A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

Figure 1.  Left panel: The positive real eigenvalue $\lambda_0(T)$ of the delayed local problem $L_\mu\Phi = \lambda\Phi$ when $N = 1$ as obtained by solving (3.4) numerically with $l = 0$. Right panel: all the paths of complex-valued spectra of $L_\mu$ in $\mbox{Re}(\lambda)\geq 0$ for $T_{\textrm{H}}^{1}\leq T \leq T_{\textrm{H}}^{5}$, as computed from (3.4) for $l = 0$. Here $T_{\textrm{H}}^{j}$ for $j\geq 1$ is the $j$-th value of $T$ where $L_\mu$ has a pure imaginary eigenvalue $\lambda = i\lambda_I$ with $\lambda_I\approx 2.1015$ (see (3.10)). The path with imaginary eigenvalue when $T = T_{\textrm{H}}^j$ is labeled by $\lambda_j$. For even larger values of $T$, these paths all tend to the origin $\lambda = 0$, but in the half-space $\mbox{Re}(\lambda)>0$. For each path, we also plot its continuation into $\mbox{Re}(\lambda) < 0$ for smaller delays

Figure 2.  The positive real eigenvalue $\lambda_0(T)$ of $L_\mu\Phi = \lambda\Phi$ when $N = 2$ (solid curve), as computed numerically from a BVP solver. The asymptotic results (3.17) for small and large delay are the dashed curves

Figure 3.  Plot of the numerically computed function ${\mathcal F}_\mu(\lambda)$, defined in (3.2), when $N = 1$ (left panel) and when $N = 2$ (right panel) on the positive real axis for $0 < \lambda < \lambda_0(T)$ for three values of the delay $T$ as indicated in the figure. We confirm that ${\mathcal F}_\mu(\lambda)$ is monotone increasing on $0 < \lambda < \lambda_0(T)$, with ${\mathcal F}_\mu(\lambda)\to +\infty$ as $\lambda\to\lambda_0(T)$ from below

Figure 4.  HB threshold for (3.32) versus the constant multiplier $\chi_0$ of the nonlocal term in (3.32). Left panel: The minimum value $T_H$ of $T$ versus $\chi_0$ where a HB occurs. Right panel: The HB frequency $\lambda_{IH}$ versus $\chi_0$. A HB occurs only on $0\leq \chi_g1$ with $\lambda_{IH}\to 0^+$ and $T_H\to +\infty$ as $\chi_0\to 1^{-}$. For $\chi_0>1$ the NLEP (3.32) does not have any HB as $T$ is increased

Figure 5.  HB threshold $\tau_H$ (left panel) and frequency $\lambda_{IH}$ (solid curve in middle panel) versus $T$, as computed from (2.6) for the shadow problem (2.3). For $\tau < \tau_H$ (shaded region), the spike solution is linearly stable. The dashed curve in the middle panel is the large-delay asymptotic result for $\lambda_{IH}$ given in (4.4). Right panel: plot of ${\tau_H/T}$ (monotone decreasing solid curve) and ${\lambda_{IH}T}$ (monotone increasing solid curve), as computed from (2.6). The asymptotes (dashed lines) are the theoretically predicted limiting values $\lim_{T\to\infty} {\tau_H/T}\approx {\sqrt{3}/\pi}$ and $\lim_{T\to\infty} \lambda_{IH} T\approx {\pi/3}$, as obtained from (4.1)

Figure 6.  HB values for the infinite-line problem (2.1), with the same caption as in Fig. 5. In the middle panel, the large-delay asymptotic result (dashed curve) is $\lambda_{IH}\sim {c_0/T}$ with $c_0\approx 0.782$. In the right panel the theoretically predicted horizontal asymptotes are $\lim_{T\to\infty} {\tau_H/T}\approx 1.99$ and $\lim_{T\to\infty} \lambda_{IH} T \approx 0.782$, as obtained from (4.8)

Figure 7.  Plot of the HB threshold $\tau_H$ versus $T$ for the 1-D finite-domain problem (4.9) on $|x|\leq L$ where $L = 0.2$ (dashed curve), $L = 1$ (dashed-dotted curve), $L = 2$ (solid curve), and $L = 10$ (heavy solid curve). The threshold was computed numerically from (2.6) with $\chi(\tau\lambda)$ as given in (4.10). The one-spike solution is linear stable when $\tau < \tau_H$. The threshold for $L = 10$ closely approximates that for the infinite-line problem, which was given in the left panel of Fig. 6

Figure 8.  Plot of the amplitude $v(0, t)$ of the spike versus $t$ for $\tau = 5.3$ (left panel), $\tau = 5.6$ (middle panel), and $\tau = 10$ (right panel), as computed numerically by discretizing (4.9) with $151$ spatial meshpoints on $[-2, 2]$ with $\varepsilon = 0.05$ and $T = 2$. The theoretical HB prediction is $\tau_H\approx 5.573$ (see Fig. 7 with $L = 2$ and $T = 2$). The numerics shows a slowly decaying (growing) oscillation when $\tau = 5.3$ ($\tau = 5.6$), respectively. A large oscillation leading to a collapse of the spike occurs when $\tau$ is well-above the HB threshold (right panel)

Figure 9.  HB thresholds for the synchronous mode computed from (5.5) and (5.7). Left panel: The HB threshold $\tau_H$ versus $\beta$ for $T = 0$ (heavy solid curve). The spot pattern is linearly stable for all $\beta <1$ and for $\tau < \tau_H$ when $\beta>1$. From bottom to top, the various dashed curves are HB thresholds for $T = 1$, $T = 2$, $T = 5$ and $T = 10$. The thin solid curve is the asymptotic scaling law (5.12) for $\tau_H$ when $T = 10$. The effect of activator delay leads to a wider parameter range for stability. Right panel: plot of the HB frequency $\lambda_{IH}$ versus $\beta$ on $\beta>1$ with the same labeling as in the left panel, except that now as $\lambda_{IH}$ decreases as $T$ increases. The thin solid curve is the asymptotic scaling law (5.12) for $\lambda_{IH}$ when $T = 10$

Figure 10.  HB threshold for the asynchronous mode computed from (5.13). Left panel: The minimum value $T_H$ of $T$ versus $\beta$ where a HB occurs. Right panel: The HB frequency $\lambda_{IH}$ versus $\beta$. We observe that a HB occurs only on $\beta>1$ with $\lambda_{IH}\to 0^+$ and $T_H\to +\infty$ as $\beta\to 1^{+}$. For $\beta>1$ the NLEP (2.7) for the asynchronous mode also has a positive real eigenvalue for any $T\geq 0$. When $\beta < 1$, we predict that the mutli-spot pattern is linearly stable for any $T\geq 0$

Figure 11.  Left panel: The minimum value $T_H$ of $T$ versus $\beta$ where a HB occurs when the time delay occurs for both the activator and inhibitor kinetics, as computed numerically from (5.14). The HB threshold occurs for any $\beta>0$. Right panel: The corresponding HB threshold when the time-delay occurs only for the inhibitor, as computed numerically from the parameterization (5.16). The HB threshold only occurs on $0 < \beta < 1$, and $T_H\to {1/2}$ with $\lambda_{IH}\to 0^+$ as $\beta\to 1^{-}$