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

Chunyi Gai, Edgardo Villar-Sepulveda, Alan Champneys, Michael J. Ward

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 24 sections, 5 theorems, 161 equations, 33 figures.

Table of Contents

  1. Introduction
  2. Three example reaction-diffusion systems
  3. The main results
  4. Outline
  5. The Schnakenberg model
  6. Asymptotic construction of quasi-equilibria: The core problem
  7. Asymptotic construction of quasi-equilibria: The outer solution
  8. Asymptotics of the outer solution for $a$ small
  9. Global bifurcation diagrams
  10. Validation of asymptotic theory: Full PDE simulations
  11. The Brusselator model
  12. Asymptotic construction of quasi-equilibria: The core problem
  13. Asymptotic construction of quasi-equilibria: The outer solution
  14. Asymptotics of the outer solution for $a$ small
  15. Global bifurcation diagrams
  16. ...and 9 more sections

Key Result

Lemma 1

Suppose that $a < b$. Then, on the range of $x$ where $a < v(x) < 2 a$, we have $R_s(v) < 0$ and, consequently, $dv/dx > 0$.

Figures (33)

  • Figure 1: Phase diagrams in parameter space indicating the type of spike-generating mechanism that will occur as the domain half-length $L$ increases for the Schnakenberg (a) and Brusselator (b) models when $\varepsilon/\sqrt{D} \ll 1$. See § \ref{['sec:sch']}--§ \ref{['sec:no_nucleation']} for the details. The solid blue line $a_c(b)$ in (a), computed numerically from \ref{['sch:integral_Bfinal']} for $\varepsilon = 0.01$ and $D = 2$, agrees closely with \ref{['sch:ac_approx']}, which neglects the $\varepsilon/\sqrt{D}$ term. For the Brusselator in (b), the thresholds in $f$ are independent of the parameter $a$ when $\varepsilon/\sqrt{D} \ll 1$.
  • Figure 2: Left panel: Bifurcation diagram of spike solutions for the core problem \ref{['sch:rep_core']} obtained using pde2pathpde2path. A saddle-node bifurcation occurs at $B = B_c \approx 1.347$. The single-spike pattern is linearly stable only on the upper branch, which we refer to as the primary branch. Right panel: Spike profile of the activator $V_0$ at the four indicated points shown in the left panel. The profile has a volcano shape on the unstable lower branch.
  • Figure 3: Plots of $C_s$ and $U_0(0)$ as $B$ varies on the upper branch of the bifurcation diagram shown in Fig. \ref{['fig:bif_core']}, computed from the numerical solution to the core problem \ref{['sch:rep_core']} with \ref{['sch:far_u0']}. Observe that $C_s > 0$ up to the fold point $B_c = 1.347$ where $C_{sc} \equiv C_s(B_c) \approx 0.247$.
  • Figure 4: Left panel: The dimple eigenfunction component $\Phi = V_{0\beta}$ at the saddle-node bifurcation point $B = B_c \approx 1.347$ that corresponds to $\lambda = 0$. Right panel: $\hbox{Re}(\lambda)$ versus $\beta \equiv U_0(0) V_0(0)$ for the numerically computed dominant eigenvalues of \ref{['schnak:eig_prob']}. We observe that $\hbox{Re}(\lambda) < 0$ along the primary solution branch where $\beta_c<\beta<1.5$ with $\beta_c\approx 1.015$. In contrast, the lower branch in the left panel of Fig. \ref{['fig:bif_core']}, corresponding to the range $\beta<\beta_c$, is unstable owing to a real positive eigenvalue for \ref{['schnak:eig_prob']}.
  • Figure 5: Plot of critical threshold $a = a_c$ versus $b$ separating whether spike self-replication or spike-nucleation will occur, as computed by setting $B=B_c\approx 1.347$ and $\mu = 2 a$ in \ref{['sch:integral_Bfinal']}. Parameters: $\varepsilon = 0.01$ and $D = 2$.
  • ...and 28 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof