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Traveling fronts of pyramidal shapes in competition-diffusion systems
Pattern forming instabilities driven by non-diffusive interactions
1. | Advanced Semiconductor Materials Lithography, ASML B.V., Office 06.C.006, 5500AH Veldhoven, Netherlands |
2. | Westfälische-Wilhelms Universität Münster, Applied Mathematics Münster, Einsteinstr. 62, D-48149 Münster, Germany |
3. | Universität Bonn, Institut für Angewandte Mathematik, Endenicher Allee 60, D-53155 Bonn, Germany |
References:
[1] |
M. S. Alber, M. A. Kiskowski and Y. Jing, Lattice gas cellular automaton model for rippling and aggregation in myxobacteria, Physica D, 191 (2004), 343-358. |
[2] |
U. Börner and M. Bär, Pattern formation in a reaction-advection model with delay: A continuum approach to myxobacterial rippling, Annalen der Physik, 13 (2004), 432-441.
doi: 10.1002/andp.200410086. |
[3] |
U. Börner, A. Deutsch, H. Reichenbach and M. Bär, Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions, Physical Review Letters, 89 (2002), 078101. |
[4] |
O. Diekmann, M. Gyllenberg, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models I. Linear Theory, J. Math. Biol., 36 (1998), 349-388.
doi: 10.1007/s002850050104. |
[5] |
O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models II. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189.
doi: 10.1007/s002850170002. |
[6] |
M. Dworkin and D. Kaiser eds., "Myxobacteria II," American Society for Microbiology (AMS) Press, 1993. |
[7] |
R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis, DCDS-B, 6 (2006), 1239-1260.
doi: 10.3934/dcdsb.2006.6.1239. |
[8] |
R. Erban and H. Othmer, From signal transduction to spatial pattern formation in E.coli : A paradigm for multi-scale modeling in biology, Multiscale Modeling and Simulation, 3 (2005), 362-394.
doi: 10.1137/040603565. |
[9] |
E. Geigant, "Nichtlineare Integro-Differential-Gleichungen zur Modellierung interaktiver Musterbildungsprozesse auf $S^{1}$," (German) [Nonlinear Integro-Differential Equations for the Modelling of Interactive Pattern Formation Processes on $S^{1}$], Bonner Mathematische Schriften 323 Ph.D thesis, University of Bonn, 1999. |
[10] |
E. Geigant, On peak and periodic solutions of an integro-differential equation on $S^1$, in "Geometric Analysis and Nonlinear Partial Differential Equations", Springer-Verlag, Berlin, (2003), 463-474. |
[11] |
E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol., 46 (2003), 537-563.
doi: 10.1007/s00285-002-0187-1. |
[12] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12, (1972), 30-39.
doi: 10.1007/BF00289234. |
[13] |
T. Hillen, A Turing model with correlated random walk, J. Math. Biol., 35 (1996), 49-72.
doi: 10.1007/s002850050042. |
[14] |
O. Igoshin, J. Neu and G. Oster, Developmental waves in Myxobacteria: A novel pattern formation mechanism, Phys. Rev. E, 7 (2004), 1-11. |
[15] |
K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, An Integro-differential equation model for alignment and orientational aggregation, J. of Differential Equations, 246 (2009), 1387-1421.
doi: 10.1016/j.jde.2008.11.006. |
[16] |
T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1980.
doi: 10.1007/978-3-642-66282-9. |
[17] |
F. Lutscher and A. Stevens, Emerging patterns in a hyperbolic model for locally interacting cell systems, J. Nonlinear Science, 12 (2002), 619-640.
doi: 10.1007/s00332-002-0510-4. |
[18] |
H. Meinhardt, Morphogenesis of lines and nets, Differentiation, 6 (1976), 117-123. |
[19] |
J. D. Murray, "Mathematical Biology I and II," Interdisciplinary Applied Mathematics 17 and 18, Springer-Verlag, New York, 2002/03. |
[20] |
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[21] |
B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser-Verlag, Basel, 2007. |
[22] |
B. Pfistner, "Ein Eindimensionales Modell Zum Schwarmverhalten der Myxobakterien Unter Besonderer Berücksichtigung der Randzonenentwicklung," (German) [A one-dimensional model on the swarming behavior of Myxobacteria, with special consideration of the development of the boundary zone], Ph.D thesis, University of Bonn, 1992. |
[23] |
I. Primi, A. Stevens and J. J. L. Velázquez, Mass-selection in alignment models with non-deterministic effects, Communication in PDE, 34 (2009), 419-456.
