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Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain
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 |
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.
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. |











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