zbMATH — the first resource for mathematics

Axisymmetric ring solutions of the 2D Gray-Scott model and their destabilization into spots. (English) Zbl 1065.35133
The work addresses instability and nonlinear evolution of axially symmetric (ring-shaped) patterns in the Gray-Scott model of the reaction-diffusion type: \[ \begin{aligned} U_t & =D_U\nabla^2U - UV^2 +A(1-U),\\ V_t & =D_V\nabla^2V + UV^2 -BV,\end{aligned} \] for real fields \(U\) and \(V\) with real control parameters \(A\) and \(V\), which is known to demonstrate quite complex behavior (spontaneous spot multiplication, etc.) in two-dimensional simulations. In this work, solutions for ring-shaped patterns in the monostable sector of the model are constructed by means of an asymptotic method (starting from the one-dimensional solution in the radial direction as the zero-order approximation), and their stability is investigated against infinitesimal azimuthal perturbations, which tend to destroy the axial symmetry and split each ring into a set of localized spots. The stability problem is solved by reducing the one for the two coupled equations to an effective problem for a single nonlocal equation, which is achieved by eliminating the field \(U\). As a result, a type of the unstable eigenmode (the most unstable particular azimuthal harmonic) is found, thus making it possible to predict the number of spots into which the ring(s) are split. The predictions are confirmed by direct numerical simulations (which show the establishment of stationary patterns consisting of several layers of necklace-shaped arrays of spots). The analysis is also performed for the same model in the bistable section, where it reduces to the consideration of an instability of the Turing type.

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K57 Reaction-diffusion equations
92C15 Developmental biology, pattern formation
35B35 Stability in context of PDEs
35B32 Bifurcations in context of PDEs
Full Text: DOI
[1] Blom, J.G.; Trompert, R.A.; Verwer, J.G., Algorithm 758: VLUGR2: a vectorizable adaptive grid solver for PDEs in 2D, Assoc. comput. Mach. tran. math. software, 22, 302-328, (1996) · Zbl 0884.65094
[2] Blom, J.G.; Zegeling, P.A., Algorithm 731: a moving-grid interface for systems of one-dimensional time-dependent partial differential equations, Assoc. comput. Mach. trans. math. software, 20, 194-214, (1994) · Zbl 0889.65099
[3] Busse, F.H., Nonlinear properties of thermal convection, Rep. prog. phys., 41, 1929-1967, (1978)
[4] Caginalp, G.; McLeod, B., The interior transition layer for an ordinary differential equation arising from solidification theory, Quart. appl. math., 44, 155-168, (1986) · Zbl 0605.34022
[5] Davies, P.; Blanchedeau, P.; Dulos, E.; Kepper, P.D., Dividing blobs, chemical flowers, and patterned islands in a reaction – diffusion system, J. phys. chem., 102, 8236-8244, (1998)
[6] Doelman, A.; Eckhaus, W.; Kaper, T.J., Slowly-modulated two-pulse solutions in the gray – scott model. I. asymptotic construction and stability, SIAM J. appl. math., 61, 1080-1102, (2001) · Zbl 0979.35074
[7] Doelman, A.; Eckhaus, W.; Kaper, T.J., Slowly-modulated two-pulse solutions in the gray – scott model. II. geometric theory, bifurcations, and splitting dynamics, SIAM J. appl. math., 61, 2036-2062, (2001) · Zbl 0989.35073
[8] Doelman, A.; Gardner, R.A.; Kaper, T.J., Stability analysis of singular patterns in the 1D gray – scott model: a matched asymptotics approach, Physica D, 122, 1-36, (1998) · Zbl 0943.34039
[9] Doelman, A.; Gardner, R.A.; Kaper, T.J., Large stable pulse solutions in reaction-diffusion equations, Indiana univ. math. J., 50, 443-507, (2001) · Zbl 0994.35058
[10] A. Doelman, R.A. Gardner, T.J. Kaper, A stability index analysis of 1D patterns of the Gray-Scott model, Mem. Am. Math. Soc. 155 (737) (2002), AMS, Providence, RI. ISSN 0065-9266. · Zbl 0994.35059
[11] Doelman, A.; Kaper, T.J., Semi-strong pulse interactions in a class of coupled reaction-diffusion equations, SIAM J. appl. dyn. syst., 2, 53-96, (2003) · Zbl 1088.35517
[12] Doelman, A.; Kaper, T.J.; Zegeling, P., Pattern formation in the one-dimensional gray – scott model, Nonlinearity, 10, 523-563, (1997) · Zbl 0905.35044
[13] Doelman, A.; van der Ploeg, H., Homoclinic stripe patterns, SIAM J. appl. dyn. syst., 1, 65-104, (2002) · Zbl 1004.35063
[14] W. Eckhaus, On modulation equations of the Ginzburg-Landau type, in: Proceedings of the Second International Conference on Industrial and Applied Mathematics ICIAM 91, 1992, pp. 83-98.
