## The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime.(English)Zbl 1145.65328

Summary: In a singularly perturbed limit of small diffusivity $$\varepsilon$$ of one of the two chemical species, equilibrium spike solutions to the Gray-Scott (GS) model on a bounded one-dimensional domain are constructed asymptotically using the method of matched asymptotic expansions. The equilibria that are constructed are symmetric k-spike patterns where the spikes have equal heights. Two distinguished limits in terms of a dimensionless parameter in the reaction-diffusion system are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime, the solution branches of k-spike equilibria are found to have a saddle-node bifurcation structure. The stability properties of these branches of solutions are analyzed with respect to the large eigenvalues $$\lambda$$ in the spectrum of the linearization. These eigenvalues, which have the property that $$\lambda =O(1)$$ as $$\varepsilon \to 0$$, govern the stability of the solution on an $$O(1)$$ time scale. Precise conditions, in terms of the nondimensional parameters, for the stability of symmetric $$k$$-spike equilibrium solutions with respect to this class of eigenvalues are obtained. In the low feed-rate regime, it is shown that a large eigenvalue instability leads either to a competition instability, whereby certain spikes in a sequence are annihilated, or to an oscillatory instability (typically synchronous) of the spike amplitudes as a result of a Hopf bifurcation. In the intermediate regime, it is shown that only oscillatory instabilities are possible, and a scaling-law determining the onset of such instabilities is derived. Detailed numerical simulations are performed to confirm the results of the stability theory. It is also shown that there is an equivalence principle between spectral properties of the GS model in the low feed-rate regime and the Gierer-Meinhardt model of morphogenesis. Finally, our results are compared with previous analytical work on the GS model.

### MSC:

 65P40 Numerical nonlinear stabilities in dynamical systems 92E20 Classical flows, reactions, etc. in chemistry

NAG; nag; d03pcf
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 [1] DOI: 10.1016/0009-2509(84)87017-7 [2] Doelman A., J. Appl. Math. 61 (3) pp 1080– (2000) [3] Doelman A., J. Appl. Math. 61 (6) pp 2036– (2000) [4] Doelman A., Physica D 122 (1) pp 1– (1998) [5] Doelman A., Memoirs of the AMS 155 (2002) [6] DOI: 10.1088/0951-7715/10/2/013 · Zbl 0905.35044 [7] 7. B. S.Kerner, and V. V.Osipov, Autosolitions:A New Approach to Problem of Self-Organization and Turbulence, Kluwer Academic Publishers, Dordrecht1994 . [8] Kerner B. S., Sov. Phys., Doklady 28 (6) pp 485– (1983) [9] DOI: 10.1038/369215a0 [10] DOI: 10.1103/PhysRevE.51.1899 [11] DOI: 10.1016/S0167-2789(01)00259-7 · Zbl 0986.34023 [12] Muratov C., J. Appl. Math. 62 (5) pp 1463– (2002) [13] DOI: 10.1088/0305-4470/33/48/321 · Zbl 1348.92178 [14] Nishiura Y., Methods and Appl. of Analysis 8 (2) pp 321– (2001) [15] Nishiura Y., Physica D. 130 (1) pp 73– (1999) [16] Nishiura Y., Physica D. 150 (3) pp 137– (2001) [17] Pearson J. E., Science 216 pp 189– (1993) [18] Petrov V., Phil. Trans. R. Soc. 347 pp 631– (1994) [19] DOI: 10.1103/PhysRevLett.72.2797 [20] DOI: 10.1103/PhysRevE.56.185 [21] Ueyama D., Hokkaido Math J. 28 (1) pp 175– (1999) · Zbl 0987.34031 [22] Kolokolnikov T., Physica D. 202 (3) pp 258– (2005) [23] DOI: 10.1007/s00332-002-0531-z · Zbl 1030.35011 [24] DOI: 10.1007/BF00289234 [25] Harrison L., Physica A. 222 pp 210– (1995) [26] 26. H.Meinhardt,Models of Biological Pattern Formation, Academic Press, London, 1982 . [27] 27. H.Meinhardt,The Algorithmic Beauty of Sea Shells, Springer-Verlag, Berlin, 1995 . · Zbl 1011.00506 [28] Bose A., Meth. Appl. Anal. 5 (4) pp 351– (1998) [29] DOI: 10.1137/S0036141098342556 · Zbl 0952.34027 [30] DOI: 10.1142/S0219530504000291 · Zbl 1050.35008 [31] Kolokolnikov T., Interfaces and Free Boundaries (2005) [32] DOI: 10.1016/j.matpur.2003.09.006 · Zbl 1107.35049 [33] Iron D., Physica D. 150 (1) pp 25– (2001) [34] Nishiura Y., Dynamics Reported: Expositions in Dynamical Systems 3 (1995) [35] Ascher U., Math. Comp. 33 pp 659– (1979) [36] DOI: 10.1016/0022-0396(88)90147-7 · Zbl 0676.35030 [37] 37. NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995 ). [38] DOI: 10.1017/S0956792599003770 · Zbl 1014.35005 [39] DOI: 10.1137/S0036139901393676 · Zbl 1019.35016 [40] DOI: 10.1111/1467-9590.t01-1-00227 · Zbl 1141.35389 [41] Sun W., J. App. Dyn. (2005)
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