ODE system (2.4.6) doesshare only a few of the characteristics observed in the bifurcation diagramof Fig. 2.12 for the fully coupled compartmental-bulk problem (2.1.1) with(2.4.1). For both the ODE model and the fully coupled model, new branchesof steady-state solutions, other than the base-state Ve = µ, are possible.However, for the fully coupled problem, the branches of periodic solutionsare isolated in the sense that they do not arise from bifurcations off of thesteady-state solution branches.39Chapter 3Nonlinear Analysis Near theDouble Hopf BifurcationPointIn the previous chapter, parameter study of the cell-bulk coupled systemwith linear coupling shows the existence of a double Hopf bifurcation pointin the β versus D plane. In the following chapter, we provide an exten-sive analyze near such point with the full PDE-ODE coupled system. Theanalysis explains and predicts certain features of the parameter study ofthe system regarding the interaction of in-phase and anti-phase modes. Forexample, in parameter regions of bistability near the double Hopf point,there is an unstable invariant torus in the dynamics whose stable manifoldforms a boundary in phase space between the stable in-phase and anti-phasemodes. We express the coupled compartment-bulk system as an evolutionequation in an infinite-dimensional space, and use center manifold theoryto reduce the evolution to a four-dimensional local invariant manifold inthe infinite-dimensional space. This latter evolution is further reduced toa normal form, which is then used to make predictions about the in-phaseand anti-phase modes and their nonlinear interaction near the double Hopfbifurcation.In the following section §3.1, we restate the model system that describesthe two diffusively coupled cells. Then in section §3.2 we find in param-eter space the location of the double Hopf point, and calculate associatedeigenvalues and eigenvectors. In section §3.3 we describe the double Hopfbifurcation analysis and its results, and describe tests of its predictions usingAUTO and simulations.403.1. The Coupled Compartment-Bulk Diffusion Model3.1 The Coupled Compartment-Bulk DiffusionModelThe model we consider describes chemical reactions in two cells, and thediffusion of a signalling chemical in the extracellular space between the cells.We will take a one-dimensional spatial domain, and let C(x, t) denote theconcentration of the signalling chemical, where x denotes the spatial locationin the interval −L ≤ x ≤ +L representing the extracellular space, and t istime. We assume the signalling chemical diffuses and degrades while inthe extracellular space, and model this with the linear partial differentialequation∂C∂t(x, t) = D∂2C∂x2(x, t)− k C(x, t), −L < x < +L, (3.1.1)where D > 0 and k > 0 are the diffusion and degradation constants. Twocells, or compartments, are located at each of the boundaries of the extracel-lular space x = −L and x = +L. Inside each cell, the signalling chemical isinvolved in chemical reactions. We assume chemicals are well-mixed insideeach cell, and let V−(t) and V+(t) denote the concentration of the signallingchemical inside the cell at x = −L and the cell at x = +L, respectively. Thecells can exchange the signalling chemical with the extracellular space, andwe model this with a flux condition at each boundary of the extracellularspace,−D∂C∂x(−L, t) = κ [V−(t)− C(−L, t)] ,+D∂C∂x(+L, t) = κ[V+(t)− C(+L, t)],(3.1.2)where κ is a positive flux constant. Thus, if the concentration V± of thesignalling chemical inside a cell is higher than the concentration C(±L, ·) atthe corresponding boundary, there is a positive flux of the chemical out ofthe cell, into the extracellular space.Inside each cell, at x = −L and at x = +L, the signalling chemicalreacts with some intermediate chemical product, whose concentrations aredenoted W−(t) and W+(t), respectively. The reactions are governed byordinary differential equationsdV−dt(t) = f(V−(t),W−(t)) + β[C(−L, t)− V−(t)],dW−dt(t) = g(V−(t), W−(t)),(3.1.3)413.2. Linearized Stabilityinside the cell at x = −L, anddV+dt(t) = f(V+(t),W+(t)) + β[C(+L, t)− V+(t)],dW+dt(t) = g(V+(t), W+(t)),(3.1.4)inside the cell at x = +L, where  is a positive constant. The influence of theoutside concentration of the signalling chemical on the reaction dynamicsinside each cell is described by the coupling terms β[C(±L, ·)− V±], with apositive constant β that represents the coupling strength. We take identicalcells, so  and β are the same for each cell, and the functions f and grepresenting the reaction kinetics inside each cell are the same. For specifickinetics, we take the Sel’kov modelf(V, W ) = −V + αW + V 2W,g(V, W ) = µ− αW − V 2W, (3.1.5)where α and µ are positive constants. These kinetics have the property thatwhen the cells are isolated (β = 0) there is a unique steady state V0 = µ,W0 = µ/(α+ V20 ) which is stable, but when the cells are coupled there canbe stable oscillating solutions, so in this sense, oscillations are “conditional”.Our model system is (3.1.1)–(3.1.5), which in more mathematical terms canbe described as a pair of identical, diffusively coupled, conditional oscillators.Since the cells are identical, the model system (3.1.1)–(3.1.5) has a reflectionsymmetry, under spatial reflection x→ −x and exchange of cells.3.2 Linearized StabilityIn this section we study the eigenvalue problem that gives the linearizedstability of the steady state solution of the model system. From this weobtain parameter values that give Hopf points for two types of marginallystable synchronized linear oscillations, which we call anti-phase and in-phaseeigenvectors. In particular, we obtain parameter values for a double Hopfpoint, where both the anti-phase and in-phase eigenvectors are marginallystable.We find the steady state, or equilibrium, of the model system (3.1.1)–(3.1.5) asC(x, t) = Ce(x), V−(t) = V e, V+(t) = V e, W−(t) = W e, W+(t) = W e,(3.2.1)423.2. Linearized StabilitywhereCe(x) = Ce0cosh(Ω0x)cosh(Ω0L), V e =µ+ βCe01 + β, W e =µ1 + (V e)2,andΩ0 =√kD, Ce0 =κµκ+DΩ0(1 + β) tanh(Ω0L).Then defining deviations from the steady state byC(x, t) = Ce(x) + c(x, t), V±(t) = V e + v±(t), W±(t) = W e + w±(t),from (3.1.1)–(3.1.4) we obtain the corresponding differential equations forthe deviations∂c∂t= D∂2c∂x2− kc,dv−dt= f(V e + v−,W e + w−)− f(V e,W e) + β[c(−L, ·)− v−],dw−dt= [g(V e + v−,W e + w−)− g(V e,W e)],dv+dt= f(V e + v+,We + w+)− f(V e,W e) + β[c(+L, ·)− v+],dw+dt= [g(V e + v+,We + w+)− g(V e,W e)],(3.2.2)with boundary conditions−D ∂c∂x(−L, ·) = κ [v− − c(−L, ·)] ,+D∂c∂x(+L, ·) = κ [v+ − c(+L, ·)] , (3.2.3)We linearize (3.2.2)–(3.2.3) about the origin (which now corresponds to thesteady state) and obtain∂c∂t= D∂2c∂x2− kc,dv±dt= feV v± + feWw± + β[c(±L, ·)− v±],dw±dt= [geV v± + geWw±],(3.2.4)with the same boundary conditions±D ∂c∂x(±L, ·) = κ [v± − c(±L, ·)] , (3.2.5)433.2. Linearized Stabilitywhere feV , geV feW and geW are the partial derivatives of f and g, evaluatedat the steady state (3.2.1).To study the linearized stability of the steady state, we make the usualansatz c(x, t) = eλtη(x), v± = eλtϕ± and w± = eλtψ± in (3.2.4)–(3.2.5), andobtain the eigenvalue problemλη = Dη′′ − kη,λϕ± = feV ϕ± + feWψ± + β[η(±L)− ϕ±],λψ± = [geV ϕ± + geWψ±],(3.2.6)with boundary conditions±Dη′(±L) = κ [ϕ± − η(±L)] . (3.2.7)If Reλ < 0 for all eigenvalues λ, then the steady state is asymptoticallystable. We seek parameter values where the steady state is marginally stable:Reλ = 0 for finitely many eigenvalues, called critical eigenvalues, and Reλ <0 for all remaining eigenvalues. Near such parameter values, we expect thenonlinear system (3.2.2)–(3.2.3) will have bifurcations of solutions near thesteady state.Due to the reflection symmetry of (3.2.6)–(3.2.7), the eigenvectors comein two types, odd or “anti-phase” withη(−x) = −η(x), v− = −v+, w− = −w+,and even or “in-phase” withη(−x) = η(x), v− = v+, w− = w+,Solving the eigenvalue problem for anti-phase eigenvectors, we haveη−(x) = η01sinh(Ωλx)sinh(ΩλL)for some constant η01, whereΩλ =√k + λD,and the boundary condition at x = +L gives[κ+DΩλ coth(ΩλL)]η01 = κϕ−.443.2. Linearized StabilityTherefore the ϕ− and ψ− components of an anti-phase eigenvector satisfythe homogeneous system of linear equations[feV − p−(λ)− λ]ϕ− + feWψ− = 0,geV ϕ− + (geW − λ)ψ− = 0,(3.2.8)wherep−(λ) =DΩλ coth(ΩλL)κ+DΩλ coth(ΩλL).Taking the determinant of the coefficient matrix of (3.2.8) we obtain a tran-scendental equation for any eigenvalue λ for anti-phase eigenvectors[feV − p−(λ)− λ](geW − λ)− feW geV = 0. (3.2.9)Similarly, for in-phase eigenvectors we haveη+(x) = η0+cosh(Ωλx)cosh(ΩλL)and λ must satisfy[feV − p+(λ)− λ](geW − λ)− feW geV = 0, (3.2.10)wherep+(λ) =DΩλ tanh(ΩλL)κ+DΩλ tanh(ΩλL).We solve the eigenvalue equations (3.2.9) and (3.2.10) numerically, usingthe mathematical software package Maple. We putk = 1, L = 1, α = 0.9,  = 0.15, κ = 1, µ = 2, (3.2.11)and seek parameter values (“Hopf points”) that give purely imaginary eigen-valuesλ = iω1 or λ = iω2with anti-phase or in-phase eigenvectors, respectively. With D and β asfree parameters, we solve (3.2.9) to find a curve of Hopf points for anti-phase eigenvectors, and solve (3.2.10) to obtain another curve of Hopf points,for in-phase eigenvectors. These curves intersect, at a double Hopf point,and this intersection can be found by simultaneously solving (3.2.9) and(3.2.10), using starting values from the preliminary explorations with Autoon the finite-difference approximation. We obtain, for the double Hopf point,parameter valuesD = 0.555509 β = 0.508394, (3.2.12)453.2. Linearized Stabilitywithω1 = 0.811618, ω2 = 0.794334.We checked, using spatially discretized finite-difference approximations ofthe eigenvalue problem (3.2.6)–(3.2.7), for parameter values at the doubleHopf point (3.2.11)–(3.2.12), that (allowing for small discretization errors)there are four critical, purely imaginary simple eigenvalues λ = ±iω1,2, andall remaining eigenvalues have negative real parts bounded away from 0.If parameters are varied continuously, the eigenvalues change continu-ously. Therefore, if parameters are near the double Hopf point, there arefour simple eigenvalues near ±iω1,2, near the imaginary axis, which we stillcall critical eigenvalues. Near the double Hopf point, by continuity the re-maining eigenvalues still have negative real parts bounded away from 0.We conclude this section by introducing a vector notation, which makesthe subsequent bifurcation calculations more convenient. We letX(t) =c(x, t)v−(t)w−(t)v+(t)w+(t), (3.2.13)where for each t, X(t) belongs to a real infinite-dimensional function space Hconsisting of vectorsX(t) whose components satisfy the boundary conditions(3.2.5). We define the linear differential operator M byMX =D ∂2c∂x2− kcfeV v− + feWw− + β[c(−L, ·)− v−]geV v− + geWw−feV v+ + feWw+β[c(+L, ·)− v+]geV v+ + geWw+, (3.2.14)for all X(t) belonging to H. Then the linearized system (3.2.4)–(3.2.5) canbe written asX˙ = MX, (3.2.15)463.2. Linearized Stabilityfor X(t) belonging to H, where the dot denotes differentiation with respectto t. Setting X(t) = eλtq, whereq =η(x)ϕ−ψ−ϕ+ψ+belongs to H, the eigenvalue problem (3.2.6)–(3.2.7) is expressed asMq = λq, (3.2.16)For complex eigenvalues λ, we seek the corresponding eigenvectors q in thecomplexification of H. In particular, at the double Hopf point (3.2.11)–(3.2.12) we haveMq1 = iω1q1, Mq2 = iω2q2.The complex eigenvectors (up to multiplication by an arbitrary complexscalar) areq1 =η01 sinh(Ω1x)/sinh(Ω1L)−1−geV /(iω1 − geW )1geV /(iω1 − geW ), q2 =η02 cosh(Ω2x)/cosh(Ω2L)1geV /(iω2 − geW )1geV /(iω2 − geW ),(3.2.17)whereΩ1 =√k+iω1D , Ω2 =√k+iω2D , η01 =κκ+DΩ1 coth(Ω1L), η02 =κκ+DΩ2 coth(Ω2L).Generally, the critical eigenspace (or center subspace) T c is the real subspaceconsisting of the span of the real and imaginary parts of the (generalized)eigenvectors corresponding to all eigenvalues λ with Reλ = 0. In our caseit is the four-dimensional subspaceT c = span {Re q1, Im q1,Re q2, Im q2}.473.3. Double Hopf BifurcationFor later computational convenience we express the critical eigenspace incomplex notation asT c = {z1q1 + z¯1q1 + z2q2 + z¯2q2 : z1, z2 ∈ C}.Since all eigenvalues other than the four critical ones ±iω1,2 have nega-tive real parts, the complementary subspace to T c in H is T s, the infinite-dimensional stable subspace. If parameters changes smoothly, then all theeigenvalues change continuously, and simple eigenvalues change smoothly.Therefore, if parameters are near a double Hopf point, there are four simpleeigenvalues near ±iω1,2, near the imaginary axis, which we still call criti-cal eigenvalues, and the remaining eigenvalues still have negative real partsbounded away from 0.3.3 Double Hopf BifurcationIn the previous section, we found a double Hopf point, i.e. parameter valueswhere the critical eigenvalues for the linearization of the model system aretwo pairs of purely imaginary eigenvalues ±iω1, ±iω2. In the nonlinearmodel system itself, for parameter values near the double Hopf point, weexpect bifurcations of nonlinear modes of oscillations that resemble the linearanti-phase and in-phase eigenvectors. This is confirmed by a bifurcationanalysis, which also tells us the stabilities of the nonlinear anti-phase andin-phase modes, and how the modes interact near the double Hopf point.Key to this analysis is the reduction of the infinite-dimensional model systemnear the steady state to a four-dimensional normal form whose dynamics canbe more easily determined.We extend the vector notation introduced in the previous section to thenonlinear problem, and write the model system (3.1.1)–(3.1.5) asX˙ = MX + 12!B(X,X) +13!C(X,X,X), (3.3.1)for X(t) belonging to H, where X(t) is given by (3.2.13) and the lineardifferential operator M is given by (3.2.14). The operators B and C are483.3. Double Hopf Bifurcationsymmetric bilinear and trilinear forms, respectively, given byB(Xa, Xb) =0b1−b1b2−b2, C(Xa, Xb, Xc) =0c1−c1c2−c2.whereb1 = 2Weva−vb− + 2Ve(va−wb− + vb−wa−), b2 = 2Weva+vb+ + 2Ve(va+wb+ + vb+wa+),c1 = 2va−vb−wc− + 2vb−vc−wa− + 2vc−va−wb−, c2 = 2va+vb+wc+ + 2vb+vc+wa+ + 2vc+va+wb+,Then12!B(X,X) =0W ev2− + V ev−w−−(W ev2− + V ev−w−)W ev2+ + Vev+w+−(W ev2+ + V ev+w+), 13!C(X,X,X) =0v2−w−−v2−w−v2+w+−v2+w+.are the quadratic and cubic terms, respectively, of the model system.At the double Hopf point (3.2.11)–(3.2.12), we recall that the linear op-erator M has the four critical eigenvalues ±iω1,2 on the imaginary axis inthe complex plane, and the remaining eigenvalues of M are in the left com-plex half-plane, bounded away from the imaginary axis. In this situation,for parameter values near the double Hopf point, the nonlinear evolutionequation (3.3.1) possesses a four-dimensional invariant local center manifoldW cloc in the function space H, that is tangent to the critical eigenspace Tc atthe double Hopf point. Furthermore, all solutions of (3.3.1) near the steadystate decay exponentially rapidly, as t increases, to the local center manifoldW cloc. Therefore, the local long-term dynamics of the entire system (3.3.1)is governed by a four-dimensional system of ordinary differential equationsthat describes the dynamics restricted to W cloc. In fact, only low-order termsin the Taylor series expansion of this system are required. Finally, a stan-dard procedure of introducing coordinate changes reduces the system of493.3. Double Hopf Bifurcationdifferential equations to a simpler but equivalent one, called a normal form.This normal form is easier to analyze, and predicts the local dynamics ofthe entire infinite-dimensional system (3.3.1).In the Appendix, we give some details of the reduction of the evolutionequation (3.3.1) to the normal form. The computations are analytical, as-sisted by the mathematical software package Maple. Near the double Hopfpoint, the normal form is a four-dimensional system of ordinary differentialequations, written in complex notation asζ˙1 = λ1ζ1 +G2100ζ21 ζ¯1 +G1011ζ1ζ2ζ¯2 +O(‖µ‖‖ζ‖3 + ‖ζ‖5),ζ˙2 = λ2ζ2 +H1110ζ1ζ¯1ζ2 +H0021ζ22 ζ¯2 +O(‖µ‖‖ζ‖3 + ‖ζ‖5),(3.3.2)whose solutions ζ1(t), ζ2(t) are complex numbers that, to leading order, rep-resent the evolving amplitudes and phases of the anti-phase and in-phaseoscillatory modes in the nonlinear system (3.3.1). The critical eigenvaluesof the linearization M near the double Hopf point are λ1 and λ2, so atthe double Hopf point itself we have λ1 = iω1, λ2 = iω2. Near the dou-ble Hopf point, the real parts of the critical eigenvalues µj = Reλj serveas “unfolding” parameters that usefully quantify small deviations from thedouble Hopf point. The higher-order Taylor series terms in the expansionsO(‖µ‖‖ζ‖3 + ‖ζ‖5), where µ = (µ1, µ2) and ζ = (ζ1, ζ¯1, ζ2, ζ¯2), are not ex-plicitly needed for our work. The four coefficients Gjklm and Hjklm of thecubic terms in the normal form are calculated with the help of Maple, andwe evaluate them at the double Hopf point,G2100 = −3.07849 + i0.00166, G1011 = −5.89627 + i2.80222,H1110 = −6.00121− i0.14896, H0021 = −2.90063 + i1.38790.(3.3.3)The analysis of the normal form (3.3.2) is described in several textbookson bifurcation theory. Here we briefly summarize the relevant parts of thetreatment in [39]. If we take polar representations ζ1 = r1eiφ1 , ζ2 = r2eiφ2 ,and truncate higher-order terms, then in polar coordinates (r1, r2, φ1, φ2)the normal form (3.3.2) can be written asr˙1 = r1(µ1 + p11r21 + p12r22),r˙2 = r2(µ2 + p21r21 + p22r22),φ˙1 = ω1,φ˙2 = ω2,(3.3.4)wherep11 = ReG2100, p12 = ReG1011, p21 = ReH1110, p22 = ReH0021.503.3. Double Hopf BifurcationWe point out that the truncated normal form (3.3.4) is an approximationof the normal form (3.3.2) due to missing higher-order terms in the Taylorseries expansions, but the approximation turns out to be sufficiently accurateto predict the existence and stability of bifurcating solutions.We see in the truncated normal form (3.3.4) that the first pair of equa-tions is independent of the second pair and thus the bifurcations of (3.3.4)are completely determined by the two equations in rj , where rj representthe amplitudes of the anti-phase and in-phase modes:r˙1 = r1(µ1 + p11r21 + p12r22),r˙2 = r2(µ2 + p21r21 + p22r22).(3.3.5)Since we have p11 = −3.07849 and p22 = −2.90063, the normal form fallsinto the “simple” case of [39] (p. 359), where p11p22 > 0 and no fifth-orderterms are needed in the amplitude equations (3.3.5). We observe that thesystem (3.3.5) has a trivial equilibrium E0 = (0, 0) for all µ1,2. Moreover,there can be as many as three nontrivial equilibria. Equilibria on the coor-dinate axesE1 = (r1, 0), r1 > 0; E2 = (0, r2), r2 > 0,wherer1 =√µ1−p11 ; r2 =√µ2−p22exist if µ1 > 0, µ2 > 0, respectively. Another equilibriumE3 = (r1, r2), r1 > 0, r2 > 0,wherer1 =√−µ1 + θµ2−p11(θδ − 1) , r2 =√δµ1 − µ2−p22(θδ − 1) , (3.3.6)andθ =p12p22= 2.03276, δ =p21p11= 1.94940, (3.3.7)exists if both −µ1 + θµ2 > 0 and δµ1−µ2 > 0. The equilibria E1,2 bifurcatefrom the origin E0 at the bifurcation linesH1 = {(µ1, µ2)| µ1 = 0}, H2 = {(µ1, µ2)| µ2 = 0}, (3.3.8)and E3 bifurcates from E2 or E1 on the bifurcation linesT1 = {(µ1, µ2)| µ1 = θµ2, µ2 > 0},T2 = {(µ1, µ2)| µ2 = δµ1, µ1 > 0},(3.3.9)513.3. Double Hopf Bifurcationrespectively. We plot the parametric portraits of (3.3.5) in Figure 3.1. Inthe left panel, the four lines H1, H2, T1 and T2 divide the (µ1, µ2) parameterplane into six open regions indicated by roman numerals. The correspondingphase portraits of the amplitude system (3.3.5) are also shown within eachopen region. The H1 (red dash-dot) line is the vertical µ2-axis, while H2(black solid) is the horizontal µ1-axis. The T1 (magenta dashed) and T2(blue dotted) lines separate regions III, IV and V. In region I, the amplitudesystem (3.3.5) has the unique equilibrium E0 and it is asymptotically stable.When entering region II (VI) from region I, crossingH1 (H2), the equilibriumE1 (E2) bifurcates from E0 and is asymptotically stable, while E0 is unstable.When entering region III (V) from region II (VI), crossing H2 (H1), another,unstable, equilibrium E2 (E1) bifurcates from E0 while E1 (E2) remains atlarge amplitude and asymptotically stable, and E0 is unstable. Finally, inregion IV, there is bistability as the two equilibria E1 and E2 are bothasymptotically stable. A fourth equilibrium E3 exists and is unstable, whileE0 is unstable. Although E3 is unstable, it has an important effect on theoverall dynamics. The unstable manifold of E3 forms the boundary betweenthe basins of attraction of the two stable equilibria E1 and E2. Thus theeventual limiting state of a generic trajectory depends on the location of itsinitial value relative to the unstable manifold of E3. For more details, see[39]. If we fix other parameters at value (3.2.11) and only change β or Dnear the double Hopf point, the curves corresponding to Hj , Tj in the (β,D)plane are shown in the right panel of Figure 3.1. The red dash-dot curvecorresponds to H1, and the black solid red curve corresponds to H2. Thedashed magenta and dotted blue lines are tangent lines at the double Hopfpoint to the curves corresponding to T1 and T2, respectively.Restoring the angular variables to (3.3.5) to recover the truncated normalform (3.3.4), the equilibria of (3.3.5) get different interpretations. The originE0 is still an equilibrium at the origin, but E1 and E2 are cycles, or periodicsolutions of (3.3.4), while E3 for (3.3.4) is a two-dimensional invariant torus.Their stability properties remain the same. Thus the lines Hj correspond toHopf bifurcations, and the lines Tj to torus (or Neimark-Sacker) bifurcations.Because nondegeneracy conditions are satisfied in our case, restoring thehigher-order terms to the truncated normal form (3.3.4) to return to (3.3.2)changes the bifurcation results only subtly. The torus bifurcation lines Tjbecome torus bifurcation curves T1 : µ1 = θµ2 + O(µ22) and T2 : µ2 =δµ1 +O(µ21) tangent at the origin to the lines (3.3.9), while solutions on theinvariant two-torus are slightly changed, but the two-torus persists as aninvariant manifold with the same stability type.Finally, transferring the bifurcation and stability results to the original523.3. Double Hopf Bifurcationµ2µ1VIIV IVIIIII0.5 0.51 0.520.50.550.6βDVIVIVIIIIIIFigure 3.1: Parametric portrait of the amplitude equations (3.3.5) in the(µ1, µ2) plane near the origin with phase portraits of the amplitude equations(left panel), and the corresponding parametric portrait in the (β,D) planenear the double Hopf point (right panel). In the left panel, the red dash-dot vertical µ2-axis is H1 : µ1 = 0, the black solid horizontal µ1-axis isH2 : µ2 = 0, the magenta dashed line is T1 : µ2 = δµ1 and the blue dottedline is T2 : µ1 = θµ2. In the right panel, the red dash-dot curve (Hopfbifurcation of anti-phase modes) corresponds to H1, the black solid curve(Hopf bifurcation of in-phase modes) corresponds to H2, the magenta dashedline (tangent at the double Hopf point to a curve of torus bifurcations fromanti-phase modes) corresponds to T1 and the blue dotted line (tangent atthe double Hopf point to a curve of torus bifurcations from in-phase modes)corresponds to T2. Other parameter values are fixed at values (3.2.11). Notethe right panel corresponds to the enlarged version of Fig. 2.2 in chapter §2near the intersection of the blue solid curve and the red dashed curve.533.3. Double Hopf Bifurcationmodel system (3.3.1), or equivalently (3.1.1)–(3.1.5), is straightforward. Theorigin E0 corresponds to the steady state (3.2.1), E1 and E2 correspond tononlinear oscillating anti-phase and in-phase modes, and E3 corresponds toan invariant two-torus or modulated oscillations, while the stability typesremain the same. The lines H1 and H2 correspond to curves of Hopf bifurca-tions from the steady state, of anti-phase and in-phase modes, respectively.The line T1 corresponds to a curve of torus bifurcations from the anti-phasemode and T2 corresponds to torus bifurcations from the in-phase mode. So-lutions on the invariant two-torus are characterized by two frequencies, onenear ω1 and the other near ω2.To check our results we consider parameter paths near the double Hopfpoint in the (β,D) plane and plot corresponding bifurcation diagrams ob-tained by using Auto on a spatially discretized finite-difference approx-imation of the model system (3.1.1)–(3.1.5). In Figure 3.2, we consideran elliptical parameter path around the double Hopf point, setting β =βc+ 0.005 cos(θ) and D = Dc+ 0.025 sin(θ), and increasing the path param-eter θ from −pi to pi. This parameter path, shown as a green curve in theinset panel, starts directly to the left of the double Hopf point, traces theellipse in a counterclockwise direction as θ increases, starting in region I andvisiting the regions II, III, IV, etc. in sequence before returning to regionI. The main panel of Figure 3.2 shows the bifurcation diagram obtained byAUTO for the finite-difference approximation of the model system, with thepath parameter θ plotted on the horizontal axis, and the V− component ofthe vector solutions on the vertical axis. As the path parameter θ increasesfrom −pi, the steady state is stable (solid curve), then first loses stability(beginning of dashed curve) as the stable anti-phase mode bifurcates (solidcircles indicating maximum and mimimum values of V−(t) on the periodicsolution) from the steady state. The anti-phase mode remains stable andwith a large amplitude as the unstable (open circles) in-phase mode bifur-cates from the unstable (dashed curve) steady state. As θ increases further,the in-phase mode gains stability (transition from open circles to closedcircles), and then there is an interval where both anti-phase and in-phasemodes are stable (the solution branches appear to cross but this is onlybecause only one component of each vector solution is plotted). Then theanti-phase mode loses stability (transition from closed circles to open cir-cles), then as θ increases there is a Hopf bifurcation back to the unstable(dashed curve) steady state. Finally there is a Hopf bifurcation of the sta-ble in-phase mode (solid circles) back to the steady state as the steady stategoes from unstable (dashed curve) to stable (solid curve). AUTO detectsstability changes of the anti-phase and in-phase modes when a pair of com-543.3. Double Hopf Bifurcationplex conjugate Floquet multipliers for the oscillating modes crosses the unitcircle in the complex plane, which is characteristic of a torus bifurcation.The sequence of stability changes and bifurcations found by AUTO is aspredicted by the normal form analysis. Although the amplitude equations(due to truncation of higher-order terms) and the AUTO bifurcation com-putations (due to finite differences) are both approximations of the modelsystem, the numerical values of the bifurcation points agree well, close tothe double Hopf point.−pi 0 pi1.551.61.651.71.751.8θV−βDFigure 3.2: Bifurcation diagram for the model system, for a parameter patharound the double Hopf point, β = βc + 0.005 cos(θ), D = Dc + 0.025 sin(θ),increasing θ from −pi to pi, with other parameters fixed at the values (3.2.11).The inset panel shows the elliptical parameter path plotted in green in the(β,D) plane, together with the bifurcation curves obtained from the nor-mal form. The parameter path starts directly to the left of the double Hopfpoint, in the region I where the steady state is stable. As θ increases (movingcounterclockwise from the leftmost point on the green ellipse), the param-eter path crosses the red dash-dot Hopf bifurcation curve, the black solidHopf bifurcation curve, the blue dotted torus bifurcation curve, the magentadashed torus bifurcation curve, the red Hopf bifurcation curve, and finallythe black Hopf bifurcation curve before returning to the starting point. Themain panel shows the bifurcation diagram for the parameter path obtainedusing AUTO on the spatially discretized finite-difference approximation ofthe model system, with the path parameter θ plotted on the horizontal axis,the V− component of the vector solution on the vertical axis. See the maintext for the coding of solution type and stability by line style.AUTO is able to detect stability changes of periodic solutions that cor-553.3. Double Hopf Bifurcationrespond to torus bifurcations, but is unable to continue along branches ofinvariant tori. To look for invariant tori where their existence is predicted bythe normal form analysis, we simulated directly the model system (3.1.1)–(3.1.5) with finite differences in both space and time. Although the tori areunstable, if initial conditions are chosen close enough to an invariant torus,the solution will stay close to a solution on the unstable torus for some timebefore the exponentially growing drift apart becomes noticeable. We takeβ = 0.509 and D = 0.55486, which, according to the normal form, is in theparameter region IV between the two torus bifurcation curves, where there isbistability due to both the anti-phase and in-phase modes being asymptoti-cally stable, and an unstable invariant torus. We choose the initial conditioncorresponding toX(0) = 2Re(r1q1 + r2q2), (3.3.10)recalling that the vector X(t) represents the deviation of variables fromthe steady state (3.2.1), and r1, r2 are the amplitudes (3.3.6)–(3.3.7) givenby the equilibrium E3 of the amplitude equations that corresponds to theinvariant torus. Since the parameters are close to the double Hopf point, wereason that neglecting higher-order terms in the amplitude equations and inthe local center manifold should not give seriously large errors, and therefore(3.3.10) represents an initial condition close to the unstable invariant torus.The simulated results appear to validate this choice of initial condition. Aplot of the time evolution of V− component of the solution is shown in theleft panel of Figure 3.3. For a reasonably long time the numerical solutionexhibits oscillations characterized by two periods which correspond to thetwo oscillating frequencies ω1 and ω2 at the double Hopf point. The signaldisplays a phenomenon similar to that of beats or amplitude modulationthat occurs when two linear oscillations with nearly the same frequenciesare added, with a fast “carrier” frequency |ω1 +ω2|/2 and a slow modulated“envelope” frequency |ω1 − ω2|. The power spectrum from an FFT analysisof the time series of the numerical solution is shown in the right panel ofFigure 3.3. The two peaks on the FFT plot indicate the two main frequencycomponents from the time series in the left panel, and the peak locations onthe horizontal axis agree with the values of ω1, ω2. This is consistent withthe predictions of the normal form analysis.563.3. Double Hopf Bifurcation0 500 1000 1500 20001.621.641.661.681.71.72TV−0 1 210−1510−1010−5100ω0.75 0.8 0.85Figure 3.3: The time evolution of the V− component of a simulated solutionof the model system with initial condition (3.3.10) (left) and its correspond-ing power spectrum obtained from FFT analysis (right). In the left panel,we can observe two periods from the series. The shorter period correspondsto angular frequency |ω1 + ω2|/2 and the longer period corresponds to an-gular frequency |ω1 − ω2|. The left inset panel shows the simulation resultson a short time scale that more clearly resolves the rapid oscillations cor-responding to the shorter period. In the right panel, the power spectrumof the solution is plotted, where the horizontal axis is angular frequency ω.Two peaks in the power spectrum at ω = 0.79 and ω = 0.81 are clearlyvisible(other peaks correspond to integer linear combinations of ω1 and ω2).The right inset panel shows a more detailed graph near ω = 0.8.57Chapter 4Generalized Model in OneDimensional SpaceThe goal of this chapter is to formulate and analyze a general class of coupledmembrane-bulk dynamics in a simplified 1-D spatial domain.We first construct a general model system that describes the couplingof two dynamically active membranes, on which n species undergo chemicalreactions, through the bulk diffusion in a one dimensional finite domain oflength 2L in §4.1.In §4.2 we construct a steady-state solution for the general model systemthat is symmetric about the midline x = L. The analytical construction ofthis symmetric steady-state solution is reduced to the problem of determin-ing roots to a nonlinear algebraic system involving both the local membranekinetics and the nonlinear feedback and flux functions. We then formulatethe linear stability problem associated with this steady-state solution. Inour stability theory, we must allow for perturbations that are either sym-metric or anti-symmetric about the midline, which leads to the possibilityof either synchronous (in-phase) or asynchronous (out-of-phase) instabilitiesin the two membranes. By using a matrix determinant lemma for rank-oneperturbations of a matrix, we show that the eigenvalue parameter associ-ated with the linearization around the steady-state satisfies a rather simpletranscendental equation for either the synchronous or asynchronous mode.In §4.3 we analyze in detail the spectrum of the linearized problem as-sociated with a one-component membrane dynamics. For the infinite-lineproblem, corresponding to the limit L→ +∞, in §4.3.1 we use complex anal-ysis together with a rigorous winding number criterion to derive sufficientconditions, in terms of properties of the reaction-kinetics and nonlinear feed-back and flux, that delineate parameter ranges where Hopf bifurcations dueto coupled membrane-bulk dynamics will occur. Explicit formulae for theHopf bifurcation values, in terms of critical values of τ , which characterizethe time scale of bulk decay in the general model system, are also obtained.In §4.3.1 further rigorous results are derived that establish parameter rangeswhere no membrane oscillations are possible. For the finite-domain prob-58Chapter 4. Generalized Model in One Dimensional Spacelem, and assuming a one-component membrane dynamics, we show in §4.3.2that some of the rigorous results for the infinite-line problem, as derived in§4.3.1, are still valid. However, in general, for the finite-domain problemnumerical computations of the winding number are needed to predict Hopfbifurcation points and to establish parameter ranges where the steady-statesolution is linearly stable.We remark that for the case of a one-component membrane dynamics,the eigenvalue problem derived in §4.3.1, characterizing the linear stabilityof the coupled membrane-bulk dynamics, is remarkably similar in form tothe spectral problem that arises in the stability of localized spike solutionsto reaction-diffusion (RD) systems of activator-inhibitor type (cf. [57] and[79] and the references therein). More specifically, on the infinite-line, thespectral problem for our coupled membrane-bulk dynamics is similar to thatstudied in §3.1 of [57] for a class of activator-inhibitor RD systems.For a one-component membrane dynamics, in §4.4 we illustrate the the-ory of §4.3 for determining Hopf bifurcation points corresponding to the on-set of either synchronous or asynchronous oscillatory instabilities. For theinfinite-line problem, where these two instability thresholds have coalescedto a common value, we illustrate the theoretical results of §4.3.1 for theexistence of a Hopf bifurcation point. For the finite-domain problem, wherethe two active membranes are separated by a finite distance 2L, numericalcomputations of the winding-number are used to characterize the onset ofeither mode of instability. The theory is illustrated for a class of feedbackmodels in §4.4.1, for an exactly solvable model problem in §4.4.2, for a kinaseactivity regulation model in §4.4.3, and for two specific biological systemsin §4.4.4. The biological systems in §4.4.4 consist of a model of hormonalactivity due to GnRH neurons in the hypothalamus (cf. [32], [47], and [17]),and a model of quorum sensing behavior of Dictyostelium (cf. [19]). For theproblems in §4.4.1, 4.4.2, 4.4.4, we supplement our analytical theory withnumerical bifurcation results, computed from the coupled membrane-bulkPDE-ODE system using the bifurcation software XPPAUT [16]. For theDictyostelium model and the model in §4.4.2, our results shows that thereis a rather large parameter range where stable synchronous membrane os-cillations occur. Full numerical computations of the PDE-ODE system ofcoupled membrane-bulk dynamics, undertaken using a method-of-lines ap-proach, are used to validate the theoretical predictions of stable synchronousoscillations.In §4.5 we consider a specific coupled membrane-bulk model, having twoactive components on each membrane. For the case where the two mem-branes are identical, and have a common value of the coupling strength to59Chapter 4. Generalized Model in One Dimensional Spacethe bulk medium, we use a numerical winding number argument to predictthe onset of either a synchronous or an asynchronous oscillatory solutionbranch that bifurcates from the steady-state solution. The numerical bifur-cation package XPPAUT [16] shows that there is a parameter range wherethe synchronous solution branch exhibits bistable behavior. In contrast,when the coupling strengths to the two membranes are different, we showthat the amplitude ratio of the oscillations in the two membranes can berather large, with one membrane remaining, essentially, in a quiescent state.For the case of a one-component membrane dynamics on a finite do-main, in §4.6 we formulate and then implement a weakly nonlinear multipletime-scale theory to derive an amplitude equation that characterizes whethera synchronous oscillatory instability is subcritical or supercritical near theHopf bifurcation point. For a specific choice of the nonlinearities, corre-sponding to the model considered in §4.4.2, theoretical predictions basedon the amplitude equation are then confirmed with full bifurcation resultscomputed using XPPAUT (cf. [16]). Similar analysis is applied to the Dic-tyostelium model. Moreover, for the model system considered in §4.4.2,time-dependent full numerical computations of the coupled membrane-bulkPDE-ODE system are performed to show a delayed triggered synchronousoscillation arising from the slow passage of a parameter through a Hopf bifur-cation point. Similar delayed bifurcation problems in a purely ODE contexthave been studied in [1], [2], [36], and [49] (see the references therein).We emphasize that the theoretical challenge and novelty of our weaklynonlinear analysis in §4.6 is that both the differential operator and theboundary condition on the membrane for the linearized problem involves theeigenvalue parameter. This underlying spectral problem, with an eigenvalue-dependent boundary condition, is not self-adjoint and is rather non-standard.Motivated by the theoretical approach developed in [18] to account foreigenvalue-dependent boundary conditions, we introduce an extended op-erator L, and an associated inner product, from which we determine thecorresponding adjoint problem. In this way, we formulate an appropriatesolvability condition in Lemma 4.6.1 that is one of the key ingredients, inour multiple time-scale analysis, for deriving the amplitude equation charac-terizing the branching behavior of synchronous oscillations near onset. Weremark that a similar methodology of introducing an extended operator totreat a transcritical bifurcation problem involving an eigenvalue-dependentboundary condition, which arises in a mathematical model of thermoelas-tic contact of disc brakes, was undertaken in [60] and [61]. However, toour knowledge, there has been no previous work for the corresponding Hopfbifurcation problem of the type considered herein.604.1. Model Formulation4.1 Model FormulationIn our simplified 1-D setting, we assume that there are two dynamicallyactive membranes, located at x = 0 and x = 2L, that can release a specificsignaling molecule into the bulk region 0 < x < 2L, and that this secretionis regulated by both the bulk concentration of that molecule together withits concentration on the membrane. In the bulk region, we assume that thesignaling molecule undergoes passive diffusion with a specified bulk decayrate. If C(x, t) represents the concentration of the signaling molecule in thebulk, then its spatial-temporal evolution in this region is governed by thedimensionless modelτCt = DCxx − C , t > 0 , 0 < x < 2L ,DCx(0, t) = G(C(0, t), u1(t)) , −DCx(2L, t) = G(C(2L, t), v1(t)) ,(4.1.1a)where τ > 0 is a time-scale for the bulk decay and D/τ > 0 is the constantdiffusivity. On the membranes x = 0 and x = 2L, the fluxes G(C(0, t), u1)and G(C(2L, t), v1) model the influx of signaling molecule into the bulk,which depends on the bulk concentrations C(0, t) and C(2L, t) at the twomembranes together with the local concentrations u1(t) and v1(t) of thesignaling molecule on the membranes. We assume that on each membrane,there are n species that can interact, and that their dynamics are describedby n-ODE’s of the formdudt= F(u) + βP(C(0, t), u1)e1 , dvdt= F(v) + βP(C(2L, t), v1)e1 ,(4.1.1b)where e1 ≡ (1, 0, . . . , 0)T . Here, u(t) ≡ (u1(t), . . . , un(t))T and v(t) ≡(v1(t), . . . , vn(t))T represents the concentration of the n species on the twomembranes and F(u) is the vector nonlinearity modeling the chemical ki-netics for these membrane-bound species. In our formulation (4.1.1b), onlyone of these internal species, labeled by u1 and v1 at the two membranes,is capable of diffusing into the bulk. The coupling to the bulk is modeledby the two feedback terms βP(C(0, t), u1) and βP(C(2L, t), v1), where thecoupling parameter β models the strength of the membrane-bulk exchange.4.2 The Steady-State Solution and theFormulation of the Linear Stability ProblemIn this section we construct a steady-state solution for (4.1.1), and thenformulate the associated linear stability problem. In (4.1.1), we have as-614.2. The Steady-State Solution and the Formulation of the Linear Stability Problemsumed for simplicity that the two membranes have the same kinetics andmembrane-bulk coupling mechanisms. As such, this motivates the construc-tion of a steady-state solution for (4.1.1) that is symmetric with respect tothe midline x = L of the bulk region. The corresponding symmetric steady-state bulk solution Ce(x) and the membrane-bound steady-state concentra-tion field ue satisfyDCexx − Ce = 0 , 0 < x < L ;Cex(L) = 0 , DCex(0) = G(Ce(0), u1e),F(ue) + βP(Ce(0), u1e)e1 = 0 .