[15] W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam, 1979. · Zbl 0421.34057
[16] W. Eckhaus, Studies in Nonlinear Stability Theory, Springer-Verlag, New York, 1965. · Zbl 0125.33101
[17] Gray, P.; Scott, S.K., Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem. eng. sci., 38, 29-43, (1983)
[18] Gray, P.; Scott, S.K., Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A+2B→3B, B→C, Chem. eng. sci., 39, 1087-1097, (1984)
[19] Hagberg, A.; Meron, E.; Passot, T., Phase dynamics of nearly stationary patterns in activator – inhibitor systems, Phys. rev. E, 61, 6471-6476, (2000)
[20] Haim, D.; Li, G.; Ouyang, Q.; McCormick, W.D.; Swinney, H.; Hagberg, A.; Meron, E., Breathing spots in a reaction-diffusion system, Phys. rev. lett., 77, 190, (1996)
[21] Hirschberg, P.; Knobloch, E., Zigzag and varicose instabilities of a localized stripe, Chaos, 3, 713-721, (1993) · Zbl 1055.37569
[22] Laing, C.; Troy, W., Two-bump solutions of amari-type models of neuronal pattern formation, Physica D, 178, 190-218, (2003) · Zbl 1011.92007
[23] Lee, K.J.; Swinney, H.L., Lamellar structures and self-replicating spots in a reaction – diffusion system, Phys. rev. E, 51, 1899-1915, (1995)
[24] Lee, K.; Swinney, H., Replicating spots in reaction – diffusion systems, Int. J. bifurcat. chaos, 7, 1149-1158, (1995) · Zbl 0910.92006
[25] Lin, K.-J.; McCormick, W.D.; Pearson, J.E.; Swinney, H.L., Experimental observation of self-replicating spots in a reaction – diffusion system, Nature, 369, 6477, 215-218, (1994)
[26] Mimura, M.; Nagayama, M., Nonannihilation dynamics in an exothermic reaction – diffusion system with mono-stable excitability, Chaos, 7, 817-826, (1997) · Zbl 0933.35094
[27] Morgan, D.S.; Doelman, A.; Kaper, T.J., Stationary periodic orbits in the 1D gray – scott model, Meth. appl. anal., 7, 105-150, (2000) · Zbl 0996.92041
[28] D.S. Morgan, On existence and stability of spatial patterns in an activator – inhibitor system exhibiting self-replication, Ph.D. Thesis, Boston University, 2001.
[29] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. · Zbl 0051.40603
[30] Muratov, C.; Osipov, V., Static spike autosolitons in the gray – scott model, J. phys. A, 33, 8893-8916, (2000) · Zbl 1348.92178
[31] Muratov, C.; Osipov, V., Stability of the static spike autosolitons in the gray – scott model, SIAM J. appl. math., 62, 1463-1487, (2002) · Zbl 1012.35042
[32] Nefedov, N., Contrast structures of spike type in nonlinear singularly perturbed elliptic equations, Russ. acad. sci. doklady math., 46, 411-413, (1993)
[33] Nishiura, Y.; Fujii, H., Stability of singularly perturbed solutions to systems of reaction – diffusion equations, SIAM J. math. anal., 18, 1726-1770, (1987) · Zbl 0638.35010
[34] Nishiura, Y.; Suzuki, H., Nonexistence of higher dimensional stable Turing patterns in the singular limit, SIAM J. math. anal., 29, 1087-1105, (1998) · Zbl 0921.35014
[35] Nishiura, Y.; Ueyama, D., A skeleton structure for self-replication dynamics, Physica D, 130, 73-104, (1999) · Zbl 0936.35090
[36] Nishiura, Y.; Ueyama, D., Spatio-temporal chaos for the gray – scott model, Physica D, 150, 137-162, (2001) · Zbl 0981.35022
[37] Pearson, J.E., Complex patterns in a simple system, Science, 261, 189-192, (1993)
[38] Petrov, V.; Scott, S.K.; Showalter, K., Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction – diffusion system, Phil. trans. roy. soc. London, ser. A, 347, 631-642, (1994) · Zbl 0867.35047
[39] H.V.D. Ploeg, Personal communication.
[40] Reynolds, W.N.; Pearson, J.E.; Ponce-Dawson, S., Dynamics of self-replicating patterns in reaction diffusion systems, Phys. rev. lett., 72, 2797-2800, (1994)
[41] Reynolds, W.N.; Ponce-Dawson, S.; Pearson, J.E., Self-replicating spots in reaction – diffusion systems, Phys. rev. E, 56, 185-198, (1997)
[42] Turing, A.M., The chemical basis of morphogenesis, Phil. trans. roy. soc. London, ser. B, 237, 37-72, (1952) · Zbl 1403.92034
[43] D. Walgraef, Spatio-temporal Pattern Formation: With Examples from Physics, Chemistry, and Materials Science, Springer-Verlag, New York, 1997.
[44] Wei, J., Pattern formation in two-dimensional gray – scott model: existence of single-spot solutions and their stability, Physica D, 148, 20-48, (2001) · Zbl 0981.35026
[45] Wei, J.; Winter, M., Existence and stability of multi-spot solutions of the gray – scott model in R2, Physica D, 176, 147-180, (2003) · Zbl 1014.37036
[46] Wei, J.; Winter, M., Asymmetric spotty patterns for the gray – scott model in R2, Stud. appl. math., 110, 63-102, (2003) · Zbl 1141.35401
[47] S. Wolfram, The Mathematica Book, 4th ed., Wolfram Media/Cambridge University Press, 1999. · Zbl 0924.65002
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.