arXiv subjects

Michael J. Ward

Publications and source records attributed to Michael J. Ward.

At least 19 recordsLinked to original sources

arXiv2026

The Asymptotic Analysis of Some PDE and Steklov Eigenvalue Problems with Partially Reactive Patches in 3-D

We consider steady-state diffusion in a three-dimensional bounded domain with a smooth reflecting boundary that is partially covered by small partially reactive patches. By using the method of matched asymptotic expansions, we investigate the competition of these patches for a diffusing particle and the crucial role of surface reactions on these targets. After a brief overview of former contributions to this field, we first illustrate our approach by considering the classical problems of the mean first-reaction time (MFRT) and the splitting probability for partially reactive patches characterized by a Robin boundary condition. For a spherical domain, we derive a three-term asymptotic expansion for the MFRT and splitting probabilities in the small-patch limit. This expansion is valid for arbitrary reactivities, and also accounts for the effect of the spatial configuration of patches on the boundary. Secondly, we consider more intricate surface reactions modeled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behavior of the eigenvalues and eigenfunctions for these spectral problems in the small-patch limit for a spherical domain. Extensions of these asymptotic results to arbitrary domains and their physical applications are discussed.

math.AP↗

arXiv2026

The Effective Reactivity for Capturing Brownian Motion by Partially Reactive Patches on a Spherical Surface

We analyze the trapping of diffusing ligands, modeled as Brownian particles, by a sphere that has $N$ partially reactive boundary patches, each of small area and arbitrary shape, on an otherwise reflecting boundary. For such a structured target, the partial reactivity of each boundary patch is characterized by a Robin boundary condition, with a local boundary reactivity $κ_i$ for $i=1,\ldots,N$. For any spatial arrangement of well-separated patches on the surface of the sphere, the method of matched asymptotic expansions is used to derive explicit results for the capacitance $C_{\rm T}$ of the structured target, which is valid for any $κ_i>0$. This target capacitance $C_{\rm T}$ is defined in terms of a Green's matrix, which depends on the spatial configuration of patches, the local reactive capacitance $C_i(κ_i)$ of each patch and another coefficient that depends on the local geometry near a patch. The analytical dependence of $C_{i}(κ_i)$ on $κ_i$ is uncovered via a spectral expansion over Steklov eigenfunctions. For circular patches, the latter are readily computed numerically and provide an accurate fully explicit sigmoidal approximation for $C_{i}(κ_i)$. In the homogenization limit of $N\gg 1$ identical uniformly-spaced patches with $κ_i=κ$, we derive an explicit scaling law for the effective capacitance and the effective reactivity of the structured target that is valid in the limit of small patch area fraction. From a comparison with numerical simulations, we show that this scaling law provides a highly accurate approximation over the full range $κ>0$, even when there is only a moderately large number of reactive patches.

math.AP↗

arXiv2025

Competition of small targets in planar domains: from Dirichlet to Robin and Steklov boundary condition

We consider steady-state diffusion in a bounded planar domain with multiple small targets on a smooth boundary. Using the method of matched asymptotic expansions, we investigate the competition of these targets for a diffusing particle and the crucial role of surface reactions on the targets. We start from the classical problem of splitting probabilities for perfectly reactive targets with Dirichlet boundary condition and improve some earlier results. We discuss how this approach can be generalized to partially reactive targets characterized by a Robin boundary condition. In particular, we show how partial reactivity reduces the effective size of the target. In addition, we consider more intricate surface reactions modeled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behavior of the eigenvalues and eigenfunctions for these spectral problems in the small-target limit. Finally, we show how our asymptotic approach can be extended to interior targets in the bulk and to exterior problems where diffusion occurs in an unbounded planar domain outside a compact set. Direct applications of these results to diffusion-controlled reactions are discussed.

math.AP↗

arXiv2025

Neural encoding of real world face perception

Social perception unfolds as we freely interact with people around us. We investigated the neural basis of real world face perception using multi electrode intracranial recordings in humans during spontaneous interactions with friends, family, and others. Computational models reconstructed the faces participants looked at during natural interactions, including facial expressions and motion, from brain activity alone. The results highlighted a critical role for the social vision pathway, a network of areas spanning parietal, temporal, and occipital cortex. This network was more sharply tuned to subtle expressions compared to intense expressions, which was confirmed with controlled psychophysical experiments. These findings reveal that the human social vision pathway encodes facial expressions and motion as deviations from a neutral expression prototype during natural social interactions in real life.