(4.2.1)We readily calculate thatCe(x) = C0ecosh[ω0(L− x)]cosh(ω0L), ω0 ≡ 1/√D , (4.2.2a)where C0e ≡ Ce(0) and ue are the solutions to the n+1 dimensional nonlinearalgebraic system−C0e tanh(ω0L) = ω0G(C0e , u1e), F(ue) + βP(C0e , u1e)e1 = 0 .(4.2.2b)In general it is cumbersome to impose sufficient conditions on F , P, andG, guaranteeing a solution to (4.2.2b). Instead, we will analyze (4.2.2b) forsome specific models below in §4.4 and in §4.5.To formulate the linear stability problem, we introduce the perturbationC(x, t) = Ce(x) + eλtη(x) , u(t) = ue + eλtφ ,into (4.1.1) and linearize. In this way, we obtain the eigenvalue problemτλη = Dηxx − η , 0 < x < L ; Dηx(0) = Gecη0 +Geu1φ1 ,Jeφ+ β(Pec η0 + Peu1φ1)e1 = λφ .(4.2.3)Here we have defined η0 ≡ η(0), Gec ≡ Gc(C0e , u1e), Geu1 ≡ Gu1(C0e , u1e),Pec ≡ Pc(C0e , u1e), and Peu1 ≡ Pu1(C0e , u1e). In addition, Je is the Jacobianmatrix of the nonlinear membrane kinetics F evaluated at ue.The formulation of the linear stability problem is complete once we im-pose a boundary condition for η at the midline x = L. There are twochoices for this boundary condition. The choice η(L) = 0 corresponds to ananti-phase asynchronization of the two membranes (asymmetric case), whilethe choice ηx(L) = 0 corresponds to an in-phase synchronization of the two624.2. The Steady-State Solution and the Formulation of the Linear Stability Problemmembranes. We will consider both anti-phase and in-phase instabilities inour analysis.For the synchronous mode we solve (4.2.3) with ηx(L) = 0 to obtain thatη(x) = η0cosh[Ωλ(L− x)]cosh(ΩλL), Ωλ ≡√1 + τλD, (4.2.4)where we have specified the principal branch of the square root if λ is com-plex. Upon substituting (4.2.4) into the boundary condition for η on x = 0in (4.2.3), we readily determine η0 in terms of φ1 asη0 = −Geu1φ1Gec +DΩλ tanh(ΩλL). (4.2.5)We then substitute (4.2.5) into the last equation of (4.2.3), and rewrite theresulting expression in the form(Je − λI)φ = p+(λ)φ1e1 ,p+(λ) ≡ β(Geu1Pec − Peu1Gec − Peu1DΩλ tanh(ΩλL)Gec +DΩλ tanh(ΩλL)).(4.2.6)Similarly, for the asynchronous mode we solve (4.2.3) with η(L) = 0 togetη(x) = η0sinh[Ωλ(L− x)]sinh(ΩλL).Upon applying the boundary condition for η at x = 0 from (4.2.3), we canwrite η0 in terms of φ1 asη0 = −Geu1φ1Gec +DΩλ coth(ΩλL). (4.2.7)Upon substituting this expression into the last equation of (4.2.3), we caneliminate η0 to obtain(Je − λI)φ = p−(λ)φ1e1 ,p−(λ) ≡ β(Geu1Pec − Peu1Gec − Peu1DΩλ coth(ΩλL)Gec +DΩλ coth(ΩλL)).(4.2.8)In summary, we conclude that an eigenvalue λ and eigenvector φ as-sociated with the linear stability of the symmetric steady-state solution(Ce(x), ue) is determined from the matrix system(Je − λI − p±(λ)E)φ = 0 , E ≡ e1eT1 , (4.2.9)634.2. The Steady-State Solution and the Formulation of the Linear Stability Problemwhere e1 ≡ (1, 0, . . . , 0)T . Here p+(λ) and p−(λ) are defined for the syn-chronous and asynchronous modes by (4.2.6) and (4.2.8), respectively. Wenow seek values of λ for which (4.2.9) admits nontrivial solutions φ 6= 0.These values of λ satisfy the transcendental equationdet(Je − λI − p±(λ)E)= 0 . (4.2.10)Since E is an n×n rank-one matrix, the transcendental equation (4.2.10)for the eigenvalue λ can be simplified considerably by using the followingwell-known Matrix Determinant Lemma:Lemma 4.2.1 Let A be an invertible n× n matrix and let a and b be twocolumn vectors. Then,det(A+ abT)=(1 + bTA−1a)det(A) . (4.2.11)Therefore, (A+abT )φ = 0 has a nontrivial solution if and only if bTA−1a =−1.Proof: We start the proof with a special choice A = I and consider thefollowing equality I 0bT 1 I + abT a0 1 I 0−bT 1 = I a0 1 + bTa .(4.2.12)If we take determinants on both sides, the right hand side gives 1 + bTa.The determinant of the left hand side is the product of the determinants ofthree matrices. Note the first and third matrices are triangle matrices withunit diagonal, then their determinants are 1. Thus it follows thatdet(I + abT ) = 1 + bTa . (4.2.13)Then for a general invertible matrix A, we havedet(A+ abT ) = det(A) det(I +A−1abT ) = det(A)(1 + bTA−1a) . (4.2.14)This completes the proof of the lemma. Applying this lemma to (4.2.10) and (4.2.9), where we identify A ≡Je − λI, a ≡ −p±e1, and b ≡ e1, we conclude that if λ is not an eigenvalueof Je, then λ must satisfy1− p±(λ)eT1 (Je − λI)−1 e1 = 0 . (4.2.15)644.3. One-Component Membrane DynamicsTo simplify (4.2.15), we write (Je − λI)−1 in terms of the cofactor matrixM as(Je − λI)−1 = 1det(Je − λI)MT ,where the entries Mij of M are the cofactors of the element ai,j of thematrix Je − λI. Since eT1 (Je − λI)−1 e1 = M11/det (Je − λI), we obtainthat (4.2.15) reduces to the following more explicit transcendental equationfor λ:1− p±(λ) M11(λ)det (Je − λI) = 0 , (4.2.16a)whereM11(λ) ≡ det∂F2∂u2∣∣∣u=ue− λ, · · · , ∂F2∂un∣∣∣u=ue· · · , · · · , · · ·∂Fn∂u2∣∣∣u=ue, · · · , ∂Fn∂un∣∣∣u=ue− λ . (4.2.16b)Here F2, . . . ,Fn denote the components of the vector F ≡ (F1, . . . ,Fn)Tcharacterizing the membrane kinetics.For the special case of a two-component membrane dynamics of the formF = (f, g)T , with f = f(u1, u2) and g = g(u1, u2), (4.2.16a) reduces to1−(gu2 − λ)det (Je − λI)p±(λ) = 0 , Je ≡ ∂f∂u1∣∣∣u=ue, ∂f∂u2∣∣∣u=ue∂g∂u1∣∣∣u=ue, ∂g∂u2∣∣∣u=ue ,(4.2.17)where p±(λ) are defined in (4.2.6) and (4.2.8). An example of this case isconsidered below in §4.5.4.3 One-Component Membrane DynamicsIn this section we study the stability of steady-state solutions when themembrane dynamics consists of only a single component. For this case, it isconvenient to label u1 = u and to define F(C(0, t), u)byF(C(0, t), u) ≡ F(u) + βP (C(0, t), u) . (4.3.1)The symmetric steady-state solution Ce(x) is given by (4.2.2a), where C0eand ue satisfy the nonlinear algebraic system−C0e tanh(ω0L) = ω0G(C0e , u1e), F(C0e , ue)= 0 , where ω0 ≡ 1/√D .(4.3.2)654.3. One-Component Membrane DynamicsIn terms of F defined in (4.3.1), the spectral problem characterizing thestability properties of this steady-state solution for either the synchronousor asynchronous mode isDΩλ tanh(ΩλL) = −Gec +F ecGeuF eu − λ, (sync) ,DΩλ coth(ΩλL) = −Gec +F ecGeuF eu − λ, (async) ,(4.3.3)where Ωλ ≡√(1 + τλ)/D is the principal branch of the square root. Wewill first derive theoretical results for the roots of (4.3.3) for the infinite-lineproblem where L→∞.4.3.1 Theoretical Results for a Hopf Bifurcation: TheInfinite-Line ProblemFor the infinite-line problem where L → ∞, (4.3.3) reduces to the limitingspectral problem of finding the roots of G(λ) = 0 in Re(λ) ≥ 0, whereG(λ) ≡ √1 + τλ− g(λ) , and g(λ) ≡ c+ aλb+ λ. (4.3.4a)Here the constants a, b, and c, are defined bya ≡ − Gec√D, b ≡ −F eu , c ≡1√D[GecFeu −GeuF ec ] . (4.3.4b)Our goal is to characterize any roots of G(λ) = 0 in Re(λ) > 0 as the coeffi-cients a, b, and c, are varied, and in particular to detect any Hopf bifurcationpoints. In (4.3.4), b represents the dependence of the local kinetics on themembrane-bound species. If b > 0, this term indicates a self-inhibiting ef-fect, whereas if b < 0 the membrane-bound species is self-activating. Thesign of Gec represents the feedback from the environment to its own secre-tion. If Gec is positive (negative) it represents negative (positive) feedback.We remark that the spectral problem (4.3.4) has the same form, but withdifferent possibilities regarding the signs of the coefficients, as the spectralproblem studied in [57] characterizing the stability of a pulse solution for asingularly perturbed reaction-diffusion on the infinite line.We first use a winding number argument to count the number N of rootsof G(λ) = 0 in Re(λ) ≥ 0 in terms of the behavior of G(λ) on the imaginaryaxis of the λ-plane. If N = 0, the symmetric steady-state solution is linearlystable, whereas if N > 0 this solution is unstable.664.3. One-Component Membrane DynamicsLemma 4.3.1 Let N be the number of zeroes of G(λ) = 0 in Re(λ) > 0,where G(λ) is defined in (4.3.4). Assume that there are no such zeroes onthe imaginary axis. Then,N =14+1pi[arg G] ∣∣ΓI++ P , (4.3.5)where P = 0 if b > 0 and P = 1 if b < 0. Here [arg G] ∣∣ΓI+denotesthe change in the argument of G(λ) along the semi-infinite imaginary axisλ = iω with 0 < ω <∞, traversed in the downwards direction.Proof: We take the counterclockwise contour consisting of the imaginaryaxis −iR ≤ Imλ ≤ iR, decomposed as ΓI+ ∪ ΓI− , where ΓI+ = iω andΓI− = −iω with 0 < ω < R, together with the semi-circle ΓR, given by|λ| = R > 0 with | arg λ| ≤ pi2 . We use the argument principle of complexanalysis to obtainlimR→∞[arg G] ∣∣C= 2pi(N − P ) , C ≡ ΓR ∪ ΓI+ ∪ ΓI− , (4.3.6)where [arg G] ∣∣Cdenotes the change in the argument of G over the contour Ctraversed in the counter-clockwise direction, and P is the number of poles ofG inside C. Clearly P = 1 if b < 0 and P = 0 if b > 0. We calculate G(λ) ∼√τReiθ/2 on ΓR as R → ∞, where θ = arg λ, so that limR→∞ [arg G]ΓR =pi/2. Moreover, since G(λ) = G(λ), we get that [arg G]ΓI+ = [arg G]ΓI− . Inthis way, we solve for N in (4.3.6) to obtain (4.3.5). Next, we set λ = iω in (4.3.4a) to calculate [arg G] ∣∣ΓI+and detect anyHopf bifurcation points. Since we have specified the principal branch of thesquare root in (4.3.4a), we must have that Re(√1 + τλ) > 0. Therefore, ifwe square both sides of the expression for G = 0 in (4.3.4a) and solve forτ , we may obtain spurious roots. We must then ensure that Re(g) > 0 atany such root. Upon setting λ = iω in (4.3.4a) and squaring both sides,we obtain that τ = i(1− [g(iω)]2) /ω. Upon taking the real and imaginaryparts of this expression we conclude thatτ =1ωIm([g(iω)]2)=2ωgR(ω)gI(ω) =2(cb+ aω2)(b2 + ω2)2 (ab− c) . (4.3.7a)Here ω > 0 is a root ofRe([g(iω)]2)= 1 , (4.3.7b)674.3. One-Component Membrane Dynamicsfor which gR(ω) > 0 and gI(ω) > 0 to ensure that Re(√1 + iτω) > 0 andτ > 0, respectively. In (4.3.7a), g(iω) has been decomposed into real andimaginary parts as g(iω) = gR(ω) + igI(ω), wheregR(ω) =bc+ aω2b2 + ω2, gI(ω) =ω(ab− c)b2 + ω2. (4.3.7c)In addition, if we separate√1 + iτω into real and imaginary parts, we readilyderive thatRe(√1 + iτω)=1√2[√1 + τ2ω2 + 1]1/2,Im(√1 + iτω)=1√2[√1 + τ2ω2 − 1]1/2.(4.3.8)We now apply the winding number criterion of Lemma 4.3.1 togetherwith (4.3.7) to determine the location of the roots of G(λ) = 0 for variousranges of a, b, and c, as the parameter τ is varied.Proposition 4.3.1 Suppose that cb < 0 and that a ≤ 0. Then, no Hopfbifurcations are possible as τ > 0 is varied. Moreover, if b > 0 we haveN = 0, so that the symmetric steady-state solution is linearly stable for allτ > 0. Alternatively, when b < 0 we have N = 1, and so the symmetricsteady-state solution is unstable for all τ > 0.Proof: We note that g(λ), defined in (4.3.4a), is a bilinear form and isreal-valued when λ is real. It does not have a pole at λ = 0 since b 6= 0.Therefore, it follows that the imaginary axis λ = iω must map to a disk Bcentered on the real axis in the (gR, gI) plane. When cb < 0 and a ≤ 0,it follows from (4.3.7c) that gR < 0, and so this disk lies in the left half-plane Re(g) < 0. When b > 0, we have that g(λ) is analytic in Re(λ) > 0,and so the region Re(λ) > 0 must map to inside the disk B. As such,since Re(√1 + τλ)> 0, it follows that there are no roots to G(λ) = 0 inRe(λ) > 0, and so N = 0.For the case b < 0, we use the winding number criterion (4.3.5). Sincecb < 0 and a ≤ 0, we have gR(ω) < 0, so thatRe[G(iω)] = Re [√1 + iτω − g(iω)] > 0 .We have arg G(iω) → pi/4 as ω → +∞ and G(0) > 0, so that arg G(0) = 0.This yields that [arg G] ∣∣ΓI+= −pi/4. In addition, since P = 1 in (4.3.5), weobtain that N = 1 for all τ > 0. 684.3. One-Component Membrane DynamicsNext, we establish the following additional result that characterizes N ,independent of the value of τ > 0.Proposition 4.3.2 When c > ab, there are no Hopf bifurcation points forany τ > 0. If in addition, we have(I) b > 0 , and c/b < 1 , then, N = 0 ∀τ > 0 ,(II) b < 0 , and c/b < 1 , then, N = 1 ∀τ > 0 ,(III) b > 0 , and c/b > 1 , then, N = 1 ∀τ > 0 ,(IV ) b < 0 , and c/b > 1 , then, N = 2 ∀τ > 0 .(4.3.9)Proof: We first observe from (4.3.8) and (4.3.7c) that Im(G(iω)) > 0 forall τ > 0 when c > ab. Therefore, there can be no Hopf bifurcations asτ is increased. To establish (I) of (4.3.9) we use G(0) > 0, since c/b <1, arg G(iω) → pi/4 as ω → +∞, and Im(G(iω)) > 0 to conclude that[arg G] ∣∣ΓI+= −pi/4. Then, since b > 0 we have P = 0, and (4.3.5) yieldsN = 0. The proof of (II) of (4.3.9) is identical except that we have P = 1in (4.3.5) since b < 0, so that N = 1. This unstable eigenvalue is locatedon the positive real axis on the interval −b < λ < ∞. To prove (III) wenote that G(0) < 0 since c/b > 1, and P = 0 since b > 0. This yields[arg G] ∣∣ΓI+= 3pi/4, and N = 1 from (4.3.5). This root is located on thepositive real axis. Finally, to prove (IV) we use G(0) < 0 and b < 0 tocalculate [arg G] ∣∣ΓI+= 3pi/4 and P = 1. This yields N = 2 from (4.3.5). Asimple plot of√1 + τλ and g(λ) on the positive real axis for this case showsthat there is a real root in 0 < λ < −b and in −b < λ <∞ for any τ > 0.Next, we consider the range ab > c and bc > 0 for which Hopf bifur-cations in τ can be established for certain subranges of a, b, and c. Toanalyze this possibility, we substitute g(iω) into (4.3.7b), to obtain that ωmust satisfy (aω2 + bc)2 − ω2(ab− c)2 = (b2 + ω2)2 ,in the region bc+ aω2 > 0. Upon defining ξ = ω2, it follows for |a| 6= 1 thatwe must find a root of the quadratic Q(ξ) = 0 with ξ ∈ S, whereQ(ξ) ≡ ξ2−a0ξ+a1 =(ξ − a0/2)2+a1−a20/4 , S ≡ {ξ | ξ > 0 and aξ > −cb } .(4.3.10a)We refer to S as the admissible set. Here a0 and a1 are defined bya0 =1a2 − 1[(ab− c)2 + 2b(b− ac)], a1 =b21− a2(b2 − c2).(4.3.10b)694.3. One-Component Membrane DynamicsFor the special case where a = ±1, we haveξ = b2(c/b− 1c/b+ 3), if a = −1 ; ξ = −b2(c/b+ 13− c/b), if a = 1 .(4.3.10c)Our first result shows shows that there are certain subranges of theregime ab > c and bc > 0 for which we again have that no Hopf bifurcationscan occur for any τ > 0.Proposition 4.3.3 Suppose that b < 0, 0 < c/b < 1, and c/b > a. Then,N = 1 for all τ > 0.Proof: We first establish, for any τ > 0, that Re(G(iω)) > 0 when ω >0. We observe from (4.3.8) that Re(√1 + iτω) is a monotone increasingfunction of ω, while gR(ω), defined in (4.3.7c), is a monotone decreasingfunction of ω when c/b > a. This implies that Re(G(iω)) is monotoneincreasing in ω when c/b > a. Since Re(G(0)) = 1− c/b > 0 when c/b < 1,we conclude that Re(G(iω)) > 0 for ω > 0. Then, since Re(G(iω)) → +∞as ω → +∞, we obtain [arg G] ∣∣ΓI+= −pi/4. Using this result in (4.3.5),together with P = 1 since b < 0, we get that N = 1 for all τ > 0. We now use Lemma 4.3.1 and (4.3.10) to identify a parameter regime inthe range ab > c with bc > 0 where there is a unique Hopf bifurcation valuefor τ :Proposition 4.3.4 Suppose that b < 0, c/b > 1 and a < 1. Then, we haveeither N = 0 or N = 2 for all τ > 0. Moreover, N = 0 for 0 < τ  1and N = 2 for τ  1. For a 6= −1, there is a unique Hopf bifurcation atτ = τH > 0 given byτH =2(cb+ aω2H)(b2 + ω2H)2 (ab− c) , ωH =√a02+ ζ√a204− a1 , (4.3.11a)where ζ = +1 if |a| < 1 and ζ = −1 if a < −1. Here a0 and a1 are definedin (4.3.10b). When a = −1, we haveτH = −2(cb− ω2H)(b2 + ω2H)2 (b+ c) , ωH = |b|√c/b− 1c/b+ 3. (4.3.11b)704.3. One-Component Membrane DynamicsProof: We first establish that, for any τ > 0, there is a unique root ω? toRe(G(iω)) = 0 in ω > 0. To prove this we follow the proof of Proposition4.3.3 to obtain that Re(G(iω)) is a monotone increasing function of ω whenc/b > a. Moreover, since Re(G(0)) = 1 − c/b < 0, as a result of c/b > 1,and Re(G(iω)) → +∞ as ω → +∞, we conclude that there is a a uniqueroot ω? to Re(G(iω)) = 0 in the region ω > 0. The uniqueness of theroot to Re(G(iω)) = 0, together with the facts that G(0) = 1 − c/b < 0and arg G(iω) → pi/4 as ω → +∞, establishes that either [arg G] ∣∣ΓI+=3pi/4 or [arg G] ∣∣ΓI+= −5pi/4 depending on whether Im(G(iω?)) > 0 orIm(G(iω?)) < 0, respectively. Therefore, since P = 1, owing to the fact thatb < 0, we conclude from (4.3.5) that either N = 0 or N = 2 for any τ > 0.To determine N when either 0 < τ  1 or when τ  1, we examine thebehavior of the unique root ω? to Re(G(iω)) = 0 for these limiting ranges ofτ . For τ  1, we readily obtain that ω? = O(1/τ), so that Im(G(iω?)) > 0from estimating Im(√1 + iτω) and gI(ω) in (4.3.8) and (4.3.7c). Thus,N = 2 for τ  1. Alternatively, if 0 < τ  1, we readily obtain thatω? = O(1), and that Im(G(iω?)) ∼ −gI(ω?) +O(τ2) < 0. Therefore, N = 0when 0 < τ  1. By continuity with respect to τ it follows that there is aHopf bifurcation at some τ > 0.To establish that the Hopf bifurcation value for τ is unique and to derivea formula for it, we now analyze the roots of Q(ξ) = 0 for ξ ∈ S, where Q(ξ)and the admissible set S are defined in (4.3.10). In our analysis, we mustseparately consider four ranges of a: (i) 0 ≤ a < 1, (ii) −1 < a < 0, (iii)a = −1, and (iv) a < −1.For (i) where 0 ≤ a < 1, the admissible set S reduces to ξ > 0 sincecb > 0. Moreover, we have Q(0) = a1 < 0 since c/b > 1 and Q → +∞ asξ → +∞. Since Q(ξ) is a quadratic, it follows that there is a unique root toQ(ξ) = 0 in ξ > 0, with the other (inadmissible) root to Q(ξ) = 0 satisfyingξ < 0. By using (4.3.10a) to calculate the largest root of Q(ξ) = 0, andrecalling (4.3.7a), we obtain (4.3.11a).The proof of (ii) for the range −1 < a < 0 is similar, but for this case theadmissible set S is the finite interval 0 < ξ < −cb/a. Since Q(0) = a1 < 0and Q is a quadratic, to prove that there is a unique root to Q(ξ) = 0on this interval it suffices to show that Q(−cb/a) > 0. A straightforwardcalculation using the expressions for a0 and a1 in (4.3.10b) yields, upon714.3. One-Component Membrane Dynamicsre-arranging terms in the resulting expression, thatQ(−cb/a) = c2b2a2− cba(1− a2)[(ab− c)2 + 2b(b− ac)]+b2(b2 − c2)1− a2 ,=c2b2a2− cba(1− a2)(ab− c)2 +b21− a2[(b− c)2 + 2cb(1− 1a)].Since cb > 0 and −1 < a < 0 all three terms in this last expression forQ(−cb/a) are positive. Thus, there is a unique root to Q(ξ) = 0 in 0 < ξ <−cb/a, which is given explicitly by (4.3.11a).When a = −1, the admissible set S is the interval 0 < ξ < cb. It isthen readily verified that the explicit formula for ξ given in (4.3.10c) whena = −1 lies in this interval. In this way, we obtain (4.3.11b).Finally, we consider the range (iv) where a < −1, where the admissibleset is 0 < ξ < −cb/a. Since c/b > 1 and a < −1 we have from (4.3.10b)that a0 > 0 and Q(0) = a1 > 0. Thus the minimum value of Q(ξ) is atsome point ξ = ξm > 0. To prove that there is a unique root to Q(ξ) = 0 on0 < ξ < −cb/a we need only prove that Q (−cb/a) < 0. By re-arranging theterms in the expression for Q(−cb/a) we obtain, after some algebra, thatQ(−cb/a) = − cb3a2(a2 − 1)cb(1 + a2)− a(c2b2+ a2)− b2a2 − 1[(b− c)2 + 2cb(a− 1)a].Since cb > 0 and a < −1, we have that the expressions inside each of the twosquare brackets are positive, while the terms multiplying the square bracketsare negative. This establishes that Q(−cb/a) < 0 and the existence of aunique root to Q(ξ) = 0 in 0 < ξ < −cb/a. By taking the smallest root ofQ(ξ) = 0 on ξ > 0 we get (4.3.11a). Our next result is for the case b > 0 on a subrange of where ab− c > 0.Proposition 4.3.5 The following results hold for the case b > 0: (I) Sup-pose that c/b < a < 1. Then, we have N = 0 for all τ > 0. (II) Supposethat c/b < 1 < a. Then, there is a Hopf bifurcation at some τ = τH > 0.If 0 < τ < τH , then N = 2, whereas if τ > τH , then N = 0. The Hopfbifurcation value τH > 0 is given byτH =2(cb+ aω2H)(b2 + ω2H)2 (ab− c) , ωH =√a02+√a204− a1 , (4.3.12)724.3. One-Component Membrane Dynamicswhere a0 and a1 are defined in (4.3.10b).Proof: We first prove (I). When c/b < a < 1, we have from (4.3.7c) thatgR(ω) is monotone increasing with c/b = gR(0) < gR(ω) < gR(∞) = a < 1.Since Re(√1 + iωτ) > 1 for all τ > 0, it follows that Re(G(iω)) > 0 on0 ≤ ω < ∞, and consequently [arg G] ∣∣ΓI+= −pi/4. Then, since P = 0,owing to b > 0, (4.3.5) yields that N = 0 for all τ > 0.To prove (II) we consider the range c ≥ 0 and c < 0 separately, and wefirst examine the roots to Q(ξ) = 0 for ξ ∈ S, as defined in (4.3.10). For thecase c ≥ 0, the admissible set is ξ > 0. Since the quadratic Q(ξ) satisfiesQ(0) = a1 < 0 when 0 < c/b < 1 < a, together with Q(ξ)→ +∞ as ξ →∞,it follows that there is a unique root to Q(ξ) = 0 on ξ > 0. This yields theunique Hopf bifurcation value τH given in (4.3.12). Alternatively, supposethat c < 0. Then the admissible set is ξ > −bc/a. We calculate Q (−bc/a)from (4.3.10), and after re-arranging the terms in the resulting expression,we obtainQ(−cb/a) = c2b2a2+cba(a2 − 1)[(ab− c)2 + 2b(b− ac)]+b2(b2 − c2)1− a2 ,=bca(ab− c)2a2 − 1 +b2(b2 − c2)1− a2 +c2b2a2(a2 − 1)[−a2 − 1 + 2abc].Since each of the three terms in the last expression is negative when c/b <1 < a, we have Q(−bc/a) < 0. It follows that there is a unique root toQ(ξ) = 0 on −bc/a < ξ < ∞, and consequently a unique Hopf bifurcationpoint.Combining the results for c ≥ 0 and c < 0, we conclude that there is aunique Hopf bifurcation point τH > 0 when c/b < 1 < a and b > 0. We nowmust prove the result that N = 0 for τ > τH and N = 2 for 0 < τ < τH . Toestablish this result, we need only prove than N = 0 for τ  1 and N = 2 for0 < τ  1. Then, by the uniqueness of τH , the continuity of λ with respectto τ , and the fact that λ = 0 cannot be eigenvalue, the result follows. Forτ  1, we obtain from the unboundedness of Re(√1 + iτω) as τ → +∞ forω > 0 fixed that Re(G(iω)) > 0 on 0 ≤ ω <∞ when τ  1. Therefore, since[arg G] ∣∣ΓI+= −pi/4 and P = 0, owing to b > 0, (4.3.5) yields that N = 0for τ  1. Next, since a > 1, we readily observe that there are exactly tworoots ω± with 0 < ω− < ω+ to Re(G(iω)) = 0 on 0 < ω < ∞, with theproperty that ω− = O(1) and ω+ = O(τ−1)  1 when 0 < τ  1. Wereadily estimate that Im(G(iω+)) > 0 and Im (G(iω−)) < 0 when τ  1.Therefore, since arg G(iω) → pi/4 as ω → +∞ and arg G(0) = 0 since734.3. One-Component Membrane Dynamicsc/b < 1, we conclude that [arg G] ∣∣ΓI+= 7pi/4 when 0 < τ  1. Finally,since P = 0, owing to b > 0, (4.3.5) yields N = 2 when 0 < τ  1. Our final result is for the range 1 < a < c/b with b < 0 where there canbe either two Hopf bifurcation values of τ or none.Proposition 4.3.6 Suppose that b < 0 and 1 < a < c/b. Then, if c/b ≤3a+ 2√2(a2− 1)1/2, we have N = 2 for all τ > 0, and consquently no Hopfbifurcation points. Alternatively, if c/b > 3a + 2√2(a2 − 1)1/2, then thereare two Hopf bifurcation values τH±, with τH− > τH+, so that N = 0 forτH+ < τ < τH− and N = 2 when either 0 < τ < τH+ or τ > τH−.Proof: Since the proof of this result is similar to those of Propositions 4.3.4and 4.3.5, we only briefly outline the derivation. First, since necessarilyc < 0, the admissible set for Q(ξ) in (4.3.10a) is ξ ≥ 0, and hence we focuson determining whether Q(ξ) = 0 has any positive real roots. For the range1 < a < c/b, we calculate Q(0) = a1 > 0 from (4.3.10). As such it followsthat there are either two real roots to Q(ξ) = 0 in ξ > 0, a real positiveroot of multiplicty two, or no real roots. From (4.3.10), there are two realroots only when a0 > 0 and a20/4 − a1 > 0, where a0 and a1 are defined in(4.3.10b).Upon using (4.3.10b) for a0 and a1, we can show after some lengthybut straightforward algebra that a0 > 0 when c/b > 2a +√3a2 − 2, anda20/4− a1 > 0 when (cb− 3a)2+ 8(1− a2) > 0 .For any a > 1, the intersection of these two ranges of c/b is c/b > 3a +2√2(a2 − 1)1/2. On this range, Q(ξ) = 0 has two positive real roots, andhence there are two Hopf bifurcation values of τ . For the range 1 < a <c/b < 3a + 2√2(a2 − 1)1/2, then either a0 < 0 or a20/4 − a1 < 0, and soQ(ξ) = 0 has no positive real roots.The determination of N follows in a similar way as in the proof of Propo-sition 4.3.5. 4.3.2 A Finite Domain: Numerical Computations of theWinding NumberFor finite domain length L, the synchronous and asychronous modes will, ingeneral, have different instability thresholds. For finite L, we use (4.3.3) to744.4. Examples of the Theory: One-Component Membrane Dynamicsconclude that we must find the roots of G(λ) = 0, where we now re-defineG(λ) asG(λ) ≡ DΩλh (Ωλ)+Gec−F ecGeuF eu − λ, h(Ωλ) ≡{tanh (ΩλL) , (synchronous)coth (ΩλL) , (asynchronous),(4.3.13a)where Ωλ =√(1 + τλ)/D. It is readily shown that (4.3.5) still holds, andsoN =14+1pi[arg G] ∣∣ΓI++ P , (4.3.13b)where P = 0 if F eu < 0 and P = 1 if Feu > 0. To determine N for aspecific membrane-bulk system, numerical computations of [arg G] ∣∣ΓI+mustbe performed separately for both the synchronous and asynchronous modes.This is illustrated below in §4.4 for some specific membrane-bulk systems.We remark that some of the results in §4.3.1 are still valid when L isfinite. To see this, we write (4.3.13a) in the form√1 + τλh(Ωλ)h(ω0)= g(λ) , g(λ) ≡ cL + aLλb+ λ, (4.3.14a)where ω0 = D−1/2, and where we have defined aL, b, and cL, byaL ≡ − Gec√Dh(ω0), b ≡ −F eu , cL ≡1√Dh(ω0)[GecFeu −GeuF ec ] .(4.3.14b)We remark that as L → ∞, (4.3.14) reduces to the eigenvalue problem(4.3.4a) for the infinite-line problem studied in §4.3.1.With this reformulation, the left-hand side of (4.3.14a) has the samequalitative properties as√1 + τλ that were used in the proofs of some ofthe propositions in §4.3.1. In particular, Propositions 4.3.1–4.3.3 and part(I) of Proposition 4.3.5 still apply provided we replace a and c in theseresults by aL and cL. We do not pursue this extension any further here.4.4 Examples of the Theory: One-ComponentMembrane DynamicsIn this section we consider some specific systems to both illustrate our sta-bility theory and to show the existence of synchronous and asynchronousoscillatory instabilities induced by coupled membrane-bulk dynamics. As-suming a one-component membrane dynamics, we determine the stability754.4. Examples of the Theory: One-Component Membrane Dynamicsof the steady-state solution by numerically computing the number N ofeigenvalues of the linearization in Re(λ) > 0 from either (4.3.5) for theinfinite-line problem, or from (4.3.13) for the finite-domain problem. Forsome subranges of the parameters in these systems, the theoretical resultsof §4.3.1 for the infinite-line problem determines N without the need for anynumerical winding number computation.To confirm our stability results for the case of a one-component mem-brane dynamics we also computed symmetric steady-state solutions of (4.1.1)and bifurcations of this solution to periodic solutions by first spatially dis-cretizing (4.1.1) with finite differences. Then, from this method of linesapproach, together with the path continuation program Auto with the in-terface provided by XPPAUT (cf. [16]), branches of steady-state and peri-odic solution branches were computed numerically. To confirm predictions ofoscillatory dynamics, full time-dependent numerical solutions of the coupledPDE-ODE system (4.1.1) were computed using the method of lines.4.4.1 A Class of Feedback ModelsWe first apply the theory of §4.3.1 to a class of membrane-bulk problems ofthe formτCt = DCxx − C , t > 0 , x > 0 ;DCx∣∣x=0= G(C(0, t), u) ; C → 0 as x→∞ ,dudt= F (C(0, t), u) , where F (C(0, t), u) ≡ F(u) + σG(C(0, t), u(t)) ,(4.4.1)for some σ > 0. For this class, the flux on x = 0 acts as a source term to themembrane dynamics. A special case of (4.4.1), which is considered below,is when the membrane-bulk coupling is linear and, for some κ > 0, has theformG(C(0, t), u) ≡ κ [C(0, t)− u] . (4.4.2)To apply the theory in §4.3.1 to (4.4.1) we first must calculate a, b, and c,from (4.3.4b). We readily obtain thatb = −F ′(ue)−σGeu , a = −Gec√D, c =1√DGecF ′(ue) , ab−c =σ√DGeuGec ,(4.4.3)where ue is a steady-state value for u. The first result for (4.4.1) showsthat a Hopf bifurcation is impossible with a linear membrane-bulk couplingmechanism.764.4. Examples of the Theory: One-Component Membrane DynamicsProposition 4.4.1 Let Ce, ue be a symmetric steady-state solution for(4.4.1) with the linear membrane-bulk coupling (4.4.2). Let N denote thenumber of unstable eigenvalues in Re(λ) > 0 for the linearization of (4.4.1)around this steady-state solution. Then, for any τ > 0, we have N = 0when F ′(ue) < FLth, and N = 1 when F ′(ue) > FLth, where FLth ≡σκ/[1 + κ/√D].Proof: Since with the coupling (4.4.2) we have ab − c = −κ2σ/√D < 0,it follows by Proposition 4.3.2 that there are no Hopf bifurcations for anyτ > 0. To determine the stability threshold, we calculate a = −κ/√D < 0,b = −F ′(ue)+σκ, and c = κF ′(ue)/√D, and apply the results of Proposition4.3.2. We separate our analysis into three ranges of F ′(ue). First supposethat F ′(ue) < 0. Then, since b > 0, c < 0, and a < 0, we have by (I)of Proposition 4.3.2 that N = 0. Next, suppose that 0 < F ′(ue) < σκ,so that b > 0 and c > 0. We calculate that c/b > 1 if F ′(ue) > FLth,where FLth, which satisfies 0 < FLth < σκ, is defined above. Since c/b > 1,(III) of Proposition 4.3.2 proves that N = 1 for all τ > 0. Alternatively, if0 < F ′(ue) < FLth, then c/b < 1, and (I) of Proposition 4.3.2 proves thatN = 0 for all τ > 0. Finally, suppose that F ′(ue) > σκ. Then, c > 0, b < 0,so that bc < 0 and a < 0. We conclude from Proposition 4.3.1 that N = 1for all τ > 0. The proof is complete by combining these results on the threeseparate ranges of F ′(ue). This result for the non-existence of oscillations for a linear membrane-bulk coupling mechanism holds only for the case of a single membrane-boundspecies. As shown in §4.5, when there are two species in the membrane,oscillatory dynamics can occur even with a linear membrane-bulk couplingmechanism. Our next result for (4.4.1) specifies a class of nonlinear couplingmechanisms G(C(0, t), u) for which no Hopf bifurcations of the steady-statesolution are possible.Proposition 4.4.2 When GecGeu < 0, then the symmetric steady-state so-lution of (4.4.1) does not undergo a Hopf bifurcation for any τ > 0. Inparticular, if Geu < 0 and Gec > 0, then for any τ > 0 we have N = 1 whenF ′(ue) > Fth, and N = 0 when F ′(ue) < Fth. Here Fth > 0 is the thresholdvalueFth ≡ − σGeu1 +Gec/√D. (4.4.4)Proof: From (4.4.3) we have ab− c < 0 when GecGeu < 0. Proposition 4.3.2proves that there are no Hopf bifurcations for any τ > 0. The second part774.4. Examples of the Theory: One-Component Membrane Dynamicsof the proof parallels that done for Proposition 4.4.1. A similar analysis can be done for the case where Geu > 0 and Gec < 0.For this case, the steady-state solution is unstable when Gec < −√D for allranges of F ′(ue). When Gec > −√D, the steady-state is linearly stable onlywhen F ′(ue) < −σGeu/[1 +Gec/√D].Our final result for (4.4.1) characterizes a class of nonlinear couplingmechanisms for which a Hopf bifurcation of the steady-state solution doesoccur for some value of τ .Proposition 4.4.3 Suppose that Gec > 0 and Geu > 0. Then, for the sym-metric steady-state solution of (4.4.1), we have:(I) If F ′(ue) > Fth , then N = 1 ∀τ > 0 ,(II) If − σGeu < F ′(ue) < Fth , then N = 2 for τ > τH , andN = 0 for 0 < τ < τH ,(III) If F ′(ue) < −σGeu , then N = 0 ∀τ > 0 .(4.4.5)Here τH > 0 is the unique Hopf bifurcation point, and Fth < 0 is defined in(4.4.4).Proof: Since Gec > 0, we have a < 0 from (4.4.3). To establish (III)we calculate from (4.4.3) that b > 0 and c < 0 when F ′(ue) < −σGeu.From the first statement of Proposition 4.3.1, we conclude that N = 0. Toestablish (II), we observe that b < 0, c < 0, and c/b > 1 when −σGeu <F ′(ue) < Fth < 0. Proposition 4.3.4 then proves that there is a uniqueHopf bifurcation value τ = τH > 0 on this range of F ′(ue), as given in(4.3.11). Finally, to establish (I), we observe that b < 0 and c/b < 1 whenF ′(ue) > Fth. For the range c < 0, where Fth < F ′(ue) < 0, we havefrom Proposition 4.3.3 that N = 1. Finally, for the range c > 0, whereF ′(ue) > 0, Proposition 4.3.1 also yields that N = 1. We now discuss the limiting behavior of τH and the corresponding Hopfbifurcation frequency ωH , as given by (4.3.11), at the two edges of theinterval for F ′(ue) in (II) of Proposition 4.4.3. First, we observe that asF ′(ue) approaches −σGeu from above, we have that b → 0−. Therefore,from (4.3.11) we have a1 → 0, and so at this lower edge of the interval wehave ωH → 0+ and τH → +∞. At the other end of the interval, whereF ′(ue) approaches Fth from below, we have that c − b → 0, so that againa1 → 0 in (4.3.11). Therefore, from (4.3.11), we conclude at this upper784.4. Examples of the Theory: One-Component Membrane Dynamicsedge of the interval that ωH → 0+. However, since b = O(1), we have from(4.3.11a) that τH → 2(1− a)/|b| = O(1) at the upper edge.4.4.2 A Phase Diagram for an Explicitly Solvable ModelNext, we consider a simple model where a phase diagram characterizingthe possibility of Hopf bifurcations can be determined analytically for theinfinite-line problem. For β > 0, γ > 0 and κ > 0, we considerτCt = DCxx − C , t > 0 , 0 < x < 2L ,DCx∣∣x=0= G(C(0, t), u) ≡ κ(C(0, t)− u)1 + β(C(0, t)− u)2 ,dudt= F (C(0, t), u) ≡ γC(0, t)− u ,(4.4.6)with identical membrane dynamics at x = 2L. We remark that in (4.4.6)the only nonlinearity arises from the flux term G(C(0, t), u). The symmetricsteady-state solution for (4.4.6) is Ce(x) given in (4.2.2a), where C0e ≡ Ce(0)satisfies the cubic equation(C0e )3β(γ−1)2 tanh(ω0L)−C0e[κω0(γ − 1)− tanh(ω0L)]= 0 , ω0 ≡√1/D0 .(4.4.7)In our analysis, we will focus on periodic solutions that bifurcate from thesteady-state solution branch where C0e is positive. From (4.4.7), the positiveroot is given explicitly byC0e =√κω0(γ − 1)− tanh(ω0L)β(γ − 1)2 tanh(ω0L) ,ue = γC0e , when κω0(γ − 1)− tanh(ω0L) > 0 .(4.4.8)We first consider the infinite-line problem where L → ∞ and we setD = 1 for convenience. Then, (4.4.8) reduces toC0e =√κ(γ − 1)− 1β(γ − 1)2 . (4.4.9)For this example, we calculate a, b, and c, in (4.3.4b) asa = −Gec = −1κ(γ − 1)2[2− κ(γ − 1)] , b = 1 ,c = (γ − 1)Gec , ab− c = −γGec .(4.4.10)794.4. Examples of the Theory: One-Component Membrane DynamicsWe now apply the theory of §4.3.1 to obtain the phase-diagram Fig. 4.1in the parameter space κ versus γ. Since C0e > 0 only when γ > 1 andκ > 1/(γ − 1), the boundary between region I and II in Fig. 4.1 is κ =1/(γ − 1). Next, we calculate that ab − c < 0 and 0 < c/b < 1 when(γ − 1)−1 < κ < 2(γ − 1)−1, which is labeled as region II in Fig. 4.1.Therefore, in this region, we conclude from condition (I) of Proposition 4.3.2that the steady-state is stable for all τ > 0. Next, we calculate from (4.4.10)that c/b < a < 1 when 2(γ − 1)−1 < κ < 2(γ − 1)−1(2 − γ)−1 and γ > 1,which is region III of Fig. 4.1. For this range, Proposition 4.3.5 proves thatthe steady-state solution is stable for all τ > 0. Finally, region IV of Fig. 4.1given by κ > 2(γ − 1)−1(2 − γ)−1 for 1 < γ < 2, is where c/b < 1 < a.At each point in this region, Proposition 4.3.5 proves that there is a Hopfbifurcation value τ = τH > 0, and that the steady-state solution is unstableif 0 < τ < τH .1 1.5 224681012γκIIIIIIIVFigure 4.1: Phase diagram for (4.4.6) in the κ versus γ plane for the infinite-line problem when D = 1. In region I, κ < (γ−1)−1 with γ > 1, and there isno steady-state solution. In region II, bounded by (γ−1)−1 < κ < 2(γ−1)−1for γ > 1, we have ab − c < 0 and b > 0, and the steady-state solution islinearly stable for all τ > 0. In region III, bounded by 2(γ − 1)−1 < κ <2(γ − 1)−1(2 − γ)−1 for γ > 1, we have b > 0 and c/b < a < 1, and so bythe first statement in Proposition 4.3.5 there is no Hopf bifurcation and thesteady-state solution is linearly stable for all τ > 0. In region IV, boundedby κ > 2(γ−1)−1(2−γ)−1 for 1 < γ < 2, we have b > 0 and c/b < 1 < a, andso by the second statement in Proposition 4.3.5 there is a Hopf bifurcationand the steady-state solution is unstable if 0 < τ < τH and is linearly stableif τ > τH , where τH > 0 is given by (4.3.12).For the finite-domain problem with L = 2, and for two values of κ, in804.4. Examples of the Theory: One-Component Membrane Dynamics1.3 1.4 1.5 1.65.55.65.75.85.96γu1.2 1.3 1.4 1.5 1.6 1.75.866.26.46.66.8γuFigure 4.2: Two typical bifurcation diagrams for u versus γ for (4.4.6) on afinite domain with L = 2, D = 1, τ = 0.1, and β = 1. Left panel: κ = 9.Right-panel: κ = 10.5. The solid and dashed lines denote linearly stableand unstable branches of steady-state solutions. The outer and inner closedloops correspond to branches of synchronous and asynchronous periodic so-lutions, respectively. The solid/open circles indicate linearly stable/unstableperiodic solutions, respectively.Fig. 4.2 we plot numerically computed bifurcation diagrams of u versus γ forboth the steady-state and bifurcating periodic solution branches. For thecorresponding infinite-line problem, this corresponds to taking a horizontalslice at fixed κ through the phase diagram of Fig. 4.1. The results in the leftpanel of Fig. 4.2 show that when κ = 9 the bifurcating branch of synchronousoscillations is linearly stable, while the asynchronous branch is unstable. Toconfirm this prediction of a stable synchronous oscillation for κ = 9 andγ = 1.45, in Fig. 4.3 we plot the full numerical solution computed from thePDE-ODE system (4.4.6). Starting from the initial condition C(x, 0) = 1,together with u1(0) = 0.04 and u2(0) = 0.5 in the left and right membranes,respectively, this plot shows the eventual synchrony of the oscillations in thetwo membranes. In the right panel of Fig. 4.2, where κ = 10.5, we show thatthe synchronous mode is stable for a wide range of γ, but that there is anarrow parameter range in γ where both the synchronous and asynchronousmodes are unstable. For the value γ = 1.28 within this dual-unstable zone,the full numerical solution of the PDE-ODE system (4.4.6), shown in Fig. 4.4reveals a phase-locking phenomena in the oscillatory dynamics of the twomembranes.814.4. Examples of the Theory: One-Component Membrane Dynamics15 20 255.55.65.75.85.96tu1, u2Figure 4.3: Full numerical solutions(left panel) of the PDE-ODE system for(4.4.6) for the finite-domain problem with L = 2, D = 1, τ = 0.1, κ = 9,γ = 1.45, and β = 1. The initial condition is C(x, 0) = 1 with u1(0) = 0.04and u2(0) = 0.5 in the left and right membranes. On the infinite line theparameter values are in region IV of Fig. 4.1. For this value of γ and κ weobserve from the left panel of the global bifurcation diagram Fig. 4.2 thatonly the synchronous mode is stable. The full numerical solutions for u1and u2 (right panel) confirm this prediction.10 15 2066.26.46.66.8tu1, u2Figure 4.4: Full numerical solutions(left panel) of the PDE-ODE system for(4.4.6) for the finite-domain problem with L = 2, D = 1, τ = 0.1, κ = 10.5,γ = 1.28, and β = 1. The initial condition is as given in Fig. 4.3. For thisvalue of γ and κ we observe from the right panel of the global bifurcationdiagram Fig. 4.2 that synchronous and asynchronous periodic solutions areboth linearly unstable. The full numerical solutions for u1 and u2 (rightpanel) reveal a phase-locking phenomenon.824.4. Examples of the Theory: One-Component Membrane Dynamics4.4.3 A Model of Kinase Activity RegulationIn the following, we study a model which describes the regulation of ofkinase activity by its diffusion in space and feedback through membrane-bound receptors from [30]. In [30], K(t, x) represent the concentration ofactive kinase, Q denotes the total concentration of the kinase, R(t) denotesthe surface concentration of the active receptors and P represents the totalsurface concentration of the ligand bound receptors(including active andinactive). In the original paper [30], a 2D model in a ball domain withradius r0 is considered. Here we import the model in 1D and consider thedomain [0, 2L]. The original system readsKt = d1∆K − b1K ,−d1Kx∣∣∣x=0= a1R(Q−K(0, t)) ,dRdt= a2K(0, t)(P −R)− b2R ,(4.4.11)where b1 > 0 denotes the dephosphorylation rate of kinase, d1 is the diffusioncoefficient and a1, a2, b2 are positive reaction rates. To transform (4.4.11)into the same form of (4.1.1), we defineC(x, t) = K(x, t) , u(t) = R(t) , τ =1b1, D =d1b1. (4.4.12)Then (4.4.11) can be rewritten toτCt = D∆C − C ,DCx(0, t) = G(C(0, t), u) ,dudt= F (C(0, t), u) ,(4.4.13)withG(C(0, t), u) =a1b1u(Q− C(0, t)) , F (C(0, t), u) = a2C(0, t)(P − u)− b2u .(4.4.14)The steady state Ce(x) of the concentration of active kinase has same ex-pression as (4.2.2) withC0e =ω0a2PQa1b1− b2 tanh(ω0L)a2 tanh(ω0L) + ω0a2Pa1b1, (4.4.15)andue =a2PC0eb2 + a2C0e. (4.4.16)834.4. Examples of the Theory: One-Component Membrane DynamicsFor the linear stability analysis, we calculate thatb = −F eu = a2C0e+b2 > 0 , a = −Gec√D= − a1ueb1√D< 0 , Geu < 0 , Fec > 0 ,(4.4.17)withc =a1b1√Da2P (b2Q− a2(C0e )2 − 2b2C0e )b2 + a2C0e,ab− c = GecFeu√D− GecFeu −GeuF ec√D=GeuFec√D< 0 .