A. Doelman, R. A. Gardner, and T. J. Kaper, Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach, Physica D: Nonlinear Phenomena, vol.122, issue.1-4, pp.1-36, 1998.
DOI : 10.1016/S0167-2789(98)00180-8

A. Doelman, P. Van-heijster, and T. J. Kaper, Pulse Dynamics in a Three-Component System: Existence Analysis, Journal of Dynamics and Differential Equations, vol.75, issue.19, pp.73-115, 2009.
DOI : 10.1007/s10884-008-9125-2

J. K. Hale, L. A. Peletier, and W. C. Troy, Exact Homoclinic and Heteroclinic Solutions of the Gray--Scott Model for Autocatalysis, SIAM Journal on Applied Mathematics, vol.61, issue.1, pp.61-102, 2000.
DOI : 10.1137/S0036139998334913

Y. Nec and M. J. Ward, The Stability and Slow Dynamics of Two-Spike Patterns for a Class of Reaction-Diffusion System, Mathematical Modelling of Natural Phenomena, vol.8, issue.5, pp.206-232, 2013.
DOI : 10.1051/mmnp/20138513

A. I. Volpert and V. A. Volpert, Applications of the rotation theory of vector fields to the study of wave solutions of parabolic equations, Trans. Moscow Math. Soc, pp.52-59, 1990.

A. Volpert and . Vit, Volpert and Vl. Volpert, Travelling wave solutions of parabolic systems, Translation of Mathematical Monographs Amer. Math. Society, vol.140, 1994.

V. Volpert and A. Volpert, Spectrum of elliptic operators and stability of travelling waves, Asymptotic Analysis, vol.23, pp.111-134, 2000.

V. Volpert and A. Volpert, Properness and topological degree for general elliptic operators, Abstract and Applied Analysis, vol.2003, issue.3, pp.129-182, 2003.
DOI : 10.1155/S1085337503204073

URL : http://doi.org/10.1155/s1085337503204073

V. Volpert, Elliptic partial differential equations Fredholm theory of elliptic problems in unbounded domains, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00653777

J. Wei and M. Winter, Stability of Spiky Solutions in a Reaction-Diffusion System with Four Morphogens on the Real Line, SIAM Journal on Mathematical Analysis, vol.42, issue.6, pp.2818-2841, 2010.
DOI : 10.1137/100792299

J. Wei and M. Winter, Existence, classification and stability analysis of multiplepeaked solutions for the Gierer-Meinhardt system in R 1, Methods Appl. Anal, vol.14, issue.2, pp.119-163, 2007.

(. M. Marion-)-institut and C. Jordan, France E-mail address: martine.marion@ec-lyon.fr (V. Volpert) Institut Camille Jordan, UMR 5585 CNRS