q-bio.NC↗

arXiv2024

An Asymptotic Analysis of Spike Self-Replication and Spike Nucleation of Reaction-Diffusion Patterns on Growing 1-D Domains

In the asymptotic limit of a large diffusivity ratio, certain two-component reaction-diffusion (RD) systems can admit localized spike solutions on a 1-D finite domain in a far-from-equilibrium nonlinear regime. It is known that two distinct bifurcation mechanisms can occur which generate spike patterns of increased spatial complexity as the domain half-length L slowly increases; so-called spike nucleation and spike self-replication. Self-replication is found to occur via the passage beyond a saddle-node bifurcation point that can be predicted through linearization around the inner spike profile. In contrast, spike nucleation occurs through slow passage beyond the saddle-node of a nonlinear boundary-value problem defined in the outer region away from the core of a spike. Here, by treating L as a static parameter, precise conditions are established within the semi-strong interaction asymptotic regime to determine which occurs, conditions that are confirmed by numerical simulation and continuation. For the Schnakenberg and Brusselator RD models, phase diagrams in parameter space are derived that predict whether spike self-replication or spike nucleation will occur first as L is increased, or whether no such instability will occur. For the Gierer-Meinhardt model with a non-trivial activator background, spike nucleation is shown to be the only possible spike-generating mechanism. From time-dependent PDE numerical results on an exponentially slowly growing domain, it is shown that the analytical thresholds derived from the asymptotic theory accurately predict critical values of L where either spike self-replication or spike-nucleation will occur. The global bifurcation mechanism for transitions to patterns of increased spatial complexity is further elucidated by superimposing time-dependent PDE simulation results on the numerically computed solution branches of spike equilibria.

math.AP↗

arXiv2024

Synchronized Memory-Dependent Intracellular Oscillations for a Cell-Bulk ODE-PDE Model in $\mathbb{R}^2$

For a cell-bulk ODE-PDE model in $\mathbb{R}^2$, a hybrid asymptotic-numerical theory is developed to provide a new theoretical and computationally efficient approach for studying how oscillatory dynamics associated with spatially segregated dynamically active ``units" or ``cells" are regulated by a PDE bulk diffusion field that is both produced and absorbed by the entire cell population. The study of oscillator synchronization in a PDE diffusion field was one of the initial aims of Yoshiki Kuramoto's foundational work. For this cell-bulk model, strong localized perturbation theory, as extended to a time-dependent setting, is used to derive a new integro-differential ODE system that characterizes intracellular dynamics in a memory-dependent bulk-diffusion field. For this nonlocal reduced system, a novel fast time-marching scheme, relying in part on the \emph{sum-of-exponentials method} to numerically treat convolution integrals, is developed to rapidly and accurately compute numerical solutions to the integro-differential system over long time intervals. For the special case of Sel'kov reaction kinetics, a wide variety of large-scale oscillatory dynamical behavior including phase synchronization, mixed-mode oscillations, and quorum-sensing are illustrated for various ranges of the influx and efflux permeability parameters, the bulk degradation rate and bulk diffusivity, and the specific spatial configuration of cells. Results from our fast algorithm, obtained in under one minute of CPU time on a laptop, are benchmarked against PDE simulations of the cell-bulk model, which are performed with a commercial PDE solver, that have run-times that are orders of magnitude larger.

nlin.PS↗

arXiv2023

Symmetry-Breaking Bifurcations for Compartmental Reaction Kinetics Coupled by Two Bulk Diffusing Species with Comparable Diffusivities in 2-D