(4.4.18)From Proposition 4.3.2 case (I) or (III), depending on the parameter choices,we observe that no Hopf bifurcation is possible for this system for any valueof τ and N = 0(I) or N = 1(case (III)).4.4.4 Two Biologically-Inspired ModelsNext, we consider two specific biologically-inspired models which undergoa Hopf bifurcation when parameters vary. The first example is a simpli-fied version of the GnRH neuron model from [17, 32, 47]. In this context,the spatial variable C(x, t) represents the GnRH concentration in the bulkmedium while u represents the membrane concentration of the activatedα-subunits of the G-protein Gi which is activated by the binding of GnRHto its receptor. As discussed in the Appendix, the functions describing theboundary flux and the membrane kinetics for this model are as follows:G(C(0, t), u) = −σ[1 + β(ι+ 1 + ζqµ+ 1 + δq)3(η +sω + u)3],F (C(0, t), u) = ([C(0, t)]2k2i + [C(0, t)]2− u),(4.4.19a)where s and q, which depend on C(0, t), are defined bys ≡ [C(0, t)]4k4s + [C(0, t)]4, q ≡ [C(0, t)]2k2q + [C(0, t)]2. (4.4.19b)The fixed parameters in this model, as discussed in [17, 32, 47], can beobtained from fitting experimental data.For the bulk diffusion process we let D = 0.003, τ = 1, and L = 1.Since L/√D ≈ 18.3 1, our analytical stability theory for the infinite-lineproblem will provide a good prediction for the stability properties associated844.4. Examples of the Theory: One-Component Membrane Dynamics0 0.05 0.100.10.2ReGImGFigure 4.5: Left figure: Numerical results, showing oscillatory dynamics,for C(x, t) in the GnRH model (4.4.19). The bulk diffusion parametersare D = 0.003, τ = 1, and L = 1. The parameters in the membrane-bulk coupling and dynamics in (4.4.19) are σ = 0.047, β = 5.256× 10−14,ι = 764.7, ζ = 3747.1, µ = 0.012, δ = 0.588, η = 0.410, ω = 0.011, = 0.0125, ki = 464, ks = 1, and kq = 61. Right figure: Plot of theimaginary part versus the real part of G(iω) when λ = iω and ω decreasesfrom 3 (black dot) to 0. This shows that the winding number [arg G] ∣∣ΓI+is7pi/4, and so N = 2 from (4.3.13a).854.4. Examples of the Theory: One-Component Membrane Dynamicswith this finite-domain problem. By using the parameter values of [32], aswritten in the caption of Fig. 4.5, we calculate thatb = −F eu =  > 0 , a = −Gec/√D > 0 , F ec > 0 , Geu > 0 .(4.4.20)In the right panel of Fig. 4.5 we show a numerical computation of the wind-ing number, which establishes that [arg G] ∣∣ΓI+= 7pi/4. Since b > 0, weconclude from (4.3.5) that N = 2. Our full numerical simulations of thePDE-ODE system in the left panel of Fig. 4.5, showing an oscillatory dy-namics, is consistent with this theoretical prediction. In fact, for the pa-rameter values in the caption of Fig. 4.5 we have a = 1.8223, b = 0.0125,and c = 0.0028. Since b > 0 and c/b < 1 < a, the second statement inProposition 4.3.5 proves that there is a Hopf bifurcation value of τ for thecorresponding infinite-line problem. We calculate τH ≈ 113.5 with frequencyωH ≈ 0.0169, which indicates a rather large period of oscillation at onset.Another specific biological system is a model of cell signaling in Dic-tyostelium (cf. [19]). In this context, the spatial variable C(x, t) is theconcentration of the cAMP in the bulk region, while u is the total fractionof cAMP receptor in the active state on the two membranes (binding ofcAMP to this state of the receptor elicits cAMP synthesis). As discussedin the Appendix, the boundary flux and nonlinear membrane dynamics forthis system are describedG(C(0, t), u) = −σ?α(Λθ + u[C(0,t)]21+[C(0,t)]2)(1 + αθ) + ( u[C(0,t)]21+[C(0,t])2)(1 + α),F (C(0, t), u) = f2(C(0, t))− u[f1(C(0, t)) + f2(C(0, t))] ,(4.4.21a)wheref1(C(0, t)) ≡ k1 + k2[C(0, t)]21 + [C(0, t)]2, f2(C(0, t)) ≡ k1L1 + k2L2c2d[C(0, t)]21 + c2d[C(0, t)]2.(4.4.21b)The fixed parameters in this model, as discussed briefly in the Appendix,are given in (cf. [19]) after fitting the model to experimental data. They arewritten in the caption of Fig. 4.6,For the bulk diffusion process we let D = 0.2, τ = 0.5, and L = 1. Forthis case where L/√D ≈ 2.2, the analytical stability results for the infinite-domain problem do not accurately predict the stability thresholds for thisfinite-domain problem. For the parameter values in Fig. 4.6, we calculatethatb ≡ −F eu > 0 , F ec < 0 , Geu < 0 , Gec < 0 .864.4. Examples of the Theory: One-Component Membrane DynamicsIn the right panel of Fig. 4.6 we show that [arg G] ∣∣ΓI+= 7pi/4. Since b > 0,we conclude from (4.3.13) that N = 2. Our full numerical simulations ofthe PDE-ODE system in the left panel of Fig. 4.6, showing an oscillatorydynamics, is consistent with this prediction. For the parameter values inthe caption of Fig. 4.6 we have a = 1.4223, b = 1.1525, and c = 0.2205. Weremark that since b > 0 and c/b < 1 < a, Proposition 4.3.5 proves that thereis a Hopf bifurcation value of τ for the corresponding infinite-line problemgiven by τH ≈ 0.5745.0 0.2 0.4 0.6−0.200.20.40.60.81ReGImGFigure 4.6: Left figure: Numerical results, showing oscillatory dynamics, forC(x, t) in the Dictyostelium model (4.4.21). The bulk diffusion parametersare D = 0.2, τ = 0.5, and L = 1. The parameters in the membrane-bulk coupling and dynamics in (4.4.21) are σ? = 32, α = 1.3, Λ = 0.005,θ = 0.1,  = 0.2, k1 = 1.125, L1 = 316.228, k2 = 0.45, L2 = 0.03, andcd = 100. Right figure: Plot of the imaginary part versus the real part ofG(iω) when λ = iω and ω decreases from 100 (black dot) to 0. This showsthat [arg G] ∣∣ΓI+= 7pi/4, and so N = 2 from (4.3.13).The parameters used in Fig. 4.6 are adopted from [19] (page 245) exceptfor the values of Λ, θ, α and σ. In Fig. 4.7 we plot the numerically computedbifurcation diagram of steady-state solutions for (4.4.21) as D is varied,together with the branches of synchronous periodic solutions. In the leftpanel of Fig. 4.7 we took Λ = 0.005, θ = 0.1 and τ = 1.3, corresponding toFig. 4.6, while in the right panel of Fig. 4.7 we took Λ = 0.01, θ = 0.01 andτ = 1.2. For the latter parameter set, the steady-state bifurcation diagramhas an S-shaped bifurcation structure.874.4. Examples of the Theory: One-Component Membrane Dynamics0.2 0.4 0.6 0.8 100.20.40.60.811.2DC(0)0.2 0.4 0.6 0.8 100.511.522.5DC(0)Figure 4.7: Bifurcation diagram of steady-state and synchronous periodicsolution branches for the Dictyostelium model (4.4.21) with respect to thediffusivity D. The vertical axis is C(0). Left panel: Λ = 0.005, θ = 0.1and τ = 1.3. Right panel: Λ = 0.01, θ = 0.01 and τ = 1.2. In bothpanels the other parameter values used are the same as in Fig. 4.6. Thesolid/dashed lines denote stable/unstable branches of steady-state solutions.The solid/open circles indicates stable/unstable periodic solution branchesof the synchronous mode. For the value D = 0.2 used in in the left panel ofFig. 4.6, we observe from the left panel above that the steady-state solutionis unstable (as expected).884.5. Two-Component Membrane Dynamics: Extension of the Basic Model4.5 Two-Component Membrane Dynamics:Extension of the Basic ModelIn our analysis so far we have assumed that the two membranes are identi-cal. We now extend our analysis to allow for the more general case wherethe two membranes have possibly different dynamics. From the laboratoryexperiments of Pik-Yin Lai [42], it was observed for a certain two-cell systemthat one cell can have oscillatory dynamics, while the other cell is essentiallyquiescent. To illustrate such a behavior theoretically, we now modify ourprevious analysis to remove the assumed symmetry of the bulk concentra-tion about the midline at x = L, and instead consider the whole system on0 < x < 2L. Allowing for the possibility of heterogeneous membranes, weconsiderτCt = DCxx − C , t > 0 , 0 < x < 2L ,DCx(0, t) = G1(C(0, t), u1) , DCx(2L, t) = G2(C(2L, t), v1) .(4.5.1a)Here C(x, t) represents the bulk concentration of the signal, while u1 andv1 are their concentrations at the two membranes x = 0 and x = 2L, re-spectively. Inside each membrane, we assume the two-component dynamicsdu1dt= f1(u1, u2) + β1P1(C(0, t), u1) , du2dt= g1(u1, u2) ,dv1dt= f2(v1, v2) + β2P2(C(2L, t), v1) , dv2dt= g2(v1, v2) ,(4.5.1b)where the functions G1, G2, P1, P2, f1, f2, g1, and g2 are given byG1(C(0, t), u1) = κ1[C(0, t)− u1(t)],G2(C(2L, t), v1) = κ2[v1(t)− C(2L, t)],f1(u1, u2) = σ1u2 − q1u1 − q2 u11 + q3u1 + q4u21, g1(u1, u2) =11 + u41− u2 ,f2(v1, v2) = σ2v2 − p1v1 − p2 v11 + p3v1 + p4v21, g2(v1, v2) =11 + v41− v2 ,P1(C(0, t), u1) =[C(0, t)− u1], P2(C(2L, t), v1) =[C(2L, t)− v1].(4.5.1c)This system, adopted from the key survey paper [74] for the design of re-alistic biological oscillators, models a gene expression process and proteinproduction for a certain biological system. With our choices of Gi and Pifor i = 1, 2, we have assumed a linear coupling between the bulk and the894.5. Two-Component Membrane Dynamics: Extension of the Basic Modeltwo membranes. The parameter values for σ, qi and pi, for i = 1, 2, 3, usedbelow in our simulations are computed using parameters given in Fig. 3 of[74].A simple calculation shows that the steady-state concentrations u1e, u2e,v1e, and v2e, satisfy the nonlinear algebraic systemσ11 + u41e− q1u1e − q2u1e1 + q3u1e + q4u21e+ β1(aeu1e + bev1e − u1e) = 0 ,σ21 + v41e− p1v1e − p2v1e1 + p3v1e + p4v21e+ β2(ceu1e + dev1e − v1e) = 0 ,(4.5.2)where we have defined ae, be, ce, de, Π1, and Π2, byae ≡ κ1δ−1[Dω0 coth(2Lω0) + κ2], be ≡ κ2δ−1Dω0 csch(2Lω0) ,ce ≡ κ1δ−1Dω0 csch(2Lω0) , de ≡ κ2δ−1[Dω0 coth(2Lω0) + κ1],δ ≡ D2ω20 +Dω0 (κ1 + κ2) coth(2Lω0) + κ1κ2 ,(4.5.3)where ω0 ≡ D−1/2. In terms of u1e, v1e, u2e, and v2e, we haveCe(0) = aeu1e + bev1e , u2e =11 + u41e,Ce(2L) = ceu1e + dev1e , v2e =11 + v41e.(4.5.4)To examine the stability of this steady-state solution, we introduce C(x, t) =Ce(x) + eλtη(x), together withu1(t) = u1e + eλtφ1 , u2(t) = u2e + eλtφ2 ,v1(t) = v1e + eλtψ1 , v2(t) = v2e + eλtψ2 .Upon linearizing (4.5.1), we obtain the eigenfunction η(x) satisfiesη(x) = η(0)sinh((2L− x)Ωλ)sinh(2LΩλ)+ η(2L)sinh(xΩλ)sinh(2LΩλ),We readily calculate the derivative of η(x)ηx(x) = −Ωλη(0)cosh((2L− x)Ωλ)sinh(2LΩλ)+ Ωλη(2L)cosh(xΩλ)sinh(2LΩλ),In addition, the boundary condition at x = 0 and 2L givesDηx(0) = η(0)Ge1c + φ1Ge1u1 , Dηx(2L) = η(2L)Ge2c + ψ1Ge2v1 .904.5. Two-Component Membrane Dynamics: Extension of the Basic ModelSubstitute the expression of η(x) into above expressions, we obtainD(−Ωλη(0) coth(2LΩλ) + Ωλη(2L) csch(2LΩλ)) = η(0)Ge1c + φ1Ge1u1 ,D(−Ωλη(0) csch(2LΩλ) + Ωλη(2L) coth(2LΩλ)) = η(2L)Ge2c + ψ1Ge2v1 ,Then we can solveη(0) = Aφ1 +Bψ1, η(2L) = Cφ1 +Dψ1 .For the local kinetics, we haveλφ1 = f1u1φ1 + f1u2φ2 + β1(P1cη(0) + P1u1φ1) ,λφ2 = g1u1φ1 + g1u2φ2 ,λψ1 = f2v1ψ1 + f2v2ψ2 + β2(P2cη(2L) + P2v1ψ1) ,λψ2 = g2v1ψ1 + g2v2ψ2 ,where we use fijk , gijk to represent the partial derivatives of fi and gi withrespect to jk, i, k = 1, 2, j = u, v. So it givesφ2 =g1u1φ1λ− g1u2 , ψ2 =g2v1ψ1λ− g2v2 ,and [λ− f1u1 − f1u2g1u1λ− g1u2 − β1P1u1 − β1P1cA]φ1 − β1P1cBψ1 = 0,−β2P2cCφ1 +[λ− f2v1 − f2v2g2v1λ− g2v2 − β2P2v1 − β2P2cD]ψ1 = 0,So the eigenvalue λ satisfies the transcendental equationdet λ− f1u1 −f1u2g1u1λ−g1u2 + β1 − β1A, −β1B−β2C, λ− f2v1 − f2v2g2v1λ−g2v2 + β2 − β2D = 0 ,(4.5.5)where we have defined A, B, C, and D, byA ≡ κ1∆−1[κ2 +DΩλ coth(2LΩλ)], B ≡ κ2∆−1DΩλ csch(2LΩλ) ,C ≡ κ1∆−1DΩλ csch(2LΩλ) , D ≡ κ2∆−1[κ1 +DΩλ coth(2LΩλ)],∆ ≡ D2Ω2λ + κ1κ2 + (κ1 + κ2)DΩλcoth(2LΩλ) .914.5. Two-Component Membrane Dynamics: Extension of the Basic Model0 0.2 0.4 0.6 0.8 1 1.2 1.401234βu1Figure 4.8: Left panel: Bifurcation diagram with respect to β in the twoidentical membrane case. The larger and smaller values of β at the twoHopf bifurcation points correspond to the synchronous and asynchronousmodes respectively. The branches of periodic solutions corresponding tosynchronous and asynchronous oscillations are shown. There are secondaryinstabilities bifurcating from these branches that are not shown. Thesolid/open circles indicates stable/unstable portions of the periodic solu-tion branches. The parameter values for bulk diffusion are D = 50, τ = 0.1,and L = 5, while the parameter values for the membrane dynamics areidentical for both membranes and are fixed at p1 = q1 = 1, p2 = q2 = 200,p3 = q3 = 10, p4 = q4 = 35, σ1 = σ2 = 20, and κ1 = κ2 = 20.0. Right panel:Full numerical solution of the PDE-ODE system (4.5.1) when β = 0.4, re-vealing a synchronous oscillatory instability.924.5. Two-Component Membrane Dynamics: Extension of the Basic ModelHere Ωλ ≡√1+τλD and fisj denote partial derivatives of fi where i = 1, 2with respect to sj , s = u, v and j = 1, 2.When there are two identical membranes, the eigenvector of the matrixin (4.5.5) corresponding to the eigenvalue at the stability threshold is either(1, 1)T (in-phase synchronization) or (1,−1)T (anti-phase synchronization).For this identical membrane case where β ≡ β1 = β2, in the left panel ofFig. 4.8 we plot the numerically computed bifurcation diagram in terms ofβ, showing the possibility of either synchronous or asynchronous oscillatorydynamics in the two membranes. In the right panel of Fig. 4.8 we plotthe full numerical solution computed from the PDE-ODE system (4.5.1)when β = 0.4, which reveals a synchronous oscillatory instability. Theparameter values used in the simulation are given in the caption of Fig. 4.8.To determine the number N of eigenvalues of the linearization in Re(λ) > 0for the identical membrane case, where f1 = f2 ≡ f and g1 = g2 ≡ g, werecall that λ must be a root of (4.2.17). As such, we seek roots of G(λ) = 0in Re(λ) > 0, whereG(λ) ≡ 1p±(λ)−(gu2 − λ)det (Je − λI) , Je ≡ ∂f∂u1∣∣∣u=ue, ∂f∂u2∣∣∣u=ue∂g∂u1∣∣∣u=ue, ∂g∂u2∣∣∣u=ue .(4.5.6)Here p+(λ) and p−(λ) are defined in (4.2.6) and (4.2.8), respectively. Forour example we find that p±(λ) is non-vanishing in Re(λ) > 0. Then, byusing the argument principle as in the proof of Lemma 4.3.1, and notingthat G(λ) is bounded as |λ| → +∞ in Re(λ) > 0, we obtain thatN = P +1pi[arg G] ∣∣ΓI+. (4.5.7)Here P is the number of roots of det (Je − λI) = 0 (counting multiplicity)in Re(λ) > 0, and [arg G] ∣∣ΓI+denotes the change in the argument of G(λ)along the semi-infinite imaginary axis λ = iω with 0 < ω <∞, traversed inthe downwards direction. In Fig. 4.9 we show a numerical computation ofthe winding number (4.5.7) near the values of β at the bifurcation points ofthe synchronous and asynchronous solution branches shown in the left panelof Fig. 4.8.However, when the two membranes are not identical, the matrix in(4.5.5) can have eigenvectors that are close to (1, 0)T or (0, 1)T , which cor-responds to a large difference in the amplitude of the oscillations in thetwo membranes. In such a case, we will observe a prominent oscillation in934.6. Weakly Nonlinear Theory for Synchronous Oscillations−4 −2 0 2 4 6 8−4−2024ReGImG  Sym Asy0 5 10 15−6−4−20246ReGImG  Sym AsyFigure 4.9: Winding number computation verifying the location of the Hopfbifurcation point of the synchronous mode (left panel β = 0.6757) and theasynchronous mode (right panel β = 0.2931) corresponding to the bifurca-tion diagram shown in the left panel of Fig. 4.8. The other parameter valuesare as given in the caption of Fig. 4.8. The formula (4.5.7) determines thenumber N of unstable eigenvalues in Re(λ) > 0. For both plots P = 2 in(4.5.7). When the change in the argument of G(iω) is −2pi, then N = 0.Otherwise if the change in the argument is 0, then N = 2.only one of the two membranes. We choose the coupling strengths β1 andβ2 to be the bifurcation parameters, and denote µ by µ ≡ β2 − β1. Theother parameter values in the model are taken to be the identical for thetwo membranes. To illustrate that a large oscillation amplitude ratio be-tween the two membranes can occur, in Fig. 4.10 we show full numericalresults from the PDE-ODE system (4.5.1) with D = 1 when β1 = 0.2 andβ2 = 0.7. From this figure we observe that the concentration of the signalingmolecule undergoes a large amplitude oscillation near one boundary and asignificantly smaller amplitude oscillation near the other boundary.4.6 Weakly Nonlinear Theory for SynchronousOscillationsIn §4.3 we showed that, depending on the nature of the membrane-bulk cou-pling mechanism, spatial-temporal oscillations are possible for a membrane-bulk model consisting of a single species on each membrane that is coupledthrough linear bulk diffusion. These oscillations originate from a Hopf bi-furcation associated with the symmetric steady-state solution branch. In944.6. Weakly Nonlinear Theory for Synchronous Oscillations0 10 20024u10 10 201.258021.25804v 1tFigure 4.10: Left panel: Contour plot of the oscillatory instability for thecase of heterogeneous membranes as computed from the PDE-ODE system(4.5.1) with D = 1, κ1 = κ2 = 0.1, and with the same parameters as inthe caption of Fig. 4.8. The two membranes differ only in their couplingstrengths with β1 = 0.2 and β2 = 0.7. The oscillation is pronounced onlyin the membrane at x = 0, with only a small-scale oscillation in the secondmembrane at x = 2L with L = 5. Right panel: similar plot showing u1(left boundary) and v1 (right boundary) versus t, showing the amplitudedifference.954.6. Weakly Nonlinear Theory for Synchronous Oscillationsthis section we develop a weakly nonlinear analysis in the vicinity of thisHopf bifurcation, which leads to an amplitude equation characterizing smallamplitude oscillations. By evaluating the coefficients in this amplitude equa-tion, we determine whether the Hopf bifurcation is supercritical or subcriti-cal. This asymptotic prediction for the stationary periodic solution near thebifurcation point is then compared favorably with full numerical results fortwo specific systems.We illustrate our weakly nonlinear theory only for the case of syn-chronous oscillations. The resulting model, assuming only one species onthe membrane, is formulated asCxx − 1DC =τDCt , t > 0 , 0 < x < L ;Cx(L, t) = 0 ; DCx∣∣x=0= G(C(0, t), u) ,(4.6.1a)with the local membrane dynamicsdudt= F (C(0, t), u(t)) . (4.6.1b)The steady-state solution (Ce(x), ue) of (4.6.1) satisfiesCexx − 1DCe = 0 , 0 < x < L ; Cex(L) = 0 ,DCex(0) = G(Ce(0), u) , F(Ce(0), ue)= 0 .(4.6.2)We choose the diffusivity D as the bifurcation parameter. We assume thatwhen D = D0 the linearization of (4.6.1) around the steady-state solutionhas a complex conjugate pair of imaginary eigenvalues, and that all the othereigenvalues of the linearization satisfy Re(λ) < 0.We will analyze the weakly nonlinear dynamics of (4.6.1) when D isclose to D0. As such, we introduce  1 and a detuning-parameter D1 byD = D0 + 2D1, with D1 = ±1 indicating the direction of the bifurcation,so thatD = D0 + 2D1 ,1D=1D0 + 2D1 +O(4) =1D0− 2D1D20+O(4) .(4.6.3)To derive the amplitude equation, we will employ a formal two time-scale asymptotic method where we introduce the slow time T = 2t, so thatd/dt = ∂/∂t + 2∂/∂T . For D −D0 = O(2), we then expand the solution964.6. Weakly Nonlinear Theory for Synchronous Oscillationsto (4.6.1) asC(x, t, T ) = Ce(x) + C1(x, t, T ) + 2C2(x, t, T ) + 3C3(x, t, T ) + . . . ,u(t, T ) = ue + u1(t, T ) + 2u2(t, T ) + 3u3(t, T ) + . . . .(4.6.4)We then substitute (4.6.4) into (4.6.1) and equate powers of .To leading-order in , we obtain the steady-state problem (4.6.2) whenD = D0. This has the solutionCe(x) = C0ecosh[ω0(L− x)]cosh(ω0L), ω0 ≡ 1/√D0 , (4.6.5a)with C0e ≡ Ce(0), where the constants C0e and ue are determined from thenonlinear algebraic system−C0e tanh(ω0L) = ω0G(C0e , ue) , F (C0e , ue) = 0 . (4.6.5b)The O() system is the linearization of (4.6.1) around the steady-statesolution, which is written asC1xx − 1D0C1 =τD0C1t , t > 0 , 0 < x < L ; C1x(L, t, T ) = 0 ,D0C1x∣∣x=0= C1Gec + u1Geu , on x = 0 ,u1t = C1Fec + u1Feu , on x = 0 .(4.6.6)Here F ej , Gej denote partial derivatives of F or G with respect to i evaluatedat the steady-state solution (Ce(0), ue) at x = 0, where j = {C, u}. AtO(2), we have that C2(x, t, T ) and u2(t, T ) satisfyC2xx − 1D0C2 =τD0C2t − D1D20Ce , t > 0 , 0 < x < L ; C2x(L, t, T ) = 0 ,D0C2x∣∣x=0= C2Gec + u2Geu +12(C21Gecc + u21Geuu + 2C1u1Gecu)− D1D0Ge ,on x = 0 ,u2t = C2Fec + u2Feu +12(C21Fecc + u21Feuu + 2C1u1Fecu), on x = 0 .(4.6.7)In a similar notation, F ecc denotes the second partial derivative of F withrespect to C evaluated at the steady-state pair (Ce(0), ue). Lastly, the O(3)974.6. Weakly Nonlinear Theory for Synchronous Oscillationssystem for C3(x, t, T ) and u3(t, T ), where resonances will first appear, isC3xx − 1D0C3 =τD0C3t − D1D20C1 − D1τD20C1t +τD0C1T , t > 0 , 0 < x < L ;C3x(L, t, T ) = 0 ,D0C3x∣∣x=0= C3Gec + u3Geu + C1C2Gecc + u1u2Geuu + (C1u2 + C2u1)Gecu+16(C31Geccc + 3C21u1Geccu + 3C1u21Gecuu + u31Geuuu)− D1D0(C1Gec + u1Geu) , on x = 0 ,u3t = −u1T+C3F ec + u3F eu + C1C2F ecc + u1u2F euu + (C1u2 + C2u1)F ecu+16(C31Feccc + 3C21u1Feccu + 3C1u21Fecuu + u31Feuuu), on x = 0 .(4.6.8)When D = D0, (4.6.6) is assumed to have a complex conjugate pair ofpure imaginary eigenvalues, and so we writeC1(x, t, T ) = A(T )eiλI tη0(x) + c.c. , u1(t, T ) = A(T )eiλI tφ0 + c.c. ,(4.6.9)for some λI > 0. Here η0(x) and φ0 is the eigenpair associated with thelinearized problem, and c.c. denotes the complex conjugate. An ODE forthe unknown complex amplitude A(T ) will be derived by imposing a non-resonance condition on the O(3) system (4.6.8). To normalize the eigenpair,we impose for convenience that η0(0) = 1.Upon substituting (4.6.9) into (4.6.6), we obtain that η0(x) and φ0 satisfyη′′0 −(1 + iλIτ)D0η0 = 0 , 0 < x < L ;D0η0x(0) = Gecη0(0) +Geuφ0 , η0x(L) = 0 ,F ec η0(0) + Feuφ0 = iλIφ0 , on x = 0 .(4.6.10)We solve this system, and impose the normalization η0(0) = 1, to obtainη0(x) =cosh[Ωλ(L− x)]cosh(ΩλL), φ0 =F eciλI − F eu, Ωλ ≡√1 + iτλID0,(4.6.11)where we must take the principal value of the square root. From the condi-tion for η0x on x = 0 in (4.6.10), we obtain that iλI is a root of the followingtranscendental equation, which occurs at the critical value D0 of D:(D0Ωλ tanh(ΩλL) +Gec)(iλI − F eu) + F ecGeu = 0 . (4.6.12)984.6. Weakly Nonlinear Theory for Synchronous OscillationsThe spectral problem (4.6.10) is a nonstandard eigenvalue problem sincethe eigenvalue parameter appears in both the differential operator as wellas in the boundary condition on x = 0. Therefore, we cannot simply definethe operator L = D0τddx2− 1τ and consider the problem as a special case ofLu = λu, owing to the fact that the domain of L depends on λ. Instead, wemust extend our definition of L, construct its adjoint and find an expansiontheorem following the approach in [18] for treating non self-adjoint spectralproblems with an eigenvalue-dependent boundary condition. This formalismwill then allow for a systematic imposition of a solvability condition on theO(3) problem (4.6.8), which leads to the amplitude equation for A(T ).Motivated by the form of (4.6.10), we define an operator L acting on atwo-component vector U ≡ (u(x), u1)T byL u(x)u1 ≡ D0τ u′′(x)− 1τ u(x)F ec u(0) + Feuu1 , (4.6.13a)where u(x) satisfies the boundary conditionsux(L) = 0 , D0ux(0) = Gecu(0) +Geuu1 . (4.6.13b)The calculation in (4.6.11) shows that LU = iλIU , with normalizationu(0) = 1, where U is given byU = cosh[Ωλ(L−x)]cosh(ΩλL)F eciλI−F eu . (4.6.14)Next, we define an inner product of two vectors U ≡ (u(x), u1)T andV ≡ (v(x), v1)T by〈U, V 〉 ≡∫ L0u(x)v(x) dx+ u1v1 , (4.6.15)where the overbar denotes complex conjugate, and where we restrict ourattention to the subspace whereux(L) = 0 , D0ux(0) = Gecu(0) +Geuu1 . (4.6.16)With this definition of the inner product, we integrate by parts to establishthat 〈LU, V 〉 = 〈U,L?V 〉, in terms of an adjoint operator L? defined byL?V ≡ D0τ v′′(x)− 1τ v(x)F euv1 −Geuv(0)/τ . (4.6.17)994.6. Weakly Nonlinear Theory for Synchronous OscillationsHere V is a two-component vector satisfying the adjoint boundary conditionsvx(L) = 0 , D0vx(0) = Gecv(0)− τF ec v1 . (4.6.18)A simple calculation shows that −iλI is also an eigenvalue of L? (asexpected), and that the eigenvector satisfying the adjoint problem L?V =−iλIV , normalized by v(0) = 1, and where Ωλ is defined in (4.6.11), isV = cosh[Ωλ(L−x)]cosh(ΩλL)Geuτ(F eu+iλI) . (4.6.19)With the determination of the solution to (4.6.6) now complete, we thenproceed to the O(2) system (4.6.7). We substitute (4.6.9) into (4.6.7) andseparate variables to conclude that C2(x, t, T ) and u2(t, T ) must have theformC2(x, t, T ) = g0(x, T ) + g1(x, T )eiλI t + g2(x, T )e2iλI t + c.c. ,u2(t, T ) = h0(T ) + h1(T )eiλI t + h2(T )e2iλI t + c.c. ,(4.6.20)where gj(x, T ) and hj(T ) for j = 0, 1, 2 are to be determined. Since theproblem for g1 and h1 is simply the linearized problem (4.6.6), withoutloss of generality we can take g1 ≡ 0 and h1 ≡ 0. By comparing termsindependent of powers of eiλI t, we conclude, upon using η0(0) = 1, that g0and h0 are real-valued and satisfyg0xx − 1D0g0 = −D1D20Ce , 0 < x < L ; g0x(L) = 0 ,Dg0x(0)−(g0(0)Gec + h0Geu)= |A|2 Π22− D1D0Ge , on x = 0 ,g0(0)Fec + h0Feu = −|A|2∆22on x = 0 .(4.6.21a)In the notation in (4.6.21a) we have suppressed the dependence of g0 on T .Here we have defined Π2 and ∆2 byΠ2 ≡ 2Gecc+2|φ0|2Geuu+4Re(φ0)Gecu , ∆2 ≡ 2F ecc+2|φ0|2F euu+4Re(φ0)F ecu ,(4.6.21b)where |z| denotes the modulus of z. In a similar way, upon comparing e2iλI t1004.6. Weakly Nonlinear Theory for Synchronous Oscillationsterms, we obtain that g2 and h2 satisfyg2xx − (1 + 2iτλI)D0g2 = 0 , 0 < x < L ; g2x(L) = 0 ,Dg2x(0)−(g2(0)Gec + h2Geu)= |A|2 Π12, on x = 0 ,g2(0)Fec + h2Feu − 2iλIh2 = −|A|2∆12on x = 0 ,(4.6.22a)and are complex-valued. Here, we have defined Π1 and ∆1 byΠ1 ≡ Gecc + φ20Geuu + 2φ0Gecu , ∆1 ≡ F ecc + φ20F euu + 2φ0F ecu . (4.6.22b)Next, we solve the problem (4.6.21) for g0(x) and h0 explicitly. Since theinhomogeneous term proportional to Ce in the differential operator for g0satisfies the homogeneous problem, we can readily determine the particularsolution for (4.6.21a). With this observation, and after some algebra, weobtain thatg0 = g10 cosh[ω0(L− x)]+P0D1L2ω0sinh[ω0(L− x)]−P0D12ω0x sinh[ω0(L− x)],(4.6.23a)where ω0 ≡√1/D0 and P0 is defined byP0 ≡ − C0eD20 cosh(ω0L). (4.6.23b)In (4.6.23a), the constant g10 is given byg10 = D1χ1 + |A|2χ2 , (4.6.23c)where χ1 and χ2 are defined in terms of ∆2 and Π2, given in (4.6.21b), byχ1 ≡ P03Geu − P02F euP01F eu − F ecGeu cosh(ω0L), χ2 ≡ 12(∆2Geu −Π2F euP01F eu − F ecGeu cosh(ω0L)).(4.6.23d)Here the three new quantities P01, P02, and P03, are defined in terms of P0of (4.6.23b), byP01 ≡ D0ω0 sinh(ω0L) +Gec cosh(ω0L) , P03 ≡ F ec(P0L2ω0)sinh(ω0L) ,P02 ≡ P0L2ω0[D0ω0 cosh(ω0L) +Gec sinh(ω0L)]+P0D02ω0sinh(ω0L)− GeD0.(4.6.23e)1014.6. Weakly Nonlinear Theory for Synchronous OscillationsIn addition, the real-valued constant h0 is given by in terms of P0, P01, P02,P03, Π2, and ∆2, byh0 = D1χ3 + |A|2χ4 , (4.6.24a)where χ3 and χ4 are defined byχ3 ≡ P02Fec cosh(ω0L)− P01P03P01F eu −GeuF ec cosh(ω0L), χ4 ≡ 12(Π2Fec cosh(ω0L)−∆2P01P01F eu −GeuF ec cosh(ω0L)).(4.6.24b)Finally, in our solvability condition for the amplitude equation to be derivedbelow, we will need to evaluate g0 at x = 0. Upon using (4.6.23a) and(4.6.23c), we can write g0(0) asg0(0) = D1g0c + g0A|A|2 ;g0c ≡ χ1 cosh(ω0L) + P0L2ω0sinh(ω0L) , g0A ≡ χ2 cosh(ω0L) .(4.6.25)Next, we solve the problem (4.6.22) for g2 and h2. We readily calculatethatg2(x) = g02cosh[Ω2λ(L− x)]cosh(Ω2λL), Ω2λ ≡√1 + 2iτλID0,where g02 and h2 satisfy the 2× 2 linear system[D0Ω2λ tanh(Ω2λL) +Gec]g02 +Geuh2 = −Π12A2 ,F ec g02 + (Feu − 2iλI)h2 = −∆12A2 .Here Π1 and ∆1 are defined in (4.6.22b). By solving this linear system, weobtain thatg2(0) ≡ g02 = χ6A2 , h2 = χ5A2 , (4.6.26a)where χ5 and χ6 are defined byχ5 ≡ 12(Π1Fec −∆1(D0Ω2λ tanh(Ω2λL) +Gec)(D0Ω2λ tanh(Ω2λL) +Gec)(F eu − 2iλI)−GeuF ec),χ6 ≡ 12(Π1(2iλI − F eu) + ∆1Geu(D0Ω2λ tanh(Ω2λL) +Gec)(F eu − 2iλI)−GeuF ec).(4.6.26b)With the solution of the O(2) system (4.6.7) complete, we now proceedto the O(3) problem (4.6.8), where the resonance term comes into play.1024.6. Weakly Nonlinear Theory for Synchronous OscillationsWe substitute the expression of C1, u1 and C2, u2 from (4.6.9) and (4.6.20),respectively, into (4.6.8), and identify all terms that are proportional to eiλI t.In order to eliminate resonance in (4.6.8), thereby ensuring that C3(x, t, T )and u3(t, T ) remain bounded on asymptotically long time intervals of ordert = O(−1), we require that the coefficients of the eiλI t terms satisfy a certaincompatibility condition. This leads to an amplitude equation for A(T ).To derive this amplitude equation, we substituteC3(x, t, T ) = C4(x, T ) + C3(x, T )eiλI t + C2(x, T )e2iλI t + C1(x, T )e3iλI t + c.c. ,u3(t, T ) = U4(T ) + U3(T )eiλI t + U2(T )e2iλI t + U2(T )e3iλI t + c.c. ,(4.6.27)together with (4.6.9) and (4.6.20) into (4.6.8), to obtain, after a lengthy butstraightforward calculation, that C3, U3 satisfyL C3U3 ≡ D0τ C3xx − 1τ C3F ec C3(0) + F euU3 = iλI C3U3+ R1A′φ0 −R3 ,0 < x < L ,(4.6.28a)where C3(x) satisfies the boundary conditionsC3x(L) = 0 , D0C3x∣∣x=0− [GecC3(0) +GeuU3] = R2 . (4.6.28b)In the notation of (4.6.28) we have suppressed the dependence of C3 on T .In (4.6.28), R1 is defined byR1 ≡ A′η0 − D1D0τ(1 + iτλI)Aη0 , (4.6.29a)and the residuals R2 and R3 have the formR2 = D1AR20 +A|A|2R21 , R3 = D1AR30 +A|A|2R31 . (4.6.29b)The coefficients R20 and R30 of the linear term in A areR20 ≡ g0cGecc + φ0χ3Geuu + φ0g0cGecu + χ3Gecu −1D0(Gec + φ0Geu) ,R30 ≡ g0cF ecc + φ0χ3F euu + φ0g0cF ecu + χ3F ecu ,(4.6.29c)where g0c, χ3, and φ0, are defined in (4.6.25), (4.6.24b), and (4.6.11), re-spectively. In addition, the coefficients R21 and R31 of the cubic term in1034.6. Weakly Nonlinear Theory for Synchronous Oscillations(4.6.29b) are given byR21 ≡12[Geccc +Geuuuφ20φ0 +Geccu(φ0 + 2φ0)+Gecuu(φ20 + 2φ0φ0)]+ g0AGecc + χ6Gecc + φ0χ4Geuu + φ0χ5Geuu+Gecu(φ0g0A + φ0χ6 + χ4 + χ5),(4.6.29d)andR31 ≡ 12[F eccc + Feuuuφ20φ0 + Feccu(φ0 + 2φ0)+ F ecuu(φ20 + 2φ0φ0)]+ g0AFecc + χ6Fecc + φ0χ4Feuu + φ0χ5Feuu + Fecu(φ0g0A + φ0χ6 + χ4 + χ5).(4.6.29e)In (4.6.29d) and (4.6.29e), the quantities g0A, χ3, χ4, χ5, and χ6 are definedin (4.6.25), (4.6.24b), and (4.6.26b).The following lemma, consisting of a compatibility relation between R1,R2, and R3, provides a necessary condition for the existence of a solutionto (4.6.28).Lemma 4.6.1 A necessary condition for (4.6.28) to have a solution is thatA(T ) satisfiesA′[∫ L0η0v dx+ φ0v1]=D1D0τ(1 + iτλI)A∫ L0η0v dx+ v1R3 −R2/τ ,(4.6.30)where V ≡ (v, v1)T is the nontrivial solution, given in (4.6.19), to the ho-mogeneous adjoint problem L?V = −iλIV .Proof: We define U ≡ (C3, U3)T , and we calculate from (4.6.28), and thedefinition of the inner product in (4.6.15), that〈LU − iλIU , V 〉 =∫ L0R1v dx+(A′φ0 −R3)v1 . (4.6.31)We then integrate by parts on the left-hand side of (4.6.31), and use theboundary conditions for v and C3 from (4.6.18) and (4.6.28b), respectively.1044.6. Weakly Nonlinear Theory for Synchronous OscillationsIn this way, we obtain〈LU − iλIU , V 〉 =∫ L0(D0τvxx − vτ)C3 dx+[F ec C3(0) + F euU3]v1+D0τ[C3(0)vx(0)− C3x(0)v(0)]− iλI〈U, V 〉=∫ L0(D0τvxx − vτ)C3 dx+(v1Feu −1τv(0)Geu)U3− v(0)τR2 − iλI〈U, V 〉 ,= 〈U,L?V + iλIV 〉 − v(0)τR2 .(4.6.32)To obtain the compatibility condition, we compare (4.6.31) with (4.6.32) anduse L?V + iλIV = 0. By substituting (4.6.29a) for R1 into this condition,and recalling that v(0) = 1, we readily obtain (4.6.30). Finally, upon substituting (4.6.29) into (4.6.30), we obtain the followingamplitude equation for A(T ):A′ = D1b1A+ b2A2A , (4.6.33a)where the complex-valued coefficients b1 and b2, which are independent ofD1, are given byb1 ≡ 1N[(1 + iτλI)D0τ∫ L0η0v dx+ v1R30 −R20/τ], b2 ≡ 1N[v1R31 −R21/τ],(4.6.33b)where we have defined N byN ≡[∫ L0η0v dx+ φ0v1]. (4.6.33c)In (4.6.33b), the coefficientsR20, R30, R21, andR31, are defined in (4.6.29c),(4.6.29d), and (4.6.29e). Moreover, v(x) and v1 are the components of theadjoint eigenfunction V , satisfying L?V = −iλIV , given in (4.6.19).The ODE (4.6.33a), commonly referred to as the Stuart-Landau equa-tion, characterizes the weakly nonlinear behavior of the oscillation near thecritical stability threshold. We write A as A = reiθ and decompose b1 andb2 into real and imaginary parts as b1 = b1R+ ib1I and b2 = b2R+ ib2I . From(4.6.33a), we obtain that r and θ satisfyr′ = r(D1b1R + b2Rr2), θ′ = D1b1I + b2Ir2 . (4.6.34)1054.6. Weakly Nonlinear Theory for Synchronous OscillationsThe fixed points in r, when they exist, correspond to periodic solutions forA. These special solutions arere =√−b1RD1b2R; θ = θ˜T , θ˜ ≡ D1b1I + b2Ir2e . (4.6.35)For → 0, and with D −D0 = 2D1, we conclude from (4.6.4), (4.6.9), and(4.6.35), that there is a periodic solution near the Hopf bifurcation point ofthe form C(x, t, T )u(t, T ) ∼ Ce(x)ue+ reei(λI+2θ˜)t η0(x)φ0+ c.c. .(4.6.36)The analysis of the amplitude equation (4.6.34) is routine, and dependson the signs of b1R and b2R. The Hopf bifurcation is supercritical whenb2R < 0 and is subcritical if b2R > 0. More precisely, if b1R > 0, thesymmetric steady-state solution (Ce(x), ue) is linearly stable if D1 < 0 andis unstable if D1 > 0. An unstable branch of periodic solutions exists inthe region D1 < 0 if b2R > 0 (subcritical Hopf). If b2R < 0, then there isa stable periodic solution branch in the region D1 > 0 (supercritical Hopf).In contrast, if b1R < 0, the symmetric steady-state solution (Ce(x), ue) islinearly stable if D1 > 0 and is unstable if D1 < 0. An unstable branch ofperiodic solutions exists in the region D1 > 0 if b2R > 0 (subcritical Hopf).If b2R < 0, there is a stable periodic solution branch for D1 < 0 (supercriticalHopf).Remark 4.6.1 A similar weakly nonlinear analysis can be done to deter-mine whether an asynchronous periodic solution branch is subcritical or su-percritical at the Hopf bifurcation point. To consider this case, we sim-ply replace the no-flux condition at x = L for η(x), v(x), and Cj(x) forj = 1, . . . , 3 with a homogeneous Dirichlet condition. We do not carry outthe details of this calculation here.4.6.1 Numerical Validation of the Weakly NonlinearTheory With the Explicitly Solvable ModelWe now apply our weakly nonlinear theory to the explicitly solvable modelsystem of §4.4.2, where G(C(0, t), u) and F (C(0, t), u) are given in (4.4.6).Since, for this example, F (C(0, t), u) is linear in its variables, the only non-linearity in (4.6.1) arises from G(C(0, t), u). In our analysis, we will focus1064.6. Weakly Nonlinear Theory for Synchronous Oscillationson periodic solutions that bifurcate from the steady-state solution branchwhere C0e ≡ Ce(0) is positive, and given explicitly in (4.4.8). For this systemwe compare predictions from the amplitude equation (4.6.33) with full nu-merical results computed from the numerical bifurcation software XPPAUT(cf. [16]). The numerical procedure used to compute these bifurcation dia-grams is described in §4.4.Treating D as the main bifurcation parameter we numerically computedsteady-state and periodic solution branches of (4.6.1) for two values of γ.In our numerical experiments, we found that a periodic solution bifurcatesvia a Hopf bifurcation from the positive steady-state solution branch. Asshown in Fig. 4.11, by tuning the parameter γ, while holding the other pa-rameters fixed, the Hopf bifurcation was found to change from supercriticalto subcritical.1.6 1.65 1.7 1.75 1.85.45.65.866.2Du0.95 1 1.05 1.166.577.58DuFigure 4.11: Bifurcation diagrams with diffusivity D as bifurcation param-eter showing either a supercritical or subcritical Hopf bifurcation structurefor (4.6.1), with coupling functions given in (4.4.6), for two values of γ. Leftpanel: γ = 1.55 (supercritical). Right panel: γ = 1.7 (subcritical). Thesolid and dashed lines represent stable and unstable steady-state solutions,respectively. Open circles indicate the max/min amplitude of unstable pe-riodic solutions, while the solid dots correspond to linearly stable periodicsolution branches. The bulk diffusion parameters are τ = 0.1 and L = 5.The membrane kinetic and coupling parameters are β = 1 and κ = 12.By using the amplitude equation (4.6.33), our weakly nonlinear asymp-totic theory predicts that the switching point from a supercritical to a sub-critical Hopf bifurcation occurs at γ = 1.628 (accurate to three decimalplaces), which agrees with the corresponding numerical result. Furthermore,the amplitude equation also allow us to approximate the solution near the1074.6. Weakly Nonlinear Theory for Synchronous OscillationsHopf bifurcation point as shown in (4.6.36). For the local variable u(t),we obtain from (4.6.36) that the amplitude of the periodic solution can bewritten as|u(t, T )− ue| = re|ei(λI+2θ˜)tφ0 + c.c.| = 2re|φ0| , (4.6.37)where re is the fixed point of the amplitude equation given in (4.6.35). Here  1 and φ0 is the eigenfunction of u(t), given explicitly in (4.6.11). Ifwe define uamp ≡ |u(t, T )− ue| and plot uamp versus the diffusivity D, thenuamp should be proportional to  ≡√D −D0 in the vicinity of the Hopfbifurcation point D0. The quantity uamp is plotted in Fig. 4.12.0.987 0.988 0.98900.010.02D|u(t)−u e|0 100 200 300 400 5000.9850.9860.9870.988SizeD 0Figure 4.12: Left panel: Comparison of bifurcation diagrams near a sub-critical Hopf bifurcation obtained from full numerics and from the weaklynonlinear analysis. Red dots represent the amplitude of the unstable pe-riodic solution uamp (see the text) obtained from the amplitude equation(4.6.33) and the black circles are from the full numerical simulations. Theblack and blue curves are the corresponding fitted parabola and the curva-ture of the two curves are 5.6 (black) and 5.0 (blue), respectively, at theHopf bifurcation point D0 = 0.9879 (red dots) and D0 = 0.9874 (black cir-cles). The computations are done with 80 interior spatial meshpoints. Rightpanel: Plot of the Hopf bifurcation point D0 versus the number of spatialmeshpoints of the discretized system. The parameter values are the sameas those used in Fig. 4.11 with γ = 1.7.The left panel of Fig. 4.12 shows a comparison between the analyticaland numerical bifurcation diagrams near a subcritical Hopf bifurcation pointD0. In our numerical experiments, since we discretized the PDE-ODE sys-tem (4.6.1), with coupling functions (4.4.6), with finite differences into asystem of ODE’s, some error is incurred in predicting the location of the1084.6. Weakly Nonlinear Theory for Synchronous OscillationsHopf bifurcation value D0. In contrast, in the implementation of the weaklynonlinear theory we solved the transcendental equation (4.6.12) for a com-plex conjugate pair of imaginary eigenvalues and D0 directly. Therefore, theD0 calculated from (4.6.12) is more accurate than the one computed fromthe numerics and it results in the shifting of the bifurcation point D0, asshown in the left panel of Fig. 4.12. The right panel of Fig. 4.12 shows howthe numerically calculated value D0 shifts towards the more accurate value,computed from (4.6.12), when we increase the number of spatial meshpointsin the discretized system. Although, there is a small difference in predictingthe value of D0, the amplitude calculated by the weakly nonlinear theoryshows good agreement with the corresponding amplitude computed fromthe numerical bifurcation software, as evidenced by the close comparison ofthe curvature of the two curves in Fig. 4.12 at D = D0.1.757 1.758 1.75900.0050.010.015D|u(t)−u e|0 100 200 300 400 5001.7581.759SizeD 0Figure 4.13: Left panel: Comparison of bifurcation diagrams near a super-critical Hopf bifurcation obtained from full numerics and from the weaklynonlinear analysis. The notations are the same as those in Fig. 4.12 exceptnow the red dots and black circles represent the stable periodic solutionbranch. The curvature of the two curves are 9.8 (black) and 9.3 (blue),respectively, at the Hopf bifurcation point D0 = 1.7591 (red dots) andD0 = 1.7583 (black circles). The computations are done with 80 interiorspatial meshpoints. Right panel: Plot of the Hopf bifurcation point D0versus the number of spatial meshpoints of the discretized system. Theparameter values are the same as those used in Fig. 4.11 with γ = 1.55.Fig. 4.13 compares the numerical bifurcation diagram with the asymp-totic prediction near a supercritical Hopf bifurcation point D0. The ampli-tude of the stable periodic orbits calculated by the weakly nonlinear theoryand the numerical simulations are seen to compare favorably. The right1094.6. Weakly Nonlinear Theory for Synchronous Oscillationspanel of Fig. 4.13 shows that the numerically calculated Hopf bifurcationpoint shifts toward the more accurate value as the number of interior mesh-points increase.In Fig. 4.11, we observe that when γ = 1.7 the system (4.6.1), withcoupling functions (4.4.6), undergoes a subcritical Hopf bifurcation at D0 =0.9879. In Fig. 4.14 we show that as D is decreased slowly below D0 onthe range from 1 to 0.95, that there is a delayed bifurcation effect wherebythe transition from stable steady-state to stable periodic orbits occurs whenD is somewhat below the critical value D0 predicted from the bifurcationanalysis. For our choice D = 1−σt, where σ = 0.0001, we plot D versus t inthe left panel of Fig. 4.14. The method of lines and the forward Euler methodis then used to solve (4.6.1), with coupling functions as given in (4.4.6). Inthe right panel of Fig. 4.14 we plot the numerically computed u(t) versusD(t), which clearly illustrates the delayed transition to the periodic state.0 100 200 300 400 5000.950.960.970.980.991tD0.94 0.96 0.98 16.46.66.877.27.47.6D(t)u(t)Figure 4.14: Left panel: Plot the diffusivity D as a function of time for D =1−σt, where σ = 0.0001. Right panel: Plot of the local variable u(t) versusD(t). As D passes slowly below the critical value D0, the periodic solutionappears when D is around 0.965, which due to the delayed bifurcation effectis less than the theoretically predicted value. The parameter values used arethe same as in the right panel of Fig. 4.11.4.6.2 Numerical Validation of the Weakly NonlinearTheory With the Dictyostelium ModelIn the following, we apply our weakly nonlinear theory to the model ofcell signaling in Dictyostelium studied in §4.4.4, where G(C(0, t), u) andF (C(0, t), u) are given in (4.4.21).1104.6. Weakly Nonlinear Theory for Synchronous OscillationsSimilar to the analysis done in §4.6.1, we will compare predictions fromthe amplitude equation (4.6.33) with full numerical results computed fromthe numerical bifurcation software XPPAUT (cf. [16]). We use the diffusioncoefficient D as the bifurcation parameters, numerically compute steadystate and the synchronous periodic solution branches. One typical bifurca-tion diagram of this system is given in the left panel of Fig. 4.7. We observethat when D is gradually decreasing from 0.65, the steady state solutionbecomes unstable at the Hopf bifurcation point D0 ≈ 0.62 and an unstablesynchronous periodic solution appears, which indicates a subcritical Hopfbifurcation.In Fig. 4.7, the vertical axis represents the value of the global variableC(x, t) near the membrane x = 0. Then from (4.6.36), we readily calculatethe amplitude of the periodic solution can be written as|C(0, t, T )− Ce(0)| = re|ei(λI+2θ˜)tη0(0) + c.c.