# American Institute of Mathematical Sciences

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

## 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

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.
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.

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##### 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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