For a 2-D coupled PDE-ODE bulk-cell model, we investigate symmetry-breaking bifurcations that can emerge when two bulk diffusing species are coupled to two-component nonlinear intracellular reactions that are restricted to occur only within a disjoint collection of small circular compartments, or "cells", of a common small radius that are confined in a bounded 2-D domain. Outside of the union of these cells, the two bulk species with comparable diffusivities and bulk degradation rates diffuse and globally couple the spatially segregated intracellular reactions through Robin boundary conditions across the cell boundaries, which depend on certain membrane reaction rates. In the singular limit of a small common cell radius, we construct steady-state solutions for the bulk-cell model and formulate a nonlinear matrix eigenvalue problem that determines the linear stability properties of the steady-states. For a certain spatial arrangement of cells for which the steady-state and linear stability analysis become highly tractable, we construct a symmetric steady-state solution where the steady-states of the intracellular species are the same for each cell. As regulated by the ratio of the membrane reaction rates on the cell boundaries, we show for various specific prototypical intracellular reactions, and for a specific two-cell arrangement, that our 2-D coupled PDE-ODE model admits symmetry-breaking bifurcations from this symmetric steady-state, leading to linearly stable asymmetric patterns, even when the bulk diffusing species have comparable or possibly equal diffusivities. Overall, our analysis shows that symmetry-breaking bifurcations can occur without the large diffusivity ratio requirement for the bulk diffusing species as is well-known from a Turing stability analysis applied to a spatially uniform steady-state for typical two-component activator-inhibitor systems.

nlin.PS↗

arXiv2023

The Emergence of Spatial Patterns for Compartmental Reaction Kinetics Coupled by Two Bulk Diffusing Species with Comparable Diffusivities

Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns can typically only occur from a spatially uniform state when a slowly diffusing "activator" species reacts with a much faster diffusing "inhibitor" species. However, from a modeling perspective, this large diffusivity ratio requirement for pattern formation is often unrealistic in biological settings since different molecules tend to diffuse with similar rates in extracellular spaces. As a result, one key long-standing question is how to robustly obtain pattern formation in the biologically realistic case where the time scales for diffusion of the interacting species are comparable. For a coupled 1-D bulk-compartment theoretical model, we investigate the emergence of spatial patterns for the scenario where two bulk diffusing species with comparable diffusivities are coupled to nonlinear reactions that occur only in localized "compartments", such as on the boundaries of a 1-D domain. The exchange between the bulk medium and the spatially localized compartments is modeled by a Robin boundary condition with certain binding rates. As regulated by these binding rates, we show for various specific nonlinearities that our 1-D coupled PDE-ODE model admits symmetry-breaking bifurcations, leading to linearly stable asymmetric steady-state patterns, even when the bulk diffusing species have equal diffusivities. Depending on the form of the nonlinear kinetics, oscillatory instabilities can also be triggered. Moreover, the analysis is extended to treat a periodic chain of compartments.

nlin.PS↗

arXiv2020

Weakly Nonlinear Analysis of Peanut-Shaped Deformations for Localized Spots of Singularly Perturbed Reaction-Diffusion Systems

Spatially localized 2-D spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio. Such localized, far-from-equilibrium, patterns are known to exhibit a wide range of different instabilities such as breathing oscillations, spot annihilation, and spot self-replication behavior. Prior numerical simulations of the Schnakenberg and Brusselator systems have suggested that a localized peanut-shaped linear instability of a localized spot is the mechanism initiating a fully nonlinear spot self-replication event. From a development and implementation of a weakly nonlinear theory for shape deformations of a localized spot, it is shown through a normal form amplitude equation that a peanut-shaped linear instability of a steady-state spot solution is always subcritical for both the Schnakenberg and Brusselator reaction-diffusion systems. The weakly nonlinear theory is validated by using the global bifurcation software {\em pde2path} [H.~Uecker et al., Numerical Mathematics: Theory, Methods and Applications, {\bf 7}(1), (2014)] to numerically compute an unstable, non-radially symmetric, steady-state spot solution branch that originates from a symmetry-breaking bifurcation point.