| = 2re|η0(0)| , (4.6.38)where re is the fixed point of the amplitude equation given in (4.6.35).  1and η0(0) is the eigenfunction of C(x, t) evaluated at x = 0. Similarly, wedefine camp ≡ |C(0, t, T ) − Ce(0)| and plot camp versus the diffusivity D,then camp should be proportional to  ≡√D −D0 in the vicinity of the Hopfbifurcation point D0. The left panel of Fig. 4.15 shows the plot of camp versusD in the neighborhood of D0. The amplitude of the unstable periodic orbitscalculated by the weakly nonlinear theory and the numerical simulationsare seen to compare favorably. In the right panel of Fig. 4.15 we observethe value of the Hopf bifurcation D0 is gradually approaching a horizontalasymptote when the number of spatial meshpoints of the discretized systemincreases.1114.6. Weakly Nonlinear Theory for Synchronous Oscillations0.621 0.622 0.62300.010.020.030.04D|C(0,t)−Ce(0)|0 100 200 300 400 5000.62120.62130.6214SizeD 0Figure 4.15: Left panel: Comparison of bifurcation diagrams near a sub-critical Hopf bifurcation obtained from full numerics and from the weaklynonlinear analysis. The notation are the same as those in Fig. 4.12. Thecurvature of the two curves are 0.63(black) and 0.77(blue), respectively atthe Hopf bifurcation point D0 = 0.62134(red dots) and D0 = 0.62132(blackdots). The computations are done with 100 interior spatial meshpoints.Right panel: Plot of the Hopf bifurcation point D0 versus the number ofspatial meshpoints of the discretized system. The parameter values are thesame as those used in the left panel of Fig. 4.7.112Chapter 5A Model of Bulk-DiffusionCoupled to ActiveMembranes With Slow-FastKineticsIn a previous work [23] by Gomez et. al, a model of coupled active mem-branes with activator-inhibitor dynamics is proposed. They use numericalsimulations together with nullcline analysis to explore different oscillatorypatterns that the system possessed and predict the stability boundary of thesteady state of the system.The goal of this chapter is to give a detailed analysis of the triggering ofsynchronous oscillations for the coupled 1-D coupled membrane-bulk modelof [23]. This chapter proceeds as follows.In §5.1, we restate the model system constructed in [23] and give a reviewof the nullcline analysis in the slow-fast limit where the time scale of theactivator and inhibitor dynamics are largely distinct. In §5.2 we constructa symmetric steady-state solution, and we formulate the linear stabilityproblem for this solution. In §5.3 we consider a one-bulk species modelwhere only the inhibitor V can diffuse in the bulk. For this case, in §5.3.1an asymptotic analysis for  → 0 of the stability problem is provided toanalyze Hopf bifurcations of the symmetric steady-state and the emergenceof asymmetric steady-states. The  = O(1) problem for one diffusing bulkspecies is studied numerically in §5.3.2. In §5.4 we extend our analysis tothe full model which consists of two diffusing bulk species.5.1 Coupled Membrane-bulk Model WithActivator-Inhibitor DynamicsThe coupled membrane-bulk model of [23] on a one-dimensional spatial do-main consists of two active membranes with activator-inhibitor dynamics1135.1. Coupled Membrane-bulk Model With Activator-Inhibitor Dynamicsat x = 0 and x = L that are coupled through passive diffusion in the bulkregion 0 < x < L. In the bulk we assume that there are two diffusing specieswith concentrations U(x, t) and V (x, t) satisfyingUt = DuUxx − σuU , Vt = DvVxx − σvV , 0 < x < L , t > 0 .(5.1.1a)Here Du and Dv are the two diffusion coefficients, while σu and σv are theconstant bulk decay rates. The kinetics on the two active membranes atx = 0 and x = L are assumed to be identical, and given byu′1 = f(u1, v1) + kuUx(0, t) , v′1 = g(u1, v1) + kvVx(0, t) ,u′2 = f(u2, v2)− kuUx(L, t) , v′2 = g(u2, v2)− kvVx(L, t) ,(5.1.1b)where ui and vi, i = 1, 2 denote the two concentrations on the membrane, sothat u1(t) = U(0, t), u2(t) = U(L, t), v1(t) = V (0, t) and v2 = V (L, t). Theparameter  that accompanies g(ui, vi) determines the relative difference inthe time-scale for the boundary kinetics, so that the time evolution of u ismuch faster than v if 0 <  1. The terms kuUx(0, t) and kvVx(0, t) accountfor the exchange of species between the membrane and the bulk, where theconstants ku and kv are the coupling strengths. The kinetics f(u, v) andg(u, v) are chosen to account for a local activator-inhibitor dynamics, andwe use the Fitzhugh-Nagumo type kinetics considered in [23], given for q > 0and z > 0 byf(u, v) = u− q(u− 2)3 + 4− v , g(u, v) = uz − v . (5.1.1c)The qualitative mechanism, as discussed in [23], for the triggering of time-periodic solutions for the coupled system (5.1.1) in the slow-fast limit → 0is based on a simple nullcline analysis, and is described in the caption ofFig. 5.1. We assume that the parameters q and z are chosen so that themembrane kinetics in the absence of any coupling to the bulk has a singlestable equilibrium point. For q = 5 and z = 3.5, this stable fixed point(ue, ve) occurs at the intersection of the two nullclines v = V(u) = u −q(u − 2)3 + 4 and v = zu where f(u,V(u)) = 0 and g(u, zu) = 0. In thelimit → 0, it is readily shown from the Jacobian of the membrane kineticsthat the equilibrium state (ue, ve) is linearly stable only when V ′(ue) < 0,and undergoes a Hopf bifurcation when V ′(ue) crosses through zero. For acoupling strength in (5.1.1b) for which ku = O() and kv = O(), it readilyfollows, to leading order in , that the steady-state of (5.1.1b) remains on thenullcline f(u, v) = 0. However, the effect of the coupling to the bulk for V isto shift the nullcline for the v-component in (5.1.1b) to v = βu, for some β1145.1. Coupled Membrane-bulk Model With Activator-Inhibitor Dynamics0 1 2 3 40246810uVFigure 5.1: Plot of the nullcline V(u) = u−q(u−2)3 +4 with q = 5 for whichf(u,V(u)) = 0. The straight lines are v = βu with β = 3.5, β = 3, andβ = 2. The straight line v = 3.5u is the nullcline of g(u, v) = 0 when z = 3.5for the uncoupled membrane-bulk problem. As the bulk-coupling strengthkv = O() increases, the effective parameter β decreases. The dotted linewith β = 3 intersects V(u) in the unstable region where V ′(u) > 0. Forβ = 2.0, the intersection again occurs in the stable region for the membranekinetics. For a coupling strength where ku = O() in (5.1.1b) the nullclinefor f(u, v) = 0 is, to leading order in , unchanged by the coupling of themembrane to the bulk.1155.2. The Steady-State Solution and the Formulation of the Linear Stability Problemthat is a monotonically decreasing function of kv. This shows that there isan intermediate range of β where the equilibrium point is unstable, such asgiven by the dotted line in Fig. 5.1. Although this mechanism of [23] doesprovide a clear qualitative reason underlying the triggering of oscillationsinduced by membrane-bulk coupling in the limit → 0, it does not providea detailed quantitative characterization of these oscillations.As an extension of this qualitative and numerical analysis of [23], we useasymptotic analysis together with bifurcation and stability theory to give adetailed theoretical analysis of the onset of oscillatory dynamics for (5.1.1).In the singular limit → 0 of slow-fast membrane dynamics, and assumingonly one diffusing species in the bulk, our stability analysis of the uniquesymmetric steady-state solution will provide a detailed phase diagram inparameter space where various types of oscillatory dynamics can occur. Inthe limit  → 0, our asymptotic analysis of the spectral problem, and inparticular the winding number, will yield asymptotic approximations forthe Hopf bifurcation thresholds in parameter space for both the synchronousand asynchronous periodic solution branches, as well as the Hopf bifurcationfrequencies near onset. In addition, zero-eigenvalue crossings correspondingto the emergence of asymmetric steady-state solutions will be studied. Inthe non-singular case, where  = O(1), a numerical study of the windingnumber together with the numerical bifurcation software XPPAUT [16] willbe used to to construct global bifurcation diagrams of steady-states andperiodic solution branches for the case of either one or two diffusing speciesin the bulk. Overall, we show that stable synchronous oscillations betweenthe two membranes is a robust feature of the dynamics that occurs in a wideparameter regime. A glimpse at some more exotic dynamics such as a torusbifurcation, arising from secondary bifurcations, is given.5.2 The Steady-State Solution and theFormulation of the Linear Stability ProblemIn this section we determine a symmetric steady-state solution for (5.1.1)and analyze the linear stability properties of this solution. Since the twomembranes are identical, it is natural to seek a steady-state solution that issymmetric about the midline x = L/2, so that for the steady-state problemof (5.1.1) we consider 0 < x < L/2 and impose zero flux conditions atx = L/2. In the steady-state analysis, for convenience we drop the subscriptsfor u and v at the left membrane. As such, the symmetric steady-state1165.2. The Steady-State Solution and the Formulation of the Linear Stability Problemsolution Ue and Ve for (5.1.1) satisfiesDuUexx − σuUe = 0 , DvVexx − σvVe = 0 , 0 < x < L/2 , (5.2.1)with boundary conditions Ue(0) = ue, Ve(0) = ve, Uex(L/2)= 0, andVex(L/2)= 0. The solution to this problem isUe(x) = uecosh[ωu(L2 − x)]cosh(ωuL/2) , ωu ≡√ σuDu,Ve(x) = vecosh[ωv(L2 − x)]cosh(ωvL/2) , ωv ≡√ σvDv.(5.2.2)Then, by using (5.1.1b) with kinetics (5.1.1c), we readily derive that u = ueis a root of the cubic H(u) = 0, given byH(u) ≡ qu3 − 6qu2 + (12q − 1 + au + β) u− (8q + 4) , (5.2.3a)where we have defined au and β, byau ≡ kuωu tanh(ωuL/2), β ≡ z/(+ av) , where av ≡ kvωv tanh(ωvL/2).(5.2.3b)In terms of any solution u = ue to the cubic, ve is given by ve = βue.We now claim that (5.1.1) has a unique positive symmetric steady-statesolution. To show this, we must verify that there is a unique root ue > 0 toH(u) = 0 when u > 0. In our proof below we will consider two cases: CaseI: au+β > 1. Case II: 0 < au+β ≤ 1. We first consider Case I. We calculatethat H(0) = −(8q + 4) < 0 and H(u) → +∞ as u → +∞. Moreover, wederive that H′(u) = 3q(u− 2)2 + (au + β − 1), so that H′(u) > 0 for u > 0in Case I. Since H(0) < 0, H(u) → +∞ as u → ∞, and H(u) is monotoneincreasing on u > 0, there is a unique root to H(u) = 0 in u > 0 in Case I.Next, we consider Case II. We conclude that H′(u) = 0 at exactly twopoints u = u±, given byu± ≡ 2± 1√3q√1− au − β . (5.2.4)Here u− < 2 < u+, with u− and u+ a local maximum and local minimumof H(u), respectively. Therefore, since 2 < u+, H(2) = −6 + 2(au + β) < 0,H(u) → +∞ as u → ∞, and u+ is a local minimum point of H(u), weconclude that there is a unique root to H(u) = 0 in u > 2. To conclude theproof for Case II, we need only show that H(u−) < 0 whenever u− is on1175.2. The Steady-State Solution and the Formulation of the Linear Stability Problemthe range 0 < u− < 2. To show this we use 3q(u− − 2)2 = 1 − (au + β) tocalculateH(u−) = q(u− − 2)3 − u− − 4 + (au + β)u−=[1− (β + au)]3(u− − 2)− u− − 4 + (au + β)u− ,=13(−2u−[1− (au + β)]− 14 + 2(au + β)) .Since the last expression shows that H(u−) < 0 for 0 < u− < 2 when0 < au + β ≤ 1, we conclude that there are no additional roots to H(u) = 0located on 0 < u < 2 whenever 0 < au + β ≤ 1. Combining the results ofCase I and Case II, we conclude that, for any au + β > 0, there is a uniquepositive symmetric steady-state solution to (5.1.1).We will assume that z and q are such that the steady-state of the mem-brane kinetics, when uncoupled to the bulk, is a linearly stable fixed point.As a result, any instability that arises in our analysis is due specifically tothe coupling of the two membranes by the bulk. For the uncoupled problem,where au = 0 and av = 0 in (5.2.3), we obtain that ue is a root of (5.2.3a)in which we set au = 0 and β = z in (5.2.3a). In terms of ue, the Jacobianof the membrane kinetics isJ0e ≡ feu , fevgeu gev = 1− 3q(ue − 2)2 , −1z − , (5.2.5)when uncoupled from the bulk. We calculate the determinant and trace ofthis matrix asdetJ0e = [z − 1 + 3q(ue − 2)2], trJ0e = 1− − 3q(ue − 2)2 . (5.2.6)To ensure that the uncoupled membrane kinetics has a stable fixed point,we will assume that z > 1, so that detJ0e > 0 for any parameter set. There-fore, the stability of the fixed point is determined by the sign of trJ0e . Todetermine the region of the (q, z) plane, with z > 1, for which trJ0e < 0,we simply locate the stability boundary where trJ0e = 0. We solve (5.2.6)for q and then (5.2.3a), with au = 0 and β = z, for z, to readily obtain aparametric description of this stability boundary in the form q = Q(u) andz = Z(u), for u ≥ 0, whereq = Q(u) ≡ (1− )3(u− 2)2 , z = Z(u) ≡ 1 +4u− (1− )3u(u− 2) . (5.2.7)1185.2. The Steady-State Solution and the Formulation of the Linear Stability Problem1 2 3 4 50246zq1 2 3 4 50246zqFigure 5.2: The stability boundary when  = 0.015 (left panel) and when = 0.3 (right panel) for the membrane dynamics when uncoupled to thebulk diffusion. The heavy solid curve is the stability boundary (5.2.7) wheretrJ0e = 0. Above this curve in the shaded region, trJ0e < 0 so that theuncoupled membrane kinetics has a stable fixed point in this region. For = 0.015, the parameter values z = 3.5 and q = 5, used in [23], correspondto the marked point.For  = 0.015 and  = 0.3 in the left and right panels of Fig. 5.2, re-spectively, we plot the stability boundary (5.2.7) for the uncoupled problemin the (q, z) plane for z > 1. In the shaded regions of this figure we havetrJ0e < 0. In our analysis below, we will assume that the pair (q, z) belongsto this region, which ensures that the fixed point associated with the uncou-pled membrane kinetics is linearly stable. As a remark, for the parametervalues z = 3.5, q = 5.0, and  = 0.015, as used in [23], we calculate from(5.2.3a) and (5.2.6) that ue ≈ 1.67 and trJ0e ≈ −0.649 < 0, so that the fixedpoint for the uncoupled dynamics is stable. This point is marked in the leftpanel of Fig. 5.2.5.2.1 Formulation of the Linear Stability ProblemNext, we derive the linear stability problem associated with the symmetricsteady-state solution. We introduceU(x, t) = Ue(x) + ϕ(x)eλt , u(t) = ue + ξeλt ,V (x, t) = Ve(x) + ψ(x)eλt , v(t) = ve + ηeλt ,1195.2. The Steady-State Solution and the Formulation of the Linear Stability Probleminto (5.1.1). Upon linearizing the resulting system we obtain thatDuϕxx − (σu + λ)ϕ = 0 , Dvψxx − (σv + λ)ψ = 0 , 0 < x < L/2 ,(5.2.8a)with ϕ(0) = ξ and ψ(0) = η, together with the linearized membrane kineticson x = 0 given byξλ = ξfeu + ηfev + kuϕ′(0) , ηλ = ξgeu + ηgev + kvψ′(0) . (5.2.8b)Since the two membranes, one at x = 0 and the other at x = L, are iden-tical, then due to reflection symmetry there are two types of eigenfunctionsfor (5.2.8). One type is the synchronous, or in-phase, mode withϕ′(L/2)= 0 , ψ′(L/2)= 0 , (5.2.9)while the other is the asynchronous, or anti-phase, mode for whichϕ(L/2)= 0 , ψ(L/2)= 0 . (5.2.10)Upon solving (5.2.8) for the in-phase, or synchronous (’+’) mode, we getϕ+(x) = ξcosh[Ωu(L2 − x)]cosh(ΩuL/2) , ψ+(x) = η cosh[Ωv(L2 − x)]cosh(ΩvL/2) , (5.2.11)where we have defined Ωu = Ωu(λ) and Ωv = Ωv(λ) byΩu ≡√σu + λDu, Ωv ≡√σv + λDv. (5.2.12)In (5.2.12) we have chosen the principal value of the square root, whichensures that ϕ+ and ψ+ are analytic in Re(λ) > 0 and decay at x = L/2when L 1. By substituting (5.2.11) into the boundary condition (5.2.8b)at x = 0, we get that ξ and η satisfy the homogeneous linear system(feu−λ−kup+Ωu)ξ+fevη = 0 , geuξ+(gev−λ−kvq+Ωv)η = 0 , (5.2.13)where we have defined p+ = p+(λ) and q+ = q+(λ) byp+ ≡ tanh(ΩuL/2), q+ ≡ tanh(ΩvL/2). (5.2.14)1205.2. The Steady-State Solution and the Formulation of the Linear Stability ProblemBy setting the determinant of the coefficient matrix of this linear system tozero, this linear system has a nontrivial solution if and only if λ is a root ofthe transcendental equation(feu − λ− kup+Ωu) (gev − λ− kvq+Ωv)− fevgeu = 0 . (5.2.15)Similarly, for the anti-phase, or asynchronous (’-’), mode we obtain from(5.2.8a) and (5.2.10) thatϕ−(x) = ξsinh[Ωu(L2 − x)]sinh(ΩuL/2) , ψ−(x) = η sinh[Ωv(L2 − x)]sinh(ΩvL/2) ,where λ satisfies(feu − λ− kup−Ωu)(gev − λ− kvq−Ωv)− fevgeu = 0 . (5.2.16)Here we have defined p− = p−(λ) and q− = q−(λ) byp− ≡ coth(ΩuL/2), q− ≡ coth(ΩvL/2). (5.2.17)The eigenvalue problems (5.2.15) and (5.2.16) for the synchronous andasynchronous modes can be written in terms of locating the roots λ ofF±(λ) = 0, whereF±(λ) ≡ 1kukvp±q±+ΩuΩvdet(Je − λI)−Ωu(gev − λ)kvq± det(Je − λI)−Ωv(feu − λ)kup± det(Je − λI) .(5.2.18)Here Je is the Jacobian matrix of the membrane dynamics (5.1.1b), evalu-ated at the steady-state solution associated with the coupled membrane-bulkmodel. Therefore, from (5.2.6), we havedet(Je − λI) = λ2 − (trJe)λ+ detJe ,trJe = 1− − 3q(ue − 2)2 , detJe = [z − 1 + 3q(ue − 2)2],(5.2.19)where ue is the unique root of H(u) = 0, with H(u) as defined in (5.2.3a).In (5.2.18) we have gev = − and feu = trJe + .To analyze the stability of the steady-state solution, we use the argumentprinciple of complex analysis to count the number N of roots of F±(λ) = 0in the right half-plane Re(λ) > 0. We take the counterclockwise contourconsisting of the imaginary axis −iR ≤ Imλ ≤ iR, denoted by Γ− ∪Γ+, andthe semicircle ΓR, given by |λ| = R > 0, for |argλ| ≤ pi/2. Since 1/p±, 1/q±,1215.3. One Diffusive Species in the BulkΩu, and Ωv are analytic functions of λ in the right half-plane Reλ > 0, anddetJe > 0 since z > 1, it follows that the number P of poles of F±(λ) in theright half-plane depends only on the sign of trJe.Since p± → 1 and q± → 1 as R → ∞ on ΓR, with λ = Reiω and|ω| ≤ pi/2, we have that the decay of F±(λ) as R→∞ is dominated by thefirst term in (5.2.18), so thatF±(λ) = 1kukv+O(R−1/2), as R→∞ ,on ΓR. Therefore, there is no change in the argument of F±(λ) over ΓR asR → ∞. By using the argument principle, together with F±(λ¯) = F±(λ),we conclude thatN = P +1pi[argF±]Γ+ , where P = 2 if trJe > 00 if trJe < 0 . (5.2.20)Here [argF±]Γ+ denotes the change in the argument of F± along the semi-infinite imaginary axis Γ+ = iλI , with 0 ≤ λI < ∞, traversed downwards.When the membrane dynamics are uncoupled with the bulk we have as-sumed trJe < 0, so that P = 0. For the coupled problem, we show belowthat P depends on the strength of the coupling between the membrane andthe bulk. We remark that, although, it is analytically intractable to calcu-late [argF±]ΓI , this quantity is easily evaluated numerically after separatingF±(iλI) into real and imaginary parts. In terms of this readily-computedquantity, the global criterion (5.2.20) yields the number of unstable eigen-values of the linearization (5.2.8) in Re(λ) > 0.In §5.3 we determine P and [argF±]ΓI for the synchronous and asyn-chronous modes in the limiting case where there is only one bulk diffusingspecies. The general case of two diffusing bulk species is considered in §5.4.5.3 One Diffusive Species in the BulkIn this section we analyze the special case, considered in [23], where there isonly one diffusing species in the bulk. As in [23], we assume that only theinhibitor V can detach from the membrane and diffuse and be degraded inthe bulk. As such, in this section we consider the limiting problem whereku = 0 in (5.1.1) so that U(x, t) has no effect on the membrane kinetics, andcan be neglected. The one-bulk species model is formulated asVt = DvVxx − σvV , 0 < x < L , t > 0 , (5.3.1a)1225.3. One Diffusive Species in the Bulkcoupled to the membrane dynamics v1(t) = V (0, t) and v2(t) = V (L, t),whereu′1 = f(u1, v1) , v′1 = g(u1, v1) + kvVx(0, t) ;u′2 = f(u2, v2) , v′2 = g(u2, v2)− kvVx(L, t) .(5.3.1b)Here f(u, v) and g(u, v) are given in (5.1.1c). For this limiting problem, thesymmetric steady-state solution satisfiesDvVexx − σvVe = 0 , 0 < x < L/2 ; Ve(0) = ve , Vex(L/2) = 0 ,(5.3.2a)f(ue, ve) = 0 , g(ue, ve) + kvVex(0) = 0 . (5.3.2b)Defining ωv ≡√σv/Dv, we obtain that ue is the unique root of H(u) = 0,whereH(u) ≡ qu3 − 6qu2 + (12q − 1 + β) u− (8q + 4) ;β ≡ z/(1 + −1av) , av ≡ kvωv tanh(ωvL/2).(5.3.3)By setting ku = 0 in the stability analysis of §5.2.1, we obtain that thelinear stability properties of this symmetric steady-state is determined bythe roots of G±(λ) = 0, defined byG±(λ) = 1kvq±− Ωv(feu − λ)det(Je − λI) , (5.3.4)where Ωv, q+, and q−, are defined in (5.2.12), (5.2.14), and (5.2.17), respec-tively.To analyze the roots of (5.3.4), it is convenient to write (5.3.3) and (5.3.4)in terms of dimensionless bifurcation parameters. To this end, we introducetwo parameter l1 and l0, and define β in (5.3.3) in terms of them asl1 = kv/ , l0 =√Dv/σv = 1/ωv , β = z[1 +l1l0tanh(L2l0)]−1.(5.3.5)The ratio L/l0 is a nondimensional measure of the distance between thecompartments to the diffusion length l0, while l1/l0 is a nondimensionalmeasure of the strength of the membrane-bulk coupling relative to the dif-fusion length.Next, we define b ≡ trJe, and use (5.2.19) to writefeu = b+ , detJe = a , a ≡ z−−b > 0 , where b = 1−−3q(ue−2)2 .(5.3.6)1235.3. One Diffusive Species in the BulkWe recall that a > 0 since we assume from §5.2 that z > 1. In this way, andupon writing q± and Ωv in terms of l0 and l1, we obtain that (5.3.4) can bewritten as G±(λ) =[1/(l0)]G0±(λ), where G0±(λ) is defined byG0±(λ) ≡l0l1q±(τvλ)− √1 + τvλ(b+ − λλ2 − bλ+ a), (5.3.7)where q±(τvλ), with τv ≡ 1/σv, is defined byq±(τvλ) = tanh[L2l0√1 + τvλ], synchronous (+) modecoth[L2l0√1 + τvλ], asynchronous (−) mode. (5.3.8)By using a winding number argument, similar to that in §5.2.1, the numberN of unstable roots of (5.3.7) in Re(λ) > 0 isN = P +1pi[argG0±]Γ+ , where P = 2 if b = trJe > 00 if b = trJe < 0 .(5.3.9)In terms of our dimensionless parameters, we remark that the symmetricsteady-state solution, and consequently b = trJe, depends on l1/l0 and L/l0.In contrast, the stability properties of this solution, as to be analyzed from(5.3.7) below in §5.3.1, depends on l1/l0, L/l0, and τv ≡ 1/σv.In our stability analysis we will consider two distinct cases. In §5.3.1we analyze (5.3.7) in the limit  → 0 where the membrane-bulk couplingparameter satisfies kv = O(), so that β = O(1) in (5.3.5). For the parametervalues given in [23]Dv = 0.5 ,  = 0.015 , σv = Dv/100 , z = 3.5 , q = 5 , L = 10 , kv = 0.0225 ,(5.3.10a)we calculate thatl1 = 1.5 , l0 = 10 , τv = 200 ,l1l0= 0.15 ,Ll0= 1 . (5.3.10b)For this parameter set, our small  stability analysis below will provide atheoretical understanding of the numerical results in [23]. In §5.3.2, weuse a numerical winding number approach together with numerically com-puted global bifurcation diagrams, obtained using the bifurcation softwareXPPAUT [16], to study the  = O(1) problem.1245.3. One Diffusive Species in the Bulk5.3.1 Stability Analysis for the → 0 Limiting ProblemIn this subsection we study the roots of (5.3.7) for various ranges of b. Todetermine the winding number [argG0±]Γ+ in (5.3.9) we must consider severaldistinct ranges of λI . For   1, we will calculate this winding numberanalytically.We first observe that Re(G0±(iλI)) → l0/l1 > 0 as λI → +∞, so thatin the (G0R±,G0I±) plane, where G0R±(λI) ≡ Re(G0±(iλI)) and G0I±(λI) ≡Im(G0±(iλI)), we begin at a point on the positive real axis. Then, sinceRe(√1 + τvλ) > 0, and both Re(tanh z) > 0 and Re(coth z) > 0 whenRe(z) > 0, we conclude for → 0 from (5.3.7) that G0R±(λI) > 0 for λI > 0with λI = O(1). We then use (5.3.7) to calculate G0±(0), for any  > 0, asG0±(0) =l0l1q±(0)− (b+ )a, where a = z − − b > 0 . (5.3.11)We conclude that G0±(0) > 0 when b+ < al0/[l1q±(0)]. Since a = z−−b >0, we solve this inequality for b to obtain that G0±(0) > 0 when b < b± andG0±(0) < 0 when b > b±, where b± is defined byb < b± ≡ z[1 +l1q±(0)l0]−1−  . (5.3.12)For the synchronous (+) mode, we now show that b < b+ always holdsfor any root to the cubic H(u) = 0, defined in (5.3.3). In particular, wewill prove that b > b+ is incompatible with a root of (5.3.3). To show this,we first observe that since q+(0) = tanh(L/2l0), the condition b > b+ isequivalent to b > β − , where β is defined in (5.3.5). We have from (5.3.6)that b > β −  and β > 0, when 0 < β < 1− 3q(u− 2)2, where u > 0 is theunique root of H(u) ≡ q(u − 2)3 + βu − (4 + u) = 0. From this inequalityon β, we calculateH(u) < q(u− 2)3 + u[1− 3q(u− 2)2]− (4 + u)= q(u− 2)3 − 4− 3qu(u− 2)2 = −2q(u− 2)2(u+ 1)− 4 < 0 ,for all u > 0. Therefore, there is no root to H(u) = 0 when b > b+. As aconsequence, we must have b < b+, so that for the synchronous mode wehave G0+(0) > 0 unconditionally.To determine the curve in the l1/l0 versus L/l0 plane where the asyn-chronous (-) mode has a zero-eigenvalue crossing, we set b = b−, and use1255.3. One Diffusive Species in the Bulk(5.3.12) together with (5.3.3) to obtain that a zero eigenvalue crossing occurswhenz[1 +l1l0tanh(L2l0)]−1= −q(u− 2)3u+4u+ 1 ,z[1 +l1l0coth(L2l0)]−1= 1− 3q(u− 2)2 .We rearrange these expressions to getl1l0tanh(L2l0)= χ1(u) , χ1(u) ≡ zu−q(u− 2)3 + 4 + u − 1 ,l1l0coth(L2l0)= χ2(u) , χ2(u) ≡ zu1− 3q(u− 2)2 − 1 .(5.3.13)For the range of u > 0 for which χ1 > 0, χ2 > 0, and χ1/χ2 < 1, we readilyderive from (5.3.13) that the curve in the l1/l0 versus L/l0 plane where theasynchronous (-) mode has a zero-eigenvalue crossing is given parametricallyin terms of u byl1l0=√χ1(u)χ2(u) ,Ll0= ln(√χ2(u) +√χ1(u)√χ2(u)−√χ1(u)). (5.3.14)For q = 5, z = 3.5, (5.3.14) yields the upward facing horseshoe-shaped curveshown in the left panel of Fig. 5.4. Below in (5.3.30), we show that this zero-eigenvalue crossing for the asynchronous mode is a bifurcation point whereasymmetric equilibria of (5.3.1) bifurcate from the symmetric steady-statesolution branch.Now that the possibility of zero-eigenvalue crossings has been analyzed,we proceed to determine [argG0±]Γ+ . Since G0R±(λI) > 0 for λI > 0 withλI = O(1), we need only analyze (5.3.7) with λ = iλI and λI near theorigin. For |b|  O(), we set λI = λ0I with λ0I = O(1) in (5.3.7) to obtainG0R± = Re(G0±(iλ0I))∼ l0l1q±(0)− baa2 + b2(λ0I)2. (5.3.15)For the synchronous mode, we conclude from (5.3.15) that G0R± > 0 whenλI = O(), and consequently [argG0+]Γ+ = 0, for any b with |b|  O(). Asa result, for the synchronous mode, we obtain from (5.3.9) thatN = P , where P = 2 if b = trJe > 00 if b = trJe < 0 , (synchronous mode) .(5.3.16)1265.3. One Diffusive Species in the BulkIn contrast, for the asynchronous mode, we conclude from (5.3.15) thatfor λI = O() and |b|  O(), we have G0R−(λI) > 0 when b < b−, andG0R−(λI) < 0 when b > b−, where b− is defined in (5.3.12). Therefore, forthe asynchronous mode, we have [argG0−]Γ+ = 0 when b < b−, so that (5.3.16)still holds when b < b− and |b|  O(). However, it remains to calculate[argG0−]Γ+ = 0 when b > b−, for which G0R−(λI) < 0. This computation isdone numerically below.2 6 10 14−4−202G0R±G0 I±  Syn Asy0 0.2 0.4 0.6 0.8 100.20.40.60.8λ   G+(λ) G−(λ) H(λ)Figure 5.3: Left panel: The path G0±(iλI) = G0R±(λI) + iG0I±(λI), for theparameter set (5.3.10a) of [23]. For both ± modes, we start at λI = 100(solid dot), which corresponds to the common value G0R± ∼ l0/l1 > 0 andG0I± = 0. As λI increases, G0R± remains positive, and the path for each modeterminates when λI = 0 at different points on the positive real axis, withoutwrapping around the origin. This establishes that [argG0±]Γ+ = 0. Rightpanel: The functions G±(λ) and H(λ), as defined in (5.3.17), when λ > 0 isreal, for the parameter set (5.3.10a). The curves for G+ and G− essentiallycoincide. Since there are no intersections between G± and H(λ), then thereare no real positive roots to G0±(λ) = 0 in (5.3.7) for either the synchronousor asynchronous modes.For any b independent of , this analysis shows for the synchronousmode that N = 0 for b < 0 and N = 2 for any b > 0. For the asynchronousmode we have N = 0 for b < 0 and N = 2 for any 0 < b < b−. As aresult, to leading order in , we conclude that both the synchronous andasynchronous modes undergo a Hopf bifurcation as b crosses through zero.For the parameter set (5.3.10a), as used in [23], in the left panel of Fig. 5.3 weplot the numerically computed path of G0±(iλI) in the plane (G0R±,G0I±). Forthis parameter set we calculate numerically that b− ≈ 2.627 and b+ ≈ 3.258from (5.3.12), and b = trJe = 1 −  − 3q(ue − 2)2 ≈ 0.2746, from (5.2.19),1275.3. One Diffusive Species in the Bulkwhere ue is the unique root of H(u) = 0 defined in (5.3.3). Since b < b−,our theoretical prediction that [argG0±]Γ+ = 0 is confirmed from the plot inthe left panel of Fig. 5.3.To determine the location of the two unstable eigenvalues of the lin-earization for the parameter set of (5.3.10a), we look for zeroes of (5.3.7) onthe positive real axis λ > 0. To this end, we rewrite (5.3.7) asG0±(λ) =H(λ)−G±(λ)[l1l0H(λ)q±(τvλ)] ,H(λ) ≡ λ2 − bλ+ a , G±(λ) ≡ l1l0(b+ − λ)q±(τvλ)√1 + τvλ .(5.3.17)For the parameter set of (5.3.10a), in the right panel of Fig. 5.3 we plotG±(λ) and H(λ) for λ > 0 real, which shows that there are no real positiveroots to G0±(λ) = 0. As a result, the two unstable eigenvalues for the param-eter set (5.3.10a) are not real-valued, and do indeed generate an oscillatoryinstability of the symmetric steady-state solution.Next, we examine the region near b = 0 where a Hopf bifurcation foreither of the two modes must occur. To determine the precise location ofthe Hopf bifurcation point we look for a root λI of (5.3.7), with λI  1,when b = O(). We Taylor-expand the right-hand side of (5.3.7) as λI → 0,and set G0±(iλI) = 0 to obtain that−λ2I − ibλI + (z − − b)∼ (c± + iλIs± +O(λ2I))(1 +i2τvλI +O(λ2I))(b+ − iλI) ,(5.3.18)where c± and s± are determined from the Taylor series of l1q±(iτvλI)/l0 asl1l0q±(iτvλI) = c±+ iλIs±+O(λ2I) ; c± ≡l1l0q±(0) , s± ≡ l1l0τvq′±(0) .(5.3.19)Upon expanding the right-hand side of (5.3.18), we obtain that−λ2I − ibλI + (z− − b) ∼ [c± + i(s± +c±2τv)λI +O(λ2I)](b+ − iλI) .(5.3.20)To determine λI and the critical value of b for a Hopf bifurcation we takethe real and imaginary parts of both sides of (5.3.20). From the imaginary1285.3. One Diffusive Species in the Bulkparts, we getb ∼ c± − (b+ )(s± +c±2τv).Upon solving this equation asymptotically for b, we obtain thatb ∼ c± − 2(c± + 1)(s± +c±2τv). (5.3.21)Next, by taking the real parts of both sides of (5.3.18), we get−λ2I + (z − − b) ∼ (b+ )c± + λ2I(s± +c±2τv).We substitute b ∼ c± into this equation, and simplify the resulting expres-sion to getλ2I ∼ z − 2(c± + 1)2 − λ2I(s± +c±2τv).For  1, we readily derive from this last expression thatλI ∼ 1/2z1/21− 2z[(c± + 1)2 + z(s± +c±2τv)]+O(2) . (5.3.22a)Upon recalling (5.3.19) and (5.3.8), we determine c± and s± asc± =l1l0tanh(L2l0), synchronous (+)l1l0coth(L2l0), asynchronous (−),s± =τv4(l1l0)(Ll0)sech2(L2l0), synchronous (+)− τv4(l1l0)(Ll0)csch2(L2l0), asynchronous (−).(5.3.22b)In summary, we conclude to leading-order in the limit → 0 that thereis a Hopf bifurcation when b ≡ trJe ∼ c± with leading-order frequencyλI ∼ 1/2z1/2. Therefore, the period T of small-amplitude oscillations at theonset of the Hopf bifurcation is long as → 0, with scaling T ∼ 2pi/√z. Ahigher-order asymptotic formulae for the Hopf bifurcation point is given in(5.3.21) and (5.3.22). We observe that the critical threshold for b, given byb ∼ c±, shows that the Hopf bifurcation threshold for the synchronous andasynchronous modes are only slightly different when  1.1295.3. One Diffusive Species in the BulkTo determine the curves in the l1/l0 versus L/l0 parameter plane whereHopf Bifurcations occur, we set b = trJe = 1 −  − 3q(ue − 2)2 = c±, andsolve for ue. This yields the two roots u1± and u2±, defined byu1± = 2− 1√3q√1− (1 + c±) , u2± = 2 + 1√3q√1− (1 + c±) .(5.3.23)By using (5.3.3), we then solve H(u1±) = 0 and H(u2±) = 0 for β to obtainthat β = Z(u1±) and β = Z(u2±), whereZ(u) ≡ 1 + 4u− (1− )3u(u− 2) . (5.3.24)Finally, we use (5.3.5) to relate β to l1/l0 and L/l0. Upon solving theresulting expression for l1/l0 we obtain that the Hopf bifurcation curves forthe synchronous and asynchronous modes are given byl1l0=[zZ(uj+)− 1]coth(L2l0), for j = 1, 2 ; synchronous (+) mode ,(5.3.25a)l1l0=[zZ(uj−)− 1]coth(L2l0), for j = 1, 2 ; asynchronous (−) mode ,(5.3.25b)where Z(u) is defined in (5.3.24) and c± is defined in (5.3.22b). We remarkthat since c± depends on l1/l0, (5.3.25) is a weakly implicit equation forl1/l0 when   1. We solve (5.3.25) when   1 for l1/l0 using one step ofa fixed point iteration.In the left panel of Fig. 5.4 we plot the Hopf bifurcation curves from(5.3.25) in the l1/l0 versus L/l0 plane when  = 0.015, q = 5, and z = 3.5.For this parameter set, we observe from this figure that the Hopf bifurcationthresholds for the synchronous and asynchronous modes almost coincide.Inside the region bounded by the curves, the symmetric steady-state solutionis unstable and there may either be stable or unstable periodic solutions.The parameter set (5.3.10a), as used in [23], corresponds to the markedpoint l1/l0 = 0.15 and L/l0 = 1 in this figure that is near the stabilityboundary. This phase diagram is comparable to the one obtained in [23].However, as discussed in §5.1, in [23] the Hopf bifurcation boundary forthe leading order theory, where the synchronous and asynchronous modeshave a common threshold, was obtained qualitatively through an analysisbased on the crossing of nullclines. No zero-eigenvalue crossing was noted1305.3. One Diffusive Species in the Bulk0 0.5 1 1.5 200.511.52L/l0l 1/l 00.2 0.3 0.4 0.500.10.20.3L/l0kvFigure 5.4: Left panel: The Hopf bifurcation boundaries for the synchronousmode (solid curves) and asynchronous mode (dotted curves), as computedfrom (5.3.25), for the one-bulk species model (5.3.1) when q = 5, z = 3.5,τv = 200, and  = 0.015. These Hopf bifurcation thresholds essentiallycoincide except when L/l0 is small. The upward horseshoe-shaped curvecorresponds to a zero-eigenvalue crossing for the asynchronous mode, asgiven parametrically by (5.3.14). The marked point corresponds to the pa-rameter set l1/l0 = 0.15 and L/l0 = 1 used in [23]. Right panel: Phasediagram in the kv versus L/l0 plane when l0 = 10 showing a clear differencebetween the Hopf bifurcation boundaries for the synchronous (outer solid)and asynchronous (inner solid) curves. Between the two outer solid curves,the synchronous mode is unstable while between the inner solid curves theasynchronous mode is unstable. The dashed horseshoe shaped-curve corre-sponds to the zero eigenvalue crossing (5.3.14). Inside the region boundedby horseshoe-shaped curve there are asymmetric steady-state solutions. TheHopf curves coincide almost exactly with full numerical results computed bysolving (5.3.4) for a pure imaginary eigenvalue λ = iω using Maple [50] (dot-ted curve).1315.3. One Diffusive Species in the Bulkin [23]. Our stability analysis for the limiting problem  → 0 has beenable to determine two-term approximations to the Hopf boundaries for boththe synchronous and asynchronous modes, to determine the Hopf bifurcationfrequencies near onset, and to detect zero-eigenvalue crossings correspondingto the emergence of asymmetric steady-state solutions of (5.3.1).In the right panel of Fig. 5.4 we plot the corresponding Hopf bifur-cation curves in the kv versus L/l0 when l0 = 10, which shows a clearerdistinction between the synchronous and asynchronous modes of instabil-ity. Between the two outer solid curves, representing the Hopf thresholdfor the synchronous mode, the synchronous mode is unstable. Similarly,between the two inner solid curves, representing the Hopf threshold forthe asynchronous mode, the asynchronous mode is unstable. The dashedhorseshoe-shaped curve corresponds to a zero eigenvalue crossing. Insidethis horseshoe-shaped region, there are asymmetric steady-state solutionsto (5.3.1). This plot shows that for a given value of L/l0 the symmetricsteady-state solution is unstable to an oscillatory instability only for someintermediate range kv− < kv < kv+ of the coupling strength between themembrane and the bulk. We remark that the analytical stability boundariesin Fig. 5.4 were all verified numerically by determining the complex roots ofG± = 0, as defined in (5.3.4), using Maple [50].−0.5 0 0.5 1 1.5 2−0.4−0.200.2G0R−G0 I−−0.5 0 0.5 1 1.5−0.200.20.4G0R−G0 I−Figure 5.5: Plot of G0−(iλI) = G0R−(λI) + iG0I−(λI) for the parameter setl1/l0 = 1.0, L/l0 = 0.4, q = 5.0, z = 3.5, that lies within the horseshoe-shaped region of the left panel of Fig. 5.4 where b > b−. Left panel: τv = 200.Right panel: τv = 1. For λI → ∞, both paths begin on the positive realaxis, and end on the negative real axis when λI = 0. For both values of τvwe have [argG0−]Γ+ = −pi, so that N = 1 from (5.3.9).Next, we numerically compute the winding number [argG0−]Γ+ to count1325.3. One Diffusive Species in the Bulkthe number of unstable eigenvalues for the asynchronous mode for parame-ter values inside the horseshoe-shaped zero-eigenvalue crossing curve in theleft panel of Fig. 5.4. Recall that within this region, we have b > b− andso G0−(0) < 0. For the particular point l1/l0 = 1.0 and L/l0 = 0.4 in thisregion, and for q = 5 and z = 3.5, in Fig. 5.5 we show that [argG0−]Γ+ = −pifor two different values of τv. From (5.3.9) this implies that N = 1, and sofor the asynchronous mode the linearization around the symmetric steady-state has an unique unstable real eigenvalue. By further similar numeri-cal computations of the winding number (not shown), we conjecture that[argG0−]Γ+ = −pi, and consequently N = 1 for the asynchronous mode,whenever b > b−.In the left panel of Fig. 5.6 we show the numerically computed spec-trum of the linearization, obtained using Maple [50] on (5.3.7), when wetake a vertical slice at fixed L/l0 = 0.3 in the right panel of Fig. 5.4 thatbegins within the horseshoe-shaped region, first traversing above the zero-eigenvalue curve, then past the asynchronous Hopf threshold, and finallybeyond the synchronous Hopf threshold. A zoom of the region in Fig. 5.4where these crossings are undertaken is shown in the right panel of Fig. 5.6.The expected transition in the spectrum as predicted by our theory, anddiscussed in the caption of Fig. 5.6, is confirmed.Next, we show analytically that the zero-eigenvalue crossing for the asyn-chronous mode at b = b− corresponds to a bifurcation point where asymmet-ric equilibria of (5.3.1) bifurcate from the symmetric steady-state solutionbranch. To show this, we first construct a more general steady-state solu-tion (5.3.1), where we remove the symmetry assumption about the midlinex = L/2. For this more general steady-state, we calculate from the steady-state system for (5.3.1) thatVe(x) = v1sinh[ωv(L− x)]sinh(ωvL)+ v2sinh(ωvx)sinh(ωvL), ωv ≡√σv/Dv , (5.3.26)where v1 = Ve(0) and v2 = Ve(L). By setting g(u1, v1) + kvVex(0) = 0 andg(u2, v2) − kvVex(L) = 0, we readily derive, in terms of a 2× 2 symmetricmatrix A, thatA v1v2 = z u1u2 , A ≡ 1 + l1l0 coth(Ll0)− l1l0 csch(Ll0)− l1l0 csch(Ll0)1 + l1l0 coth(Ll0) .(5.3.27a)Upon setting f(uj , vj) = 0, j = 1, 2, we obtain a nonlinear algebraic system1335.3. One Diffusive Species in the Bulk0 0.05 0.1 0.15 0.2−0.1−0.0500.050.10 0.05 0.1 0.15−0.2−0.100.10.2−0.05 0 0.05−0.2−0.100.10.2−0.1 −0.08 −0.06 −0.04 −0.02 0−0.2−0.100.10.20.29 0.3 0.31 0.320.250.260.270.280.29L/l0kvIIIIIIIVFigure 5.6: The spectrum Im(λ) versus Re(λ) (left panel) near the origin forthe asynchronous (solid dots) and synchronous (diamonds) modes for a fixedL/l0 = 0.3, l0 = 10, q = 5, z = 3.5, τv = 200, and  = 0.015, as the couplingstrength kv crosses various stability boundaries as shown in the right panel,representing a zoom of a portion of the kv versus L/l0 plane of the rightpanel of Fig. 5.4. Top left kv = 0.26 (in the horseshoe-shaped region I):N = 1 and N = 2 for the anti-phase and in-phase modes, respectively.Top right: kv = 0.265 (outside the horseshoe, but before the asynchronousHopf boundary: region II): N = 2 for both anti-phase and in-phase modes.Bottom Left: kv = 0.275 (between the asynchronous and synchronous Hopfboundaries: region III): N = 0 for the anti-phase and N = 2 for the in-phasemodes. Bottom Right: kv = 0.282 (above the synchronous Hopf boundary:region IV): N = 0 for both the anti-phase and in-phase modes. Thesespectral results are all consistent with our stability theory.1345.3. One Diffusive Species in the Bulkfor u ≡ (u1, u2)T given by N (u) = 0, whereN (u) ≡ −q (u1 − 2)3(u2 − 2)3+ 4 11+ u− zA−1u . (5.3.27b)Since the matrix A is symmetric and has a constant row sum, it followsthat q1 ≡ (1, 1)T (in-phase) and q2 ≡ (1,−1)T (anti-phase) are its twoeigenvectors. After some algebra, we obtain that the two correspondingeigenvalues areAqj = µjqj ; µ1 = 1 + l1l0tanh(L2l0), µ2 = 1 +l1l0coth(L2l0).(5.3.28)To recover the construction of the symmetric steady-state branch we useq1 ≡ (1, 1)T , and look for a solution to (5.3.27b) with u1 = u2 ≡ ue. SincezA−1q1 = zµ−11 q1, and zµ−11 = β, we readily identify that (5.3.27b) reducesto (5.3.3), where β is defined in (5.3.5). To determine whether there are anybifurcation points from this symmetric branch, we write u = ue(1, 1)T + δφ,where δ  1 and φ is a 2-vector. Upon linearizing (5.3.27b), we readilyobtain thatAφ =(z1− 3q(ue − 2)2)φ . (5.3.29)Bifurcation points correspond to where (5.3.29) has a nontrivial solution.Such points occur whenever1− 3q(ue − 2)2 = β , (in-phase);1− 3q(ue − 2)2 = z[1 +l1l0coth(L2l0)]−1, (anti-phase) .