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June  2018, 23(4): 1431-1458. doi: 10.3934/dcdsb.2018158

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

 1 Oxford Center for Industrial and Applied Mathematics (OCIAM), Oxford University, Oxford, UK 2 Dept. of Mathematics, University of British Columbia, Vancouver, B.C., Canada

*Corresponding author: Michael Ward

Received  November 2016 Revised  January 2018 Published  April 2018

We study the spectrum of a new class of nonlocal eigenvalue problems (NLEPs) that characterize the linear stability properties of localized spike solutions to the singularly perturbed two-component Gierer-Meinhardt (GM) reaction-diffusion (RD) system with a fixed time-delay $T$ in only the nonlinear autocatalytic activator kinetics. Our analysis of this model is motivated by the computational study of Seirin Lee et al. [Bull. Math. Bio., 72(8), (2010)] on the effect of gene expression time delays on spatial patterning for both the GM and some related RD models. For various limiting forms of the GM model, we show from a numerical study of the associated NLEP, together with an analytical scaling law analysis valid for large delay $T$, that a time-delay in only the activator kinetics is stabilizing in the sense that there is a wider region of parameter space where the spike solution is linearly stable than when there is no time delay. This enhanced stability behavior with a delayed activator kinetics is in marked contrast to the de-stabilizing effect on spike solutions of having a time-delay in both the activator and inhibitor kinetics. Numerical results computed from the RD system with delayed activator kinetics are used to validate the theory for the 1-D case.

Citation: Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158
##### References:
 [1] S. Chen and J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonlinear Analysis: Real World Applic, 14 (2013), 1871-1886.  doi: 10.1016/j.nonrwa.2012.12.004.  Google Scholar [2] S. Chen, J. Shi and J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Nonl. Sci., 23 (2013), 1-38.  doi: 10.1007/s00332-012-9138-1.  Google Scholar [3] E. N. Dancer, On stability and Hopf bifurcations for chemotaxis systems, Meth. Applic. of Anal., 8 (2001), 245-256.  doi: 10.4310/MAA.2001.v8.n2.a3.  Google Scholar [4] A. Doelman, A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotic approach, Physica D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [5] A. Doelman, R. A. Gardner and T. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana U. Math. Journ., 50 (2001), 443-507.  doi: 10.1512/iumj.2001.50.1873.  Google Scholar [6] N. T. Fadai, M. J. Ward and J. Wei, Delayed reaction-kinetics and the stability of spikes in the Gierer-Meinhardt Model, SIAM J.Appl.Math., 77 (2017), 664-696.  doi: 10.1137/16M1063460.  Google Scholar [7] E. A. Gaffney and N. A. M. Monk, Gene expression time delays and Turing pattern formation systems, Bull. Math. Bio., 68 (2006), 99-130.  doi: 10.1007/s11538-006-9066-z.  Google Scholar [8] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar [9] I. Gokhberg and M. Krein, Fundamental theorems on deficiency indices, root numbers and indices of linear operators, Amer Math Soc. Transls., 2 (1960), 185-264.   Google Scholar [10] D. Iron and M. J. Ward, A metastable spike solution for a non-local reaction-diffusion model, SIAM J. Appl. Math., 60 (2000), 778-802.  doi: 10.1137/S0036139998338340.  Google Scholar [11] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar [12] D. Iron and M. J. Ward, The dynamics of multi-spike solutions for the one-dimensional Gierer-Meinhardt model, SIAM J.Appl.Math., 62 (2002), 1924-1951.  doi: 10.1137/S0036139901393676.  Google Scholar [13] T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976.   Google Scholar [14] T. Koloklonikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Sys., 5 (2006), 313-363.  doi: 10.1137/050635080.  Google Scholar [15] S. Lee, E. A. Gaffney and N. A. M. Monk, The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems, Bull. Math. Bio., 72 (2010), 2139-2160.  doi: 10.1007/s11538-010-9532-5.  Google Scholar [16] S. Lee and E. A. Gaffney, Aberrant behaviors of reaction-diffusion self-organization models on growing domains in the presence of gene expression time delays, Bull. Math. Bio., 72 (2010), 2161-2179.  doi: 10.1007/s11538-010-9533-4.  Google Scholar [17] S. Lee, E. A. Gaffney and R. E. Baker, The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays, Bull. Math. Bio., 73 (2011), 2527-2551.  doi: 10.1007/s11538-011-9634-8.  Google Scholar [18] P. Maini, T. Woolley, R. E. Baker, E. A. Gaffney and S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496.  doi: 10.1098/rsfs.2011.0113.  Google Scholar [19] I. Moyles and M. J. Ward, Existence, stability, and dynamics of ring and near-ring Solutions to the Gierer-Meinhardt model in the semi-strong regime, SIAM J.Appl.Dyn.Sys., 16 (2017), 597-639.  doi: 10.1137/16M1060327.  Google Scholar [20] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, U.K., 1966.   Google Scholar [21] A. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 327 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar [22] M. J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model, Europ.J.Appl.Math., 14 (2003), 677-711.  doi: 10.1017/S0956792503005278.  Google Scholar [23] M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonl. Sci., 13 (2003), 209-264.  doi: 10.1007/s00332-002-0531-z.  Google Scholar [24] J. Wei, On single interior spike solutions for the Gierer-Meinhardt system: uniqueness and spectrum estimates, Europ.J.Appl.Math., 10 (1999), 353-378.  doi: 10.1017/S0956792599003770.  Google Scholar [25] J. Wei and M. Winter, Critical threshold and stability of cluster solutions for large reaction-diffusion systems in $\mathbb{R}^1$, SIAM J. Math. Anal., 33 (2002), 1058-1089.  doi: 10.1137/S0036141000381704.  Google Scholar [26] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonl. Sci., 11 (2001), 415-458.  doi: 10.1007/s00332-001-0380-1.  Google Scholar