nlin.PS↗

arXiv2020

Spot Patterns in the 2-D Schnakenberg Model with Localized Heterogeneities

A hybrid asymptotic-numerical theory is developed to analyze the effect of different types of localized heterogeneities on the existence, linear stability, and slow dynamics of localized spot patterns for the two-component Schnakenberg reaction-diffusion model in a 2-D domain. Two distinct types of localized heterogeneities are considered: a strong localized perturbation of a spatially uniform feed rate and the effect of removing a small hole in the domain, through which the chemical species can leak out. Our hybrid theory reveals a wide range of novel phenomena such as, saddle-node bifurcations for quasi-equilibrium spot patterns that otherwise would not occur for a homogeneous medium, a new type of spot solution pinned at the concentration point of the feed rate, spot self-replication behavior leading to the creation of more than two new spots, and the existence of a creation-annihilation attractor with at most three spots. Depending on the type of localized heterogeneity introduced, localized spots are either repelled or attracted towards the localized defect on asymptotically long time scales. Results for slow spot dynamics and detailed predictions of various instabilities of quasi-equilibrium spot patterns, all based on our hybrid asymptotic-numerical theory, are illustrated and confirmed through extensive full PDE numerical simulations.

nlin.PS↗

arXiv2020

An Asymptotic Analysis of Localized 3-D Spot Patterns for Gierer-Meinhardt Model: Existence, Linear Stability and Slow Dynamics

Localized spot patterns, where one or more solution components concentrates at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. In an arbitrary bounded 3-D domain, the existence, linear stability, and slow dynamics of localized multi-spot patterns is analyzed for the well-known singularly perturbed Gierer-Meinhardt (GM) activator-inhibitor system in the limit of a small activator diffusivity $\varepsilon^2\ll 1$. Our main focus is to classify the different types of multi-spot patterns, and predict their linear stability properties, for different asymptotic ranges of the inhibitor diffusivity $D$. For the range $D={\mathcal O}(\varepsilon^{-1})\gg 1$, although both symmetric and asymmetric quasi-equilibrium spot patterns can be constructed, the asymmetric patterns are shown to be always unstable. On this range of $D$, it is shown that symmetric spot patterns can undergo either competition instabilities or a Hopf bifurcation, leading to spot annihilation or temporal spot amplitude oscillations, respectively. For $D={\mathcal O}(1)$, only symmetric spot quasi-equilibria exist and they are linearly stable on ${\mathcal O}(1)$ time intervals. On this range, it is shown that the spot locations evolve slowly on an ${\mathcal O}(\varepsilon^{-3})$ time scale towards their equilibrium locations according to an ODE gradient flow, which is determined by a discrete energy involving the reduced-wave Green's function. The central role of the far-field behavior of a certain core problem, which characterizes the profile of a localized spot, for the construction of quasi-equilibria in the $D={\mathcal O}(1)$ and $D={\mathcal O}(\varepsilon^{-1})$ regimes, and in establishing some of their linear stability properties, is emphasized.

nlin.PS↗

arXiv2020

Pattern Formation and Oscillatory Dynamics in a Two-Dimensional Coupled Bulk-Surface Reaction-Diffusion System

On a two-dimensional circular domain, we analyze the formation of spatio-temporal patterns for a class of coupled bulk-surface reaction-diffusion models for which a passive diffusion process occurring in the interior bulk domain is linearly coupled to a nonlinear reaction-diffusion process on the domain boundary. For this coupled PDE system we construct a radially symmetric steady state solution and from a linearized stability analysis formulate criteria for which this base state can undergo either a Hopf bifurcation, a symmetry-breaking pitchfork (or Turing) bifurcation, or a codimension-two pitchfork-Hopf bifurcation. For each of these three types of bifurcations, a multiple time-scale asymptotic analysis is used to derive normal form amplitude equations characterizing the local branching behavior of spatio-temporal patterns in the weakly nonlinear regime. Among the novel aspects of this weakly nonlinear analysis are the two-dimensionality of the bulk domain, the systematic treatment of arbitrary reaction kinetics restricted to the boundary, the bifurcation parameters which arise in the boundary conditions, and the underlying spectral problem where both the differential operator and the boundary conditions involve the eigenvalue parameter. The normal form theory is illustrated for both Schnakenberg and Brusselator reaction kinetics, and the weakly nonlinear results are favorably compared with numerical bifurcation results and results from time-dependent PDE simulations of the coupled bulk-surface system. Overall, the results show the existence of either subcritical or supercritical Hopf and symmetry-breaking bifurcations, and mixed-mode oscillations characteristic of codimension-two bifurcations. Finally, the formation of global structures such as large amplitude rotating waves is briefly explored through PDE numerical simulations.