(5.3.30)As shown previously, the in-phase equation above is inconsistent with anyroot of the cubic (5.3.3). In contrast, the anti-phase equation in (5.3.30) isprecisely the condition b = b−. Therefore, zero eigenvalue crossings for theasynchronous mode correspond to where branches of asymmetric steady-state solutions bifurcate from the symmetric steady-state branch.In the left panel of Fig. 5.7 we plot a global bifurcation diagram ver-sus the coupling strength kv showing only the symmetric and asymmetricsteady-state solution branches and the two bifurcation points off of the sym-metric branch. This plot corresponds to taking a slice at fixed L/l0 throughthe phase diagram in the right panel of Fig. 5.4. It also corresponds to1355.3. One Diffusive Species in the Bulkthe solution set of the nonlinear algebraic system (5.3.27b). Although thebifurcation diagram can be obtained from (5.3.27b), we used the numericalbifurcation software XPPAUT [16] after first discretizing (5.3.1) into a largeset of ODE’s. In this way, the stability properties of the asymmetric steady-state branch was determined numerically. Our computations show that theasymmetric solution branch is unstable except in a narrow window of kv. Inthe right panel of Fig. 5.7 we show results from full numerical solutions tothe PDE-ODE system (5.3.1), computed using a method of lines approach,that verify this prediction of a stable window in kv for the stability of theasymmetric steady-state solutions.0.1 0.12 0.14 0.16 0.18 0.21.822.22.4kvu0 100 2001.822.22.4Tu1, u2Figure 5.7: Left panel: Global bifurcation diagram of u1 and u2, computedusing XPPAUT [16], for the asymmetric and symmetric steady-state so-lutions to (5.3.1) showing the two bifurcation points off of the symmetricsteady-state branch. The parameter values are l0 = 10, L/l0 = 0.4, q = 5,z = 3.5, τv = 200, and  = 0.015. Thin curves represent unstable steady-state solutions while thick curves indicate stable ones. Right panel: Timeevolution of u1, u2, as computed from the full PDE-ODE system (5.3.1)using a method of lines approach. The parameter values are in left panelexcept that kv = 0.13 is chosen so that the asymmetric steady-state solutionis stable. Initial conditions for (5.3.1) are chosen close to the stable asym-metric solution. As expected, the two ui approach their steady-state valuesafter a transient period.In the left panel of Fig. 5.8, we plot a global bifurcation diagram of u1versus kv, computed using XPPAUT [16], showing only the local branchingbehavior of the synchronous periodic solution branch near the Hopf bifurca-tion point where it emerges from the symmetric steady-state branch. In theright panel of Fig. 5.8 we plot the corresponding period of the synchronous1365.3. One Diffusive Species in the Bulk0.015 0.02 0.0251.41.82.22.6kvu10.015 0.02 0.02520406080100kvPeriodFigure 5.8: Left panel: The global bifurcation diagram of u1 versus kv, com-puted using XPPAUT [16], for in-phase perturbations, showing the periodicsolution branch of synchronous oscillations near the lower Hopf boundaryfor the synchronous mode in the right panel of Fig. 5.4. The other pa-rameter values are L/l0 = 1.0, l0 = 10, q = 5, z = 3.5, τv = 200, and = 0.015 corresponding to the parameter set (5.3.10a) of [23]. Right panel:The period of oscillations along the synchronous branch. The solid and opencircles in both plots represent stable and unstable periodic solutions, respec-tively. The solid and dashed lines in the left panel are stable and unstablesymmetric steady-state solutions, respectively.1375.3. One Diffusive Species in the Bulkoscillations. The periodic solution branch is found to be supercritical atonset, with a narrow range of kv where the branch is unstable. Stabilityis regained at some larger value of kv. From the right panel of Fig. 5.8the period of oscillations at onset is 27.99, which agrees rather well withthe asymptotic result T = 2pi/λI ≈ 30.42, where we used λI ≈ 0.2065 ascomputed from (5.3.22a) for the synchronous mode.Figure 5.9: Full numerical simulations of the PDE-ODE system (5.3.1) forV (x, t), with time running from bottom to top and space represented hor-izontally. The fixed parameter values are q = 5, z = 3.5, τv = 200, and = 0.015. The initial conditions used in the simulation are V (x) = 0.5,u1 = 1, u2 = 5. Left panel: L = 5 with l1/l0 = 0.5 and L/l0 = 0.5 showingstable synchronous oscillations. Middle panel: L = 10 with l1/l0 = 0.25 andL/l0 = 1 showing stable asynchronous oscillations. Right panel: L = 10with l1/l0 = 0.15 and L/l0 = 1 corresponding to the parameter set (5.3.10a)of [23].Finally, to illustrate the oscillatory dynamics, full time-dependent nu-merical solutions for V (x, t) from the coupled PDE-ODE system (5.3.1)were computed using a method of lines approach. We choose three param-eter sets that are inside the region of the left panel of Fig. 5.4 where bothoscillatory modes are unstable with N = 2 unstable eigenvalues for eachmode. In the left and middle panels in Fig. 5.9, representing contour plotsof V (x, t), we show the clear possibility of either stable synchronous or sta-ble asynchronous oscillatory instabilities, depending on the particular pointchosen within the instability region. For the right panel in Fig. 5.9, we use1385.3. One Diffusive Species in the Bulkthe parameter set (5.3.10a) of [23], for which kv = 0.0225, which is closeto the stability boundary where a Hopf bifurcation occurs (see the markedpoint in the left panel of Fig. 5.4). For this parameter set, we observe fromthe right panel of Fig. 5.8 that the synchronous periodic solution is stableand that the period of oscillations is T ≈ 64. The corresponding full numer-ical results computed from the PDE-ODE system (5.3.1) shown in the rightplot of Fig. 5.9 reveal stable synchronous oscillations with a period close tothis predicted value.5.3.2 Stability Analysis for the  = O(1) ProblemNext, we study oscillatory dynamics for (5.3.1) when  = 0.3, which is atwenty-fold increase over the value used in §5.3.1. We use a combination ofa numerical winding number computation, based on (5.3.9), to determinethe stability properties of the symmetric steady-state, and Maple [50] tofind the roots of (5.3.7) determining the eigenvalues of the linearization of(5.3.1) around the symmetric steady-state solution. Since for this largervalue of  the PDE-ODE system (5.3.1) is not as computationally stiff aswhen  = 0.015, we are able to use XPPAUT [16] to calculate global branchesof synchronous and asynchronous periodic solutions. Asymmetric steady-state branches and their bifurcations are also computed.In the left panel of Fig. 5.10 we plot the Hopf bifurcation curves, com-puted from the roots of (5.3.7), in the l1/l0 versus L/l0 plane when q = 5,z = 3.5, τv = 200, and  = 0.3. In contrast to the similar plot in the leftpanel of Fig. 5.4 where  = 0.015, we observe from the left panel of Fig. 5.10that the Hopf bifurcation thresholds for the synchronous and asynchronousmodes are now rather distinct when L/l0 < 0.5. The left horseshoe-shapedcurve is the zero-eigenvalue crossing boundary for the asynchronous mode,as parametrized by (5.3.14).For a fixed L/l0 = 0.4, and with q = 5, z = 3.5, τv = 200, and  = 0.3, inthe right panel of Fig. 5.10 we plot the bifurcation diagram of u with respectto l1/l0, showing the primary solution branches and some secondary bifurca-tions. This plot corresponds to taking a vertical slice in the phase diagramgiven in the left panel of Fig. 5.10. There are several key features in thisplot. Firstly, as l1/l0 is increased from zero, the first bifurcation is to syn-chronous temporal oscillations. Our computations show that, except in verynarrow ranges of l1/l0, the global branch of synchronous oscillations betweenthe two membranes is stable. Secondly, we observe that the asynchronousbranch of oscillations that bifurcates from the symmetric steady-state atl1/l0 ≈ 0.41 is mostly unstable, but exhibits a small region of bistability.1395.3. One Diffusive Species in the Bulk0.3 0.4 0.5 0.60.20.611.41.8L/l0l 1/l 00.4 0.6 0.8 1 1.2 1.41.522.5l1/l0uFigure 5.10: Left panel: The Hopf bifurcation boundaries for the syn-chronous (solid curves) and asynchronous (dotted curves) modes for the one-bulk species model (5.3.1), as computed from solving (5.3.7) with Maple [50]when q = 5, z = 3.5, τv = 200, and  = 0.3. These Hopf bifurcation thresh-olds are distinct when L/l0 < 0.5. The dashed curve is the zero-eigenvaluecrossing for the asynchronous mode, given parametrically by (5.3.14). Rightpanel: Bifurcation diagram of u with respect to l1/l0 for a fixed L/l0 = 0.4.The solid and dashed curves indicate stable and unstable steady-states, re-spectively. The solid and open circles correspond to stable and unstableperiodic solutions, respectively. The synchronous and asynchronous peri-odic solution branches first bifurcate from the symmetric steady-state atl1/l0 ≈ 0.38 and l1/l0 ≈ 0.41, respectively. Asymmetric steady-state solu-tion branches, that bifurcate from the symmetric steady-state solutions atthe zero eigenvalue crossings, are also shown. Additional periodic solutionbranches, arising from Hopf bifurcations off of these asymmetric steady-states, also occur.1405.3. One Diffusive Species in the BulkThirdly, the asymmetric steady-state solution branch bifurcates from thesymmetric steady-state branch at two values of l1/l0. These asymmetricsteady-states are mostly unstable, but there is a range of l1/l0 where theyare stabilized. Unstable periodic solution branches, emerge from, and ter-minate on, the asymmetric steady-state branch. Overall, the bifurcationdiagram is rather intricate, and it is beyond the scope of this study to clas-sify and study all of these secondary bifurcations.5.85.960 50 1005.85.966.1TV 1, V 2Figure 5.11: Left panel: Contour plot of V (x, t) computed numerically fromthe PDE-ODE system (5.3.1) for l1/l0 ≈ 0.406 (kv = 1.22) (left) and forl1/l0 ≈ 0.41 (kv = 1.23) (right). The initial condition is V (x, 0) = 0.5,u1 = 1, and u2 = 5, at t = 0. The system exhibits synchronized oscilla-tions with unequal amplitude for l1/l0 ≈ 0.406, and synchronized period-doubling oscillations for l1/l0 ≈ 0.41. The other parameters are the sameas in Fig. 5.10. Right panel: Time series of the inhibitor concentration v1and v2 in the two membranes for l1/l0 ≈ 0.406 (top) and for l1/l0 ≈ 0.41(bottom).From the right panel in Fig. 5.10, we observe that although the syn-chronous periodic solution branch is stable in a large range of l1/l0, thereis a narrow region 0.413 < l1/l0 < 0.430 predicted by XPPAUT [16] wherethe symmetric steady-state and the synchronous periodic solution branchare both unstable. As a result, in this narrow region we anticipate that(5.3.1) will exhibit rather complex dynamics. Due to the small numericalerror associated with using XPPAUT on the spatially discretized version of(5.3.1), in our numerical simulations of the PDE-ODE system (5.3.1) shownin Fig. 5.11 we have observed qualitatively interesting dynamics in a slightlyshifted interval of l1/l0. The results in Fig. 5.11 are for l1/l0 = 0.406 and for1415.4. Two Diffusive Species in the Bulkl1/l0 = 0.41. From Fig. 5.11, we conclude that as l1/l0 is increased throughthe narrow zone where the synchronous branch is unstable, the two mem-branes first exhibit synchronized oscillations with a significant distinction intheir oscillating amplitudes. This is followed by period-doubling behavior.As l1/l0 increases further, the periodic-doubling behavior disappears and thetwo membranes return to synchronous oscillations with unequal amplitudes.5.4 Two Diffusive Species in the BulkIn §5.3 we considered the case where only the inhibitor can detach from themembrane and diffuse in the bulk. In this section, we consider the full model(5.1.1) where both the activator and inhibitor undergo bulk diffusion. Topartially restrict the wide parameter space for (5.1.1), we will study (5.1.1)for the fixed parameter valuesDv = 0.5 , Du = 1.5 , σv = 0.008 , σu = 0.01 , z = 3.5 , q = 5 ,(5.4.1a)and we will impose equal coupling strengths so that k ≡ ku = kv. We willvary the domain length L and k. We then introduce the diffusion lengths l0and lu, the ratio of diffusion lengths αu, and the two time-constants τv andτu, byl0 ≡√Dvσv≈ 7.9057 , lu ≡√Duσu≈ 12.247 ,αu ≡ l0lu≈ 0.6455 , τv ≡ 1σv= 125 , τu ≡ 1σu= 100 .(5.4.1b)To determine the stability of the unique symmetric steady-state solution,we first nondimensionalize (5.2.18), in a similar way as was done in (5.3.7)for the one-bulk species model. We obtain that the number N of eigenvaluesλ of the linearization of (5.1.1) in Re(λ) > 0 isN = P +1pi[argF0±]Γ+ , (5.4.2a)whereP = 2 if b = trJe > 00 if b = trJe < 0 , b ≡ 1− − 3q(ue − 2)2 , (5.4.2b)1425.4. Two Diffusive Species in the Bulkwhere ue is the unique root of the cubic (5.2.3a), and where F0±(λ) is definedbyF0±(λ) ≡l0l1q±− √1 + τvλ (b+ − λ)det(Je − λI) +αu√1 + τuλ (+ λ)det(Je − λI)(p±q±)+(2αul1l0)p±√1 + τvλ√1 + τuλdet(Je − λI) .(5.4.3a)Here det(Je − λI) = λ2 − bλ+ (z − − b), q± are given in (5.3.8), and p±are defined byp+ = tanh(αu√1 + τuλL2l0), p− = coth(αu√1 + τuλL2l0).(5.4.3b)We observe that the first two-terms in (5.4.3a) are the same as in (5.3.7),with the last two terms arising from the additional coupling with the ac-tivator. As in §5.3.1, it is possible to study (5.4.3) in the limit  → 0 todetermine N analytically for both the synchronous and asynchronous modes,and to asymptotically calculate the Hopf bifurcation frequencies near onset.However, in this section, we will consider the finite  problem with  = 0.15and use Maple [50] to numerically compute both the roots of (5.4.3) and thewinding number in (5.4.2a), which gives N .In this way, in the left panel of Fig. 5.12 we show a phase diagram inthe l1/l0 versus L/l0 parameter space, where l1 = k/ and  = 0.15, withk ≡ kv = ku. The Hopf bifurcation boundaries for the synchronous andasynchronous modes are the solid and dashed curves, respectively. Insidethe region bounded by the disjoint solid curves, the synchronous mode is un-stable with two unstable eigenvalues. Inside the open loop bounded by thedashed curve, the asynchronous mode is unstable with N = 2. In contrast tothe phase diagrams for the one-bulk species case, no zero-eigenvalue cross-ings were detected for the parameter set (5.4.1). This aspect is discussedfurther at the end of this section.By using XPPAUT [16], in the right panel of Fig. 5.12 we plot the globalbifurcation diagram of u with respect to l1/l0 for a fixed vertical slice withL = 4 through the phase diagram in the left panel of Fig. 5.12, so thatL/l0 ≈ 0.505. This plot shows that the synchronous mode first loses stabilityto a stable periodic solution at l1/l0 ≈ 0.36, and that there is a subsequentHopf bifurcation to the asynchronous mode at l1/l0 ≈ 0.39. The key featurein this plot is that the synchronous branch of periodic solutions is almost1435.4. Two Diffusive Species in the Bulk0.4 0.5 0.6 0.7 0.8 0.9 100.511.5L/l0l 1/l 00.2 0.4 0.6 0.8 1 1.21.41.61.822.22.42.6l1/l0uFigure 5.12: Left panel: The Hopf bifurcation boundaries for the syn-chronous mode (solid curves) and asynchronous mode (dashed curves) forthe two-bulk species model (5.1.1), as computed from (5.4.3) with Maple[50] when  = 0.15 and for the parameters of (5.4.1). Inside the regionbounded by the solid curves, the synchronous mode is unstable, while insidethe region bounded by the dashed loop the asynchronous mode is unstable.Right panel: Global bifurcation diagram of u with respect to l1/l0 for fixedL = 4 so that L/l0 ≈ 0.505. The solid/dashed lines are stable/unstablesymmetric steady-states. The outer loop, which is almost entirely stable,corresponds to the branch of synchronous oscillations. The inner loop is theunstable branch of asynchronous oscillations.1445.4. Two Diffusive Species in the Bulkentirely stable, while the asynchronous branch is unstable. No asymmetricsteady-state solutions bifurcating from the symmetric steady-state branchwere detected. As a partial confirmation of these theoretical predictions,in Fig. 5.13 we show a contour plot of V (x, t) computed from the PDE-ODE system (5.1.1) for the parameters of (5.4.1) and with  = 0.15, L = 4,k = 0.7, so that L/l0 ≈ 0.505 and l1/l0 ≈ 0.59. With a random initialcondition, this plot shows the eventual synchronization of the oscillationsin the two membranes. This simulation confirms the prediction of the rightpanel of Fig. 5.12 that only the synchronous mode is stable for this parameterchoice.Figure 5.13: Full numerical solution of V (x, t) computed from the PDE-ODEsystem (5.1.1) showing stable synchronous oscillations for the parametervalues of (5.4.1) and with  = 0.15, L = 4, k = 0.7, so that L/l0 ≈ 0.505and l1/l0 ≈ 0.59. The initial condition for (5.1.1) was V (x, 0) = 0.8 andU(x, 0) = 2.We remark that richer spatio-temporal dynamics can occur if we choosea vertical slice through the phase diagram in the left panel of Fig. 5.12for the larger value L = 5.5, so that L/l0 ≈ 0.70. In the right panel ofFig. 5.14 we plot the global bifurcation diagram of u versus l1/l0 for thisvertical slice. In contrast to the case where L = 4 in the right panel ofFig. 5.12, we observe from Fig. 5.14 that, as we increase the value of l1/l0from 0.2, the asynchronous mode loses its stability before the synchronousmode. However, as l1/l0 is decreased from 0.8, the synchronous mode loses1455.4. Two Diffusive Species in the Bulkits stability at l1/l0 ≈ 0.76 before the asynchronous mode at l1/l0 ≈ 0.70.We again find that the synchronous mode is stable for a wide range of l1/l0.However, in the rather narrow parameter range 0.290 < l1/l0 < 0.305 boththe synchronous and asynchronous modes are unstable. To examine thebehavior of the full PDE-ODE system (5.1.1) in this range of l1/l0, in theleft panel of Fig. 5.14 we plot the numerically computed time evolution ofv2(t) = V (L, t) for (5.1.1) when l1/l0 = 0.30. The initial conditions usedin the simulation are the same as given in the caption of Fig 5.13. Theresulting time-series for v2(t) shows the presence of two distinct periods,which is indicative of a torus bifurcation. We expect similar analysis asshown in chapter §3 could be applied here for the initiation of the torus butwe will omit it here.0 100 200 300 400 5005.755.85.855.95.956TV 20.2 0.4 0.6 0.81.522.5l1/l0uFigure 5.14: Left panel: Plot v2(t) = V (L, t) versus t computed fromfull PDE-ODE system (5.1.1) for the parameter values of (5.4.1) and with = 0.15, L = 5.5, k = 0.35, so that L/l0 ≈ 0.7 and l1/l0 ≈ 0.30. The re-sulting time series has two distinct periods, which indicates a possible torusbifurcation where the in-phase and anti-phase periodic solutions change fromstable (unstable) to unstable (stable). Right panel: Global bifurcation dia-gram of u with respect to l1/l0 for fixed L = 5.5, so that L/l0 ≈ 0.7. Thelabeling of the branches and their stability is the same as in the right panelof Fig. 5.12. The closed, primarily outer, loop is the synchronous branch,while the other closed loop is the asynchronous branch, which is mostlyunstable. The synchronous branch is again mostly stable. In the window0.290 < l1/l0 < 0.305 both the synchronous and asynchronous branches ofperiodic solutions are unstable. No secondary bifurcations are shown.Finally, we confirm theoretically that there are no zero-eigenvalue cross-ings, corresponding to the bifurcation of asymmetric steady-state solutions1465.4. Two Diffusive Species in the Bulkfrom the symmetric steady-state branch. Proceeding similarly as in (5.3.26)–(5.3.27b) of §5.3.1, we obtain in place of (5.3.27b) that, for any asymmetricsteady-state solution of (5.1.1), u = (u1, u2)T now satisfiesN (u) ≡ −q (u1 − 2)3(u2 − 2)3+ 4 11+ u− zA−1u− kluBu = 0 . (5.4.4)where A is defined in (5.3.27a) and the symmetric matrix B is defined byB ≡ coth(Llu)−csch(Llu)−csch(Llu)coth(Llu) . (5.4.5)Since B is symmetric with a constant row sum, the eigenvectors q1 = (1, 1)Tand q2 = (1,−1)T are common to both A and B. We readily calculatethat the two eigenvalues ξj for j = 1, 2 of B are ξ1 = tanh(L/(2lu))andξ2 = coth(L/(2lu)).The symmetric steady-state solution is recovered by seeking a solutionto (5.4.4) of the form u = ueq1. By using the explicit expressions for theeigenvalues ξ1 and µ1 of B and A, respectively, where µ1 is given in (5.3.28),we readily derive that ue satisfies the cubic (5.2.3a). To determine whetherthere are any bifurcation points from this branch we write u = ue(1, 1)T +δφfor δ  1, and linearize (5.4.4). We conclude that bifurcation points occurwhenever−3q(ue − 2)2φ+ φ− zA−1φ− kluBφ = 0 , (5.4.6)has a nontrivial solution φ. For the in-phase mode φ = q1, we use theexplicit expressions for ξ1 and µ1 to derive from (5.4.6) that any such abifurcation point must satisfy β + au − 1 = −3q(ue − 2)2, where β and auare defined in (5.2.3b). By an identical proof as in §5.3.1, this condition isinconsistent with any root ue to the cubic (5.2.3a). Thus, no zero-eigenvaluecrossing of the in-phase mode from the symmetric steady-state branch canoccur.For the anti-phase mode φ = q2, we use the explicit expressions for theeigenvalues ξ2 and µ2 in (5.4.6) to obtain that there is a bifurcation pointto an asymmetric steady-state whenever−3q(ue − 2)2 + 1− klucoth(L2lu)=z1 + l1l0 coth(L2l0) ,1475.4. Two Diffusive Species in the Bulkhas a solution, where ue is a root of the cubic (5.2.3a). For the parameterset (5.4.1), we verified numerically that no such solution exists for any pointin the phase diagram in the left panel of Fig. 5.12. However, we remark thatfor other parameter sets, notably when  is decreased from  = 0.15, suchbifurcation points should be possible.148Chapter 6Models in aTwo-Dimensional DomainIn previous chapters we have studied coupled bulk-membrane models wherethe membranes or cells are coupled by diffusive signals in a one-dimensionaldomain. In this chapter, we formulate a class of coupled cell-bulk modelsin a two-dimensional domain, which provides a more realistic geometry formodeling experimental observations relating to quorum sensing behavior inthin petri dishes. In our analysis we will not focus on a specific biologicalapplication, but rather will formulate and provide a theoretical framework toconstruct the steady-states and to analyze their linear stability properties fora general class of cell-bulk models, with the aim to predict when synchronousoscillations can be triggered by the cell-bulk coupling. The theory will beillustrated for some specific choices of the intracellular kinetics.The outline of this chapter is as follows. We first formulate the cou-pled cell-bulk model with one signaling compartment in a two-dimensionalbounded domain in §6.1. Our modeling framework is closely related to thestudy of quorum sensing behavior in bacteria done in [54] and [55] throughthe formulation and analysis of similar coupled cell-bulk models in R3. Forthis 3-D case, in [54] and [55] steady-state solutions were constructed andlarge-scale dynamics studied in the case where the signaling compartmentshave small radius of order O(). However, due to the rapid 1/r decay ofthe free-space Green’s function for the Laplacian in 3-D, it was shown in[54] and [55] that the release of the signaling molecule leads to only a ratherweak communication between the cells of the same O() order of the cellradius. As a result, small cells in 3-D are primarily influenced by their ownsignal, and hence no robust mechanism to trigger collective synchronous os-cillations in the cells due to Hopf bifurcations was observed in [54] and [55].We emphasize that the models of [54] and [55] are based on postulating adiffusive coupling mechanism between distinct, spatially segregated, dynam-ically active sites. Other approaches for studying quorum sensing behavior,such as in [58], are based on reaction-diffusion (RD) systems, which adopta homogenization theory approach to treat large populations or colonies of149Chapter 6. Models in a Two-Dimensional Domainindividual cells as a continuum density, rather than as discrete units as in[54] and [55].Next in §6.2 we use the method of matched asymptotic expansions to con-struct steady-state solutions to our 2-D multi-cell-bulk model (6.2.1), and wederive a globally coupled eigenvalue problem whose spectrum characterizesthe stability properties of the steady-state. In our 2-D analysis, the interac-tion between the cells is of order ν ≡ −1/ log , where  1 is the assumedcommon radius of the small circular cells. In the distinguished limit wherethe bulk diffusion coefficient D is of the asymptotic order D = O(ν−1), in§6.3 we show that the leading order approximate steady-state solution andthe associated linear stability problem are both independent of the spatialconfigurations of cells and the shape of the domain. In this regime, wethen show that the steady-state solution can be destabilized by either a syn-chronous perturbation in the cells or by m− 1 possible asynchronous modesof instability. In §6.3 leading-order-in-ν limiting spectral problems whenD = D0/ν, with ν  1, for both these classes of instabilities are derived. In§6.4, we illustrate our theory for various intracellular dynamics. When thereis only a single dynamically active intracellular component, we show thatno triggered oscillations can occur. For two specific intracellular reactionkinetics involving two local species, modeled either by Sel’kov or Fitzhugh-Nagumo (FN) dynamics, in §6.4 we perform detailed analysis to obtain Hopfbifurcation boundaries, corresponding to the onset of either synchronous orasynchronous oscillations, in various parameter planes. In addition to thisdetailed stability analysis for the D = O(ν−1) regime, in §6.5 we show forthe case of one cell that when D  O(ν−1) the coupled cell-bulk modelis effectively well-mixed and its solutions can be well-approximated by afinite-dimensional system of nonlinear ODEs. The analytical and numericalstudy of these limiting ODEs in §6.5 reveals that their steady-states canbe destabilized through a Hopf bifurcation. Numerical bifurcation softwareis then used to show the existence of globally stable time-periodic solu-tion branches that are intrinsically due to the cell-bulk coupling. For theD = O(1) regime, where the spatial configuration of the cells in the domainis an important factor, in §6.6 we perform a detailed stability analysis fora ring-shaped pattern of cells that is concentric within the unit disk. Forthis simple spatial configuration of cells, phase diagrams in the τ versus Dparameter space, for various ring radii, characterizing the existence of ei-ther synchronous or asynchronous oscillatory instabilities, are obtained forthe case of Sel’kov intracellular dynamics. These phase diagrams show thattriggered synchronous oscillations can occur when cells become more spa-tially clustered. In §6.6 we also provide a clear example of quorom sensing1506.1. Formulation of a 2-D Coupled Cell-Bulk Systembehavior, characterized by the triggering of collective dynamics only whenthe number of cells exceeds a critical threshold. Next in §6.7, we constructthe steady state solution for a system with two small identical circular cellsin the infinite plane and formulate the linear stability problem.Our analysis of synchronous and asynchronous instabilities for (6.2.1) inthe D = O(ν−1) regime, where the stability thresholds are, to leading-order,independent of the spatial configuration of cells, has some similarities withthe stability analysis of [77], [78], [65], and [9] (see also the references therein)for localized spot solutions to various activator-inhibitor RD systems withshort range activation and long-range inhibition. In this RD context, whenthe inhibitor diffusivity is of the order O(ν−1), localized spot patterns canbe destabilized by either synchronous or asynchronous perturbations, withthe stability thresholds being, to leading-order in ν, independent of the spa-tial configuration of the spots in the domain. The qualitative reason forthis similarity between the coupled cell-bulk and localized spot problems isintuitively rather clear. In the RD context, the inhibitor diffusion field isthe long-range “bulk” diffusion field, which mediates the interaction betweenthe “dynamically active units”, consisting of m spatially segregated localizedregions of high activator concentration, each of which is is self-activating.In this RD context, asynchronous instabilities lead to asymmetric spot pat-terns, while synchronous oscillatory instabilities lead to collective temporaloscillations in the amplitudes of the localized spots (cf. [77], [78], [65], and[9]).Finally, we remark that the asymptotic framework for the constructionof steady-state solutions to the cell-bulk model (6.2.1), and the analysis oftheir linear stability properties, relies heavily on the methodology of stronglocalized perturbation theory (cf. [76]). Related problems where such tech-niques are used include [37], [63], [45], and [46].6.1 Formulation of a 2-D Coupled Cell-BulkSystemWe first formulate and non-dimensionalize our coupled cell-bulk model as-suming that there is only one signalling compartment Ω0 inside the two-dimensional domain Ω. We assume that the cell can release a specific sig-naling molecule into the bulk region exterior to the cell, and that this se-cretion is regulated by both the extracellular concentration of the moleculetogether with its number density inside the cell. If U(X, T ) represents theconcentration of the signaling molecule in the bulk region Ω\Ω0, then its1516.1. Formulation of a 2-D Coupled Cell-Bulk Systemspatial-temporal evolution in this region is assumed to be governed by thePDE modelUT = DB∆XU − kBU , X ∈ Ω\Ω0 ,∂nXU = 0 , X ∈ ∂Ω ,DB∂nXU = β1U − β2µ1 , X ∈ ∂Ω0 ,(6.1.1a)where, for simplicity, we assume that the signalling compartment Ω0 ∈ Ω isa disk of radius σ centered at some X0 ∈ Ω.Next, we suppose that inside the cell there are n interacting speciesµ ≡ (µ1, . . . , µn) whose dynamics are governed by n-ODEs of the formdµdT= kRµcF(µ/µc)+ e1∫∂Ω0(β1U − β2µ1)dSX , (6.1.1b)where e1 ≡ (1, 0, . . . , 0)T . Here µ = (µ1, . . . , µn)T represents the totalamount of the n species inside the cell, while kR > 0 is the reaction ratefor the intracellular dynamics. The scaling constant µc > 0 is a dimen-sional constant measuring a typical value for µ inside the compartment.The dimensionless function F (u) models the reaction dynamics for the lo-cal species within the cell. We remark that the integration in (6.1.1b) isover the boundary dSX of the cell.In this coupled cell-bulk model, DB > 0 is the diffusion coefficient forthe bulk process, kB is the rate at which the signalling molecule is degradedin the bulk, while β1 > 0 and β2 > 0 are the dimensional influx (eflux) con-stants modeling the permeability of the cell wall. In addition, ∂nX denoteseither the outer normal derivative of Ω, or the outer normal to Ω0 (whichpoints inside the bulk region). The flux β1U − β2µ1 on the cell membranemodels the influx of the signaling molecule into the extracellular bulk region,which depends on both the external bulk concentration U(X, T ) at the cellmembrane ∂Ω0 as well as on the intracellular concentration µ1 within thecell. Only one of the intracellular species, µ1, is capable of leaving the cellinto the bulk.If we let [γ] denote the dimensions of the variable γ, then the dimensionsof the various quantities in (6.1.1) are as follows:[U ] = moles(length)2, [µ] = moles , [µc] = moles , [DB] =(length)2time,[kB] = [kR] =1time, [β1] =lengthtime, [β2] =1length× time .(6.1.2)1526.1. Formulation of a 2-D Coupled Cell-Bulk SystemWe now nondimensionalize this model by introducing the dimensionlessvariables t, x, U , and u defined byt = T/tR , x = X/L , U =L2µcU , u = µµc, (6.1.3)where L is a typical radius of Ω. In terms of these variables, (6.1.1) becomes1kBtRUt =DBkBL2∆xU − U , x ∈ Ω˜\Ω˜0 ,∂nxU = 0 , x ∈ ∂Ω˜ ,DBL3∂nxU =β1L2U − β2u1 , x ∈ ∂Ω˜0 ,(6.1.4a)which is coupled to the intracelluar dynamics1kRtRdudt= F (u) +Le1kR∫∂Ω˜0(β1L2U − β2u1)dSx . (6.1.4b)Here Ω˜0 is a sphere centered at some x0 of radius σ/L.We then choose tR based on the time-scale of the reaction kinetics, andintroduce an effective dimensionless diffusivity D, both which are definedbytR ≡ 1kR, D ≡ DBkBL2. (6.1.5)Then, (6.1.4) can be written askRkBUt = D∆xU − U , x ∈ Ω˜\Ω˜0 ,∂nxU = 0 , x ∈ ∂Ω˜ ,D∂nxU =β1kBLU − β2LkBu1 , x ∈ ∂Ω˜0 ,(6.1.6a)which is coupled to the intracellular dynamicsdudt= F (u) +kBe1kR∫∂Ω˜0(β1kBLU − β2LkBu1)dSx . (6.1.6b)We now introduce our scaling assumption that the radius of the cell issmall compared to the radius of the domain, so that  ≡ σ/L 1. However,in order that the signalling compartment has a non-negligible effect on the1536.1. Formulation of a 2-D Coupled Cell-Bulk Systembulk process, we need to assume that β1 and β2 are both O(−1)  1 as→ 0. In this way, (6.1.6) reduces to the dimensionless coupled systemτUt = D∆xU − U , x ∈ Ω˜\Ω˜0 ,∂nxU = 0 , x ∈ ∂Ω˜ ,D∂nxU = d1U − d2u1 , x ∈ ∂Ω˜0 ,(6.1.7a)where Ω˜0 is a disk of radius   1 centered at some x0 ∈ Ω˜. This bulkprocess is coupled to the intracellular dynamics described bydudt= F (u) +e1τ∫∂Ω˜0(d1U − d2u1)dSx . (6.1.7b)The key O(1) dimensionless parameters in (6.1.7) are τ , D, d1, and d2,defined byτ ≡ kRkB, D ≡ DBkBL2, β1 ≡ (kBL) d1, β2 ≡(kBL)d2.(6.1.8)We remark that the limit τ  1 (τ  1) corresponds to when the intracellu-lar dynamics is very slow (fast) with respect to the time-scale of degradationof the signalling molecule in the bulk. The limit D  1 corresponds to whenthe bulk diffusion length√DB/kB is large compared to the length-scale Lof the overall domain. Finally, we remark that upon using the divergencetheorem, we can readily establish thatddt(∫Ω˜\Ω˜0U dx+ eTu)= −1τ∫Ω˜\Ω˜0U dx+ eTF (u) , (6.1.9)where e ≡ (1, . . . , 1)T . The left-hand side of this expression is the totalamount of material inside the cells and in the bulk, while the right-handside characterizes the bulk degradation and production within the cell.Our analysis of the 2-D coupled cell-bulk model (6.1.7), and its multi-cellcounterpart (6.2.1), which extends the 3-D modeling paradigm of [54] and[55], has the potential of providing a theoretical framework to model quorumsensing behavior in experiments performed in petri dishes, where cells liveon a thin substrate. In contrast to the assumption of only one active intra-cellular component used in [54] and [55], in our study we will allow for msmall spatially segregated cells with multi-component intracellular dynam-ics. We will show for our 2-D case that the communication between small1546.2. Analysis of the Dimensionless 2-D Cell-Bulk SystemFigure 6.1: Schematic diagram showing the internal molecule reactions andexternal passive diffusion of the signal. The small shaded regions are thesignalling compartments or cells.cells through the diffusive medium is much stronger than in 3-D, and leadsin certain parameter regimes to the triggering of synchronous oscillations,which otherwise would not be present in the absence of any cell-bulk cou-pling. In addition, when D = O(1), we show that the spatial configurationof small cells in the domain is an important factor in triggering collectivesynchronous temporal instabilities in the cells.6.2 Analysis of the Dimensionless 2-D Cell-BulkSystemWith the motivation provided in §6.1, in this section we analyze a gen-eral class of dimensionless coupled cell-bulk dynamics, in a two-dimensionalbounded domain Ω that contains m small, disjoint, cells or compartmentsthat are scattered inside Ω. We assume that each cell is a small region ofradius O() 1, and that both the diameter of Ω and the distance betweenany two cells is O(1) as  → 0. A schematic plot of the geometry is shownin Fig. 6.1.In analogy with the dimensional reasoning provided in §6.1, if U(x, t)represents the dimensionless concentration of the signaling molecule in the1556.2. Analysis of the Dimensionless 2-D Cell-Bulk Systembulk region between the cells, then its spatial-temporal evolution in thisregion is governed by the dimensionless PDE modelτUt = D∆U − U , x ∈ Ω\ ∪mj=1 Ωj ,∂nU = 0 , x ∈ ∂Ω ,D∂njU = d1U − d2u1j , x ∈ ∂Ωj , j = 1, . . . ,m .(6.2.1a)Here D > 0 is the diffusion coefficient of the bulk, d1 > 0, d2 > 0 are thedimensionless influx (eflux) constants modeling the permeability of the cellwall, ∂n denotes the outer normal derivative of Ω, and ∂nj denotes the outernormal to each Ωj , which points inside the bulk region. We assume thatΩj is a domain of radius O() that shrinks uniformly to a point xj ∈ Ω as → 0 for each j = 1, . . . ,m. The signalling cell, or compartment, Ωj isassumed to lie entirely within Ω. The flux d1U−d2u1j on each cell membranemodels the influx of the signaling molecule into the extracellular bulk region,which depends on both the external bulk concentration U(x, t) at the cellmembrane ∂Ωj as well as on the amount u1j of the intracellular specieswithin the j-th cell.We suppose that inside each of the m cells there are n species that caninteract with each other, and that their dynamics are governed by n-ODEsof the formdujdt= F j(uj) +1τ∫∂Ωj(d1U − d2u1j ) ds e1 , (6.2.1b)where e1 ≡ (1, 0, . . . , 0)T . Here uj = (u1j , . . . , unj )T represents the concentra-tion of the n species inside the j-th cell and F j(uj) is the vector nonlinearitymodeling the chemical dynamics for these local species within the j-th cell.We remark that the integration in (6.2.1b) is over the boundary of the com-partment. Since the perimeter of the compartmental boundary ∂Ωj haslength |∂Ωj | = O(), the source term for the ODE in (6.2.1b), arising fromthe integration over the perimeter, is O(1).The dimensionless constants D, τ , d1, and d2, are related to their di-mensional counterparts by (6.1.8). From (6.1.8), we conclude that whenτ  1 the intracellular dynamics occurs on a much slower time-scale thanthe time-scale associated with bulk decay. The limit D  1 corresponds towhere the diffusion length induced by the bulk diffusivity and the bulk decayrate is much larger than the overall length-scale of the confining domain Ω.These simple qualitative will serve as a guide for interpreting our stabilityresults obtained below.1566.2. Analysis of the Dimensionless 2-D Cell-Bulk System6.2.1 The Steady-State Solution for the m Cells SystemWe first construct a equilibrium solution to (6.2.1) with m cells scattered inthe two-dimensional domain Ω. In the next sub-section we will formulatethe linear stability problem for this steady-state solution. We assume thatthe cells are well-separated in the sense that dist(xi,xj) = O(1) for i 6= jand dist(xj , ∂Ω) = O(1) for j = 1, . . . ,m.In our analysis below, for simplicity we assume that each Ωj is a circulardomain of radius  centered at xj ∈ Ω for each j = 1, . . . ,m. With suchan assumption we are able to provide an explicit approximate solution inthe vicinity of each cell. However, as we will remark below, this assumptionthat each cell has a circular shape is readily extended to the case where thesmall cells have an arbitrary shape.Since in an O() neighborhood near each cell the solution U changesrapidly and has a sharp spatial gradient, we will use the method of matchedasymptotic expansions to construct the equilibrium solution to (6.2.1). Inthe inner region near the j-th cell, we introduce the local variables Uj andy, which are defined byy = −1(x− xj) , Uj(y) = U(xj + y) . (6.2.2)In terms of these local variables (6.2.1a) transforms toD∆yUj − 2Uj = 0 , |y| > 1 ,D∂njUj = d1Uj − d2u1j , |y| = 1 .(6.2.3)We look for a radially symmetric solution to (6.2.3) of the form Uj = Uj(ρ),where ρ ≡ |y| and ∆y = ∂ρρ + ρ−1∂ρ denotes the radially symmetric partof the Laplacian. Then, at the leading order, we seek a radially symmetricsolution to∂ρρUj + ρ−1∂ρUj = 0 , 1 < ρ <∞ , (6.2.4a)subject to the boundary conditionD∂Uj∂ρ= d1Uj − d2u1j , ρ = 1 . (6.2.4b)The solution to (6.2.4a), which has logarithmic growth at infinity, isUj = Sj log ρ+ χj , (6.2.5)where χj = Uj(1) is to be determined. We refer to Sj as the source strengthof the j-th cell. By satisfying the boundary condition (6.2.4b) at ρ = 1, we1576.2. Analysis of the Dimensionless 2-D Cell-Bulk Systemdetermine χj in terms of Sj and u1j asχj =1d1(DSj + d2u1j ) , j = 1, . . . ,m . (6.2.6)With the inner dynamics (6.2.1b) inside each cell, we find that the sourcestrength Sj and the steady-state solution uj satisfy the nonlinear algebraicsystem of equationsF j(uj) +2piDτSje1 = 0 , (6.2.7)which, with F j ≡ (F 1j , . . . , Fnj )T , is equivalent to the following system writ-ten in component form:F 1j (u1j , . . . , unj ) +2piDτSj = 0 ; Fij (u1j , . . . , unj ) = 0 , i = 2, . . . , n .In principle, we can determine u1j from this system in terms of the unknownSj as u1j = u1j (Sj). The remaining steady-state values uj2, . . . , ujn also dependon Sj . Next, in terms of u1j , we derive a system of algebraic equations forS1, . . . , Sm, which is then coupled to (6.2.7).Upon matching the far-field behavior of the inner solution (6.2.5) to theouter solution, we obtain that the outer problem for U is∆U − ϕ20U = 0 , x ∈ Ω\{x1, . . . ,xm} ,∂nU = 0, x ∈ ∂Ω,U ∼ Sj log∣∣x− xj∣∣+ Sjν+ χj , as x→ xj , j = 1, . . . ,m ,(6.2.8)where we have defined ϕ0 and ν  1 byϕ0 ≡ 1/√D , ν ≡ −1/ log  . (6.2.9)We remark that the singularity condition in (6.2.8) for U as x → xj isderived by matching the outer solution for U to the far field behavior ofthe inner solution (6.2.5). To solve (6.2.8), we introduce the reduced-waveGreen’s function G(x;xj), which satisfies∆G−ϕ20G = −δ(x−xj) , x ∈ Ω ; ∂nG = 0 , x ∈ ∂Ω . (6.2.10a)As x→ xj , this Green’s function has the local behaviorG(x;xj) ∼ − 12pilog |x− xj |+Rj + o(1) , x→ xj , (6.2.10b)1586.2. Analysis of the Dimensionless 2-D Cell-Bulk Systemwhere Rj = Rj(xj) is called the regular part of G(x;xj) at x = xj . In termsof G(x;xj) we can represent the outer solution for U in (6.2.8) asU(x) = −2pim∑i=1SiG(x,xi) . (6.2.11)By expanding U as x→ xj , and equating the resulting expression with therequired singularity behavior in (6.2.8), we obtain the following nonlinearalgebraic system for S1, . . . , Sm:Sj +νd1(DSj + d2u1j)+ 2piνSjRj + m∑i 6=jSiGji = 0 , j = 1, . . . ,m ,(6.2.12)where Gji ≡ G(xj ;xi). This system is coupled to (6.2.7), which determinesu1j in terms of Sj .It is convenient to write (6.2.12) in matrix form. To do so, we define theGreen’s matrix G, the vector of source strengths S and the vector u1, whosej-th element is the first local variable in the j-th cell, byG ≡R1 G12 · · · G1mG21 R2 · · · G2m....... . ....Gm1 Gm2 · · · Rm, S ≡S1S2...Sm, u1 ≡u11u12...u1m.By the reciprocity of the Green’s function, we have that Gji = Gij , so thatG is a symmetric matrix. In terms of this matrix notation, (6.2.12) can bewritten as (1 +Dνd1)S + 2piνGS = −d2d1νu1 . (6.2.13)Together with (6.2.7), (6.2.13) provides an approximate steady-state so-lution for u, which is coupled to the source strengths S. It is rather in-tractable analytically to write general conditions on the nonlinear kineticsto ensure the existence of a solution to the coupled algebraic system (6.2.7)and (6.2.13). As such, in subsequent sub-sections below we will analyze indetail the solvability of this system for some specific choices for the nonlinearkinetics. We remark that even if we make the assumption that the nonlinearkinetics in the cells are identical, so that F j = F for j = 1, . . . ,m, we still1596.2. Analysis of the Dimensionless 2-D Cell-Bulk Systemhave that Sj and u1 depend on j through as a result of the Green’s interac-tion matrix G, which depends on the spatial configuration {x1, . . . ,xm} ofthe cells.In summary, after solving the nonlinear algebraic system (6.2.7) and(6.2.13), the approximate steady-state solution for U is given by (6.2.11)in the outer region, defined at O(1) distances from the cells, and (6.2.5) inthe neighborhood of each cell. We remark that this approximate steady-state solution is accurate to all orders in ν, and our analysis has effectivelysummed an infinite order logarithmic expansion in powers of ν for the steady-state solution. Related 2-D problems involving logarithmic expansions, suchas the calculation of the mean first passage time for diffusion in 2-D domains,are studied in [37] and [63] (see also the references therein).Lastly, we remark on how the analysis can be extended to study thecase where the small cells have an arbitrary shape. In this case, the innerproblem near the j-th cell is∆yUj = 0 , y /∈ Ωj ; D∂njUj = d1Uj − d2u1j , y ∈ ∂Ωj ,Uj ∼ Sj log |y| , |y| → ∞ .