## The stability of spike solutions to the one-dimensional Gierer-Meinhardt model.(English)Zbl 0983.35020

Summary: The stability properties of an $$N$$-spike equilibrium solution to a simplified form of the Gierer-Meinhardt activator-inhibitor model in a one-dimensional domain is studied asymptotically in the limit of small activator diffusivity $$\varepsilon$$. The equilibrium solution consists of a sequence of spikes of equal height. The two classes of eigenvalues that must be considered are the $$O(1)$$ eigenvalues and the $$O(\varepsilon^2)$$ eigenvalues, which are referred to as the large and small eigenvalues, respectively. The spike pattern is stable when the parameters in the Gierer-Meinhardt model are such that both sets of eigenvalues lie in the left half-plane. For a certain range of these parameters and for $$N\geq 2$$ and $$\varepsilon\to 0$$, it is shown that the $$O(1)$$ eigenvalues are in the left half-plane only when $$D< D_N$$, where $$D_N$$ is some explicit critical value of the inhibitor diffusivity $$D$$. For $$N\geq 2$$ and $$\varepsilon\to 0$$, it is also shown that the small eigenvalues are real and that they are negative only when $$D< D^*_N$$, where $$D^*_N$$ is another critical value of $$D$$, which satisfies $$D^*_N< D_N$$. Thus, when $$N\geq 2$$ and $$\varepsilon\ll 1$$, the spike pattern is stable only when $$D< D^*_N$$. An explicit formula for $$D^*_N$$ is given. For the special case $$N= 1$$, it is shown that a one-spike equilibrium solution is stable when $$D< D_1(\varepsilon)$$, where $$D_1(\varepsilon)$$ is exponentially large as $$\varepsilon\to 0$$, and is unstable when $$D> D_1(\varepsilon)$$. An asymptotic formula for $$D_1(\varepsilon)$$ is given. Finally, the dynamics of a one-spike solution is studied by deriving a differential equation for the trajectory of the center of the spike.

### MSC:

 35B35 Stability in context of PDEs 35B25 Singular perturbations in context of PDEs 35K57 Reaction-diffusion equations

NAG; nag; d03pcf
Full Text:

### References:

  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, Preprint, 1999.  A. Doelman, Private communication.  Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Kybernetik, 12, 30-39, (1972)  Gui, C.; Wei, J., Multiple interior peak solutions for some singularly perturbed Neumann problems, J. diff. eq., 158, 1, 1-27, (1999) · Zbl 1061.35502  Harrison, L.; Holloway, D., Order and localization in reaction – diffusion pattern, Physica A, 222, 210-233, (1995)  D. Iron, M.J. Ward, A metastable spike solution for a non-local reaction – diffusion equation, SIAM J. Appl. Math. 60 (3) (2000) 778-802. · Zbl 0956.35011  D. Iron, Metastability of the Gierer-Meinhardt equations, M.Sc. Thesis, Institute for Applied Mathematics, University of British Columbia, Vancouver, BC, 1997.  A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics, Vol. 15, Cambridge University Press, Cambridge, 1996, pp. 197-199.  Keener, J., Activators and inhibitors in pattern formation, Stud. appl. math., 59, 1-23, (1978) · Zbl 0407.92023  Kowalczyk, M., Multiple spike layers in the shadow gierer – meinhardt system: existence of equilibria and approximate invariant manifold, Duke M. J., 98, 1, 59-111, (1999) · Zbl 0962.35063  C.-S. Lin, W.-M. Ni, On the Diffusion Coefficient of a Semilinear Neumann Problem in Calculus of Variations and Partial Differential Equations (Trento, 1986), Lecture Notes in Mathematics, Vol. 1340, Springer, Berlin, 1988, pp. 160-174.  NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, UK, 1995.  Ni, W., Diffusion, cross-diffusion, and their spike-layer steady-states, Not. AMS, 45, 1, 9-18, (1998) · Zbl 0917.35047  W. Ni, I. Takagi, E. Yanagida, Preprint, Tohoku Math J., 1999, submitted for publication.  Y. Nishiura, Coexistence of infinitely many stable solutions to reaction – diffusion equations in the singular limit, in: C.K.R.T. Jones, U. Kirchgraber (Eds.), Dynamics Reported: Expositions in Dynamical Systems, Vol. 3, Springer, New York, 1995.  Takagi, I., Point-condensation for a reaction – diffusion system, J. diff. eq., 61, 208-249, (1986) · Zbl 0627.35049  Turing, A., The chemical basis of morphogenesis, Phil. trans. roy. soc. B, 327, 37-72, (1952) · Zbl 1403.92034  Ward, M.J., An asymptotic analysis of localized solutions for some reaction – diffusion models in multi-dimensional domains, Stud. appl. math., 97, 2, 103-126, (1996) · Zbl 0932.35059  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  Wei, J., On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku math. J., 50, 159-178, (1998) · Zbl 0918.35024  E. Yanagida, Stability of stationary solutions of the Gierer-Meinhardt system, in: Proceedings of the China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (Shanghai, 1994), World Science Publishing, River Edge, NJ, 1997, pp. 191-198. · Zbl 0966.35018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.