nlin.PS↗

arXiv2020

Weakly Nonlinear Theory for Oscillatory Dynamics in a One-Dimensional PDE-ODE Model of Membrane Dynamics Coupled by a Bulk Diffusion Field

We study the dynamics of systems consisting of two spatially segregated ODE compartments coupled through a one-dimensional bulk diffusion field. For this coupled PDE-ODE system, we first employ a multi-scale asymptotic expansion to derive amplitude equations near codimension-one Hopf bifurcation points for both in-phase and anti-phase synchronization modes. The resulting normal form equations pertain to any vector nonlinearity restricted to the ODE compartments. In our first example, we apply our weakly nonlinear theory to a coupled PDE-ODE system with Sel'kov membrane kinetics, and show that the symmetric steady state undergoes supercritical Hopf bifurcations as the coupling strength and the diffusivity vary. We then consider the PDE diffusive coupling of two Lorenz oscillators. It is shown that this coupling mechanism can have a stabilizing effect, characterized by a significant increase in the Rayleigh number required for a Hopf bifurcation. Within the chaotic regime, we can distinguish between synchronous chaos, where both the left and right oscillators are in-phase, and chaotic states characterized by the absence of synchrony. Finally, we compute the largest Lyapunov exponent associated with a linearization around the synchronous manifold that only considers odd perturbations. This allows us to predict the transition to synchronous chaos as the coupling strength and the diffusivity increase.

nlin.CD↗

arXiv2020

Synchrony and Oscillatory Dynamics for a 2-D PDE-ODE Model of Diffusion-Sensing with Small Signaling Compartments

We analyze a class of cell-bulk coupled PDE-ODE models, motivated by quorum and diffusion sensing phenomena in microbial systems, that characterize communication between localized spatially segregated dynamically active signaling compartments that have a permeable boundary. Each cell secretes a signaling chemical into the bulk region at a constant rate and receives a feedback of the bulk chemical from the entire collection of cells. This global feedback, which activates signaling pathways within the cells, modifies the intracellular dynamics according to the external environment. The cell secretion and global feedback are regulated by permeability parameters across the cell membrane. For arbitrary reaction-kinetics within each cell, the method of matched asymptotic expansions is used in the limit of small cell radius to construct steady-state solutions of the PDE-ODE model, and to derive a globally coupled nonlinear matrix eigenvalue problem (GCEP) that characterizes the linear stability properties of the steady-states. In the limit of large bulk diffusivity an asymptotic analysis of the PDE-ODE model leads to a limiting ODE system for the spatial average of the concentration in the bulk region that is coupled to the intracellular dynamics within the cells. Results from the linear stability theory and ODE dynamics are illustrated for Sel'kov reaction-kinetics, where the kinetic parameters are chosen so that each cell is quiescent when uncoupled from the bulk medium. For various specific spatial configurations of cells, the linear stability theory is used to construct phase diagrams in parameter space characterizing where a switch-like emergence of intracellular oscillations can occur through a Hopf bifurcation.

nlin.PS↗

arXiv2020

Localized Signaling Compartments in 2-D Coupled by a Bulk Diffusion Field: Quorum Sensing and Synchronous Oscillations in the Well-Mixed Limit

We analyze oscillatory instabilities for a coupled PDE-ODE system modeling the communication between localized spatially segregated dynamically active signaling compartments that are coupled through a passive extracellular bulk diffusion field in a bounded 2-D domain. Each signaling compartment is assumed to secrete a chemical into the extracellular medium (bulk region) and it can also sense the concentration of this chemical in the region around its boundary. This feedback from the bulk region, resulting from the entire collection of cells, in turn modifies the intracellular dynamics within each cell. In the limit where the signaling compartments are circular disks with a small common radius $ε\ll 1$ and where the bulk diffusivity is asymptotically large, a matched asymptotic analysis is used to reduce the dimensionless PDE-ODE system into a nonlinear ODE system with global coupling. For Sel'kov reaction kinetics, this ODE system for the intracellular dynamics and the spatial average of the bulk diffusion field is then used to investigate oscillatory instabilities in the dynamics of the cells that are triggered due to the global coupling. In particular, numerical bifurcation software on the ODEs is used to study the overall effect of coupling defective cells (cells that behave differently from the remaining cells) to a group of identical cells. Moreover, when the number of cells is large, the Kuramoto order parameter is computed to predict the degree of phase synchronization of the intracellular dynamics. Quorum sensing behavior, characterized by the onset of collective behavior in the intracellular dynamics as the number of cells increases above a threshold, is also studied. Our analysis shows that the cell population density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics.

