Citation: |
[1] |
D. Avitabile, D. Lloyd, J. Burke, E. Knobloch and B. Sandstede, To snake or not to snake in the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 9 (2010), 704-733.doi: 10.1137/100782747. |
[2] |
T. Bánsági, V. K. Vanag and I. R. Epstein, Tomography of reaction-diffusion microemulsions reveals three-dimensional turing patterns, Science, 331 (2011), 1309-1312.doi: 10.1126/science.1200815. |
[3] |
M. Baurmann, W. Ebenhöh and U. Feudel, Turing instabilities and pattern formation in a benthic nutrient-microorganism system, Math. Biosci. Eng., 1 (2004), 111-130.doi: 10.3934/mbe.2004.1.111. |
[4] |
C. Beaume, E. Knobloch and A. Bergeon, Nonsnaking doubly diffusive convectons and the twist instability, Physics of Fluids, 25 (2013), 114102.doi: 10.1063/1.4826978. |
[5] |
M. Beck, J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972.doi: 10.1137/080713306. |
[6] |
I. Berenstein, L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, Superlattice turing structures in a photosensitive reaction-diffusion system, Physical review letters, 91 (2003), 058302.doi: 10.1103/PhysRevLett.91.058302. |
[7] |
A. Bergeom, J. Burke, E. Knobloch and I. Mercader, Eckhaus instability and homoclinic snaking, Phys.Rev.E, 78 (2008), 046201, 16 pp.doi: 10.1103/PhysRevE.78.046201. |
[8] |
K. Bosselmann, M. E. Böttcher, M. Billerbeck, E. Walpersdorf, A. Theune, D. De Beer, M. Hüttel, H. J. Brumsack and B. B. Jørgensen, Iron-sulfur-manganese dynamics in intertidal surface sediments of the north sea, Ber. Forschungsz. Terramare, 12 (2003), 32-35. |
[9] |
A. Bruns, H. Cypionka and J. Overmann, Cyclic amp and acyl homoserine lactones increase the cultivation efficiency of heterotrophic bacteria from the central baltic sea, Applied and Environmental Microbiology, 68 (2002), 3978-3987.doi: 10.1128/AEM.68.8.3978-3987.2002. |
[10] |
A. Bruns, U. Nübel, H. Cypionka and J. Overmann, Effect of signal compounds and incubation conditions on the culturability of freshwater bacterioplankton, Applied and environmental microbiology, 69 (2003), 1980-1989.doi: 10.1128/AEM.69.4.1980-1989.2003. |
[11] |
J. Burke and E. Knobloch, Localized states in the generalized Swift-Hohenberg equation, Phys. Rev. E, 73 (2006), 056211, 15pp.doi: 10.1103/PhysRevE.73.056211. |
[12] |
J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability, Chaos, 17 (2007), 037102, 15pp.doi: 10.1063/1.2746816. |
[13] |
T. Callahan and E. Knobloch, Symmetry-breaking bifurcations on cubic lattices, Nonlinearity, 10 (1997), 1179-1216.doi: 10.1088/0951-7715/10/5/009. |
[14] |
T. Callahan and E. Knobloch, Pattern formation in three-dimensional reaction-diffusion systems, Physica D: Nonlinear Phenomena, 132 (1999), 339-362.doi: 10.1016/S0167-2789(99)00041-X. |
[15] |
S. Camazine, Self-organization in Biological Systems, Princeton University Press, 2003. |
[16] |
V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953-2956.doi: 10.1103/PhysRevLett.64.2953. |
[17] |
S. Chapman and G. Kozyreff, Exponential asymptotics of localised patterns and snaking bifurcation diagrams, Physica D, 238 (2009), 319-354.doi: 10.1016/j.physd.2008.10.005. |
[18] |
J. Dawes, Localized pattern formation with a large-scale mode: Slanted snaking, SIAM J. Appl. Dyn. Syst., 7 (2008), 186-206.doi: 10.1137/06067794X. |
[19] |
