• Previous Article
    Electrical-thermal analytical modeling of monopolar RF thermal ablation of biological tissues: determining the circumstances under which tissue temperature reaches a steady state
  • MBE Home
  • This Issue
  • Next Article
    Directional entropy based model for diffusivity-driven tumor growth
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

T. Callahan and E. Knobloch, Symmetry-breaking bifurcations on cubic lattices,, Nonlinearity, 10 (1997), 1179. doi: 10.1088/0951-7715/10/5/009. Google Scholar

[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. Google Scholar

[15]

S. Camazine, Self-organization in Biological Systems,, Princeton University Press, (2003). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Biological Cybernetics, 12 (1972), 30. doi: 10.1007/BF00289234. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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 (). Google Scholar

[38]

J. Murray, Mathematical Biology,, Springer-Verlag, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar

[39]

L. Pismen, Patterns and Interfaces in Dissipative Dynamics,, Springer, (2006). Google Scholar

[40]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics,, Physica D, 23 (1986), 3. doi: 10.1016/0167-2789(86)90104-1. Google Scholar

[41]

U. Prüfert, Oopde - an object oriented toolbox for finite elements in matlab. quickstart guide. tu bergakademie freiberg. r 2015,, , (). Google Scholar

[42]

B. Sandstede, Stability of travelling waves,, Handbook of Dynamical Systems, 2 (2002), 983. doi: 10.1016/S1874-575X(02)80039-X. Google Scholar

[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. Google Scholar

[44]

L. J. Stal, Microphytobenthos, their extracellular polymeric substances, and the morphogenesis of intertidal sediments,, Geomicrobiology Journal, 20 (2003), 463. doi: 10.1080/713851126. Google Scholar

[45]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Physical Review A, 15 (1977). doi: 10.1103/PhysRevA.15.319. Google Scholar

[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. Google Scholar

[47]

A. M. Turing, The chemical basis of morphogenisis,, Philosophical transaction of the Royal Society of London - B, 237 (1952), 37. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[52]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

T. Callahan and E. Knobloch, Symmetry-breaking bifurcations on cubic lattices,, Nonlinearity, 10 (1997), 1179. doi: 10.1088/0951-7715/10/5/009. Google Scholar

[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. Google Scholar

[15]

S. Camazine, Self-organization in Biological Systems,, Princeton University Press, (2003). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

A. Gierer and H. Meinhardt, A theory of biological pattern formation,, Biological Cybernetics, 12 (1972), 30. doi: 10.1007/BF00289234. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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 (). Google Scholar

[38]

J. Murray, Mathematical Biology,, Springer-Verlag, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar

[39]

L. Pismen, Patterns and Interfaces in Dissipative Dynamics,, Springer, (2006). Google Scholar

[40]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics,, Physica D, 23 (1986), 3. doi: 10.1016/0167-2789(86)90104-1. Google Scholar

[41]

U. Prüfert, Oopde - an object oriented toolbox for finite elements in matlab. quickstart guide. tu bergakademie freiberg. r 2015,, , (). Google Scholar

[42]

B. Sandstede, Stability of travelling waves,, Handbook of Dynamical Systems, 2 (2002), 983. doi: 10.1016/S1874-575X(02)80039-X. Google Scholar

[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. Google Scholar

[44]

L. J. Stal, Microphytobenthos, their extracellular polymeric substances, and the morphogenesis of intertidal sediments,, Geomicrobiology Journal, 20 (2003), 463. doi: 10.1080/713851126. Google Scholar

[45]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Physical Review A, 15 (1977). doi: 10.1103/PhysRevA.15.319. Google Scholar

[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. Google Scholar

[47]

A. M. Turing, The chemical basis of morphogenisis,, Philosophical transaction of the Royal Society of London - B, 237 (1952), 37. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[52]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

Thomas März, Andreas Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems & Imaging, 2016, 10 (4) : 1087-1110. doi: 10.3934/ipi.2016033

[2]

Juan Manuel Reyes, Alberto Ruiz. Reconstruction of the singularities of a potential from backscattering data in 2D and 3D. Inverse Problems & Imaging, 2012, 6 (2) : 321-355. doi: 10.3934/ipi.2012.6.321

[3]

A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769

[4]

Alexander Balandin. The localized basis functions for scalar and vector 3D tomography and their ray transforms. Inverse Problems & Imaging, 2016, 10 (4) : 899-914. doi: 10.3934/ipi.2016026

[5]

Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559

[6]

Ferdinando Auricchio, Elena Bonetti. A new "flexible" 3D macroscopic model for shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 277-291. doi: 10.3934/dcdss.2013.6.277

[7]

Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507

[8]

Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109

[9]

Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-12. doi: 10.3934/dcdsb.2019075

[10]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575

[11]

Julijana Gjorgjieva, Jon Jacobsen. Turing patterns on growing spheres: the exponential case. Conference Publications, 2007, 2007 (Special) : 436-445. doi: 10.3934/proc.2007.2007.436

[12]

Kousuke Kuto, Tohru Tsujikawa. Stationary patterns for an adsorbate-induced phase transition model I: Existence. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1105-1117. doi: 10.3934/dcdsb.2010.14.1105

[13]

Kazuhiro Kurata, Kotaro Morimoto. Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 139-164. doi: 10.3934/dcds.2011.31.139

[14]

Carolin Kreisbeck. A note on $3$d-$1$d dimension reduction with differential constraints. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 55-73. doi: 10.3934/dcdss.2017003

[15]

Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881

[16]

Hyeong-Ohk Bae, Bum Ja Jin. Estimates of the wake for the 3D Oseen equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 1-18. doi: 10.3934/dcdsb.2008.10.1

[17]

Indranil SenGupta, Weisheng Jiang, Bo Sun, Maria Christina Mariani. Superradiance problem in a 3D annular domain. Conference Publications, 2011, 2011 (Special) : 1309-1318. doi: 10.3934/proc.2011.2011.1309

[18]

Giovanny Guerrero, José Antonio Langa, Antonio Suárez. Biodiversity and vulnerability in a 3D mutualistic system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4107-4126. doi: 10.3934/dcds.2014.34.4107

[19]

Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685

[20]

Ka Kit Tung, Wendell Welch Orlando. On the differences between 2D and QG turbulence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 145-162. doi: 10.3934/dcdsb.2003.3.145

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]