(6.2.14)Here Ωj = −1Ωj denotes the j-th cell region when magnified by −1, while∂nj now denotes the outward normal derivative to Ωj as written in the yvariable. The solution to (6.2.14) can be conveniently decomposed asUj =d2d1u1j + VjSj ,where Vj is the unique solution to∆yVj = 0 , y /∈ Ωj ; D∂njVj = d1Vj , y ∈ ∂Ωj ,Vj ∼ log |y| , |y| → ∞ .(6.2.15a)In terms of this solution, we identify the constant ξ in the far-field behaviorof Vj , which depends on the shape of Ωj and the ratio d1/D, byVj ∼ log |y| − log ξ + o(1) , |y| → ∞ . (6.2.15b)In general ξ must be calculated numerically from a boundary integral formu-lation of (6.2.15a). This yields the following far-field behavior of the innersolution Uj when written in outer variables:Uj ∼ Sj log |x− xj |+ Sjν˜+d2u1jd1, ν˜ ≡ −1/ log(ξ) . (6.2.16)1606.2. Analysis of the Dimensionless 2-D Cell-Bulk SystemBy using the divergence theorem in the region between ∂Ωj and a large disk,we can readily verify that (6.2.7) still holds, and that (6.2.8) also holds if wereplace ν and χj in (6.2.8) with ν˜ and d2u1j/d1, respectively. By repeatingour previous analysis, it is clear that we can readily determine a system verysimilar to (6.2.29). The only essential difference is that one must numericallycompute the constant ξ from the PDE (6.2.15a). This constant defines thegauge function ν˜ from (6.2.16). Rather than proceed with this general case,in the next sub-section we study the linearized stability problem for thesimpler case where all the cells are circular with a common radius .6.2.2 Formulation of the Linear Stability ProblemNext, we consider the linear stability of the steady-state solution constructedin the previous subsection. We perturb this steady-state solution, denotedhere by Ue(x) in the bulk region, and ue,j in the j-th cell asU = Ue + eλtη(x) , uj = ue,j + eλtφj .Upon substituting this perturbation into (6.2.1), we obtain in the bulk regionthatτλη = D∆η − η , x ∈ Ω\ ∪mj=1 Ωj ,∂nη = 0 , x ∈ ∂Ω ,D∂njη = d1η − d2φ1j , x ∈ ∂Ωj .(6.2.17a)Within the j-th cell the linearized problem isλφj = Jjφj +1τe1∫∂Ωj(d1η − d2φ1j)ds , (6.2.17b)where Jj denotes the Jacobian matrix of the nonlinear kinetics F j evaluatedat ue,j . We now study (6.2.17) in the limit  → 0 using the method ofmatched asymptotic expansions. The analysis will provide a limiting glob-ally coupled eigenvalue problem for λ, from which we can investigate possibleinstabilities.In the inner region near the j-th cell, we introduce the local variablesy = −1(x − xj), with ρ = |y|, and let ηj(y) = η(xj + y). We will lookfor the radially symmetric eigenfunction ηj in the inner variable ρ. Then,from (6.2.17a), upon neglecting higher order algebraic terms in , the innerproblem becomes∂ρρηj + ρ−1∂ρηj = 0 , 1 < ρ <∞ , (6.2.18a)1616.2. Analysis of the Dimensionless 2-D Cell-Bulk Systemwith boundary conditionD∂ηj∂ρ= d1ηj − d2φ1j , ρ = 1 . (6.2.18b)The solution to this problem isηj = cj log ρ+Bj , (6.2.19)where cj and Bj both are unknown constants. Upon satisfying the boundarycondition in (6.2.18b), we determine Bj in terms of cj asBj =1d1(Dcj + d2φ1j ) . (6.2.20)Then, upon substituting (6.2.19) and (6.2.20) into (6.2.17b), we obtain that(Jj − λI)φj + 2piDτcje1 = 0 , j = 1, . . . ,m . (6.2.21)In the outer region, defined at O(1) distances from the cells, the outerproblem for the eigenfunction η(x) is∆η − (1 + τλ)Dη = 0, x ∈ Ω\{x1, . . . ,xm} ,∂nη = 0, x ∈ ∂Ω ,η ∼ cj log∣∣x− xj∣∣+ cjν+Bj , x→ xj , j = 1, . . . ,m ,(6.2.22)where ν ≡ −1/ log . We remark that the singularity condition in (6.2.22)as x → xj is derived by matching the outer solution for η to the far fieldbehavior of the inner solution (6.2.19). To solve (6.2.22), we introduce theGreen’s function Gλ(x;xj) which satisfies∆Gλ − ϕ2λGλ = −δ(x− xj), x ∈ Ω ,∂nGλ = 0, x ∈ ∂Ω,Gλ(x;xj) ∼ − 12pilog∣∣x− xj∣∣+Rλ,j + o(1), x→ xj , (6.2.23)where Rλ,j ≡ Rλ(xj) is the regular part of Gλ at xj . Here we have definedϕλ byϕλ ≡√1 + τλD. (6.2.24)1626.2. Analysis of the Dimensionless 2-D Cell-Bulk SystemWe will choose the principal branch of ϕλ, which ensures that ϕλ is analyticin Re(λ) > 0. For the case of an asymptotically large domain Ω, this choicefor the branch cut also ensures that Gλ decays as |x− xj | → ∞.In terms of Gλ(x;xj), we can represent the outer solution η(x), whichsatisfies (6.2.22), asη(x) = −2pim∑i=1ciGλ(x,xi) . (6.2.25)By matching the singularity condition at x → xj , we obtain a system ofequations for cj ascjν+Bj = −2picjRλj + m∑i 6=jciGλ,ij , j = 1, . . . ,m , (6.2.26)where Gλ,ij ≡ Gλ(xj ;xi). Upon recalling that Bj = 1d1 (Dcj + d2φ1j ) from(6.2.20), we can rewrite (6.2.26) ascjν+1d1(Dcj + d2φ1j ) + 2pi(cjRλ,j +m∑i 6=jciGλ,ji) = 0 . (6.2.27)It is convenient to write (6.2.27) in matrix notation. To this end, wedefine the Green’s matrix Gλ and the vectors c and φ1 byGλ ≡Rλ,1 Gλ,12 · · · Gλ,1mGλ,21 Rλ,2 · · · Gλ,2m....... . ....Gλ,m1 Gλ,m2 · · · Rλ,m, c ≡c1c2...cm, φ1 ≡φ11φ12...φ1m.(6.2.28)We observe that the j-th entry in the vector φ1 = (φ11, · · · , φ1m)T is simplythe first element in the eigenvector for the j-th cell. In this way, (6.2.27)can be written in matrix form as(1 +Dνd1)c+d2d1νφ1 + 2piνGλc = 0 . (6.2.29)Together with (6.2.21), (6.2.29) will yield an eigenvalue problem for λ witheigenvector c.1636.2. Analysis of the Dimensionless 2-D Cell-Bulk SystemWe now determine this eigenvalue problem in a more explicit form byfirst calculating φ1 in terms of c from (6.2.21) and then substituting theresulting expression into (6.2.29). If λ is not an eigenvalue of Jj , we obtainfrom (6.2.21) thatφj =2piDτ(λI − Jj)−1cje1 ,where e1 is the n-vector e1 = (1, 0, . . . , 0)T . Upon taking the dot productwith e1, we isolate φ1j asφ1j =2piDτcje1T (λI − Jj)−1e1 .This then yields thatφ1 = 2piDτKc , (6.2.30a)where K = K(λ) is the m×m diagonal matrix with diagonal entriesKj = e1T (λI−Jj)−1e1 = 1det(λI − Jj)e1MTj e1 =Mj,11det(λI − Jj) . (6.2.30b)Here Mj is the n × n matrix of cofactors of the matrix λI − Jj , with Mj,11denoting the matrix entry in the first row and first column of Mj , givenexplicitly byMj,11 = Mj,11(λ) ≡ detλ− ∂F2j∂u2∣∣∣u=ue,j, · · · , −∂F2j∂un∣∣∣u=ue,j· · · , · · · , · · ·−∂Fnj∂u2∣∣∣u=ue,j, · · · , λ− ∂Fnj∂un∣∣∣u=ue,j .(6.2.31)Here F 2j , . . . , Fnj denote the components of the vector F j ≡ (F 1j , . . . , Fnj )T ,characterizing the membrane kinetics.Next, upon substituting (6.2.30a) into (6.2.29), we obtain the homoge-neous m×m linear systemMc = 0 , (6.2.32a)where M =M(λ) is defined byM≡(1 +Dνd1)I + 2piνd2d1τDK + 2piνGλ , (6.2.32b)1646.3. The Distinguished Limit of D = O(ν−1) 1where the diagonal matrix K has diagonal entries (6.2.30b), and Gλ is theGreen’s interaction matrix defined in (6.2.28), which depends on λ as wellas on the spatial configuration {x1, . . . ,xm} of the small cells.We refer to (6.2.32) as the globally coupled eigenvalue problem (GCEP).In the limit  → 0, we conclude that λ is a discrete eigenvalue of the lin-earized problem (6.2.17) if and only if λ is a root of the transcendentalequationdetM = 0 . (6.2.33)To determine the region of stability, we must seek conditions on the param-eters for which any such eigenvalue satisfies Re(λ) < 0. The correspondingeigenvector c of (6.2.32) associated with a root of (6.2.33) gives the spatialinformation for the eigenfunction in the bulk via (6.2.25).We now make some remarks on the form of the GCEP. We first observefrom (6.2.32b) that when D = O(1), then to leading-order in ν  1, wehave that M ∼ I + O(ν). As such, when D = O(1), we conclude that toleading order in ν there are no discrete eigenvalues of the linearized problemwith λ = O(1), and hence no O(1) time-scale instabilities. However, sinceν = −1/ log  is not very small unless  is extremely small, this prediction ofno instability in the D = O(1) regime may be somewhat misleading at smallfinite . Instead of finding the roots of (6.2.33) using numerical methods,without first assuming ν  1, in the next sub-section we will consider thedistinguished limit D = O(ν−1) 1 for (6.2.32b) where the linearized sta-bility problem becomes highly tractable analytically. In another, somewhatrelated, context this large D distinguished limit regime was found in [77],[78], and [65] (see the references therein) to play a central role in calculat-ing stability thresholds associated with localized spot patterns for certaintwo-component reaction-diffusion systems in the plane.6.3 The Distinguished Limit of D = O(ν−1) 1In the previous section we considered the general case where there are mdynamically active cells in a two dimensional bounded domain Ω with bulkdiffusion. For → 0, we constructed the steady-state solution for this prob-lem and derived the spectral problem that characterizes the linear stabilityof this solution.In this section, we consider the distinguished limit where the signalingmolecule in the bulk diffuses rapidly, so that D  1. More specifically, wewill consider the distinguished limit where D = O(ν−1), and hence for some1656.3. The Distinguished Limit of D = O(ν−1) 1D0 = O(1), we setD =D0ν. (6.3.1)Under this assumption, we will obtain a leading order approximation tothe steady-state solution. We will then analyze the corresponding spectralproblem.Since the reduced-wave Green’s function depends on D, we first approx-imate it for large D. We write (6.2.10a) as∆G− νD0G = −δ(x− xj) , x ∈ Ω ; ∂nG = 0, x ∈ ∂Ω . (6.3.2)This problem has no solution when ν = 0. Therefore, we expand G =G(x;xj) for D  1 asG =1νG−1 +G0 + νG1 + . . . . (6.3.3)Upon substituting (6.3.3) into (6.3.2), and equating powers of ν, we obtaina sequence of problems for Gi for i = −1, 0, 1. To leading-order O(ν−1), weget that G−1 satisfies∆G−1 = 0 , x ∈ Ω ; ∂nG−1 = 0 , x ∈ ∂Ω . (6.3.4)This gives that G−1 is a constant. The O(1) system for G0 is∆G0 =1D0G−1 − δ(x− xj) , x ∈ Ω ; ∂nG0 = 0 , x ∈ ∂Ω .(6.3.5)Similarly the O(ν) problem satisfied by G1 is∆G1 =1D0G0 , x ∈ Ω ; ∂nG1 = 0 , x ∈ ∂Ω . (6.3.6)The divergence theorem applied to (6.3.5) and (6.3.6) yields thatG−1 =D0|Ω| ,∫ΩG0 dx = 0 .In this way, we obtain the following two-term expansion for the reduced-wave Green’s function and its regular part Rj in the limit D = D0/ν  1:G(x;xj) =D0ν|Ω| +G0(x;xj) + · · · , Rj =D0ν|Ω| +R0,j + · · · . (6.3.7)1666.3. The Distinguished Limit of D = O(ν−1) 1Here G0(x;xj), with regular part R0j , is the Neumann Green’s functiondefined as the unique solution to∆G0 =1|Ω| − δ(x− xj) , x ∈ Ω ;∂nG0 = 0 , x ∈ ∂Ω ;∫ΩG0 dx = 0 ,G0(x;xj) ∼ − 12pilog∣∣x− xj∣∣+R0,j , x→ xj .(6.3.8)We then substitute the expansion (6.3.7) and D = D0/ν into the non-linear algebraic system (6.2.7) and (6.2.13), which is associated with thesteady-state problem, to obtain that(1 +D0d1)S +2pimD0|Ω| ES + 2piνG0S = −d2d1νu1 ;F j(uj) +2piD0τνSje1 = 0 , j = 1, . . . ,m ,(6.3.9)where E and the Neumann Green’s matrix G0 are the m×m matrices definedbyE ≡ 1meeT =1m1 · · · 1.... . ....1 · · · 1 , G0 ≡R0,1 G0,12 · · · G0,1mG0,21 R0,2 · · · G0,2m....... . ....G0,m1 G0,m2 · · · R0,m.(6.3.10)Here e is the m× 1 vector e ≡ (1, . . . , 1)T .The leading-order solution to (6.3.9) when ν  1 has the formS = νS0 +O(ν2) , uj = uj0 +O(ν) . (6.3.11)From (6.3.9) we conclude that S0 and uj0 satisfy the limiting leading-ordernonlinear algebraic system(1 +D0d1)S0 +2pimD0|Ω| ES0 = −d2d1u10 ;F j(u0j) +2piD0τS0je1 = 0 , j = 1, . . . ,m .(6.3.12)1676.3. The Distinguished Limit of D = O(ν−1) 1Since this leading order system does not involve the Neumann Green’s ma-trix G0, we conclude that S0 is independent of the spatial configuration ofthe cells in Ω.For the special case where the reaction kinetics F j is identical for eachcell, so that F j = F for j = 1, . . . ,m, we will look for a solution to (6.3.12)with identical source strengths, so that S0j is independent of j and u0j = u0is independent of j. Therefore, we writeS0 = S0ce , (6.3.13)where S0c is the common source strength. From (6.3.12), where we useEe = e, this yields that S0c and u0 satisfy the m+ 1 dimensional nonlinearalgebraic system(1 +D0d1+2pimD0|Ω|)S0c = −d2d1u10 , F (u0) +2piD0τS0ce1 = 0 ,(6.3.14)where u10 is the first component of u0. This simple limiting system willbe studied in detail in the next section for various choices of the nonlinearkinetics F (u0).Next, we will simplify the globally coupled eigenvalue problem (GCEP),given by (6.2.32), when D = D0/ν  1, and under the assumption that thereaction kinetics are the same in each cell. In the same way as was derivedin (6.3.2)–(6.3.7), we let D = D0/ν  1 and approximate the λ-dependentreduced Green’s function Gλ(x;xj), which satisfies (6.2.23). For τ = O(1),we calculate, in place of (6.3.7), thatGλ(x;xj) =D0ν(1 + τλ)|Ω| +G0(x;xj) +O(ν) ,Rλ,j =D0ν(1 + τλ)|Ω| +R0,j +O(ν) ,where G0(x;xj), with regular part R0,j , is the Neumann Green’s functionsatisfying (6.3.8). It follows that for D = D0/ν  1 and τ = O(1), we haveGλ = mD0ν(1 + τλ)|Ω|E + G0 +O(ν) , (6.3.15)where E , and the Neumann Green’s matrix G0, are defined in (6.3.10).We substitute (6.3.15) into (6.2.32b), and set D = D0/ν. In (6.2.32b),we calculate to leading order in ν that the matrix K(λ), defined in (6.2.30b),reduces toK ∼ M11det(λI − J) +O(ν) , (6.3.16)1686.3. The Distinguished Limit of D = O(ν−1) 1where J is the Jacobian of F evaluated at the solution u0 to the limitingproblem (6.3.14), and M11 is the cofactor of λI − J associated with its firstrow and first column. The O(ν) correction in K(λ) arises from the higherorder terms in the Jacobian resulting from the solution to the full system(6.3.9). In this way, the matrixM, defined in (6.2.32b), associated with theGCEP reduces to leading order toM = a(λ)I + b(λ)E +O(ν) , (6.3.17a)where a(λ) and b(λ) are defined bya(λ) = 1 +D0d1+2pid2d1τD0M11det(λI − J) , b(λ) =2pimD0(1 + τλ)|Ω| . (6.3.17b)We remark that the O(ν) correction terms in (6.3.17a) arises from both2piνG0, which depends on the spatial configuration of the cells, and theO(ν) term in K as written in (6.3.16).Therefore, when D = D0/ν, it follows from (6.3.17a) and the criterion(6.2.33) of the GCEP that λ is a discrete eigenvalue of the linearization ifand only if there exists a nontrivial solution c to(a(λ)I + b(λ)E) c = 0 , (6.3.18)where a(λ) and b(λ) are defined in (6.3.17b). Any such eigenvalue withRe(λ) > 0 leads to a linear instability of the steady-state solution on theregime D = O(ν−1).Explicit equations for λ are readily derived from (6.3.18) by using thekey property that Ee = e and Eqj = 0 for j = 2, . . . ,m, where qj forj = 2, . . . ,m are an orthogonal basis of the m−1 dimensional perpendicularsubspace to e, i.e qTj e = 0.In this way, we obtain that λ is a discrete eigenvalue for the synchronousmode, corresponding to c = e, whenever λ satisfiesa(λ) + b(λ) ≡ 1 + D0d1+2pid2d1τD0M11det(λI − J) +2pimD0(1 + τλ)|Ω| = 0 . (6.3.19)This expression can be conveniently written asM11det(λI − J) = −τ2pid2(κ1τλ+ κ2τλ+ 1),where κ1 ≡ d1D0+1 , κ2 ≡ κ1+2mpid1|Ω| .(6.3.20)1696.4. Examples of the Theory: Finite Domain With D = O(ν−1)In contrast, λ is a discrete eigenvalue for the asynchronous or competitionmodes, corresponding to c = qj for j = 2, . . . ,m, whenever λ satisfies a(λ) =0, which yieldsM11det(λI − J) = −τ2pid2(d1D0+ 1). (6.3.21)We remark that the synchronous mode depends on the total number m ofcells. In contrast, the asynchronous mode is independent of m, but doesrequire that m ≥ 2.Any discrete eigenvalue for either the synchronous or asynchronous modesthat satisfies Re(λ) > 0 leads to an instability. If all such eigenvalues sat-isfy Re(λ) < 0, then the steady-state solution for the regime D = D0/ν islinearly stable on an O(1) time-scale.6.4 Examples of the Theory: Finite DomainWith D = O(ν−1)In this section we will study the leading-order steady-state problem (6.3.14),and its associated spectral problem (6.3.19) and (6.3.21), for various specialcases and choices of the reaction kinetics F . We would like to investigate thestability properties of the steady-state (6.3.14) as parameters are varying,and in particular, find conditions for Hopf bifurcations to occur.6.4.1 Example 1: m Cells; One Local ComponentTo illustrate the asymptotic theory, we first consider a system with m cellsin a two-dimensional bounded domain Ω such that the local dynamics insideeach cell consists of only a single component with arbitrary scalar kinetics F .For this case, the steady-state problem for u0 = u0 and S, given by (6.3.14),reduces to two algebraic equations. We now show that no matter what F (u)is, the steady-state can never be destablilized by a Hopf bifurcation.We will study the stability problem for both the synchronous and asyn-chronous modes. In the one component case, we calculate M11 = 1 anddet(λI − J) = λ − F eu , where F eu is defined as the derivative of F (u) eval-uated at the steady-state u0. It is easily shown that the spectral problem(6.3.20) for the synchronous mode reduces toλ2 − λp1 + p2 = 0 , p1 ≡ F eu −γτ− ζτ, p2 ≡ 1τ(γτ− ζF eu)= 0 ,(6.4.1a)1706.4. Examples of the Theory: Finite Domain With D = O(ν−1)whereγ ≡ 2pid2D0d1 +D0> 0 , ζ ≡ 1 + 2pimd1D0|Ω|(d1 +D0) > 1 . (6.4.1b)For a Hopf bifurcation to occur we need p1 = 0 and p2 > 0. Upon settingp1 = 0, we getF eu =1τ(γ + ζ) > 0 . (6.4.2)Upon substituting (6.4.2) into the expression for p2 in (6.4.1a) we getp2 =1τ(γτ− ζF eu)=1τ2(γ(1− ζ)− ζ2)< 0 , (6.4.3)since τ > 0, γ > 0 and ζ > 1. Therefore, there can be no Hopf bifurcationfor the synchronous mode. The following result characterizes the stabilityproperties for the synchronous mode:Proposition 6.4.1 Consider the synchronous mode. Suppose thatF eu <γζτ=2pid2τ[1 +d1D0+2pimd1|Ω|]−1, (6.4.4)then we have Re(λ) < 0, and so the steady-state is linearly stable to syn-chronous perturbations. If F eu > γ/ [ζτ ], the linearization has exactly onepositive eigenvalue.To show this, we note that the steady-state solution is stable to syn-chronous perturbations if and only if p1 < 0 and p2 > 0. From (6.4.1a),p1 < 0 and p2 > 0 whenτF eu < ζ + γ , τFeu < γ/ζ , (6.4.5)respectively, which implies that we must have τF eu < min(ζ + γ, γ/ζ). Sinceζ > 1, the two inequalities in (6.4.5) hold simultaneously only when τF eu <γ/ζ. This yields that Re(λ) < 0 when (6.4.4) holds. Finally, when F eu >γ/ [ζτ ], then p2 < 0, and so there is a unique positive eigenvalue.This result shows that the effect of cell-bulk coupling is that the steady-state of the coupled system can be linearly stable even when the reactionkinetics is self-activating in the sense that F eu > 0. We observe that thestability threshold γ/ζ is a monotone increasing function of D0, with γ/ζ →0 as D0 → 0 and γ/ζ tending to a limiting value as D0 → ∞. This showsthat as D0 is decreased, corresponding to when the cells are effectively more1716.4. Examples of the Theory: Finite Domain With D = O(ν−1)isolated from each other, there is a smaller range of F eu > 0 where stabilitycan still be achieved.Next, we will consider the spectral problem for the asynchronous mode.From (6.3.21), we get1λ− F eu= −τγ, (6.4.6)where γ is defined in (6.4.1b). Therefore, λ = F eu − γ/τ , and so λ is realand no Hopf bifurcation can occur. This asynchronous mode is stable ifF eu < γ/τ . Since ζ > 1, we observe, upon comparing this threshold withthat for the synchronous mode in (6.4.4), that the stability criterion for thesynchronous mode is the more restrictive of the two stability thresholds.In summary, we conclude that a Hopf bifurcation is impossible for (6.2.1a)in the parameter regime D = D0/ν when there is only one dynamically ac-tive species inside each of m small cells. Moreover, if F eu < γ/[ζτ ], where γand ζ are defined in (6.4.1b), then the steady-state solution is linearly stableto both the synchronous and asynchronous modes.6.4.2 Example 2: m Cells; Two Local ComponentsNext we consider m cells in Ω, but now we assume that there are two dynam-ically active local species inside each cell. Without causing any confusion,we write the intracellular variable as u = (v, w)T and the local kineticsas F (v, w) = (F (v, w), G(v, w))T . In this way, the steady-state problem(6.3.14) becomes(1 +D0d1+2pimD0|Ω|)S0c = −d2d1ve , F (ve, we)+2piD0τS0c = 0 , G(ve, we) = 0 .(6.4.7)Given specific forms for F and G, we can solve the steady-state problem(6.4.7) either analytically or numerically.To analyze the stability problem, in a similar way as for the one-speciescase, we first calculate the cofactor M11 as M11 = λ−Gew and det(λI−J) =λ2− tr(J)λ+ det(J), where tr(J) and det(J) are the trace and determinantof the Jacobian of F , given bytr(J) = F ev +Gew , det(J) = FevGew − F ewGev . (6.4.8)Here F ei , Gei are partial derivatives of F , G with respect to i, with i ∈ (v, w),evaluated at the solution to (6.4.7).Next, we analyze the stability of the steady-state solution with respectto either synchronous or asynchronous perturbations. For the synchronous1726.4. Examples of the Theory: Finite Domain With D = O(ν−1)mode, we obtain, after some algebra, that (6.3.20) can be reduced to thestudy of the following cubic polynomial in λ:H(λ) ≡ λ3 + λ2p1 + λp2 + p3 = 0 , (6.4.9a)where we have defined p1, p2, and p3 byp1 ≡ γτ+ζτ− tr(J) ,p2 ≡ det(J)− γτGew +1τ(γτ− ζtr(J)) ,p3 ≡ 1τ(ζ det(J)− γτGew) .(6.4.9b)Here γ and ζ are defined in (6.4.1b).To determine whether there is any eigenvalue lying in the right-half ofthe complex λ-plane, and to detect any Hopf bifurcation boundary in the pa-rameter space, we will use the Routh-Hurwitz criterion for a cubic function.This criterion gives necessary and sufficient conditions for all of the rootsof the cubic polynomial with real coefficients to lie in Re(λ) < 0. Given acubic polynomial H(λ), the criterion states that a necessary and sufficientcondition for all its roots to satisfy Re(λ) < 0 is that the following threeconditions on the coefficients hold:p1 > 0 , p3 > 0 , p1p2 > p3 . (6.4.10)To find the conditions that parameters should satisfy on a Hopf bifurcationboundary, we need only consider a special cubic equation which has rootsλ1 = a < 0 and λ2,3 = ±iω. Thus, such a cubic equation has the form(λ− a)(λ− iω)(λ+ iω) = λ3 − aλ2 + ω2λ− aω2 = 0 . (6.4.11)Comparing this expression with (6.4.9a) and together with Routh-Hurwitzcriterion, we conclude that the Hopf bifurcation boundary lies in the pa-rameter regime wherep1 > 0 , p3 > 0 , (6.4.12a)with the Hopf bifurcation boundary given byp1p2 = p3 . (6.4.12b)We will return to criterion in the next two subsections when we study twospecific models for the local kinetics (F,G).1736.4. Examples of the Theory: Finite Domain With D = O(ν−1)Next, we consider the spectral problem for the asynchronous mode.Upon substituting the expressions of M11 and det(λI − J) into (6.3.21)and reorganizing it, (6.3.21) becomes a quadratic equation in λ given byλ2 − λq1 + q2 = 0 , q1 ≡ tr(J)− γτ, q2 ≡ det(J)− γτGew . (6.4.13)For a Hopf bifurcation to occur, we require that q1 = 0 and q2 > 0, whichyields thatγτ= tr(J) = F ev +Gew , (6.4.14a)provided the inequalitydet(J)− γτGew = −GevF ew − (Gew)2 > 0 , (6.4.14b)holds. Finally, we conclude that Re(λ) < 0 for the asynchronous modes ifand only iftr(J) < γ/τ , and det(J)− γτGew > 0 . (6.4.15)To write the stability problem for the asynchronous mode in terms of D0,we use (6.4.1b) for γ in terms of D0 to obtain from the conditions (6.4.14a)and (6.4.14b) that the Hopf bifurcation threshold for the asynchronous modeis given by the transcendental equationD0 =τd1tr(J)2pid2 − τtr(J) , (6.4.16a)provided that the inequalityD0(2pid2τGew − det(J))< d1 det(J) , (6.4.16b)is satisfied. We observe that in this formulation, both tr(J) and det(J)depend on the form of the local kinetics and the steady-state solution, whichdepends on D0 and τ .In the next two subsections we study in some detail two specific choicesfor the local kinetics, and we show phase diagrams where oscillatory insta-bilities can occur.1746.4. Examples of the Theory: Finite Domain With D = O(ν−1)Local Kinetics Described by the Sel’kov ModelWe first consider the Sel’kov model, use in simple models of glycolysis, wherethe functions F (v, w) and G(v, w) are given in terms of parameters α > 0,µ > 0, and 0 > 0 byF (v, w) = αw + wv2 − v , G(v, w) = 0(µ− (αw + wv2)). (6.4.17)First, we determine the approximate steady-state solution by substituting(6.4.17) into (6.4.7). This system can be solved analytically to obtain thesteady-state solutionve =µ[1 + 2piD0β/τ] , we = µα+ v2e, S0c = βve , (6.4.18a)where we have defined β > 0 byβ ≡ d2d1 +D0 + 2pimd1D0/|Ω| . (6.4.18b)As needed below, we first calculate the partial derivatives of F and G eval-uated at the steady-state solution asF ev = 2vewe−1 , F ew = α+v2e , Gev = −20vewe , Gew = −0(α+v2e) ,(6.4.19a)which yieldsdet(J) = 0(α+ v2e)> 0 , tr(J) = 2vewe− 1− 0(α+ v2e). (6.4.19b)To study possible synchronous oscillations of the m cells, we compute theHopf bifurcation boundaries as given in (6.4.12), where we use (6.4.19). Forthe parameter set τ = 1, D0 = 1, |Ω| = 10, µ = 2, α = 0.9, and 0 = 0.15, weobtain the Hopf bifurcation boundary in the d1 versus d2 parameter planeas shown by the solid curves in Fig. 6.2 for m = 1, 2, 3.Next, to obtain instability thresholds corresponding to asynchronousmode, we substitute (6.4.19a) into (6.4.14) and obtain that, the Hopf bi-furcation boundary is given byγ = τtr(J) ≡ τ[2vewe − 1− (α+ v2e)], (6.4.20)provided that det(J) − tr(J)Gew > 0. This latter condition can be written,using (6.4.19a), as 0(α + v2e)(1 + tr(J))> 0, and so is satisfied provided1756.4. Examples of the Theory: Finite Domain With D = O(ν−1)that tr(J) > −1. Since γ > 0 from (6.4.20), we must have tr(J) > 0, whichguarantees that det(J) − tr(J)Gew > 0 always holds at a Hopf bifurcation.In this way, and by substituting (6.4.18) for we into (6.4.20), we obtain thatthe asynchronous mode has a pure imaginary pair of complex conjugateeigenvalues whenγ = τ[2veµα+ v2e− 1− 0(α+ v2e)], where ve =µ[1 + 2piD0β/τ] ,(6.4.21)where γ and β, depending on d1, d2, m, |Ω|, and D0, are defined in (6.4.1b)and (6.4.18b), respectively. By using these expressions for γ and β, we canreadily determine a parametric form for the Hopf bifurcation boundary inthe d1 versus d2 plane as the solution to a linear algebraic system for d1and d2 in terms of the parameter ve with 0 < ve < µ. Some simple algebrayields thatd1 =D0(a12 − a22)a11a22 − a21a12 , d2 =D0(a21 − a11)a11a22 − a21a12 , (6.4.22a)where a11, a12, a22, and a21, are defined in terms of the parameter ve bya11 ≡ 1 + 2pimD0|Ω| , a12 ≡ −1β(ve), a21 ≡ 1 , a22 ≡ −2piD0γ(ve),(6.4.22b)whereγ(ve) ≡ τ[2veµα+ v2e− 1− 0(α+ v2e)], β(ve) ≡ τ (µ− ve)2piD0ve. (6.4.22c)By varying ve, with 0 < ve < µ, and retaining only the portion of thecurve for which d1 > 0 and d2 > 0, we obtain a parametric form for theHopf bifurcation boundary for the asynchronous mode in the d1 versus d2parameter plane. For m = 2 and m = 3, these are the dashed curves shownin Fig. 6.2.1766.4. Examples of the Theory: Finite Domain With D = O(ν−1)0 0.05 0.1 0.15 0.2 0.2500.511.522.5d2d1m=1m=2m=3m=2m=3Figure 6.2: Hopf bifurcation boundaries for the synchronous (solid curve)and asynchronous (dashed curve) modes for the Sel’kov model with variousnumber m of cells in the d1 versus d2 parameter plane. The synchronousmode for m = 1 between the two black lines is unstable. For m = 2 andm = 3 the synchronous mode is unstable in the horseshoe-shaped regionbounded by the blue and red solid curves, respectively. Inside the dottedregions for m = 2 and m = 3 the asynchronous mode is unstable. For theasynchronous mode, the boundary of these regions is given parametricallyby (6.4.22). The parameters used are µ = 2, α = 0.9, 0 = 0.15, τ = 1,D0 = 1, and |Ω| = 10.We now discuss qualitative aspects of the Hopf bifurcation boundaries forboth synchronous and asynchronous modes for various choices of m as seenin Fig. 6.2. For m = 1, we only need to consider the synchronous instability.The Hopf bifurcation boundary is given by the two black lines, and the regionwith unstable oscillatory dynamics is located between these two lines. Form = 2, inside the region bounded by the blue solid curve, the synchronousmode is unstable and under the blue dashed curve, the asynchronous modeis unstable. Similar notation applies to the case with m = 3, where the Hopfbifurcation boundaries for synchronous/asynchronous mode are denoted byred solid/dashed curves.One key qualitative feature we can observe from Fig. 6.2 is that the oscil-latory region for a larger value of m lies completely within the unstable re-gion for smaller m for both the synchronous and asynchronous modes. Thissuggests that if a coupled system with m1 cells is unstable to synchronousperturbations, then a system with m2 < m1 cells will also be unstable tosuch perturbations. However, if a two-cell system is unstable, it is possible1776.4. Examples of the Theory: Finite Domain With D = O(ν−1)that a system with three cells, with the same parameter set, can be stable.Finally, we observe qualitatively that the Hopf bifurcation boundary of theasynchronous mode always lies between that of the synchronous mode. Thissuggests that as we vary d1 and d2 from a stable parameter region into anunstable parameter region, we will always first trigger a synchronous os-cillatory instability rather than an asynchronous instability. It is an openproblem to show that these qualitative observations still hold for a widerange of other parameter sets.0 0.5 1012345D0τm=1m=2m=30 5 10 15012345D0τFigure 6.3: Left panel: Hopf bifurcation boundaries for the synchronous(solid curves) and asynchronous (dashed curves) modes for the Sel’kov modelwith various number m of cells in the τ versus D0 plane. Only inside theregion bounded by the two black solid curves, the synchronous mode isunstable for m = 1. Similarly, in the lobe formed by the blue solid and redsolid curves the synchronous mode is unstable for m = 2 (blue) and m = 3(red), respectively. In the region enclosed by the blue (red) dashed curve,the asynchronous mode is unstable for m = 2 (m = 3). Right panel: Hopfbifurcation boundaries for the synchronous mode with m = 1 shown in alarger region of the τ versus D0 plane. Parameters used are µ = 2, α = 0.9,0 = 0.15, d1 = 0.5, d2 = 0.2 and |Ω| = 10. With these choices of µ, αand 0, the uncoupled system has a stable steady-state for the intracellularkinetics.Next, we show the region where oscillatory instabilities can occur in theτ versus D0 parameter plane for the synchronous and asynchronous modes.We fix the Sel’kov parameter values as µ = 2, α = 0.9, and 0 = 0.15, so thatthe uncoupled intracelluar kinetics has a stable steady-state. We then taked1 = 0.5, d2 = 0.2, and |Ω| = 10. For this parameter set, we solve the Hopfbifurcation conditions (6.4.12) by a root finder. In this way, in the left panel1786.4. Examples of the Theory: Finite Domain With D = O(ν−1)of Fig. 6.3 we plot the Hopf bifurcation boundaries for the synchronous modein the τ versus D0 plane for m = 1, 2, 3. Similarly, upon using (6.4.16), inthe left panel of Fig. 6.3 we also plot the Hopf bifurcation boundaries for theasynchronous mode. In the right panel of Fig. 6.3, where we plot in a largerregion of the τ versus D plane, we show that the instability lobe for them = 1 case is indeed closed. We observe for m = 2 and m = 3 that, for thisparameter set, the lobes of instability of the asynchronous mode are almostentirely contained inside the instability lobes for the synchronous mode.Finally, we consider the effect of changing d1 and d2 to d1 = 0.1 andd2 = 0.2, while fixing the Sel’kov parameters as µ = 2, α = 0.9, and0 = 0.15, and keeping |Ω| = 10. In Fig. 6.4 we plot the Hopf bifurcationcurve for the synchronous mode when m = 1, computed using (6.4.12), inthe τ versus D0 plane. We observe that there is no longer any closed lobeof instability. In this figure we also show the two Hopf bifurcation values,derived below in §6.5, that corresponds to taking the limit D0  1. Theselatter values are Hopf bifurcation points associated with the linearization ofthe ODE system (6.5.16) around its steady-state value. This ODE system(6.5.16), which will be derived in the next section, describes large-scale cell-bulk dynamics in the regime D  O(ν).Local Kinetics Described by a Fitzhugh-Nagumo Type SystemNext, we will consider another form for the local kinetics, taken from [23],that is of Fitzhugh-Nagumo (FN) type. The functions F (v, w) and G(v, w),have the formF (v, w) = 0(wz − v) , G(v, w) = w − q(w − 2)3 + 4− v , (6.4.23)where the parameters satisfy 0 > 0, q > 0, and z > 1.Upon substituting (6.4.23) into (6.4.7) we calculate that the steady-statesolution we > 0 is given by the unique real positive root of the cubic C(w) = 0given byC(w) ≡ qw3 − 6qw2 + w (12q − 1 + Λ)− (8q + 4) , (6.4.24a)where Λ is defined asΛ ≡ 0z[0 + 2piD0β/τ] , (6.4.24b)and β is defined in (6.4.18b). The uniqueness of the positive root of thiscubic for any Λ > 0 was proved previously in our 1-D analysis of membrane-bulk coupling with FN membrane dynamics. In terms of the solution we > 01796.4. Examples of the Theory: Finite Domain With D = O(ν−1)0 1 2051015D0τFigure 6.4: Hopf bifurcation boundaries for the synchronous mode for theSel’kov model with m = 1 in the τ versus D0 plane when d1 = 0.1 andd2 = 0.2. The other parameters are the same as in Fig. 6.3. Inside theregion bounded by the two black solid curves, which were computed using(6.4.12), the synchronous mode is unstable. The instability region is nolonger a closed lobe as in Fig. 6.3. The dashed lines represent the two Hopfbifurcation points that are obtained by a numerical path following usingXPPAUT [16] of the steady-states of the ODE system (6.5.16). We observethat as D0 increases, the Hopf bifurcation thresholds in τ gradually approachthat obtained by the large D approximation.1806.4. Examples of the Theory: Finite Domain With D = O(ν−1)to the cubic equation, we can calculate ve by ve = Λwe and the commonsource strength by S0c = −βΛwe.As needed below, we first calculate the partial derivatives of F and G atthe steady-state solution asF ev = −0 , F ew = 0z , Gev = −1 , Gew = 1− 3q(we − 2)2 ,(6.4.25a)which yieldsdet(J) = 0[z − 1 + 3q(we − 2)2]> 0 , tr(J) = −0 + 1− 3q(we − 2)2 .(6.4.25b)To determine conditions for which the synchronous oscillation mode hasa Hopf bifurcation we first substitute (6.4.25a) into (6.4.9b). Then, fromthe conditions in (6.4.12), to obtain purely imaginary roots of (6.4.9a), weneed to numerically compute the parameter regime where p1p2 = p3, withp1 > 0 and p3 > 0.Similarly, to study instabilities associated with the asynchronous oscil-latory mode we substitute (6.4.25a) into (6.4.14) to obtain that the Hopfbifurcation boundary is given byγ = τ[−0 + 1− 3q(we − 2)2], (6.4.26a)provided thatdet(J)− γτGew = −GevF ew−(Gew)2 = 0z−[1− 3q(we − 2)2]2> 0 , (6.4.26b)where we is the positive root of the cubic (6.4.24a). In a similar way as wasdone for the Sel’kov model in §6.4.2, the Hopf bifurcation boundary for theasynchronous mode in the d1 versus d2 parameter plane can be parametrizedas in (6.4.22a) where where a11, a12, a22, and a21, are now defined in termsof the parameter we > 0 bya11 ≡ 1 + 2pimD0|Ω| , a12 ≡ −1β(we), a21 ≡ 1 , a22 ≡ − 2piD0γ(we),(6.4.27a)whereβ(we) ≡ τ 02piD0(zΛ(we)− 1), with Λ(we) ≡ −q(we − 2)3we+ 1 +4we,γ(we) ≡ τ[−0 + 1− 3q(we − 2)2].(6.4.27b)1816.4. Examples of the Theory: Finite Domain With D = O(ν−1)0 0.1 0.2 0.30510152025d1m=10 0.1 0.2 0.30510152025d2m=20 0.1 0.2 0.30510152025m=3Figure 6.5: Hopf bifurcation boundaries for the synchronous (solid curve)and asynchronous (dashed curve) modes for the FN system (6.4.23) withvarious number m of cells in the d1 versus d2 parameter plane. Between thesolid lines the synchronous mode is unstable, whereas between the dashedlines the asynchronous mode is unstable. Notice that the region of instabilityfor the asynchronous mode is contained within the instability region for thesynchronous mode. Parameters used are z = 3.5, q = 5, 0 = 0.5, τ = 1,D0 = 1, and |Ω| = 10.By varying we > 0 and retaining only the portion of the curve for whichd1 > 0 and d2 > 0, and ensuring that the constraint (6.4.26b) holds, weobtain a parametric form for the Hopf bifurcation boundary for the asyn-chronous mode in the d1 versus d2 parameter plane. For m = 2 and m = 3,these are the dashed curves shown in Fig. 6.5.In this way, in Fig. 6.5 we plot the Hopf bifurcation boundaries for thesynchronous mode (solid curves) and asynchronous mode (dashed curves)for various values of m for the parameter set z = 3.5, q = 5, 0 = 0.5, τ = 1,D0 = 1, and |Ω| = 10. As compared to Fig. 6.2, we notice that the unstableregions for both modes are not only shrinking but also the boundaries shiftas the number m of cells increases. This feature does not appear in theprevious Sel’kov model.Next, in Fig. 6.6 we show the region of oscillatory instabilities for thesynchronous and asynchronous modes for m = 1, 2, 3 in the τ versus D0plane. These Hopf bifurcation boundaries are computed by finding roots of(6.4.12) for the synchronous mode or (6.4.26) for the asynchronous modefor various values of D0. The other parameter values are the same as thoseused for Fig. 6.5 except d1 = 10 and d2 = 0.2. Inside the region bounded bythe solid curves the synchronous mode is unstable, while inside the regionbounded by the dashed curves, the asynchronous mode is unstable. Similar1826.5. Finite Domain: Reduction to ODEs for D  O(ν−1)to that shown in Fig. 6.5, the regions of instability are shrinking, at thesame time as the Hopf bifurcation boundaries shift, as m increases. Forthese parameter values of d1 and d2, the Hopf bifurcation still occurs forlarge value of D0.0 2 4 6 80123τm=10 2 4 6 80123D0m=20 2 4 6 80123m=3Figure 6.6: Hopf bifurcation boundaries for the synchronous (solid curves)and asynchronous (dashed curves) modes for the FN system (6.4.23) withvarious number m of cells in the τ versus D0 parameter plane. Between thesolid lines the synchronous mode is unstable, whereas between the dashedlines the asynchronous mode is unstable. Parameters used are z = 3.5,q = 5, 0 = 0.5, d1 = 10, d2 = 0.2, and |Ω| = 10.6.5 Finite Domain: Reduction to ODEs forD  O(ν−1)In this section we will consider our basic model with one small cell circularΩ centered in a bounded two-dimensional domain Ω under the assumptionthat the diffusion coefficient D satisfies D  O(ν−1), with ν = −1/ log . Inthis limit, in which the bulk region is effectively well-mixed, we show thatwe can reduce the dynamics in the coupled cell-bulk model to a system ofnonlinear ODEs for the time history of the bulk and cell concentrations.For the case of one cell, the basic model is formulated asτUt = D∆U − U , x ∈ Ω\Ω ; ∂nU = 0 , x ∈ ∂Ω ,D∂nU = d1U − d2u1 , x ∈ ∂Ω ,(6.5.1a)which is coupled to the cell dynamics bydudt= F (u) +1τ∫∂Ω(d1U − d2u1) ds e1 , (6.5.1b)1836.5. Finite Domain: Reduction to ODEs for D  O(ν−1)where e1 ≡ (1, 0, . . . , 0)T . Here u = (u1, . . . , un)T represents the concen-tration of the n species inside the cell with intracellular reaction kineticsF (u).We will assume that D  1, and in the bulk region expandU = U0 +1DU1 + · · · . (6.5.2)Upon substituting this expansion into (6.5.1a), and noting that Ω → x0 as → 0, we obtain to leading order in 1/D that ∆U0 = 0 with ∂nU0 = 0 on∂Ω. As such, we have U0 = U0(t). At next order, we have that U1 satisfies∆U1 = U0 + τU0t, x ∈ Ω\{x0} ; ∂nU1 = 0, x ∈ ∂Ω . (6.5.3)The formulation of this problem is complete once we determine a matchingcondition for U1 as x→ x0.To obtain this matching condition, we must consider the inner regiondefined at O() distances outside the cell. In this inner region we introducethe new variables y = −1(x − x0) and Uˆ(y, t) = U(x0 + y, t). From(6.5.1a), we obtain thatτUˆt =D2∆yUˆ − Uˆ , ρ = |y| ≥ 1 ; D∂Uˆ∂ρ= d1Uˆ − d2u1 , on ρ = 1 .For D  1, we then seek a radially symmetric solution to this inner problemin the formUˆ(ρ, t) = Uˆ0(ρ, t) +1DUˆ1(ρ, t) + · · · . (6.5.4)To leading order we obtain that∆ρUˆ0 = 0, ρ ≥ 1 ; ∂Uˆ0∂ρ= 0, ρ = 1 ,subject to the matching condition to the bulk that Uˆ0 → U0 as ρ→∞. Weconclude that Uˆ0 = U0. At next order, the problem for Uˆ1 is that∆ρUˆ1 = 0, ρ ≥ 1 ; ∂Uˆ1∂ρ= d1U0 − d2u1, ρ = 1 . (6.5.5)Allowing for logarithmic growth at infinity, the solution to this problem isUˆ1 = (d1U0 − d2u1) log ρ+ C , (6.5.6)1846.5. Finite Domain: Reduction to ODEs for D  O(ν−1)where C is a constant to be found. Then, by writing (6.5.6) in outer vari-ables, and recalling (6.5.4), we obtain that the far field behavior of the innerexpansion isUˆ ∼ U0 + 1D[(d1U0 − d2u1) log |x− x0|+ 1ν(d1U0 − d2u1) + C]+ · · · .