Pattern forming instabilities driven by non-diffusive interactions

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March 2013, 8(1): 397-432. doi: 10.3934/nhm.2013.8.397

Pattern forming instabilities driven by non-diffusive interactions

Ivano Primi ^1, , Angela Stevens ^2, and Juan J. L. Velázquez ^3,

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

Received January 2012 Revised March 2013 Published April 2013

Abstract
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In analogy to the analysis of minimal conditions for the formation of diffusion driven instabilities in the sense of Turing, in this paper minimal conditions for a class of kinetic equations with mass conservation are discussed, whose solutions show patterns with a characteristic wavelength. The related linearized systems are analyzed, and the minimal number of equations is derived, which is needed for specific patterns to occur.

Keywords: rippling dynamics of myxobacteria., pattern formation, Hyperbolic system, perturbation of spectra of matrices.

Mathematics Subject Classification: Primary: 35L45, 35P20, 35B20, 35B36; Secondary: 92C15, 92C1.

Citation: Ivano Primi, Angela Stevens, Juan J. L. Velázquez. Pattern forming instabilities driven by non-diffusive interactions. Networks and Heterogeneous Media, 2013, 8 (1) : 397-432. doi: 10.3934/nhm.2013.8.397

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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