nlin.PS↗

arXiv2020

Stable Asymmetric Spike Equilibria for the Gierer-Meinhardt Model with a Precursor Field

Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction-kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1-D Gierer-Meinhardt (GM) model we show that a simple precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric, two-spike patterns, corresponding to localized peaks in the activator of different heights. This is a qualitatively new phenomena for the GM model, in that asymmetric spike patterns are all unstable in the absence of the precursor field. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach we analyze the spectrum of the linearization of the GM model around the two-spike steady-state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis a new class of vector-valued nonlocal eigenvalue problem (NLEP) is derived and analyzed.

nlin.PS↗

arXiv2019

Simulation and Optimization of Mean First Passage Time Problems in 2-D using Numerical Embedded Methods and Perturbation Theory

We develop novel numerical methods and perturbation approaches to determine the mean first passage time (MFPT) for a Brownian particle to be captured by either small stationary or mobile traps inside a bounded 2-D confining domain. Of particular interest is to identify optimal arrangements of small absorbing traps that minimize the average MFPT. Although the MFPT, and the associated optimal trap arrangement problem, has been well-studied for disk-shaped domains, there are very few analytical or numerical results available for general star-shaped domains or for thin domains with large aspect ratio. Analytical progress is challenging owing to the need to determine the Neumann Green's function for the Laplacian, while the numerical challenge results from a lack of easy-to-use and fast numerical tools for first computing the MFPT and then optimizing over a class of trap configurations. In this direction, and for the stationary trap problem, we develop a simple embedded numerical method, based on the Closest Point Method (CPM), to perform MFPT simulations on elliptical and star-shaped domains. For periodic mobile trap problems, we develop a robust CPM method to compute the average MFPT. Optimal trap arrangements are identified numerically through either a refined discrete sampling approach or from a particle-swarm optimization procedure. To confirm some of the numerical findings, novel perturbation approaches are developed to approximate the average MFPT and identify optimal trap configurations for a class of near-disk confining domains or for an arbitrary thin domain of large aspect ratio.

math.NA↗

arXiv2018

The Linear Stability of Symmetric Spike Patterns for a Bulk-Membrane Coupled Gierer-Meinhardt Model

We analyze a coupled bulk-membrane PDE model in which a scalar linear 2-D bulk diffusion process is coupled through a linear Robin boundary condition to a two-component 1-D reaction-diffusion (RD) system with Gierer-Meinhardt (nonlinear) reaction kinetics defined on the domain boundary. For this coupled model, in the singularly perturbed limit of a long-range inhibition and short-range activation for the membrane-bound species, asymptotic methods are used to analyze the existence of localized steady-state multi-spike membrane-bound patterns, and to derive a nonlocal eigenvalue problem (NLEP) characterizing $\mathcal{O}(1)$ time-scale instabilities of these patterns. A central, and novel, feature of this NLEP is that it involves a membrane Green's function that is coupled nonlocally to a bulk Green's function. When the domain is a disk, or in the well-mixed shadow-system limit corresponding to an infinite bulk diffusivity, this Green's function problem is analytically tractable, and as a result we will use a hybrid analytical-numerical approach to determine unstable spectra of this NLEP. This analysis characterizes how the 2-D bulk diffusion process and the bulk-membrane coupling modifies the well-known linear stability properties of steady-state spike patterns for the 1-D Gierer-Meinhardt model in the absence of coupling. In particular, phase diagrams in parameter space for our coupled model characterizing either oscillatory instabilities due to Hopf bifurcations, or competition instabilities due to zero-eigenvalue crossings are constructed. Finally, linear stability predictions from the NLEP analysis are confirmed with full numerical finite-element simulations of the coupled PDE system.

nlin.PS↗