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2016, 13(2): 303-332. doi: 10.3934/mbe.2015004

Pattern analysis in a benthic bacteria-nutrient system

1. 

Institut für Mathematik, Universität Oldenburg, 26111 Oldenburg, Germany

Received  August 2014 Revised  October 2015 Published  November 2015

We study steady states in a reaction-diffusion system for a benthic bacteria-nutrient model in a marine sediment over 1D and 2D domains by using Landau reductions and numerical path following methods. We point out how the system reacts to changes of the strength of food supply and ingestion. We find that the system has a stable homogeneous steady state for relatively large rates of food supply and ingestion, while this state becomes unstable if one of these rates decreases and Turing patterns such as hexagons and stripes start to exist. One of the main results of the present work is a global bifurcation diagram for solutions over a bounded 2D domain. This bifurcation diagram includes branches of stripes, hexagons, and mixed modes. Furthermore, we find a number of snaking branches of stationary states, which are spatial connections between homogeneous states and hexagons, homogeneous states and stripes as well as stripes and hexagons in parameter ranges, where both corresponding states are stable. The system under consideration originally contains some spatially varying coefficients and with these exhibits layerings of patterns. The existence of spatial connections between different steady states in bistable ranges shows that spatially varying patterns are not necessarily due to spatially varying coefficients.
    The present work gives another example, where these effects arise and shows how the analytical and numerical observations can be used to detect signs that a marine bacteria population is in danger to die out or on its way to recovery, respectively.
    We find a type of hexagon patches on a homogeneous background, which seems to be new discovery. We show the first numerically calculated solution-branch, which connects two different types of hexagons in parameter space. We check numerically for bounded domains whether the stability changes for hexagons and stripes, which are extended homogeneously into the third dimension. We find that stripes and one type of hexagons have the same stable range over bounded 2D and 3D domains. This does not hold for the other type of hexagons. Their stable range is shorter for the bounded 3D domain, which we used here. We find a snaking branch, which bifurcates when the hexagonal prisms loose their stability. Solutions on this branch connects spatially between hexagonal prisms and a genuine 3D pattern (balls).
Citation: Daniel Wetzel. Pattern analysis in a benthic bacteria-nutrient system. Mathematical Biosciences & Engineering, 2016, 13 (2) : 303-332. doi: 10.3934/mbe.2015004
References:
[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. 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. 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. 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). 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. 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). 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). 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.

[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. 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. doi: 10.1128/AEM.69.4.1980-1989.2003.

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J. Burke and E. Knobloch, Localized states in the generalized Swift-Hohenberg equation,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.056211.

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J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability,, Chaos, 17 (2007). doi: 10.1063/1.2746816.

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T. Callahan and E. Knobloch, Symmetry-breaking bifurcations on cubic lattices,, Nonlinearity, 10 (1997), 1179. 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. 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. 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. doi: 10.1016/j.physd.2008.10.005.

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J. Dawes, Localized pattern formation with a large-scale mode: Slanted snaking,, SIAM J. Appl. Dyn. Syst., 7 (2008), 186. doi: 10.1137/06067794X.

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J. Dawes, Modulated and localized states in a finite domain,, SIAM J. Appl. Dyn. Syst., 8 (2009), 909. doi: 10.1137/080724344.

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A. Dean, P. Matthews, S. Cox and J. King, Exponential asymptotics of homoclinic snaking,, Nonlinearity, 24 (2011), 3323. 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. 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. 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. 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. 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). 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. doi: 10.1103/PhysRevE.51.2046.

[27]

S. M. Houghton and E. Knobloch, Homoclinic snaking in bounded domains,, Phys.Rev.E, 80 (2009). doi: 10.1103/PhysRevE.80.026210.

[28]

R. Hoyle, Pattern Formation,, Cambridge University Press., (2006). doi: 10.1017/CBO9780511616051.

[29]

G. Kozyreff, P. Assemat and S. Chapman, Influence of boundaries on localized patterns,, Phys. Rev. Letters, 103 (2009). 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). 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. 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. 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, (): 91.

[34]

B. Malomed, A. Nepomnyashchy and M. Tribelsky, Domain boundaries in convection patterns,, Phys.Rev.A, 42 (1990), 7244. 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, 19 (2004), 367. doi: 10.1016/S0960-0779(03)00049-3.

[36]

J. Monod, The growth of bacterial cultures,, Annual Review of Microbiology, 3 (1949), 371.

[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, (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. 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,, , ().

[42]

B. Sandstede, Stability of travelling waves,, Handbook of Dynamical Systems, 2 (2002), 983. 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. 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. doi: 10.1080/713851126.

[45]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Physical Review A, 15 (1977). 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). 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.

[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. 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.

[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. 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). doi: 10.1103/PhysRevLett.87.198101.

[52]

N. Wai-Leung and B. Bassler, Bacterial quorum-sensing network architectures,, Annu. Rev. Genet., 43 (2009), 197.

[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. 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. 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). 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. doi: 10.1016/j.physa.2005.05.026.

show all references

References:
[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. 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. 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. 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). 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. 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). 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). 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.

[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. 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. 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). doi: 10.1103/PhysRevE.73.056211.

[12]

J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability,, Chaos, 17 (2007). doi: 10.1063/1.2746816.

[13]

T. Callahan and E. Knobloch, Symmetry-breaking bifurcations on cubic lattices,, Nonlinearity, 10 (1997), 1179. 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. 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. 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. 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. doi: 10.1137/06067794X.

[19]

J. Dawes, Modulated and localized states in a finite domain,, SIAM J. Appl. Dyn. Syst., 8 (2009), 909. doi: 10.1137/080724344.

[20]

A. Dean, P. Matthews, S. Cox and J. King, Exponential asymptotics of homoclinic snaking,, Nonlinearity, 24 (2011), 3323. 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. 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. 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. 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. 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). 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. doi: 10.1103/PhysRevE.51.2046.

[27]

S. M. Houghton and E. Knobloch, Homoclinic snaking in bounded domains,, Phys.Rev.E, 80 (2009). doi: 10.1103/PhysRevE.80.026210.

[28]

R. Hoyle, Pattern Formation,, Cambridge University Press., (2006). doi: 10.1017/CBO9780511616051.

[29]

G. Kozyreff, P. Assemat and S. Chapman, Influence of boundaries on localized patterns,, Phys. Rev. Letters, 103 (2009). 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). 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. 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. 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, (): 91.

[34]

B. Malomed, A. Nepomnyashchy and M. Tribelsky, Domain boundaries in convection patterns,, Phys.Rev.A, 42 (1990), 7244. 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, 19 (2004), 367. doi: 10.1016/S0960-0779(03)00049-3.

[36]

J. Monod, The growth of bacterial cultures,, Annual Review of Microbiology, 3 (1949), 371.

[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, (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. 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,, , ().

[42]

B. Sandstede, Stability of travelling waves,, Handbook of Dynamical Systems, 2 (2002), 983. 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. 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. doi: 10.1080/713851126.

[45]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Physical Review A, 15 (1977). 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). 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.

[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. 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.

[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. 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). doi: 10.1103/PhysRevLett.87.198101.

[52]

N. Wai-Leung and B. Bassler, Bacterial quorum-sensing network architectures,, Annu. Rev. Genet., 43 (2009), 197.

[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. 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. 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). 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. doi: 10.1016/j.physa.2005.05.026.

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