doi: 10.1080/03605300902797171. |
[24] |
M. Rotenberg, Transport theory for growing cell populations, J. Theor. Biology, 103 (1983), 181-199.
doi: 10.1016/0022-5193(83)90024-3. |
[25] |
J. Scheuer, "Pattern Formation in Reaction-Drift and Diffusion Systems," Diploma thesis, University of Heidelberg, 2009. |
[26] |
H. R. Thieme, "Mathematics in Population Biology," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. |
[27] |
A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72. |
show all references
References:
[1] |
M. S. Alber, M. A. Kiskowski and Y. Jing, Lattice gas cellular automaton model for rippling and aggregation in myxobacteria, Physica D, 191 (2004), 343-358. |
[2] |
U. Börner and M. Bär, Pattern formation in a reaction-advection model with delay: A continuum approach to myxobacterial rippling, Annalen der Physik, 13 (2004), 432-441.
doi: 10.1002/andp.200410086. |
[3] |
U. Börner, A. Deutsch, H. Reichenbach and M. Bär, Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions, Physical Review Letters, 89 (2002), 078101. |
[4] |
O. Diekmann, M. Gyllenberg, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models I. Linear Theory, J. Math. Biol., 36 (1998), 349-388.
doi: 10.1007/s002850050104. |
[5] |
O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models II. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189.
doi: 10.1007/s002850170002. |
[6] |
M. Dworkin and D. Kaiser eds., "Myxobacteria II," American Society for Microbiology (AMS) Press, 1993. |
[7] |
R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis, DCDS-B, 6 (2006), 1239-1260.
doi: 10.3934/dcdsb.2006.6.1239. |
[8] |
R. Erban and H. Othmer, From signal transduction to spatial pattern formation in E.coli : A paradigm for multi-scale modeling in biology, Multiscale Modeling and Simulation, 3 (2005), 362-394.
doi: 10.1137/040603565. |
[9] |
E. Geigant, "Nichtlineare Integro-Differential-Gleichungen zur Modellierung interaktiver Musterbildungsprozesse auf $S^{1}$," (German) [Nonlinear Integro-Differential Equations for the Modelling of Interactive Pattern Formation Processes on $S^{1}$], Bonner Mathematische Schriften 323 Ph.D thesis, University of Bonn, 1999. |
[10] |
E. Geigant, On peak and periodic solutions of an integro-differential equation on $S^1$, in "Geometric Analysis and Nonlinear Partial Differential Equations", Springer-Verlag, Berlin, (2003), 463-474. |
[11] |
E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol., 46 (2003), 537-563.
doi: 10.1007/s00285-002-0187-1. |
[12] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12, (1972), 30-39.
doi: 10.1007/BF00289234. |
[13] |
T. Hillen, A Turing model with correlated random walk, J. Math. Biol., 35 (1996), 49-72.
doi: 10.1007/s002850050042. |
[14] |
O. Igoshin, J. Neu and G. Oster, Developmental waves in Myxobacteria: A novel pattern formation mechanism, Phys. Rev. E, 7 (2004), 1-11. |
[15] |
K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, An Integro-differential equation model for alignment and orientational aggregation, J. of Differential Equations, 246 (2009), 1387-1421.
doi: 10.1016/j.jde.2008.11.006. |
[16] |
T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1980.
doi: 10.1007/978-3-642-66282-9. |
[17] |
F. Lutscher and A. Stevens, Emerging patterns in a hyperbolic model for locally interacting cell systems, J. Nonlinear Science, 12 (2002), 619-640.
doi: 10.1007/s00332-002-0510-4. |
[18] |
H. Meinhardt, Morphogenesis of lines and nets, Differentiation, 6 (1976), 117-123. |
[19] |
J. D. Murray, "Mathematical Biology I and II," Interdisciplinary Applied Mathematics 17 and 18, Springer-Verlag, New York, 2002/03. |
[20] |
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[21] |
B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser-Verlag, Basel, 2007. |
[22] |
B. Pfistner, "Ein Eindimensionales Modell Zum Schwarmverhalten der Myxobakterien Unter Besonderer Berücksichtigung der Randzonenentwicklung," (German) [A one-dimensional model on the swarming behavior of Myxobacteria, with special consideration of the development of the boundary zone], Ph.D thesis, University of Bonn, 1992. |
[23] |
I. Primi, A. Stevens and J. J. L. Velázquez, Mass-selection in alignment models with non-deterministic effects, Communication in PDE, 34 (2009), 419-456.
doi: 10.1080/03605300902797171. |
[24] |
M. Rotenberg, Transport theory for growing cell populations, J. Theor. Biology, 103 (1983), 181-199.
doi: 10.1016/0022-5193(83)90024-3. |
[25] |
J. Scheuer, "Pattern Formation in Reaction-Drift and Diffusion Systems," Diploma thesis, University of Heidelberg, 2009. |
[26] |
H. R. Thieme, "Mathematics in Population Biology," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. |
[27] |
A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237 (1952), 37-72. |
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