J. Dawes, Modulated and localized states in a finite domain, SIAM J. Appl. Dyn. Syst., 8 (2009), 909-930.doi: 10.1137/080724344. |
[20] |
A. Dean, P. Matthews, S. Cox and J. King, Exponential asymptotics of homoclinic snaking, Nonlinearity, 24 (2011), 3323-3351.doi: 10.1088/0951-7715/24/12/003. |
[21] |
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, Propagation of hexagonal patterns near onset, European J. Appl. Math., 14 (2003), 85-110.doi: 10.1017/S095679250200503X. |
[22] |
S. U. Gerbersdorf, W. Manz and D. M. Paterson, The engineering potential of natural benthic bacterial assemblages in terms of the erosion resistance of sediments, FEMS Microbiology Ecology, 66 (2008), 282-294.doi: 10.1111/j.1574-6941.2008.00586.x. |
[23] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Biological Cybernetics, 12 (1972), 30-39.doi: 10.1007/BF00289234. |
[24] |
M. Golubitsky, J. W. Swift and E. Knobloch, Symmetries and pattern selection in Rayleigh-Bénard convection, Physica D: Nonlinear Phenomena, 10 (1984), 249-276.doi: 10.1016/0167-2789(84)90179-9. |
[25] |
K. Gowda, H. Riecke and M. Silber, Transitions between patterned states in vegetation models for semiarid ecosystems, Phys. Rev. E, 89 (2014), 022701.doi: 10.1103/PhysRevE.89.022701. |
[26] |
M. F. Hilali, S. Métens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg model, Phys. Rev. E, 51 (1995), 2046-2052.doi: 10.1103/PhysRevE.51.2046. |
[27] |
S. M. Houghton and E. Knobloch, Homoclinic snaking in bounded domains, Phys.Rev.E, 80 (2009), 026210.doi: 10.1103/PhysRevE.80.026210. |
[28] |
R. Hoyle, Pattern Formation, Cambridge University Press., Cambridge, UK, 2006.doi: 10.1017/CBO9780511616051. |
[29] |
G. Kozyreff, P. Assemat and S. Chapman, Influence of boundaries on localized patterns, Phys. Rev. Letters, 103 (2009), 164501.doi: 10.1103/PhysRevLett.103.164501. |
[30] |
M. Leda, V. K. Vanag and I. R. Epstein, Instabilities of a three-dimensional localized spot, Physical Review E, 80 (2009), 066204.doi: 10.1103/PhysRevE.80.066204. |
[31] |
D. Lloyd and H. O'Farrell, On localised hotspots of an urban crime model, Physica D, 253 (2013), 23-39.doi: 10.1016/j.physd.2013.02.005. |
[32] |
D. Lloyd, B. Sandstede, D. Avitabile and A. Champneys, Localized hexagon patterns of the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 7 (2008), 1049-1100.doi: 10.1137/070707622. |
[33] |
S. Madani, F. Meysman and J. Middelburg, Biogeochemical modeling of sediments from the santa barbara basin (california), BioGeoChemistry of Tidal Flats (J. Rullkötter, ed.). Forschungszentrum Terramare, Wilhelmshaven, 91-93. |
[34] |
B. Malomed, A. Nepomnyashchy and M. Tribelsky, Domain boundaries in convection patterns, Phys.Rev.A, 42 (1990), 7244-7263.doi: 10.1103/PhysRevA.42.7244. |
[35] |
E. Meron, E. Gilad, J. von Hardenberg, M. Shachak and Y. Zarmi, Vegetation patterns along a rainfall gradient, Chaos, Solitons, and Fractals, 19 (2004), 367-376, URL http://www.sciencedirect.com/science/article/pii/S0960077903000493, Fractals in Geophysics.doi: 10.1016/S0960-0779(03)00049-3. |
[36] |
J. Monod, The growth of bacterial cultures, Annual Review of Microbiology, 3 (1949), 371-394. |
[37] |
Z. J. Mudryk, B. Podgorska, A. Ameryk and J. Bolalek, The occurrence and activity of sulphate-reducing bacteria in the bottom sediments of the gulf of gdańsk, Oceanologia, 42. |
