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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

Nabil T. Fadai 1, , Michael J. Ward 2,, and  Juncheng Wei 2,

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  June 2018 Early access  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.

Keywords: Activator delay, spike, nonlocal eigevalue problem, Hopf bifurcation.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.

[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. 

[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.

[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.

[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.

[13] T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976. 
[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.

[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.

[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.

[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.

[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.

[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.

[20] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, U.K., 1966. 
[21]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 327 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[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.

[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.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.

[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. 

[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.

[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.

[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.

[13] T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976. 
[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.

[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.

[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.

[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.

[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.

[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.

[20] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, U.K., 1966. 
[21]

A. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 327 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[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.

[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.

[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.

[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.

[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.

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
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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
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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
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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
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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)
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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)
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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
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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)
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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 $T = 10$
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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 $T\geq 0$
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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 $\beta\to 1^{-}$
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