(6.5.7)From (6.5.7) we observe that the term proportional to 1/D is smallerthan the first term provided that D  O(ν−1). This specifies the asymptoticrange of D for which our analysis will hold. From asymptotic matching ofthe bulk and inner solutions, the far-field behavior of the inner solution(6.5.7) provides the required singular behavior as x→ x0 for the outer bulksolution. In this way, we find from (6.5.7) and (6.5.2) that U1 satisfies (6.5.3)subject to the singular behaviorU1 ∼ (d1U0 − d2u1) log |x− x0| , as x→ x0 . (6.5.8)Then, (6.5.3) together with (6.5.8) determines U1 uniquely. Finally, in termsof this solution, we identify that the constant C in (6.5.7) and (6.5.6) isobtained fromlimx→x0[U1 − (d1U0 − d2u1) log |x− x0|]= ν−1 (d1U0 − d2u1) + C . (6.5.9)We now carry out the details of the analysis. We can write the problem(6.5.3) and (6.5.8) for U1 as∆U1 = U0+τU0t+2pi (d1U0 − d2u1) δ(x−x0) , x ∈ Ω ; ∂nU1 = 0, x ∈ ∂Ω .(6.5.10)By the divergence theorem, this problem has a solution only if (U0 + τU0t) |Ω| =−2pi(d1U0 − d2u1). This leads to the following ODE for the leading-orderbulk solution U0(t):U ′0 = −1τ(1 +2pid1|Ω|)U0 +2pid2τ |Ω| u1 . (6.5.11)Without loss of generality we can impose that∫Ω U1 dx = 0 so that U0describes the spatial average of U . The solution to (6.5.10) is then simplyU1 = −2pi (d1U0 − d2u1)G0(x;x0) , (6.5.12)whereG0(x;x0) is the Neumann Green’s function defined uniquely by (6.3.8).We then expand (6.5.12) as x→ x0, and use (6.5.9) to identify C asC = − (d1U0 − d2u1)(ν−1 + 2piR0), (6.5.13)1856.5. Finite Domain: Reduction to ODEs for D  O(ν−1)where R0 is the regular part of the Neumann Green’s function defined in(6.3.8).In summary, by using (6.5.4), (6.5.6), and (6.5.13), the two-term innerexpansion near the cell is given byUˆ ∼ U0 + 1D(d1U0 − d2u1)(log ρ− 1ν− 2piR0)+ · · · . (6.5.14)From (6.5.2) and (6.5.12), we obtain the corresponding two-term expansionof the outer bulk solutionU ∼ U0 − 2piD(d1U0 − d2u1)G0(x;x0) , (6.5.15)where U0(t) satisfies the ODE (6.5.11).The final step in the analysis is to use (6.5.1b) to derive the dynamicsinside the cell. We readily calculate that1τ∫∂Ω(d1U − d2u1) ds ∼ 2piτ(d1U0 − d2u1) ,which determines the dynamics inside the cell from (6.5.1b).This leads to our main result that, for D  O(ν−1), the coupled PDEmodel (6.5.1) reduces to the study of the coupled (n+1) dimensional ODEsystem for the leading-order average bulk concentration U0(t) and cell dy-namics u given byU ′0 = −1τ(1 +2pid1|Ω|)U0 +2pid2τ |Ω| u1 , u′ = F (u) +2piτ[d1U0 − d2u1] e1 .(6.5.16)Before studying (6.5.16) for some specific reaction kinetics, we first ex-amine conditions for the existence of steady-state solutions for (6.5.16) andwe derive the spectral problem characterizing the linear stability of thesesteady-states.A steady-state solution U0e and ue of (6.5.16), if it exists, is a solutionto the nonlinear algebraic systemF (ue) +2piτ(d1U0e − d2u1e) e1 = 0 , rU0e = su1e , (6.5.17a)where r and s are defined byr ≡ 1 + 2pid1|Ω| , s ≡2pid2|Ω| . (6.5.17b)1866.5. Finite Domain: Reduction to ODEs for D  O(ν−1)To examine the linearized stability of such a steady-state, we substituteU0 = U0e + eλtη , u = ue + eλtφ .into (6.5.16) and linearize. This yields that η and φ satisfyλφ = Jφ+2piτ(d1η − d2φ1) e1 , τλη = −rη + sφ1 ,where J is the Jacobian of F evaluated at the steady-state u = ue. Uponsolving the second equation for η, and substituting the resulting expressioninto the first equation, we readily derive the homogeneous linear system[(λI − J)− µE1]φ = 0 , where µ ≡ 2piτ(d1sτλ+ r− d2), (6.5.18)where E1 is the rank-one matrix E1 = e1eT1 .By using the matrix determinant lemma we conclude that λ is an eigen-value of the linearization if and only if λ satisfies eT1 (λI − J)−1 e1 = 1/µ,where µ is defined in (6.5.18). From this expression, and by using d1s−d2r =−d2 as obtained from (6.5.17b), we conclude that λ must be a root ofQ(λ) = 0, whereQ(λ) ≡ − τ(r + τλ)2pid2 (1 + τλ)− M11det(λI − J) , (6.5.19)where r is defined in (6.5.17b). Here M11 is the cofactor of the element inthe first row and first column of λI − J .We use the argument principle to determine the number N of roots ofQ(λ) in Re(λ) > 0. We first observe that [argQ(λ)]ΓR→ 0 as R → ∞,where ΓR is the semi-circle λ = Reiθ with |θ| ≤ pi/2 and the square bracketsindicate the change in the argument over ΓR. Assuming that there are nozeroes on the imaginary axis, we readily obtain thatN =1pi[argQ(λ)]ΓI++ P , (6.5.20)where P is the number of zeroes of det(λI − J) in Re(λ) > 0, and ΓI+denotes the positive imaginary axis λ = iλI with λI > 0 traversed in thedownwards direction.Next, we show that (6.5.19), which characterizes the stability of a steady-state solution of the ODE dynamics (6.5.16), can also be derived by takingthe limit D0  1 in the stability results obtained in (6.3.19) of §6.3 for the1876.5. Finite Domain: Reduction to ODEs for D  O(ν−1)D = O(ν−1) regime where we set D = D0/ν. Recall from the analysis in §6.3for D = D0/ν, that when m = 1 only the synchronous mode can occur, andthat the linearized eigenvalue satisfies (6.3.20). By formally letting D0 →∞in (6.3.20) we readily recover (6.5.19).We now gives some examples of our stability theory.6.5.1 Large D Theory: Analysis of Reduced DynamicsWe first consider the case where there is exactly one dynamical species in thecell so that n = 1. From (6.5.17) with n = 1 we obtain that the steady-stateue is any solution ofF (ue)− 2pid2τ[1 +2pid1|Ω|]−1ue = 0 , U0e =2pid2|Ω|[1 +2pid1|Ω|]−1ue = 0 .(6.5.21)In the stability criterion (6.5.19) we set M11 = 1 and det(λI − J) = λ− F ′e,where F eu ≡ dF/du|u=ue , to conclude that the stability of this steady-stateis determined by the roots of the quadraticλ2 − λp1 + p2 = 0 , (6.5.22a)where p1 and p2 are defined byp1 = −1τ(1 +2pid1|Ω|)+ F eu −2pid2τ, p2 = −Feuτ(1 +2pid1|Ω|)+2pid2τ2.(6.5.22b)We now establish the following simple result based on (6.5.22).Proposition 6.5.1 Let n = 1. Then, no steady-state solution of (6.5.16)can undergo a Hopf bifurcation. Furthermore, ifF eu < Fth ≡2pid2τ[1 +2pid1|Ω|]−1, (6.5.23)then Re(λ) < 0, and so the steady-state is linearly stable. If F eu > Fth, thelinearization has exactly one positive eigenvalue.We first prove that no Hopf bifurcations are possible for the steady-state. From (6.5.22a) it is clear that there exists a Hopf bifurcation if andonly if p1 = 0 and p2 > 0 in (6.5.22b). Upon setting p1 = 0, we get1886.5. Finite Domain: Reduction to ODEs for D  O(ν−1)F eu = 2pid2τ−1 + τ−1(1 + 2pid1/|Ω|). Upon substituting this expression into(6.5.22b) for p2, we get thatp2 =1τ[−4pi2d22τ |Ω| −1τ(1 +2pid1|Ω|)(1 +2pid2|Ω|)]< 0 .Since p2 < 0 whenever p1 = 0, we conclude that no Hopf bifurcations forthe steady-state are possible.The second result follows by establishing that p1 < 0 and p2 > 0 whenF eu < Fth. From (6.5.22b) it follows that p1 < 0 and p2 > 0 when2pid2τ− F eu +1τ(1 +2pid1|Ω|)> 0 ,2pid2τ− F eu −2pid1|Ω| Feu > 0 . (6.5.24)These two inequalities hold simultaneously only when the second relation issatisfied. This yields that when (6.5.23) holds, we have Re(λ) < 0. Finally,when F eu > Fth, we have p2 < 0, and so there is a unique positive eigenvalue.This result qualitatively shows that the effect of cell-bulk coupling isthat the steady-state of the ODE dynamics (6.5.16) can be linearly stableeven when the reaction kinetics is self-activating in the sense that F eu > 0.Moreover, we observe that as τ increases, corresponding to the situationwhere the membrane kinetics has faster dynamics than the time scale forbulk decay, then the stability threshold Fth decreases. Therefore, for fastcell dynamics there is a smaller parameter range where self-activation ofthe intracelluar dynamics can occur while still maintaining stability of thesteady-state to the coupled system.Next, we consider the case n = 2, where F (u) = (F (u1, u2), G(u1, u2))T .We readily derive that any steady-state of the ODEs (6.5.16) must satisfyF (u1e, u2e)− 2pid2rτu1e = 0 , G(u1e, u2e) = 0 , U0e =sru1e , (6.5.25)where r and s are defined in (6.5.17b). We then observe that the roots ofQ(λ) = 0 in (6.5.19) are roots to a cubic polynomial in λ. Since M11 =λ−Geu2 , det(λI − J) = λ2 − tr(J)λ+ det J , wheretr(J) = F eu1 +Geu2 , det J = Feu1Geu2 − F eu2Geu1 , (6.5.26)and F ev , Gev are partial derivatives of F or G with respect to v ∈ (u1, u2)evaluated at the steady-state, we conclude that the linear stability of thesteady-state is characterized by the roots of the cubicλ3 + p1λ2 + p2λ+ p3 = 0 , (6.5.27a)1896.5. Finite Domain: Reduction to ODEs for D  O(ν−1)where p1, p2 and p3 are defined asp1 ≡ 2pid2τ+1τ(1 +2pid1|Ω|)− tr(J) ,p2 ≡ det J − 2pid2τGeu2 +1τ(2pid2τ−(1 +2pid1|Ω|)tr(J)),p3 ≡ 1τ((1 +2pid1|Ω|)det J − 2pid2τGeu2).(6.5.27b)By taking m = 1 and letting D0 → ∞ in (6.4.9b), it is readily verifiedthat the expressions above for pi, for i = 1, 2, 3, agree exactly with those in(6.4.9b). Then, by satisfying the Routh-Hurwitz conditions (6.4.12), we canplot the Hopf bifurcation boundaries in different parameter planes.1906.5. Finite Domain: Reduction to ODEs for D  O(ν−1)Example: One Cell With Sel’kov Dynamics0 0.05 0.1 0.1500.511.52d2d1D0 ≫ 1/νD0 = 5D0 = 1D0 = 5011.522.50.40.50.60.70.800.010.020.030.040.05u1u2U 0Figure 6.7: Left panel: Comparison of the Hopf bifurcation boundaries forthe synchronous mode for the Sel’kov model (6.4.17) in the d1 versus d2parameter plane with D0 = 1, 5, 50 (solid), as obtained from (6.4.9), andthe large D approximation (dashed), as obtained from (6.5.27). Betweenthe outer two black curves, the synchronous mode is unstable for D0 = 1,whereas in the region bounded by the solid/dashed curve the synchronousmode is unstable. We observe that as D0 increases, the Hopf boundariesobtained from (6.4.9) gradually approaches the one obtained from (6.5.27)from the large D approximation. Right panel: Numerical simulation for theODE system (6.5.16), showing sustained oscillations. In the left and rightpanels we fixed µ = 2, α = 0.9, 0 = 0.15, τ = 1, and |Ω| = 10, and in theright panel we took d1 = 0.8 and d2 = 0.05 corresponding to a point wherethe steady-state solution of the ODEs (6.5.16) is unstable.Next, we apply our theory for the D  O(ν−1) regime to the case where thelocal kinetics is described by the Sel’kov model, where the nonlinearities Fand G are given in (6.4.17). From (6.5.25) we obtain that the steady-statesolution of the ODE dynamics (6.5.16) under Sel’kov kinetics isue1 =rµ[r + 2pid2/τ] , ue2 = µα+ (ue1)2 , U0e = sµr + 2pid2 , (6.5.28)where r and s are defined in (6.5.17b). The partial derivatives of F and Gcan be calculated as in (6.4.19a).In the left panel of Fig. 6.7 we plot the Hopf bifurcation boundary inthe d1 versus d2 plane associated with linearizing the ODEs (6.5.16) aboutthis steady-state solution. In this figure we also plot the Hopf bifurcation1916.5. Finite Domain: Reduction to ODEs for D  O(ν−1)boundary for different values of D0, with D = D0/ν, as obtained from ourstability formulation (6.4.9) of §6.4 for the D = O(ν−1) regime. We observefrom this figure that the stability boundary with D0 = 50 closely approxi-mates that obtained from (6.5.27), which corresponds to the D0 →∞ limitof (6.4.9).We emphasize that since in the distinguished limit D  O(ν−1) we canapproximate the original coupled PDE system by the system (6.5.16) ofODEs, a numerical solution of the approximate system to show large-scaletime dynamics away from the steady-state becomes possible. In the rightpanel of Fig. 6.7, we plot the numerical solution to (6.5.16) with Sel’kovdynamics (6.4.17), where the initial condition is u1(0) = 0.01, u2(0) = 0.2and U0(0) = 0.5. We observe that by choosing d1 and d2 inside the regionbounded by the dashed curve in the left panel of Fig. 6.7, where the steady-state is unstable, the full ODE system (6.5.16) exhibits a stable periodicorbit, indicating a limit cycle behavior.0.020.030.040.05U 0340 360 380 4000.40.81.21.62tu1,u20 1 2 3 400.51u1u2Figure 6.8: Left: Plot of u1, u2 and U0 versus time showing sustained os-cillatory dynamics. Parameters used are µ = 2, α = 0.9, 0 = 0.15, τ = 1,|Ω| = 10, d1 = 0.8 and d2 = 0.05. Right: Plot of u1 versus u2 when the localkinetics is decoupled from the bulk. There is decaying oscillations towardsthe stable steady-state solution at u1 = µ and u2 = µ/(α+ u21). The initialcondition is u1(0) = 0.01 and u2(0) = 0.2. The parameter values of µ, 0and α are the same as that used for the left panel.In addition, in the left panel of Fig. 6.8 we plot the time evolution of u1,u2 and U0, showing clearly the sustained periodic oscillations. For compar-ison, fixing the same parameter set for the Sel’kov kinetics (6.4.17), in theright panel of Fig. 6.8 we plot the phase plane of u2 versus u1 when there isno coupling between the local kinetics and the bulk. We now observe that1926.5. Finite Domain: Reduction to ODEs for D  O(ν−1)without this cell-bulk coupling the Sel’kov model (6.4.17) exhibits transientdecaying oscillatory dynamics, with a spiral behavior in the phase-planetowards the linearly stable steady-state.0.02 0.04 0.06 0.08 0.1 0.1211.21.41.61.822.2d2u1Figure 6.9: Global bifurcation diagram of u1 with respect to d2 at a fixedd1 = 0.8 as computed using XPPAUT [16] from the ODE system (6.5.16)for the Sel’kov kinetics (6.4.17). The thick/thin solid line represents sta-ble/unstable steady-state solutions of u1, while the solid dots indicate sta-ble synchronous periodic solution. The parameters used are µ = 2, α = 0.9,0 = 0.15, τ = 1, and |Ω| = 10.Finally, we use the numerical bifurcation software XPPAUT [16] to con-firm the existence of a stable large amplitude periodic solution to (6.5.16)with Sel’kov kinetics when d1 and d2 are in the unstable region of the leftpanel of Fig. 6.7. In Fig. 6.9 we plot a global bifurcation diagram of u1versus d2 for d1 = 0.8, corresponding to taking a horizontal slice at fixedd2 = 0.8 through the stability boundaries in the d2 versus d1 plane shownin Fig. 6.7. The two computed Hopf bifurcation points at d2 ≈ 0.0398 andd1 ≈ 0.1098 agree with the theoretically predicted values in Fig. 6.7.Example: One Cell With Fitzhugh-Nagumo DynamicsFinally, we apply our large D theory to the case where the intracellulardynamics is governed by the FN kinetics (6.4.23). From (6.5.25) we obtainthat the steady-state solution of the ODEs (6.5.16) with the kinetics (6.4.23)isue1 = Λue2 , U0e =sue1r, where Λ ≡ 0zr[0r + 2pid2/τ] . (6.5.29)1936.5. Finite Domain: Reduction to ODEs for D  O(ν−1)Here r and s are defined in (6.5.17b), and u2e > 0 is the unique root of thecubic (6.4.24a) where Λ in (6.4.24a) is now defined in (6.5.29). The partialderivatives of F and G can be calculated as in (6.4.25).0 0.1 0.2 0.301020304050D0 ≫ 1/νD0 = 50D0 = 5D0 = 1d2d15.866.26.422.22.40.050.060.070.08u1u2U 0Figure 6.10: Left panel: Comparison of the Hopf bifurcation boundaries forthe synchronous mode with FN kinetics (6.4.23) in the d1 versus d2 parame-ter plane with D0 = 1, 5, 50 (solid), as obtained from (6.4.9), and the largeD approximation (dashed), as obtained from (6.5.27). In the wedge-shapedregions bounded by the solid curves the synchronous mode is unstable forthe finite values of D0. As D0 increases, the Hopf boundaries obtainedfrom (6.4.9) becomes rather close to the dashed one obtained from (6.5.27)from the large D approximation. Right panel: Numerical simulation for theODE system (6.5.16), showing sustained oscillations, with initial conditionsu1(0) = 6.0, u2(0) = 2.3, and U0(0) = 0.05. In the left and right panelswe fixed z = 3.5, q = 5, 0 = 0.5, τ = 1, and |Ω| = 10, and in the rightpanel we took d1 = 15.6 and d2 = 0.2 corresponding to a point where thesteady-state solution of the ODEs (6.5.16) is unstable.In the left panel of Fig. 6.10 we plot by the dashed curve the Hopfbifurcation boundary in the d1 versus d2 plane associated with linearizingthe ODEs (6.5.16) about this steady-state solution. This curve was obtainedby setting p1p2 = p3 with p1 > 0 and p3 > 0 in (6.5.27). In this figure theHopf bifurcation boundaries for different values of D0, with D = D0/ν, arealso shown. These latter curves were obtained from our stability formulation(6.4.9) of §6.4. Similar to what we found for the Sel’kov model, the stabilityboundary for D0 = 50 is very close to that for the infinite D result obtainedfrom (6.5.27). In the right panel of Fig. 6.10 we plot the numerical solutionto (6.5.16) with FN dynamics (6.4.23) for the parameter set d1 = 15.6and d2 = 0.2, which is inside the unstable region bounded by the dashed1946.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimecurves in the left panel of Fig. 6.10. With the initial condition u1(0) = 6.0,u2(0) = 2.3 and U0(0) = 0.05, the numerical computations of the full ODEsystem (6.5.16) reveal a sustained and stable periodic solution.Finally, we use XPPAUT [16] on (6.5.16) to compute a global bifurcationof u1 versus d1 for fixed d2 = 0.2 for FN kinetics. This plot corresponds totaking a vertical slice at fixed d2 = 0.2 through the stability boundaries inthe d2 versus d1 plane shown in Fig. 6.10. The two computed Hopf bifur-cation points at d1 ≈ 15.389 and d1 ≈ 42.842 agree with the theoreticallypredicted values in Fig. 6.10. These results confirm the existence of a stableperiodic solution branch induced by the cell-bulk coupling.10 20 30 4055.566.57d1u1Figure 6.11: Global bifurcation diagram of u1 versus d1 at a fixed d2 = 0.2,as computed using XPPAUT [16] from the ODE system (6.5.16) for theFN kinetics (6.4.23). The thick/thin solid line represents stable/unstablesteady-state solutions of u1, while the solid dots indicate a stable periodicsolution. The other parameter values are z = 3.5, q = 5, 0 = 0.5, τ = 1,and |Ω| = 10.6.6 The Effect of the Spatial Configuration of theSmall Cells: The D = O(1) RegimeIn this section we construct steady-state solutions and study their linearstability properties in the D = O(1) regime, where both the number ofcells and their spatial distribution in the domain are important factors. Forsimplicity, we consider a special spatial configuration of the cells inside theunit disk Ω for which the Green’s matrix G has a cyclic structure. Morespecifically, on a ring of radius r0, with 0 < r0 < 1, we place m equally-1956.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimespaced cells whose centers are atxj = r0(cos(2pijm), sin(2pijm))T, j = 1, . . . ,m . (6.6.1)This ring of cells is concentric with respect to the unit disk Ω ≡ {x | |x| ≤ 1 }.A schematic diagram for this system with m = 5 is shown in Fig. 6.12.r0ΩFigure 6.12: Schematic diagram showing five cells evenly spaced on 2D ringwith radius r0 = 0.3 inside the unit disk. The red dots denotes signalingcompartments or cellsWe also assume that the intracellular kinetics is the same within each cell,so that F j = F for j = 1, . . . ,m. A related type of analysis characterizingthe stability of localized spot solutions for the Gray-Scott RD model, wherelocalized spots are equally-spaced on a ring concentric with the unit disk,was performed in [9].For the unit disk, the Green’s function G(x; ξ) satisfying (6.2.10) can bewritten as an infinite sum involving the modified Bessel functions of the firstand second kind In(z) and Kn(z), respectively, in the form (see AppendixA.1 of [9])G(x; ξ) =12piK0(θ0|x− ξ|)− 12pi∞∑n=0σn cos(n(ψ − ψ0)) K ′n(θ0)I ′n(θ0)In (θ0r) In (θ0r0) ;σ0 = 1 , σn = 2 , n ≥ 1 .(6.6.2)1966.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) RegimeHere θ0 ≡ D−1/2, x = reiψ, ξ = r0eiψ0 , and|x− ξ| =√r2 + r20 − 2rr0 cos(ψ − ψ0).By using the local behavior K0(z) ∼ − log z + log 2 − γe + o(1) as z → 0+,where γe is Euler’s constant, we can extract the regular part R of G(x; ξ)as x→ ξ, as identified in (6.2.10b), asR =12pi(log 2− γe + 12logD)− 12pi∞∑n=0σnK ′n(θ0)I ′n(θ0)[In (θ0r0)]2. (6.6.3)For this spatial configuration of cells, the Green’s matrix G is obtainedby a cyclic permutation of its first row vector a ≡ (a1, . . . , am)T , which isdefined term-wise bya1 ≡ R ; aj = Gj1 ≡ G(xj ;x1) , j = 2, . . . ,m . (6.6.4)We can numerically evaluate Gj1 for j = 2, . . . ,m and R by using (6.6.2)and (6.6.3), respectively. Since G is a cyclic matrix, it has an eigenpair,corresponding to a synchronous perturbation, given byGe = ω1e ; e ≡ (1, . . . , 1)T , ω1 ≡m∑j=1aj = R+m∑j=1Gji . (6.6.5)When D = O(1), the steady-state solution is determined by the solutionto the nonlinear algebraic system (6.2.7) and (6.2.13). Since F j = F forj = 1, . . . ,m, and e is an eigenvector of G with eigenvalue ω1, we can look fora solution to (6.2.7) and (6.2.13) having a common source strength, so thatS = Sce, uj = u for all j = 1, . . . ,m, and u1 = u1e. In this way, we obtainfrom (6.2.7) and (6.2.13), that the steady-state problem is to solve the n+ 1dimensional nonlinear algebraic system for Sc and u = (u1, u2, . . . , un)Tgiven byF (u) +2piDτSce = 0 ; Sc = −βu1 , β ≡ d2νd1 + 2piνd1ω1 +Dν,(6.6.6)where ν ≡ −1/ log  and ω1 is defined in (6.6.5). We remark that ω1 dependson D, r0, and m.To study the linear stability of this steady-state solution, we write theGCEP, given in (6.2.32), in the formGλc = − 12piν[1 +Dνd1+2piνd2Dd1τM11det(λI − J)]c , (6.6.7)1976.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimewhere J is the Jacobian of F evaluated at the steady-state. In terms of thematrix spectrum of Gλ, written asGλvj = ωλ,jvj , j = 1, . . . ,m , (6.6.8)we conclude from (6.6.7) that the set of discrete eigenvalues λ of the lin-earization of the steady-state are the union of the roots of the m transcen-dental equations, written as Fj(λ) = 0, whereFj(λ) ≡ ωλ,j + 12piν(1 +Dνd1)+(d2Dd1τ)M11det(λI − J) , j = 1, . . . ,m .(6.6.9)Any such root of Fj(λ) = 0 with Re(λ) > 0 leads to an instability ofthe steady-state solution on an O(1) time-scale. If all such roots satisfyRe(λ) < 0, then the steady-state is linearly stable on an O(1) time-scale.To study the stability properties of the steady-state using (6.6.9), andidentify any possible Hopf bifurcation values, we must first calculate thespectrum (6.6.8) of the cyclic and symmetric matrix Gλ, whose entries aredetermined by the λ-dependent reduced-wave Green’s function Gλ(x; ξ),with regular part Rλ(ξ), as defined by (6.2.23). Since Gλ is not a Hermitianmatrix when λ is complex, its eigenvalues ωλ,j are in general complex-valuedwhen λ is complex. Then, by replacing θ0 in (6.6.2) and (6.6.3) with θλ ≡√(1 + τλ)/D, we readily obtain thatGλ(x; ξ) =12piK0(θλ|x− ξ|)− 12pi∞∑n=0σn cos(n(ψ − ψ0)) K ′n(θλ)I ′n(θλ)In (θλr) In (θλr0) ;σ0 = 1 , σn = 2 , n ≥ 1 ,(6.6.10)with regular partRλ =12pi[log 2− γe + 12logD − 12log(1 + τλ)]− 12pi∞∑n=0σnK ′n(θλ)I ′n(θλ)[In (θλr0)]2,(6.6.11)where we have specified the principal branch for θλ. The Green’s matrix Gλis obtained by a cyclic permutation of its first row aλ ≡ (aλ,1, . . . , aλ,m)T ,which is defined term-wise byaλ,1 ≡ Rλ ; aλ,j = Gλ,j1 ≡ Gλ(xj ;x1) , j = 2, . . . ,m . (6.6.12)1986.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) RegimeAgain we can numerically evaluate Gλ,j1 for j = 2, . . . ,m and Rλ by using(6.6.10) and (6.6.11), respectively.Next, we must determine the full spectrum (6.6.8) of the cyclic andsymmetric matrix Gλ. For the m×m cyclic matrix Gλ, generated by permu-tations of the row vector aλ, it is well-known that its eigenvectors vj andeigenvalues ωλ,j areωλ,j =m−1∑n=0aλ,n+1e2pii(j−1)n/m , vj = (1, e2pii(j−1)/m , . . . , e2pii(j−1)(m−1)/m)T ,j = 1, . . . ,m .(6.6.13)Since G is also necessarily a symmetric matrix it follows thataλ,j = aλ,m+2−j , j = 2, . . . , dm/2e , (6.6.14)where the ceiling function dxe is defined as the smallest integer not less thanx. This relation can be used to simplify the expression (6.6.13) for ωλ,j , intothe form as written below in (6.6.16). Moreover, as a result of (6.6.14), itreadily follows thatωλ,j = ωλ,m+2−j , for j = 2, . . . , dm/2e , (6.6.15)so that there are dm/2e−1 eigenvalues of multiplicity two. For these multipleeigenvalues the two independent real-valued eigenfunctions are readily seento be Re(vj) = (vj + vm+2−j)/2 and Im(vj) = (vj − vm+2−j)/(2i). Inaddition to ω1, we also observe that there is an additional eigenvalue ofmultiplicity one when m is even.In this way, our result for the matrix spectrum of Gλ is as follows: Thesynchronous eigenpair of Gλ isωλ,1 =m∑n=1aλ,n , v1 = (1, . . . , 1)T , (6.6.16a)while the other eigenvalues, corresponding to the asynchronous modes, areωλ,j =m−1∑n=0cos(2pi(j − 1)nm)aλ,n+1 , j = 2, . . . ,m , (6.6.16b)where ωλ,j = ωλ,m+2−j for j = 2, . . . , dm/2e. When m is even, we no-tice that there is an eigenvalue of multiplicity one given by ωλ,m2+1 =1996.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regime∑m−1n=0 (−1)nan+1. The corresponding eigenvectors for j = 2, . . . , dm/2e canbe written asvj =(1, cos(2pi(j − 1)m), . . . , cos(2pi(j − 1)(m− 1)m))T,vm+2−j =(0, sin(2pi(j − 1)m), . . . , sin(2pi(j − 1)(m− 1)m))T.(6.6.16c)Finally, when m is even, there is an additional eigenvector given by vm2+1 =(1,−1, . . . ,−1)T .With the eigenvalues ωλ,j , for j = 1, . . . ,m, determined in this way, anyHopf bifurcation boundary in parameter space is obtained by substitutingλ = iλI with λI > 0 into (6.6.9), and requiring that the real and imaginaryparts of the resulting expression vanish. This yields, for each j = 1, . . . ,m,thatRe(ωλ,j)+12piν(1 +Dνd1)+d2Dd1τRe(M11det(λI − J))= 0 ,Im(ωλ,j)+d2Dd1τIm(M11det(λI − J))= 0 .(6.6.17)Finally, we can use the winding number criterion of complex analysis on(6.6.9) to count the number of eigenvalues of the linearization when the pa-rameters are off any Hopf bifurcation boundary. This criterion is formulatedbelow in §6.6.1.We remark that in the limit D  1, we can use K0(z) ∼ − log z to-gether with I0(z) ∼ 1 + z2/4 as z → 0, to estimate from the n = 0term in (6.6.10) and (6.6.11) that −(2pi)−1K ′0(θλ)/I ′0(θλ) ∼ D/[pi(1 + τλ)]as D → ∞. Therefore, for D → ∞, the Green’s matrix Gλ satisfiesGλ → DmE/[pi(1 + τλ)], where E = eeT /m and e ≡ (1, . . . , 1)T . Thisyields for D  1 that ω1 = Dm/[pi(1 + τλ)]and ωj = O(1) for j = 2, . . . , n.By substituting these expressions into (6.6.17), we can readily recover thespectral problems (6.3.20) and (6.3.21), considered in §6.3, associated withthe regime D = O(ν−1). Therefore, (6.6.17) provides a smooth transitionto the leading-order spectral problems considered in §6.3 for D = O(ν−1).6.6.1 Example: The Sel’kov ModelWe now use (6.6.17) to compute phase diagrams in the τ versus D parameterspace associated with m equally-spaced cells of radius  on a ring of radius2006.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimer0, with 0 < r0 < 1, concentric within the unit disk. For the intracellulardynamics we let n = 2, so that u = (u1, u2)T , and we consider the Sel’kovdynamics F = (F (u1, u2), G(u1, u2))T as given in (6.4.17). For this choice,(6.6.6) yields the steady-state solution (u1e, u2e)T for the coupled cell-bulksystem given byu1e =µ1 + 2piDβ/τ, u2e =µα+ u21e, (6.6.18a)where β is defined in (6.6.6). Upon using (6.4.19) we calculate thatdet(J) = 0(α+ u21e)> 0 ,tr(J) =1α+ u21e[2u1eµ− (α+ u21e)− 0(α+ u21e)2].(6.6.18b)In this subsection we fix the Sel’kov parameters µ, α, and 0, the permeabil-ities d1 and d2, and the cell radius  asµ = 2 , α = 0.9 , 0 = 0.15, d1 = 0.8 , d2 = 0.2 ,  = 0.05 .(6.6.19)With these values for µ, α, and 0, the intracellular dynamics has a stablesteady-state when uncoupled from the bulk.Then, to determine the Hopf bifurcation boundary for the coupled cell-bulk model we set M11 = λ − Geu2 in (6.6.17), and use Geu2 = −det(J) asobtained from (6.4.19). By letting λ = iλI in the resulting expression, weconclude that any Hopf bifurcation boundary, for each mode j = 1, . . . ,m,must satisfyRe(ωλ,j)+12piν(1 +Dνd1)−(d2Dd1τ) [λ2Itr(J) + det(J) (λ2I − det(J))][(det(J)− λ2I)2+(λItr(J))2] = 0 ,Im(ωλ,j)+(d2Dd1τ) [λI (det(J)− λ2I)+ det(J)tr(J)λI][(det(J)− λ2I)2+(λItr(J))2] = 0 .(6.6.20)For a specified value of D, we view (6.6.20) as a coupled system for the Hopfbifurcation value of τ and the corresponding eigenvalue λI , which we solveby Newton’s method.For parameter values off of any Hopf bifurcation boundary, we can usethe winding number criterion on Fj(λ) in (6.6.9) to count the number of2016.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimeunstable eigenvalues Nj of the linearization for the j-th mode. By using theargument principle, we obtain that the number Nj of roots of Fj(λ) = 0 inRe(λ) > 0 isNj =12pi[argFj ]Γ + P , (6.6.21)where P is the number of poles of Fj(λ) in Re(λ) > 0, and the square bracketdenotes the change in the argument of Fj over the contour Γ. Here the closedcontour Γ is the limit as R → ∞ of the union of the imaginary axis, whichcan be decomposed as ΓI+ = iλI and ΓI− = −iλI , for 0 < λI < R, and thesemi-circle ΓR defined by |λ| = R with |arg(λ)| ≤ pi/2. Since ωλj is analyticin Re(λ) > 0, it follows that P is determined by the number of roots ofdet(λI − J) = 0 in Re(λ) > 0. Since det(J) > 0, as shown in (6.6.18b),we have that P = 2 if tr(J) > 0 and P = 0 if tr(J) < 0. Next, we letR → ∞ on ΓR and calculate [argFj ]ΓR . It is readily seen that the Green’smatrix Gλ tends to a multiple of a diagonal matrix on ΓR as R  1, of theform Gλ → Rλ,∞I, where Rλ,∞ is the regular part of the free-space Green’sfunction Gf (x;x0) = (2pi)−1K0(θλ|x− x0|)at x = x0, given explicitly bythe first term in the expression (6.6.11) for Rλ. Since ωλ,j → Rλ,∞ forj = 1, . . . ,m, we estimate on ΓR as R  1 thatFj(λ) ∼ − 12pilog√1 + τλ+ c0 +O(1/λ),for some constant c0. It follows that Fj(λ) ∼ O(lnR) − i/8 as R → ∞,so that limR→∞[argFj ]ΓR = 0. Finally, since [argFj ]ΓI+ = [argFj ]ΓI− , asa consequence of Fj being real-valued when λ is real, we conclude from(6.6.21) thatNj =12pi[argFj ]ΓI+ + P , P = 2 if trJ > 00 if trJ < 0 . (6.6.22)By using (6.6.20) for the real and imaginary parts of Fj , [argFj ]ΓI+ is easilycalculated numerically by a line sweep over 0 < λI < R. Then, by using(6.6.18b) to calculate tr(J), P is readily determined. In this way, (6.6.22)leads to a highly tractable numerical procedure to calculate Nj . This crite-rion was used for all the results below to identify regions in parameter spacewhere instabilities occur away from any Hopf bifurcation boundary.In Fig. 6.13 we plot the Hopf bifurcation boundaries when m = 2 andr0 = 0.25. From the left panel of this figure, the synchronous mode is un-stable in the larger lobe shaped region, whereas the asynchronous mode is2026.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regime0 2 4 600.20.40.60.81Dτ0 1 2 3 4 5 6 700.20.40.60.81DτFigure 6.13: Hopf bifurcation boundaries in the τ versus D plane for m = 2,r0 = 0.25, and with parameters as in (6.6.19), computed from (6.6.20). Leftpanel: the heavy solid curve and the solid curve are the Hopf bifurcationboundaries for the synchronous and asynchronous modes, respectively. In-side the respective lobes the corresponding mode is linearly unstable, asverified by the winding number criterion (6.6.22). Right panel: same plotexcept that we include the Hopf bifurcation boundary for the synchronousmode from the leading-order D = D0/ν  1 theory, computed from (6.3.20).2036.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimeunstable only in the small lobe for small D, which is contained within theinstability lobe for the synchronous mode. In the right panel of Fig. 6.13 weshow the Hopf bifurcation boundary for the synchronous mode, as obtainedfrom (6.3.20), corresponding to the leading-order D = D0/ν  1 theory.Since the instability lobe occurs for only moderate values of D, and  = 0.05is only moderately small, the leading-order theory from the D = D0/νregime is, as expected, not particularly accurate in determining the Hopf bi-furcation boundary. The fact that we have stability at a fixed D for τ  1,which corresponds to very fast intracellular dynamics, is expected since inthis limit the intracellular dynamics becomes decoupled from the bulk diffu-sion. Alternatively, if τ  1, then for a fixed D, the intracellular reactionsproceed too slowly to create any instability. Moreover, in contrast to thelarge region of instability for the synchronous mode as seen in Fig. 6.13, weobserve that the lobe of instability for the asynchronous mode only occursfor small values of D, where the diffusive coupling, and communication, be-tween the two cells is rather weak. Somewhat more paradoxically, we alsoobserve that the synchronous lobe of instability is bounded in D. This issueis discussed in more detail below.In Fig. 6.14 we show the effect of changing the ring radius r0 on theHopf bifurcation boundaries. By varying r0, we effectively are modulatingthe distance between the two cells. From this figure we observe that as r0is decreased, the lobe of instability for the asynchronous mode decreases,implying, rather intuitively, that at closer distances the two cells are betterable to synchronize their oscillations than when they are farther apart. Weremark that results from the leading-order theory of §6.3 for the D = O(ν−1)regime would be independent of r0. We further observe from this figure thata synchronous instability can be triggered from a more clustered spatialarrangement of the cells inside the domain. In particular, for D = 5 andτ = 0.6, we observe from Fig. 6.14 that we are outside the lobe of instabilityfor r0 = 0.5, but inside the lobe of instability for r0 = 0.25 and r0 = 0.75.We remark that due to the Neumann boundary conditions the cells on thering with r0 = 0.75 are close to two image cells outside the unit disk, whichleads to a qualitatively similar clustering effect of these near-boundary cellsas when they are on the small ring of radius r0 = 0.25.In Fig. 6.15 we plot the Hopf bifurcation boundaries when m = 3 andr0 = 0.5. For m = 3, we now observe that the region where the synchronousmode is unstable is unbounded in D. The lobe of instability for the asyn-chronous mode still exists only for small D, as shown in the right panel ofFig. 6.15. In this case, we observe that the Hopf bifurcation boundary forthe synchronous mode, corresponding to the leading-order D = D0/ν  12046.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regime0 2 4 600.20.40.60.81DτFigure 6.14: Hopf bifurcation boundaries for the synchronous mode (largerlobes) and the asynchronous mode (smaller lobes) in the τ versus D plane form = 2 and for three values of r0, with r0 = 0.5 (heavy solid curves), r0 = 0.75(solid curves), and r0 = 0.25 (dashed curves). The other parameters aregiven in (6.6.19). We observe that as r0 decreases, where the two cellsbecome more closely spaced, the lobe of instability for the asynchronousmode decreases.theory and computed from (6.3.20), now agrees rather well with resultscomputed from (6.6.20).In the left panel of Fig. 6.16 we plot the Hopf bifurcation boundaries forthe synchronous mode for m = 5 when r0 = 0.5 (heavy solid curves) andfor r0 = 0.25 (solid curves). We observe that for moderate values of D, theHopf bifurcation values do depend significantly on the radius of the ring. Thesynchronous mode is unstable only in the infinite strip-like domain betweenthese Hopf bifurcation boundaries. Therefore, only in some intermediaterange of τ , representing the ratio of the rates of the intracellular reactionand bulk decay, is the synchronous mode unstable. As expected, the twocurves for different values of r0 coalesce as D increases, owing to the factthat the leading-order stability theory for D = D0/ν  1, as obtained from(6.3.20), is independent of r0. In the right panel of Fig. 6.16 we compare theHopf bifurcation boundaries for the synchronous mode with r0 = 0.5 withthat obtained from (6.3.20), corresponding to the leading-order theory in theD = D0/ν  1 regime. Rather curiously, we observe upon comparing thesolid curves in the left and right panels in Fig. 6.16 that the Hopf bifurcationboundaries from the D = O(1) theory when r0 = 0.25, where the five cellsare rather clustered near the origin, agree very closely with the leading2056.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regime0 1 2 3 4 500.20.40.60.81Dτ0 0.5 100.20.40.60.81DτFigure 6.15: Left panel: Hopf bifurcation boundaries in the τ versus D planefor the synchronous mode for m = 3 equally-spaced cells on a ring of radiusr0 = 0.50 (heavy solid curves), as computed from (6.6.20), with parametersas in (6.6.19). The dashed curve is the Hopf bifurcation boundary from theleading-order D = D0/ν theory computed from (6.3.20). Right panel: TheHopf bifurcation boundaries for the asynchronous mode (solid curve) andthe synchronous mode (heavy solid curve) shown in a magnified region ofD. The asynchronous mode is linearly unstable only inside this small lobe,which lies within the unstable region for the synchronous mode.2066.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regime0 0.5 1 1.5 2 2.500.20.40.60.8Dτ0 0.5 1 1.5 2 2.500.20.40.60.8DτFigure 6.16: Left panel: Hopf bifurcation boundaries in the τ versus Dplane for the synchronous mode for m = 5 equally-spaced cells on a ringof radius r0 = 0.25 (solid curves) and radius r0 = 0.5 (heavy solid curves)concentric with the unit disk, as computed from (6.6.20), with parameters(6.6.19). Right panel: Comparison of the Hopf bifurcation boundaries forthe synchronous mode with r0 = 0.5 (heavy solid curves), as computed from(6.6.20), with that obtained from (6.3.20) for the leading-order D = D0/νtheory (solid curves). These curves agree well when D is large.2076.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimeorder theory from the D = D0/ν  1 regime. Since the clustering of cells iseffectively equivalent to a system with a large diffusion coefficient, this resultabove indicates, rather intuitively, that stability thresholds for a clusteredspatial arrangement of cells will be more closely approximated by resultsobtained from a large D approximation than for a non-clustered spatialarrangement of cells. In Fig. 6.17 we plot the Hopf bifurcation boundariesfor the distinct asynchronous modes when m = 5 for r0 = 0.5 (left panel)and r0 = 0.75 (right panel), as computed from (6.6.20) with j = 2, 5 (largerlobe) and with j = 3, 4 (smaller lobe). The asynchronous modes are onlylinearly unstable within these small lobes.0.05 0.1 0.15 0.200.10.20.30.40.5Dτ0.1 0.2 0.300.20.40.6DτFigure 6.17: Hopf bifurcation boundaries for the two distinct asynchronousmodes when m = 5 for r0 = 0.5 (left panel) and r0 = 0.75 (right panel),as computed from (6.6.20) with j = 2, 5 (larger solid curve lobe) and withj = 3, 4 (smaller dashed curve lobe). The heavy solid curves are the Hopfbifurcation boundaries for the synchronous mode. The parameters are asin (6.6.19). The asynchronous mode for j = 2, 5 and j = 3, 4 is linearlyunstable only inside the larger and smaller lobe, respectively.To theoretically explain the observation that the instability region inthe τ versus D plane for the synchronous mode is bounded for m = 2, butunbounded for m ≥ 3, we must first extend the large D analysis of §6.5 tothe case of m small cells. We readily derive, assuming identical behavior ineach of the m cells, that the reduced cell-bulk dynamics (6.5.16) for one cellmust be replaced byU ′0 = −1τ(1 +2pimd1|Ω|)U0+2pid2mτ |Ω| u1 , u′ = F (u)+2piτ[d1U0 − d2u1] e1 ,(6.6.23)2086.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimewhen there are m cells. This indicates that the effective domain area is|Ω|/m = pi/m when there are m cells. Therefore, to examine the stabilityof the steady-state solution of (6.6.23) for the Sel’kov model, we need onlyreplace |Ω| with |Ω|/m in the Routh-Hurwitz criteria for the cubic (6.5.27).With this approach, in Fig. 6.18 we show that there are two Hopf bifur-cation values of τ for the steady-state solution of (6.6.23) when m = 3 andm = 5. These values correspond to the horizontal asymptotes as D →∞ inFig. 6.15 for m = 3 and in Fig. 6.16 for m = 5. The numerical results fromXPPAUT [16] in Fig. 6.18 then reveal the existence of a stable periodic solu-tion branch connecting these Hopf bifurcation points for m = 3 and m = 5.A qualitatively similar picture holds for any m ≥ 3. In contrast, for m = 2,we can verify numerically using (6.5.27), where we replace |Ω| with |Ω|/2,that the Routh-Hurwitz stability criteria p1 > 0, p3 > 0, and p1p2 > p3 holdfor all τ > 0 when m = 2 (and also m = 1). Therefore, for m = 2, thereare no Hopf bifurcation points in τ for the steady-state solution of (6.6.23).This analysis suggests why there is a bounded lobe of instability for thesynchronous mode when m = 2, as was shown in Fig. 6.13.For the permeability values d1 = 0.8 and d2 = 0.2 used, we now sug-gest a qualitative reason for our observation that the lobe of instability forthe synchronous mode is bounded in D only when m ≤ mc, where mc issome integer threshold. We first observe that the diffusivity D serves a dualrole. Although larger values of D allows for better communication betweenspatially segregated cells, suggesting that synchronization of their dynamicsshould be facilitated, it also has the competing effect of spatially homoge-nizing any perturbation in the diffusive signal. We suggest that only if thenumber of cells exceeds some threshold mc, i.e. if some quorum is achieved,will the enhanced communication between the cells, resulting from a de-crease in the effective domain area by |Ω|/m, be sufficient to overcome theincreased homogenizing effect of the diffusive signal at large values of D,and thereby lead to a synchronized time-periodic response.Finally, fixing d2 = 0.2, we use the Routh-Hurwitz stability criteriafor the cubic (6.5.27) to determine how the quorum-sensing threshold mcdepends on the permeability d1. Recall that if m > mc there is a range ofτ for which the steady-state solution of (6.6.23) is destabilized due to Hopfbifurcations at some τ− and τ+, with a periodic solution branch existingon the range τ− < τ < τ+. Our computations using the Routh-Hurwitzconditions yield for the unit disk that mc = 3 if d1 = 0.5, mc = 4 if d1 = 0.4,mc = 6 if d1 = 0.3, mc = 9 if d1 = 0.2, mc = 12 if d1 = 0.15, and mc = 19 ifd1 = 0.1. We observe that by decreasing d1, the number of cells required toattain a quorum increases rather steeply. This suggests that small changes2096.