[38] |
J. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.doi: 10.1007/978-3-662-08539-4. |
[39] |
L. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer, 2006. |
[40] |
Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11.doi: 10.1016/0167-2789(86)90104-1. |
[41] |
U. Prüfert, Oopde - an object oriented toolbox for finite elements in matlab. quickstart guide. tu bergakademie freiberg. r 2015, http://www.mathe.tu-freiberg.de/files/personal/255/oopde-quickstart-guide-2015.pdf. |
[42] |
B. Sandstede, Stability of travelling waves, Handbook of Dynamical Systems, Elsevier Science, 2 (2002), 983-1055.doi: 10.1016/S1874-575X(02)80039-X. |
[43] |
K. Siteur, E. Siero, M. B. Eppinga, J. D. M. Rademacher, A. Doelman and M. Rietkerk, Beyond Turing: The response of patterned ecosystems to environmental change, Ecological Complexity, 20 (2014), 81-96.doi: 10.1016/j.ecocom.2014.09.002. |
[44] |
L. J. Stal, Microphytobenthos, their extracellular polymeric substances, and the morphogenesis of intertidal sediments, Geomicrobiology Journal, 20 (2003), 463-478.doi: 10.1080/713851126. |
[45] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Physical Review A, 15 (1977), p319.doi: 10.1103/PhysRevA.15.319. |
[46] |
L. Tsimring, H. Levine, I. Aranson, E. Ben-Jacob, I. Cohen, O. Shochet and W. N. Reynolds, Aggregation patterns in stressed bacteria, Physical review letters, 75 (1995), p1859.doi: 10.1103/PhysRevLett.75.1859. |
[47] |
A. M. Turing, The chemical basis of morphogenisis, Philosophical transaction of the Royal Society of London - B, 237 (1952), 37-72. |
[48] |
H. Uecker and D. Wetzel, Numerical Results for Snaking of Patterns over Patterns in Some 2D Selkov-Schnakenberg Reaction-Diffusion Systems., SIAM J. Appl. Dyn. Syst., 13 (2014), 94-128.doi: 10.1137/130918484. |
[49] |
H. Uecker, D. Wetzel and J. Rademacher, pde2path - a Matlab package for continuation and bifurcation in 2D elliptic systems, Numer. Math. Theor. Meth. Appl., 7 (2014), 58-106. |
[50] |
G. J. C. Underwood and D. M. Paterson, The importance of extracellular carbohydrate productionby marine epipelic diatoms, Advances in botanical research, 40 (2003), 183-240.doi: 10.1016/S0065-2296(05)40005-1. |
[51] |
J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett., 87 (2001), 198101.doi: 10.1103/PhysRevLett.87.198101. |
[52] |
N. Wai-Leung and B. Bassler, Bacterial quorum-sensing network architectures, Annu. Rev. Genet., 43 (2009), 197-222. |
[53] |
P. Williams, K. Winzer, W. C. Chan and M. Cámara, Look who's talking: Communication and quorum sensing in the bacterial world, Phil. Trans. R. Soc. B, 362 (2007), 1119-1134.doi: 10.1098/rstb.2007.2039. |
[54] |
P. Woods and A. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation, Physica D, 129 (1999), 147-170.doi: 10.1016/S0167-2789(98)00309-1. |
[55] |
L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, Turing patterns beyond hexagons and stripes, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16 (2006), 037114.doi: 10.1063/1.2214167. |
[56] |
H. Yizhaq, E. Gilad and E. Meron, Banded vegetation: Biological productivity and resilience, Physica A: Statistical Mechanics and its Applications, 356 (2005), 139-144, URL http://EconPapers.repec.org/RePEc:eee:phsmap:v:356:y:2005:i:1:p:139-144.doi: 10.1016/j.physa.2005.05.026. |