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Spike pinning for the Gierer-Meinhardt model. (English) Zbl 0979.35142

Summary: The pinning effect induced by two different types of spatial inhomogeneities on the dynamics and equilibria of one-spike solution to the one-dimensional Gierer-Meinhardt (GM) activator-inhibitor model of morphogenesis is studied. The first problem that is treated is the shadow problem that results from taking the infinite inhibitor diffusivity limit in the GM model. For this problem, we show that an exponentially weak spatially varying activator diffusivity can stabilize an equilibrium spike-layer solution that would necessarily be unstable when the activator diffusivity was spatially uniform. The second problem that is treated is the full GM model in the presence of a spatially varying inhibitor decay rate. For this problem, we show that the equilibrium location of a one-spike solution depends on certain global properties of the inhibitor decay rate over the domain.

MSC:

35Q80 Applications of PDE in areas other than physics (MSC2000)
92D10 Genetics and epigenetics

Software:

nag; NAG; d03pcf
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References:

[1] Chapman, S.J.; Richardson, G., Vortex pinning by inhomogeneities in type-2 superconductors, Physica D, 108, 397-407, (1997) · Zbl 1039.82510
[2] X. Chen, M. Kowalczyk, Slow Dynamics of Interior Spikes in the Shadow Gierer-Meinhardt System, Center for Nonlinear Analysis Report No. 99-CNA-002, Carnegie-Mellon University, 1999, preprint.
[3] Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Kybernetik, 12, 30-39, (1972)
[4] Harrison, L.; Holloway, D., Order and localization in reaction – diffusion pattern, Physica A, 222, 210-233, (1995)
[5] Iron, D.; Ward, M.J., A metastable spike solution for a non-local reaction-diffusion model, SIAM J. appl. math., 60, 3, 778-802, (2000) · Zbl 0956.35011
[6] Lin, F.H.; Du, Q., Ginzburg – landau vortices: dynamics, pinning and hysteresis, SIAM J. math. anal., 28, 1265-1293, (1997) · Zbl 0888.35054
[7] H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, 1982.
[8] NAG Fortran Library Mark 17, Routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom, UK, 1995.
[9] Ni, W., Diffusion, cross-diffusion, and their spike-layer steady-states, Notices AMS, 45, 1, 9-18, (1998) · Zbl 0917.35047
[10] Sun, X.; Ward, M.J., Metastability and pinning for convection-diffusion-reaction equations in thin domains, Methods appl. anal., 6, 4, 451-476, (2000)
[11] Wei, J., On single interior spike solutions for the gierer – meinhardt system: uniqueness and stability estimates, Eur. J. appl. math., 10, 4, 353-378, (1999) · Zbl 1014.35005
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