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regime0.3 0.4 0.5 0.6 0.7 0.811.21.41.61.82τu10.2 0.4 0.6 0.80.81.21.62τu1Figure 6.18: Global bifurcation diagram of u1e versus τ for the Sel’kov model(6.4.17) as computed using XPPAUT [16] from the ODE system (6.6.23)characterizing the limiting problem as D →∞ with m small cells in the unitdisk Ω. Left panel: m = 3. Right panel: m = 5. The Sel’kov parametersare µ = 2, α = 0.9, and 0 = 0.15, while d1 = 0.8, and d2 = 0.2. Thethick/thin solid line represents stable/unstable steady-state solutions, whilethe solid dots indicate a stable synchronous periodic solution in the cells.For m = 5 (right panel), there are two Hopf bifurcation points at τ = 0.2187and τ = 0.6238. For m = 3 (right panel), the two Hopf bifurcation pointsare at τ = 0.3863 and τ = 0.6815. These points correspond to the horizontalasymptotes as D →∞ in Fig. 6.16 for m = 5 and in Fig. 6.15 for m = 3.2106.6. The Effect of the Spatial Configuration of the Small Cells: The D = O(1) Regimein the permeabilities can have a rather dramatic influence on increasing thenumber of cells needed to reach a quorum, and thereby initiate synchronousoscillations. For d1 ≤ 0.15, a new effect is observed in that there is alower threshold value m− of m needed to ensure that the synchronous lobeis bounded. In particular, we obtain for d1 = 0.15 and d1 = 0.1 thatthe synchronous lobe is bounded only for 2 ≤ m ≤ 12 and 2 ≤ m ≤ 19,respectively, but is unbounded if m = 1. The explanation for the lower limitis that the steady-state can be unstable when d1 = 0 and d2 = 0.2 for somerange of τ . By ensuring p1p2 = p3 at some τ = τc > 0 with p1p2 > p3 forτ 6= τc, together with p1 > 0 and p3 > 0, in the Routh-Hurwitz criteria, inFig. 6.19 we plot the numerically computed quorum-sensing threshold mcand the lower threshold m− as continuous functions of d1.0 0.5 1 1.505101520d1mcFigure 6.19: For d2 = 0.2, and as d1 is varied, the instability lobe for thesynchronous mode is bounded inD for any integer valuem between the lowerand upper curves. The domain is the unit disk, and the Sel’kov parametersare given in (6.6.19). The upper curve gives the quorum-sensing thresholdfor the number of cells m in the well-mixed D  O(ν−1) regime that areneeded to initiate a collective synchronous oscillatory instability betweenthe cells for some range of τ .Owing to the fact that the key parameter in the cubic (6.5.27) is |Ω|/m,in order to determine thresholds for other domains we need only multiply thethresholds given above in Fig. 6.19 for the unit disk by the factor |Ω|/pi. Theexistence of this lower threshold m− as d1 decreases explains the observationseen in Fig. 6.4 for d1 = 0.1 and d2 = 0.2 where one cell on a domain ofarea |Ω| = 10 lead to an unbounded lobe of instability for the synchronousmode. For this domain area, and by using the real-valued thresholds in2116.7. Infinite Domain: Two Identical CellsFig. 6.19 for the unit disk together with our scaling law, we predict thatthere is a bounded lobe of instability for the synchronous mode only when5 ≤ m ≤ 61. Therefore, for m = 1, as in Fig. 6.4, we correctly predict thatthe lobe should be unbounded. In addition, for d1 = 0.5 and d2 = 0.2, andwith |Ω| = 10, the lower threshold is slightly below unity, while the upperquorum-sensing threshold is mc = 12. This confirms the bounded lobes ofinstability for the synchronous mode for m = 1, 2, 3 as seen in Fig. 6.3.6.7 Infinite Domain: Two Identical CellsIn the previous sections we studied the case where m small disjoint cellsare located inside a bounded domain Ω. For this case, we showed thatthe construction of the steady-state solution, and the analysis of the linearstability of this steady-state, simplifies considerably when D = O(ν−1) 1.For this large D regime, a stability analysis based on retaining leading-orderterms in ν was performed. The fact that the domain was bounded wasessential to the large D analysis.In this section, we consider the case where there are two small circularcells of a common radius  centered symmetrically at (x0, 0) and (−x0, 0)in the infinite plane. For this problem, we will construct a steady-statesolution and we formulate the linear stability problem. We emphasize thatour analysis below is not a leading-order-in-ν analysis, but incorporates allorders in ν = −1/ log . The geometry is shown in Fig. 6.20.(-x0,0)Figure 6.20: Schematic plot of the geometry of two cells in the infinite planeTo formulate our model, we take the two cells as the circular regions2126.7. Infinite Domain: Two Identical CellsΩ± = {x∣∣|x− (±x0)| ≤ } of radius . Let U(x, t) represent the concen-tration of the signaling molecule in the bulk. Then, similar to (6.2.1), itsspatial-temporal dynamics can be described by the PDEτUt = ∆U − U , x ∈ R2\Ω± ; U → 0 as |x| → ∞ ,∂n+ U = d1U − d2u+1 , x ∈ ∂Ω+ ,∂n− U = d1U − d2u−1 , x ∈ ∂Ω− ,(6.7.1a)where n± denotes the outer normal to the cell Ω± , and so points into thebulk region.Within each cell we assume, as before, that there are n dynamicallyinteracting species. Assuming that the reaction kinetics are the same withineach of the two cells, the cell dynamics are governed bydu±dt= F (u±) +1τ∫∂Ω±(d1U − d2u±1)ds e1 , (6.7.1b)where u± = (u±1 , . . . , u±n )T represents the concentration of the n species inthe two cells Ω± and e1 ≡ (1, 0, . . . , 0)T . In our formulation we assumethat only one species, labeled by u±1 inside the cell, is capable of beingtransported across the cell membrane into the bulk region.To derive (6.7.1), which has unit diffusivity, starting from our originaldimensional formulation (6.1.1), we proceed as follows: We introduce thedimensionless variablest = kRT , x = X/LB , U =L2BµcU , u = µµc, (6.7.2a)where LB is the diffusion length LB ≡√DB/kB. We assume that L/LB 1, where L is the radius of Ω, so that effectively Ω can be replaced by R2.In terms of the x-variable, the radius of the cells are  ≡ σ/LB  1. Wethen introduce O(1) constants d1 and d2 defined byβ1 ≡ (kBLB) d1, β2 ≡(kBLB)d2,  ≡ σLB, LB ≡√DBkB.(6.7.2b)In terms of (6.7.2), we readily derive (6.7.1), where τ ≡ kR/kB. In (6.7.1),the key parameters are τ , the half-distance d between the centers of thecells, and the permeability constants d1 and d2.We now study (6.7.1) in the limit → 0.2136.7. Infinite Domain: Two Identical Cells6.7.1 The Steady-State SolutionWe now construct the steady-state solution to (6.7.1) when   1. Sincethe two cells have identical kinetics and coupling mechanisms, we seek asteady-state solution that is symmetric about the midplane x = 0. Thissteady-state solution satisfies∆U − U = 0 , x > 0\Ω ; U → 0 as |x| → ∞ ,∂nU = d1U − d2u1 , x ∈ ∂Ω ; Ux = 0 on x = 0 ,(6.7.3)where Ω denotes the circular disk of radius  centered at x0 ≡ (x0, 0). Herewe have dropped the subscript ′+′ for convenience of notation. The Neu-mann boundary condition at x = 0 arises from the symmetry assumption.We use the method of matched asymptotic expansions to study (6.7.3)for  1. In the neighbourhood of the cell region Ω, we introduce the localvariables y = −1(x − x0) and U0(y) = U(x0 + y), and we let ρ = |y|. Interms of these local variables, and neglecting algebraic terms in , we obtainfrom (6.7.3) that U0 satisfies∂ρρU0 + ρ−1∂ρU0 = 0 , 1 < ρ <∞ , (6.7.4a)subject to the boundary condition∂U0∂ρ= d1U0 − d2u1, ρ = 1 . (6.7.4b)We readily solve (6.7.4), in terms of the as yet unknown source strength S,to getU0 = S log ρ+ χ , where χ ≡ 1d1(S + d2u1) . (6.7.5)Within the cell, the steady-state u of the membrane dynamics satisfiesF (u) +1τ∫∂Ω(d1U0 − d2u1) ds e1 = 0 . (6.7.6)By evaluating the integral over the perimeter, we readily find thatF (u) +2piSτe1 = 0 . (6.7.7)Next, we analyze the outer region. We match the far-field behavior ofthe inner solution (6.7.5) to the bulk solution, which yields a singularity2146.7. Infinite Domain: Two Identical Cellscondition for the bulk solution as x → x0. In this way, we obtain that theouter bulk solution must satisfy∆U − U = 0 , x > 0\{x0} ; U → 0 as |x| → ∞ ,Ux = 0 on x = 0 ,U ∼ S log |x− x0|+ S/ν + χ , as x→ x0 ,(6.7.8)where χ is given in (6.7.5) and ν ≡ −1/ log .To solve (6.7.8), we introduce the reduced-wave Green’s functionG(x;x0)satisfying∆G−G = δ(x− x0) , x > 0 ; G→ 0 as |x| → ∞ ,Gx = 0 on x = 0 .(6.7.9)It is well-known that the free space Green’s function Gf for this operator isGf (x;x0) =12pi(K0(|x− x0|) , (6.7.10)where K0(z) is the modified Bessel function of the second kind of order zero.Then, by the method of images, the Green’s function satisfying (6.7.9) isG(x;x0) =12pi[K0(|x− x0|) +K0(|x− x∗0|)], (6.7.11)where x∗ ≡ (−x0, 0). We recall that as z → 0, K0(z) has the local behaviorK0(z) ∼ − log(z/2)(1+z2/4+O(z4))+(−γe+(1−γe)z2/4+· · · ) , as z → 0 ,(6.7.12)where γe is Euler’s constant. This yields the singular behavior of the Green’sfunction (6.7.11) as x→ x0 given byG ∼ − 12pilog |x− x0|+R+ o(1) , as x→ x0 , (6.7.13)where the regular part R isR =12pi(log 2− γe +K0(2d)), (6.7.14)and where 2d ≡ |x0 − x∗0| is the distance between the centers of the twocells.In terms of G(x;x0), the solution to (6.7.8) can be represented asU = −2piSG(x,x0) = −S[K0(|x− x0|) +K0(|x− x∗0|)].2156.7. Infinite Domain: Two Identical CellsTherefore, as x→ x0, we have thatU → S (log |x− x0| − log 2 + γe −K0(2d) + · · · ) . (6.7.15)By matching the regular part of the singularity behavior in (6.7.8) with thatin (6.7.15), we obtain that S satisfiesS(− log 2 + γe −K0(2d)) = Sν+ χ , (6.7.16)where χ is defined in (6.7.5). Upon substituting for χ in (6.7.16), and re-calling (6.7.7), we obtain the following n+1 dimensional nonlinear algebraicsystem for S and u = (u1, . . . , un):S(1ν+1d1+ log 2− γe +K0(2d))= −d2u1d1, F (u) +2piSτe1 = 0 ,(6.7.17)where ν ≡ −1/ log  and γe is Euler’s constant. This completes the asymp-totic approximation of the steady-state solution to (6.7.1) that is symmetricabout the midplane x = 0.6.7.2 Linear Stability AnalysisTo formulate the linear stability problem, we first introduce the perturba-tionsU(x, t) = Ue(x) + eλtΦ(x) , u(t) = ue + eλtφ ,into (6.7.1) and linearize. By symmetry, we will only consider the regionx > 0, and will impose suitable boundary conditions on x = 0 for Φ asdiscussed below. This leads to∆Φ− (1 + τλ)Φ = 0 , x > 0\Ω ; Φ→ 0 as |x| → ∞ ,∂nΦ = d1Φ− d2φ1, x ∈ ∂Ω ,(6.7.18a)which is coupled to the linearized cell dynamicsλφ = Jφ+1τ∫∂Ω(d1Φ− d2φ1) ds e1 (6.7.18b)Here J is the Jacobian of F at the steady-state solution u = ue, while φ1 isthe first element in the eigenvector φ = (φ1, . . . , φn)T .To complete the formulation of the stability problem we must impose aboundary condition for Φ on the midplane x = 0. There are two choices2166.7. Infinite Domain: Two Identical Cellsfor this boundary condition. The choice Φ(0, y) = 0 corresponds to an anti-phase synchronization of the two cells, while Φx(0, y) = 0 corresponds to anin-phase synchronization of the two cells. We will consider both types ofperturbation in our analysis.We now study (6.7.18) by the method of matched asymptotic expansions.In the inner region near the cell we introduce the local variables y = −1(x−x0) and Φ0(y) = Φ(x0 + y). Upon neglecting algebraic terms in , we lookfor a radially symmetric solution in terms of ρ = |y| to∂ρρΦ + ρ−1∂ρΦ = 0 , 1 < ρ <∞ ; ∂Φ∂ρ= d1Φ− d2φ1, ρ = 1 .(6.7.19)In terms of a constant C to be determined, the solution to (6.7.19) isΦ = C log ρ+B , where B =1d1(C + d2φ1) . (6.7.20)By substituting (6.7.20) into (6.7.18b) we obtain that(J − λI)φ+ 2piCτe1 = 0 . (6.7.21)Next, we formulate the outer problem by matching the far-field behaviorof the inner solution (6.7.20) to the bulk solution, which yields a singularitycondition for the outer solution as x→ x0. This yields the outer problem∆Φ− ϕ2λΦ = 0 , x > 0\{x0} ; Φ→ 0 as |x| → ∞ ,Φ ∼ C log |x− x0|+ Cν+B , as x→ x0 ,(6.7.22a)subject, for either symmetric (+) or asymmetric perturbations, to the bound-ary conditionΦ = 0 on x = 0 , (-) (async) , or Φx = 0 on x = 0 , (+) (sync) .(6.7.22b)Here B is defined in (6.7.20), and ϕλ ≡√1 + τλ, where we have chosen theprincipal branch of the square root.The solution to (6.7.22) is written in terms of the λ−dependent Green’sfunction Gλ(x,x0) asΦ = −2piCGλ(x,x0) , (6.7.23)where Gλ(x;x0) satisfies∆Gλ − ϕ2λGλ = −δ(x− x0) , x > 0 ; Gλ → 0 as |x| → ∞ ,(6.7.24a)2176.7. Infinite Domain: Two Identical Cellssubject to either of the two possible boundary conditionsGλ = 0 on x = 0 , (-) , or Gλx = 0 on x = 0 , (+) .(6.7.24b)By the method of images, the solution to (6.7.24) isGλ(x;x0) =12pi(K0(ϕλ|x− x0|)±K0(ϕλ|x− x∗0|)), (6.7.25)for either the synchronous (+) or asynchronous (−) mode. As x → x0 weobtain the local behaviorGλ(x;x0) ∼ − 12pilog |x− x0|+ 12pi(− log(ϕλ2)− γe ±K0(2dϕλ)),as x→ x0 ,(6.7.26)where γe is Euler’s constant, and 2d ≡ |x0−x∗0| is the distance between thecenters of the two cells. Upon using (6.7.26) in (6.7.23), we can calculatethe local behavior of Φ as x→ x0. Then, we match this local behavior withthe required singular behavior in (6.7.22a). This yields thatCν+B = −C(− log(ϕλ2)− γe ±K0(2dϕλ)). (6.7.27)Next, we substitute (6.7.20) for B into (6.7.27), and solve the resultingexpression for C to obtainC =d2d1A±λφ1 , where A±λ ≡ log(ϕλ2)+ γe − (±)K0(2dϕλ)− 1ν− 1d1,(6.7.28)where the + and − signs denote the synchronous and asynchronous modes,respectively. Upon substituting (6.7.28) into (6.7.21), we readily derive that[(Je − λI) + 2piτd2d1A±λe1eT1]φ = 0 , (6.7.29)where e1 = (1, 0, . . . , 0)T . We conclude that λ = O(1) is an eigenvalue of thelinearized problem (6.7.18) if and only if there is a nontrivial φ to (6.7.29).To derive an explicit transcendental equation for λ we use the matrixdeterminant lemma det(A+abT ) =(1 + bTA−1a)det(A), to conclude that2186.7. Infinite Domain: Two Identical Cellsif λ is not an eigenvalue of J then there is a nontrivial solution to (6.7.29)if and only if1− 2piτd2d1A±λeT1 (λI − J)−1e1 = 0 .Finally, by calculating (λI−J)−1, we conclude that λ is a discrete eigenvalueof the linearization if and only if λ is a root of Q±(λ) = 0, where Q± isdefined byQ±(λ) = A±λ −2pid2d1τM11det(λI − J) . (6.7.30a)Here M11 is the cofactor of the element in the first row and first column ofλI − J , and A±λ is defined byA±λ ≡ log(ϕλ2)+ γe − (±)K0(2dϕλ)− 1ν− 1d1, ϕλ ≡√1 + τλ ,(6.7.30b)where 2d is the distance between the centers of the two cells, ν = −1/ log ,γe is Euler’s constant, while the + and − signs indicate the synchronous andasynchronous modes, respectively. In (6.7.30a), the Jacobian of the mem-brane kinetics is evaluated at ue, where ue is obtained from the nonlinearalgebraic system (6.7.17) associated with the steady-state solution.Next we will use a winding number criterion to compute the roots ofQ±(λ) in Re(λ) > 0. By using the argument principle, we obtain that thenumber N of roots of Q±(λ) = 0 in Re(λ) > 0 isN =12pi[argQ±]Γ + P , (6.7.31)where P is the number of poles of Q±(λ) in Re(λ) > 0, and the squarebrackets denote the change in the argument of Q± over the contour Γ. Theclosed contour Γ is the limit as R→∞ of the union of the imaginary axis,which can be decomposed as ΓI+ = iλI and ΓI− = −iλI , for 0 < λI < R,and the semi-circle ΓR defined by |λ| = R with |arg(λ)| ≤ pi/2. SinceA±λ is analytic in Re(λ) > 0, it follows that P is the number of roots ofdet(λI − J) = 0 in Re(λ) > 0. Now if we let R → ∞ on ΓR, we calculateusing (6.7.30) that Q±(λ) ∼ O(lnR) + ipi/4, so that as R → ∞ we have[argQ±]ΓR = 0. Further, since [argQ±]ΓI+ = [argQ±]ΓI− , then (6.7.31)becomesN =1pi[argQ±]ΓI+ + P , (6.7.32)where P is the number of roots of det(λI − J) = 0 in Re(λ) > 0.2196.7. Infinite Domain: Two Identical CellsWith this framework we have formulated a hybrid asymptotic-numericalmethod to determine whether there can be any triggered oscillations dueto Hopf bifurcations for the two-cell infinite line problem as parameters arevaried. One future goal is to use this framework to compute phase diagramsin parameter space where different types of instabilities can occur for variousspecific reaction kinetics.220Chapter 7Conclusion and Future WorkIn this chapter we first give a brief summary of the main results presentedin this thesis and then list a few open problems for possible directions offuture work.In general, on a one-dimensional spatial domain we have introduced andanalyzed a class of models that couple two dynamically active compartments,either cell or membranes, separated spatially by a distance 2L, through alinear bulk diffusion field. For this class of models, we have shown both ana-lytically and numerically that bulk diffusion can trigger a stable synchronousoscillatory instability in the temporal dynamics associated with the two ac-tive compartments. Qualitatively, our results also show that oscillatorydynamics in the two compartments will only occur for some intermediaterange of the compartment-bulk coupling strength and the parameter rangewhere stable synchronous oscillations between the two compartments occuris much larger than that for asynchronous oscillations. This suggests thatstable synchronized oscillations between two dynamically active compart-ments coupled by passive bulk diffusion can be a robust feature in coupledcompartment-bulk dynamics.For one particular form of local kinetics, we use center manifold andnormal form theory to reduce the local dynamics of the model system toa normal form for a double Hopf bifurcation, which predicts the patternsof Hopf bifurcation and the stability of both synchronous and asynchronousmodes near the double Hopf point. In the study of coupled oscillators, doubleHopf bifurcations often appear in delay-coupled systems, e.g. [5, 69]. In ourmodel, there is no explicit delay term, but the communication between thetwo oscillators is through spatial diffusion of a signalling chemical in theextracellular medium. For spatially separated cells, this can be a morerealistic way to describe the connections among individuals and at the sametime diffusion serves effectively as a time delay, reflecting the time needed fora chemical to change concentration at a distant location. In fact, diffusioncan be explicitly represented as a distributed delay through the variation ofconstants formula [6], and this sometimes has practical advantages.For the case of a single local component in each compartment, and in2217.1. Future Workthe limit of L → ∞ we derive rigorous spectral results to characterize thepossibility of Hopf bifurcations. Also, a weakly nonlinear theory is developedto predict the local branching behavior near the Hopf bifurcation point forfinite L. In addition, we give a detailed theoretical analysis of the onset ofoscillatory dynamics for a model system from [23] using asymptotic analysistogether with bifurcation and stability theory.In §6, we have formulated and studied a general class of coupled cell-bulkproblems with the primary goal of establishing whether such a class of prob-lems can lead to the initiation of oscillatory instabilities due to the couplingbetween the cell and bulk. Our analysis, formulated in an arbitrary boundeddomain, relies on the assumption that the signalling compartments have aradius that is asymptotically small as compared to the length-scale of thedomain. In this limit → 0 of small cell radius we have used a singular per-turbation approach to determine the steady-state solution and to formulatethe eigenvalue problem associated with linearizing around the steady-state.In the limit for which the diffusivity D of the bulk is asymptotically large oforder D = O(ν−1), we have derived eigenvalue problems characterizing thepossibility of either synchronous and asynchronous instabilities triggered bythe cell-bulk coupling. Phase diagrams in parameter space, showing whereoscillatory instabilities can be triggered, were calculated for a few specificchoices of the intracellular kinetics. Our analysis shows that triggered oscil-lations are not in general possible when the intracellular dynamics has onlyone species. For the regime D  O(ν−1), where the bulk can be effectivelytreated as a well-mixed system, for a one-cell geometry we have reducedthe cell-bulk PDE system to a finite dimensional ODE system for the spa-tially constant bulk concentration field coupled to intracellular dynamics.This ODE system was shown to have triggered oscillations due to cell-bulkcoupling, and global bifurcation diagrams were calculated for some specificreaction kinetics, showing that the branch of oscillatory solutions is globallystable. For the regime D = O(1), where the spatial configuration of cells isan important factor, we have determined phase-diagrams for the initiationof synchronous temporal instabilities associated with a ring pattern of cellsinside the unit disk, showing that such instabilities can be triggered from amore clustered spatial arrangement of the cells inside the domain.7.1 Future WorkThis kind of compartment-bulk coupling model is relatively new and thereare still many open problems that require further investigations. In the2227.1. Future Workfollowing, we list a few possible directions for further work.A biologically relevant direction related to our work in chapter §2 thatwarrants further investigation is to introduce different, more detailed, mod-els for the coupling strength between the compartment and the bulk. Itwould be interesting to analyze triggered oscillations that result when thecompartment-bulk coupling strength β varies dynamically in time, or iscoupled to some slow dynamics, so as to create periodic bursts of syn-chronous oscillatory behavior, followed by intervals of quiescent behavior,in the two compartments. Such bursting and triggered dynamics have beenwell-studied in a purely ODE context (cf. [1], [2], [29], [49], see also thereference therein). A related, but rather challenging direction, would be toinvestigate the possibility of synchronized oscillations when β is allowed toswitch stochastically in time between an ON and OFF state. Such stochas-tic switching behavior is a characteristic feature of channels in biologicalmembranes. The resulting model is a stochastic hybrid system that consistsof both continuous PDE-ODE dynamics, punctuated by discrete stochasticevents. A mathematical analysis of a class of related stochastic hybrid sys-tem, whereby the boundary condition for a heat equation on a finite domainswitches randomly between Dirichlet and Neumann, is analyzed in [43] andanother example of switching boundary conditions is considered in [3].Another direction is to consider more thoroughly the case of multi-speciesmembrane dynamics in the one-dimensional system. More specifically, al-though a numerical winding number computation is readily implemented formulti-species membrane dynamics, there is a need to extend the theoreticalspectral results in chapter §4 to the case of more than a single membrane-bound species. Furthermore, it would be interesting to extend the weaklynonlinear analysis in chapter §4 to the case of multiple membrane-boundspecies.As a further extension to the studies in chapter §4 and chapter §5, itwould be interesting to use numerical bifurcation software to give a detailedinvestigation of secondary instabilities arising from bifurcations of asymmet-ric steady-state solutions or either the synchronous or asynchronous periodicsolution branch. Our preliminary results show that such secondary bifurca-tions can lead to more exotic dynamics such a quasi-periodic solutions orperiod-doubling behavior. In particular, it would be interesting to explorewhether there can be any period-doubling route to chaotic dynamics such aswas observed computationally in [56] for a related model consisting of twodiffusing bulk species that are subject to nonlinear fluxes at fixed latticesites.In addition, It would also be worthwhile to study large-scale oscillations2237.1. Future Workby representing the bulk diffusion field in terms of a time-dependent Green’sfunction with memory. Coupling to the membrane dynamics leads to a con-tinuously distributed delay equation for the dynamically active membranecomponents.An open direction relates to our assumption that the bulk diffusion fieldhas a constant diffusivity and undergoes a linear bulk degradation. It wouldbe worthwhile to extend our analysis to allow for either a nonlinear degrada-tion of the signaling molecule in the bulk, a nonlinear diffusivity, or to allowfor a sub-diffusive bulk diffusion process. Either of these three additional ef-fects could be important in various biological applications. Particularly, forour study of the two-dimensional model in chapter §6, for the large D regimewhere D  O(ν−1), it is possible to readily analyze the case where the bulkdegradation is nonlinear, with possibly a Michaelis-Menton saturation of thebulk decay of the form τUt = D∆U − σB(U), where σB(U) = U/(1 + cU).With this modification, we can readily derive in place of (6.5.16) that thecoupled cell-bulk dynamics reduces to the finite-dimensional dynamicsU ′0 = −1τ(σB(U0) +2pid1|Ω| U0)+2pid2τ |Ω| u1 , u′ = F (u)+2piτ[d1U0 − d2u1] e1 .(7.1.1)It would be interesting to explore the effect of this nonlinear bulk decay onthe possibility of Hopf bifurcations. In addition, in the large D regime, it isreadily possible to derive an extended system of ODE’s for the case wherethere are multiple, and not just one, small signalling compartment.224Bibliography[1] S. M. Baer, T. Erneux, J. Rinzel, The Slow Passage Through a Hopfbifurcation: Delay, Memory Effects, and Resonance, SIAM J. 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Assume that the signaling molecule diffuses out ofthe cell at a certain rate. The local chemical species inside the cell, denotedby u = (u1, u2, . . . , un)T , are assumed to satisfy the following systemut = Duxx + F (u) , − < x <  , t > 0 ,Dux(, t) = e1G1(C(, t), u1(, t)) , Dux(−, t) = e1G2(C(−, t), u1(−, t)) ,(A.0.1)where  1 and e1 = (1, 0, . . . , 0)T . Here for simplicity we assume that alllocal chemicals share the same diffusivity D  1, with D = O(1), which isasymptotically small as compared to the reaction rate of the kinetics.We now derive a reduced model from (A.0.1) in the limit  1 to obtainthe approximate behavior of this system. To do so, we first introduce thelocal variable y = −1x, so that in terms of the y variable (A.0.1) becomesut = −1Duyy + F (u) , −1 < y < 1 , t > 0 ,Duy(1, t) = e1G1(C(, t), u1(1, t)) , Duy(−1, t) = e1G2(C(−, t), u1(−1, t)) .(A.0.2)We then expand the local specifies u asu = u0 + u1 + · · · . (A.0.3)Substituting this expansion into (A.0.2), and linearizing, we obtain to lead-ing order that u0 satisfiesu0yy = 0 , −1 < y < 1 ; u0y(±1, t) = 0 . (A.0.4)The solution to (A.0.4), which is independent of the spatial variable y, isu0 = u0(t). We then proceed to the next order to determine the equation232Appendix A. Formulation of the PDE-ODE System for a Periodic Chainthat u0 satisfies. At the next order, u1 satisfiesDu1yy = u0t − F (u0) , −1 < y < 1 ,Du1y(1, t) = e1G1(C(, t), u01(t)) , Du1y(−1, t) = e1G2(C(−, t), u01(t)) ,(A.0.5)where e1 = (1, 0, . . . , 0)T , and u01 denotes the first component of u0. For thisO() system, we invoke the divergence theorem to obtain that∫ 1−1Du1yy dy =∫ 1−1(u0t − F (u0))dy. Upon evaluating this expression, and using (A.0.5),we getDu1y(1, t)−Du1y(−1, t) = e1(G1(C(, t), u01(t))−G2(C(−, t), u01(t)))= 2(u0t − F (u0)) .(A.0.6)Upon rewriting this equation we obtain a system of ODEs for u0 given byu0t = F (u0) +e12[G1(C(, t), u01(t))−G2(C(−, t), u01(t))]. (A.0.7)Now letting the width of the cell approach 0, or equivalently  → 0, weobtain the limiting systemu0t = F (u0) +e12[G1(C(0+, t), u01(t))−G2(C(0−, t), u01(t))]. (A.0.8)If we consider the case of linear coupling for which G1 and G2 have the formsG1(C(0+, t), u01) = κ(C(0+, t)−u01) , G2(C(0−, t), u01) = −κ(C(0−, t)−u01) ,(A.0.9)then (A.0.8) becomesu0t = F (u0) + e1[κ2(C(0+, t) + C(0−, t))− κu01]. (A.0.10)This specifies ODEs for the time evolution of the leading order term for thelocal species inside the cell, and in this way approximately characterizingthe local dynamics. In §2.3, we drop the superscript in u0 and use (A.0.10)to describe the local dynamics inside each cell.233Appendix BAn Alternative PDE-ODEFormulation for a PeriodicChainIn this appendix we briefly discuss the implications of an alternative for-mulation of the periodic cell problem (2.3.1). In this simpler formulation,we assume that C(x, t) is continuous on the ring, but has jumps in the fluxDCx across each cell. This alternative formulation isCt = DCxx − kC , t > 0 , x ∈ (−L, (2m− 1)L) ,with x 6= 2jL , j = 0, . . . ,m− 1 ,C(−L, t) = C(2mL− L, t) , Cx(−L, t) = Cx(2mL− L, t) ,[DCx]∣∣x=2jL= 2κ[C(2jL, t)− u1j], j = 1, . . . ,m ,(B.0.1a)where [ux]∣∣x0≡ ux(x+0 ) − ux(x−0 ). This bulk field is then coupled to theinternal cells dynamics bydujdt= F (uj) + e1[κC(2jL, t)− κu1j], j = 0, . . . ,m− 1 . (B.0.1b)For (B.0.1), we again obtain the symmetric steady-state solution as in§2.3.1. However, in contrast to the analysis in §2.3.2, in the linear stabilityanalysis for (B.0.1) the perturbations in the bulk diffuson field must nowbe continuous across each cell. From an analysis similar to that in §2.3.2,we readily derive for the Sel’kov kinetics that the eigenvalue parameter λsatisfies (2.3.12a), where in place of (2.3.12b), we have∆λ ≡ −1κ+1DΩλsinh (2ΩλL)Re(z)− cosh (2ΩλL) , Re(zl) = cos(2pilm), (B.0.2)where Ωλ is defined in (2.3.9). As a remark if we set zl = 1 (in-phase)and zl = −1 (anti-phase) in (B.0.2), we can readily show that (2.3.12a)234Appendix B. An Alternative PDE-ODE Formulation for a Periodic Chainwith (B.0.2) reduces, as expected, to the two-cell spectral problem (2.2.5)of §2.2.1 for either in-phase or anti-phase modes, respectively, upon settingβ = κ in (2.2.5).235Appendix CCalculation of Normal FormCoefficientsIn this section, we describe the calculations to evaluate the four cubic coef-ficients Gjklm, Hjklm in the normal form (3.3.2) that governs the dynamicsnear a double Hopf point. To evaluate the coefficients, it is sufficient to takeparameters at the double Hopf point, thus µ1 = µ2 = 0 and we haveζ˙1 = iω1ζ1 +G2100ζ21 ζ¯1 +G1011ζ1ζ2ζ¯2 +O(‖(ζ1, ζ¯1, ζ2, ζ¯2)‖5),ζ˙2 = iω2ζ2 +H1110ζ1ζ¯1ζ2 +H0021ζ22 ζ¯2 +O(‖(ζ1, ζ¯1, ζ2, ζ¯2)‖5).(C.0.1)At the double Hopf point, the nonlinear system (3.2.2)–(3.2.3), writtenas (3.3.1), is reduced to a system on a four-dimensional center manifoldthat is tangent, in the infinite-dimensional function space H, to the criticaleigenspace T c. Since this center manifold reduction is standard and followsclosely the analogous procedure at a simple Hopf bifurcation described indetail for reaction-diffusion systems in the textbook [39], we give only ashort description together with some details specific to our system. Thiscenter manifold system is further reduced to the normal form (C.0.1).We first construct a projection P c of the space H, onto the criticaleigenspace T c. This requires an inner product, and two adjoint eigenvectors.For a pair of complex vectorsp =ξ(x)χ−ϑ−χ+ϑ+, q =η(x)ϕ−ψ−ϕ+ψ+,we define their inner product to be〈p, q〉 =∫ +L−Lξ(x)η(x) dx+ χ−ϕ− + ϑ−ψ− + χ+ϕ+ + ϑ+ψ+.236Appendix C. Calculation of Normal Form CoefficientsWith respect to this inner product, the adjoint to the linear differentialoperator M is the linear differential operator M∗, given byM∗ξ(x)χ−ϑ−χ+ϑ+=D ξ′′(x)− k ξ(x)fV χ− + gV ϑ− − βχ− + κ ξ(−L)fWχ− + gWϑ−fV χ+ + gV ϑ+ − βχ+ + κ ξ(+L)fWχ+ + gWϑ+,with adjoint boundary conditions−D ξ(−L) = βχ− − κ ξ(−L),+D ξ(+L) = βχ+ − κ ξ(+L).We solve for two adjoint eigenvectorspj =ξj(x)χj,−ϑj,−χj,+ϑj,+,j = 1, 2, satisfyingM∗p1 = −iω1p1, M∗p2 = −iω2p2,with normalizations such that〈p1, q1〉 = 1, 〈p2, q2〉 = 1, (C.0.2)where q1, q2 are the eigenvectors given in Section 3.2. We note that theorthogonality conditions〈p1, q2〉 = 0, 〈p1, q1〉 = 0, 〈p1, q2〉 = 0〈p2, q1〉 = 0, 〈p2, q1〉 = 0, 〈p2, q2〉 = 0,237Appendix C. Calculation of Normal Form Coefficientsare automatically satisfied. We obtainp1 = a01ξ10 sinh Ω3x/sinh Ω3L−1fw/(iω1 + gw)1−fw/(iω1 + gw), p2 = a02ξ20 cosh Ω4x/cosh Ω4L1−fw/(iω2 + gw)1−fw/(iω2 + gw),where the constantsa01 = 0.250508− i0.172379, a02 = 0.253847− i0.181974,are chosen so that the normalization conditions (C.0.2) hold, andΩ3 =√k−iω1D , Ω4 =√i−iω2D , ξ10 =βκ+DΩ3 coth Ω3L, ξ20 =βκ+DΩ4 tanh Ω4L.We define the projection P c, of H onto the critical eigenspace T c, byP cX = z1q1 + z¯1q1 + z2q2 + z¯2q2,for any X ∈ H, where z1, z2 are complex numbers given by the innerproductsz1 = 〈p1, X〉, z2 = 〈p2, X〉.Now we can use the projection P c to split any vector X ∈ H into twopartsX = Xc + Y,where the “center” partXc = P cX = z1q1 + z¯1q1 + z2q2 + z¯2q2belongs to the four-dimensional critical eigenspace T c and the complemen-tary partY = (I − P c)X = X − 〈p1, X〉q1 − 〈p1, X〉q1 − 〈p2, X〉q2 − 〈p2, X〉q2,where I denotes the identity operator, belongs to the infinite-dimensionalstable subspace T s. Correspondingly, the system (3.3.1) splits into two partsX˙c = MXc+ 12PcB(Xc+Y,Xc+Y )+ 16PcC(Xc+Y,Xc+Y,Xc+Y ), (C.0.3)238Appendix C. Calculation of Normal Form CoefficientsY˙ = MY + 12(I−P c)B(Xc+Y,Xc+Y )+ 16(I−P c)C(Xc+Y,Xc+Y,Xc+Y ).(C.0.4)By center manifold theory, there is an invariant, exponentially attracting,four-dimensional local center manifold in H that is tangent to the criticaleigenspace T c, and the center manifold can be expanded in a Taylor seriesasY = Y (z1, z¯1, z2, z¯2) =∑j+k+l+m=21j!k!l!m!wjklmzj1z¯k1zl2z¯m2 +O(‖(z1, z¯1, z2, z¯2)‖3).(C.0.5)Substituting the expansion (C.0.5) into (C.0.3)–(C.0.4) and using the in-variance of the center manifold, we collect terms of like powers and obtainnonhomogeneous linear boundary value problems for each of the ten coeffi-cient vectors wjklm at second order (j + k + l +m = 2; j, k, l,m ≥ 0),(2iω1I −M)w2000 = (I − P c)B(q1, q1),−Mw1100 = (I − P c)B(q1, q1),(iω1I + iω2I −M)w1010 = (I − P c)B(q1, q2),(iω1I − iω2I −M)w1001 = (I − P c)B(q1, q2),etc.Using the explicit expressions (3.2.17) for q1 and q2, we use matrix algebraand the method of undetermined coefficients, assisted by the mathematicalsoftware package Maple, to solve for the wjklm that we require. It is helpfulto use symmetry to reduce the number of explicit solutions needed.Substituting (C.0.5) into each of the components of (C.0.3), we obtaina four-dimensional ordinary differential equation that gives the dynamicsrestricted to the invariant local center manifold,z˙1 = iω1z1 + g(z1, z¯1, z2, z¯2),z˙2 = iω2z2 + h(z1, z¯1, z2, z¯2).(C.0.6)Expanding in Taylor seriesg(z1, z¯1, z2, z¯2) =∑j+k+l+m≥2gjklmzj1z¯k1zl2z¯m2 ,h(z1, z¯1, z2, z¯2) =∑j+k+l+m≥2hjklmzj1z¯k1zl2z¯m2 ,239Appendix C. Calculation of Normal Form Coefficientsthe ten quadratic coefficients of the center manifold system (C.0.6) are givenbyg2000 =12〈p1, B(q1, q1)〉,g1100 = 〈p1, B(q1, q1)〉,g1010 = 〈p1, B(q1, q2)〉,g1001 = 〈p1, B(q1, q2)〉,etc.Note that several of these coefficients vanish due to symmetry. We needexplicitly only four of the cubic coefficients of the centre manifold system(C.0.6),g2100 = 〈p1, B(q1, w1100 + 12B(q1, w2000) + 12C(q1, q1, q1)〉,g1011 = 〈p1, B(q1, w0011) +B(q2, w1001) +B(q2, w1010) + C(q1, q2, q2)〉,h1110 = 〈p2, B(q2, w1100) +B(q1, w0110) +B(q1, w1010) + C(q1, q1, q2)〉,h0021 = 〈p2, B(q2, w0011) + 12B(q2, w0020) + 12C(q2, q2, q2)〉.Finally, a near-identity coordinate transformation of the formz1 = ζ1 +O(‖(ζ1, ζ¯1, ζ2, ζ¯2)‖2), z2 = ζ2 +O(‖(ζ1, ζ¯1, ζ2, ζ¯2)‖2),takes the the center manifold system (C.0.6) into the normal form (C.0.1).The procedure to construct the coordinate transformation is lengthy butstandard, and is described in textbooks. For example, see [39] for moredetails. In the end, there are formulas derived for the cubic coefficients inthe normal form (C.0.1), in terms of the quadratic and cubic coefficients ofthe center manifold system (C.0.6): see equations (8.90)–(8.93) in [39]. Weuse Maple to evaluate these coefficients numerically, obtaining (3.3.3).240Appendix DTwo Specific BiologicalModelsD.1 The Dictyostelium ModelThe amoeba Dictyostelium discoideum is one of the most studied organismin biology. There are many stages in the life cycle of each such amoebacell. When nutrient is readily available, they live as single cell organisms.However, when food becomes scarce, each cell starts to release cyclic AMP(cAMP) in order to attract other cells, and at the same time themselves areattracted by the cAMP signal emitted by others. This secretion results inan aggregation of individual amoeba to form aggregate centers [19]. Thisintercellular communication mechanism presents some similarities with theendocrine system in higher organisms. In [19] a two-variable model was pro-posed to describe the cAMP (cyclic adenosine monophosphate) oscillationsin Dictyostelium cells. This minimal model was obtained from a reductionof a more elaborate model based on desensitization of the cAMP recep-tor which consists of variables representing molecules such as the active(R)and desensitized(D) forms of the receptor, free(C) and active form(E) ofadenylate cyclase, intracellular(Pi) and extracellular(P) cAMP, and sub-strate ATP(S). In [19] this minimal model was used to analyze the burstingand birhythmicity observed in experiments with amoeba cells. The modelis formulated asdρtdt= f2(γ)− ρt(f1(γ) + f2(γ)) , dγdt= σ∗ψ(ρt, γ)− keγ , (D.1.1a)wheref1(γ) ≡ k1 + k2γ21 + γ2, f2(γ) ≡ k1L1 + k2L2c2dγ21 + c2dγ2,ψ(ρt, γ) ≡α(Λθ + ρtγ21+γ2)(1 + αθ) +(ρtγ21+γ2)(1 + α).(D.1.1b)241D.2. The GnRH ModelHere ρt is the total fraction of receptor in the active state, α and γ denotethe normalized concentration of intracellular ATP and extracellular cAMP,θ is the ratio of Michaelis constants for the E and C forms of adenylatecyclase, Λ is the ratio of catalytic constants of forms C and E of adenylatecyclase,  is the coupling constant for activation of C by cAMP-receptorcomplex in active state, k1 is the rate constant for the modification stepfrom R to D, L1 is the equilibrium ratio of the states R and D, k2 is the rateconstant for modification step from R to D in the presence of cAMP, L2is the corresponding equilibrium ratio, ke is the ratio of maximum activityfor extracellular phosphodiestease and the Michaelis constant of extracellu-lar phosphodiesterase for cAMP, cd is the ratio of dissociation constants ofcAMP-receptor complex in R and D states, σ∗ is calculated as some combi-nation of other constants. For a more detailed discussion of this model see[19] (pp. 195–258).Since the cAMP molecules can diffuse in space, in our model we assumethat the extracellular cAMP is also a function of location, so that γ = γ(x, t).We assume that it can diffuse freely in space, with some bulk decay, but thatall the reactions occur on the boundaries of amoeba cells. In this way, ourmodel for cAMP, given a cell at x = 0 and at x = 2L, and with τ ≡ 1/ke isτdγdt= Dd2γdx2− γ , t > 0 , 0 < x < L ;γx(L, t) = 0 , Dγx(0) = −σ?ψ(ρt, γ(0, t)) ,dρtdt= f2(γ(0, t))− ρt[f1(γ(0, t)) + f2(γ(0, t))].(D.1.2)D.2 The GnRH ModelGonadotropin-releasing hormone (GnRH) is a decapeptide secreted by GnRHneurons in the hypothalamus that regulates the reproductive function inmammals. There are about 800-2000 GnRH neurons scattered in a fewareas of the hypothalamus. Each GnRH neuron releases GnRH to portalblood in an oscillatory profile with a period of several minutes and they syn-chronize to produce large GnRH pulses with a period ranging from twentyminutes to one hour. Experiments reveal that GnRH neurons express GnRHreceptors. Based on these biological facts, a possible synchronization mech-anism of GnRH neurons was proposed in [17, 32, 47]. In this model, it isassumed the the GnRH neurons are coupled through GnRH in the extracel-lular environment. This model was able to predict that oscillations occurover a one hour period. Assuming two neurons, one each at x = 0 and at242D.2. The GnRH Modelx = 2L, this model system isgt = Dgxx− g , t > 0 , 0 < x < L ; gx(L, t) = 0 , Dgx(0) = −σj ,(D.2.1a)with the three-component membrane dynamicsαt = φα([g(0, t)]nαknαα + [g(0, t)]nα, α), α = {s, q, i} . (D.2.1b)with coefficients ns = 4, nq = 2 and ni = 2. In (D.2.1a), σ reflects thesecretion efficiency, and the boundary flux isj ≡ 1 + β(ι+ 1 + ζqµ+ 1 + δq)3(η +sω + i)3, (D.2.1c)(see [32, 47] for further details and definition of the parameters). In (D.2.1a),s, q and i represent the concentration of three G-proteins, GS , GQ andGI , respectively. It is postulated that the release of GnRH is mediatedthrough activation (GS , GQ) and inhibition (GI) of these proteins. Withthe assumption that the time scales of s and q are much faster than i, we usea quasi-steady state approximation to fix s and q at (approximately) theirsteady-state values. This leads to the following reduced coupled systemgt = Dgxx−g , t > 0 , 0 < x < L ; gx(L, t) = 0 , Dgx(0) = G(g(0, t), i),(D.2.2a)with the one-component membrane dynamics and boundary flux given byit = ([g(0, t)]2k2i + [g(0, t)]2− i),G(g(0, t), i)= −σ[1 + β(ι+ 1 + ζqµ+ 1 + δq)3(η +sω + i)3],(D.2.2b)Here, s and q, which depend on g(0, t), are given bys =[g(0, t)]4k4s + [g(0, t)]4, q =[g(0, t)]2k2q + [g(0, t)]2. (D.2.2c)243

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