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##### References:
 [1] S. Chen and J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonlinear Analysis: Real World Applic, 14 (2013), 1871-1886.  doi: 10.1016/j.nonrwa.2012.12.004.  Google Scholar [2] S. Chen, J. Shi and J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Nonl. Sci., 23 (2013), 1-38.  doi: 10.1007/s00332-012-9138-1.  Google Scholar [3] E. N. Dancer, On stability and Hopf bifurcations for chemotaxis systems, Meth. Applic. of Anal., 8 (2001), 245-256.  doi: 10.4310/MAA.2001.v8.n2.a3.  Google Scholar [4] A. Doelman, A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotic approach, Physica D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [5] A. Doelman, R. A. Gardner and T. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana U. Math. Journ., 50 (2001), 443-507.  doi: 10.1512/iumj.2001.50.1873.  Google Scholar [6] N. T. Fadai, M. J. Ward and J. Wei, Delayed reaction-kinetics and the stability of spikes in the Gierer-Meinhardt Model, SIAM J.Appl.Math., 77 (2017), 664-696.  doi: 10.1137/16M1063460.  Google Scholar [7] E. A. Gaffney and N. A. M. Monk, Gene expression time delays and Turing pattern formation systems, Bull. Math. Bio., 68 (2006), 99-130.  doi: 10.1007/s11538-006-9066-z.  Google Scholar [8] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar [9] I. Gokhberg and M. Krein, Fundamental theorems on deficiency indices, root numbers and indices of linear operators, Amer Math Soc. Transls., 2 (1960), 185-264.   Google Scholar [10] D. Iron and M. J. Ward, A metastable spike solution for a non-local reaction-diffusion model, SIAM J. Appl. Math., 60 (2000), 778-802.  doi: 10.1137/S0036139998338340.  Google Scholar [11] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62.  doi: 10.1016/S0167-2789(00)00206-2.  Google Scholar [12] D. Iron and M. J. Ward, The dynamics of multi-spike solutions for the one-dimensional Gierer-Meinhardt model, SIAM J.Appl.Math., 62 (2002), 1924-1951.  doi: 10.1137/S0036139901393676.  Google Scholar [13] T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976.   Google Scholar [14] T. Koloklonikov, W. Sun, M. J. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Sys., 5 (2006), 313-363.  doi: 10.1137/050635080.  Google Scholar [15] S. Lee, E. A. Gaffney and N. A. M. Monk, The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems, Bull. Math. Bio., 72 (2010), 2139-2160.  doi: 10.1007/s11538-010-9532-5.  Google Scholar [16] S. Lee and E. A. Gaffney, Aberrant behaviors of reaction-diffusion self-organization models on growing domains in the presence of gene expression time delays, Bull. Math. Bio., 72 (2010), 2161-2179.  doi: 10.1007/s11538-010-9533-4.  Google Scholar [17] S. Lee, E. A. Gaffney and R. E. Baker, The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays, Bull. Math. Bio., 73 (2011), 2527-2551.  doi: 10.1007/s11538-011-9634-8.  Google Scholar [18] P. Maini, T. Woolley, R. E. Baker, E. A. Gaffney and S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496.  doi: 10.1098/rsfs.2011.0113.  Google Scholar [19] I. Moyles and M. J. Ward, Existence, stability, and dynamics of ring and near-ring Solutions to the Gierer-Meinhardt model in the semi-strong regime, SIAM J.Appl.Dyn.Sys., 16 (2017), 597-639.  doi: 10.1137/16M1060327.  Google Scholar [20] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, U.K., 1966.   Google Scholar [21] A. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 327 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar [22] M. J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model, Europ.J.Appl.Math., 14 (2003), 677-711.  doi: 10.1017/S0956792503005278.  Google Scholar [23] M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonl. Sci., 13 (2003), 209-264.  doi: 10.1007/s00332-002-0531-z.  Google Scholar [24] J. Wei, On single interior spike solutions for the Gierer-Meinhardt system: uniqueness and spectrum estimates, Europ.J.Appl.Math., 10 (1999), 353-378.  doi: 10.1017/S0956792599003770.  Google Scholar [25] J. Wei and M. Winter, Critical threshold and stability of cluster solutions for large reaction-diffusion systems in $\mathbb{R}^1$, SIAM J. Math. Anal., 33 (2002), 1058-1089.  doi: 10.1137/S0036141000381704.  Google Scholar [26] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonl. Sci., 11 (2001), 415-458.  doi: 10.1007/s00332-001-0380-1.  Google Scholar
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
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
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
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
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)
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)
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
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)
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 $T = 10$
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 $T\geq 0$
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 $\beta\to 1^{-}$
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