American Institute of Mathematical Sciences

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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

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 & 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. Google Scholar

[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. doi: 10.1002/andp.200410086. Google Scholar

[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). Google Scholar

[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. doi: 10.1007/s002850050104. Google Scholar

[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. doi: 10.1007/s002850170002. Google Scholar

[6]

M. Dworkin and D. Kaiser eds., "Myxobacteria II,", American Society for Microbiology (AMS) Press, (1993). Google Scholar

[7]

R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis,, DCDS-B, 6 (2006), 1239. doi: 10.3934/dcdsb.2006.6.1239. Google Scholar

[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. doi: 10.1137/040603565. Google Scholar

[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 $S^{1}$], 323 (1999). Google Scholar

[10]

E. Geigant, On peak and periodic solutions of an integro-differential equation on $S^1$,, in, (2003), 463. Google Scholar

[11]

E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model,, J. Math. Biol., 46 (2003), 537. doi: 10.1007/s00285-002-0187-1. Google Scholar

[12]

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

[13]

T. Hillen, A Turing model with correlated random walk,, J. Math. Biol., 35 (1996), 49. doi: 10.1007/s002850050042. Google Scholar

[14]

O. Igoshin, J. Neu and G. Oster, Developmental waves in Myxobacteria: A novel pattern formation mechanism,, Phys. Rev. E, 7 (2004), 1. Google Scholar

[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. doi: 10.1016/j.jde.2008.11.006. Google Scholar

[16]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1980). doi: 10.1007/978-3-642-66282-9. Google Scholar

[17]

F. Lutscher and A. Stevens, Emerging patterns in a hyperbolic model for locally interacting cell systems,, J. Nonlinear Science, 12 (2002), 619. doi: 10.1007/s00332-002-0510-4. Google Scholar

[18]

H. Meinhardt, Morphogenesis of lines and nets,, Differentiation, 6 (1976), 117. Google Scholar

[19]

J. D. Murray, "Mathematical Biology I and II,", Interdisciplinary Applied Mathematics 17 and 18, (2002). Google Scholar

[20]

H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[21]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007). Google Scholar

[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, (1992). Google Scholar

[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. doi: 10.1080/03605300902797171. Google Scholar

[24]

M. Rotenberg, Transport theory for growing cell populations,, J. Theor. Biology, 103 (1983), 181. doi: 10.1016/0022-5193(83)90024-3. Google Scholar

[25]

J. Scheuer, "Pattern Formation in Reaction-Drift and Diffusion Systems,", Diploma thesis, (2009). Google Scholar

[26]

H. R. Thieme, "Mathematics in Population Biology,", Princeton Series in Theoretical and Computational Biology, (2003). Google Scholar

[27]

A. M. Turing, The chemical basis of morphogenesis,, Philosophical Transactions of the Royal Society of London. Series B, 237 (1952), 37. Google Scholar

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. Google Scholar

[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. doi: 10.1002/andp.200410086. Google Scholar

[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). Google Scholar

[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. doi: 10.1007/s002850050104. Google Scholar

[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. doi: 10.1007/s002850170002. Google Scholar

[6]

M. Dworkin and D. Kaiser eds., "Myxobacteria II,", American Society for Microbiology (AMS) Press, (1993). Google Scholar

[7]

R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis,, DCDS-B, 6 (2006), 1239. doi: 10.3934/dcdsb.2006.6.1239. Google Scholar

[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. doi: 10.1137/040603565. Google Scholar

[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 $S^{1}$], 323 (1999). Google Scholar

[10]

E. Geigant, On peak and periodic solutions of an integro-differential equation on $S^1$,, in, (2003), 463. Google Scholar

[11]

E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model,, J. Math. Biol., 46 (2003), 537. doi: 10.1007/s00285-002-0187-1. Google Scholar

[12]

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

[13]

T. Hillen, A Turing model with correlated random walk,, J. Math. Biol., 35 (1996), 49. doi: 10.1007/s002850050042. Google Scholar

[14]

O. Igoshin, J. Neu and G. Oster, Developmental waves in Myxobacteria: A novel pattern formation mechanism,, Phys. Rev. E, 7 (2004), 1. Google Scholar

[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. doi: 10.1016/j.jde.2008.11.006. Google Scholar

[16]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1980). doi: 10.1007/978-3-642-66282-9. Google Scholar

[17]

F. Lutscher and A. Stevens, Emerging patterns in a hyperbolic model for locally interacting cell systems,, J. Nonlinear Science, 12 (2002), 619. doi: 10.1007/s00332-002-0510-4. Google Scholar

[18]

H. Meinhardt, Morphogenesis of lines and nets,, Differentiation, 6 (1976), 117. Google Scholar

[19]

J. D. Murray, "Mathematical Biology I and II,", Interdisciplinary Applied Mathematics 17 and 18, (2002). Google Scholar

[20]

H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[21]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007). Google Scholar

[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, (1992). Google Scholar

[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. doi: 10.1080/03605300902797171. Google Scholar

[24]

M. Rotenberg, Transport theory for growing cell populations,, J. Theor. Biology, 103 (1983), 181. doi: 10.1016/0022-5193(83)90024-3. Google Scholar

[25]

J. Scheuer, "Pattern Formation in Reaction-Drift and Diffusion Systems,", Diploma thesis, (2009). Google Scholar

[26]

H. R. Thieme, "Mathematics in Population Biology,", Princeton Series in Theoretical and Computational Biology, (2003). Google Scholar

[27]

A. M. Turing, The chemical basis of morphogenesis,, Philosophical Transactions of the Royal Society of London. Series B, 237 (1952), 37. Google Scholar

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