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December  2013, 18(10): 2513-2536. doi: 10.3934/dcdsb.2013.18.2513

## Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model

 1 Centre for Mathematical Biology, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G2G1, Canada 2 Department of Mathematics and Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom

Received  January 2013 Revised  May 2013 Published  October 2013

In a recent study (K.J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (4), 363-375, 2011) a model for chemotaxis incorporating logistic growth was investigated for its pattern formation properties. In particular, a variety of complex spatio-temporal patterning was found, including stationary, periodic and chaotic. Complicated dynamics appear to arise through a sequence of merging and emerging'' events: the merging of two neighbouring aggregates or the emergence of a new aggregate in an open space. In this paper we focus on a time-discrete dynamical system motivated by these dynamics, which we call the merging-emerging system (MES). We introduce this new class of set-valued dynamical systems and analyse its capacity to generate similar pattern formation'' dynamics. The MES shows remarkably close correspondence with patterning in the logistic chemotaxis model, strengthening our assertion that the characteristic length scales of merging and emerging are responsible for the observed dynamics. Furthermore, the MES describes a novel class of pattern-forming discrete dynamical systems worthy of study in its own right.
Citation: Thomas Hillen, Jeffery Zielinski, Kevin J. Painter. Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2513-2536. doi: 10.3934/dcdsb.2013.18.2513
##### References:
 [1] M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453-474. doi: 10.1112/S0024610706023015. [2] V. Andasari, A. Gerisch, G. Lolas, A. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1. [3] S. Banerjee, A. P. Misra and L. Rondoni, Spatiotemporal evolution in a (2+ 1)-dimensional chemotaxis model, Physica A: Statistical Mechanics and its Applications, 391 (2012), 4061-4062. doi: 10.1016/j.physa.2011.12.054. [4] D. Barkley, Simplifying the complexity of pipe flow, Physical Review E, 84 (2011), 016309 [8 pages]. doi: 10.1103/PhysRevE.84.016309. [5] R. Baronas and R. Šimkus, Modelling the bacterial self-organization in circular container along the contact line as detected by bioluminescence imaging,, Nonlinear Anal. Model. Control, 16 (): 270. [6] J. T. Bonner, The Social Amoebae: The Biology of Cellular Slime Molds, Princeton University Press, 2008. [7] E. O. Budrene and H. C. Berg, et al, Complex patterns formed by motile cells of Escherichia coli, letters to Nature, 349 (1991), 630-633. doi: 10.1038/349630a0. [8] G. de Vries, Bursting as an emergent phenomenon in coupled chaotic maps, Phys. Rev. E, 64, 051914, 2001. doi: 10.1103/PhysRevE.64.051914. [9] Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170. doi: 10.1007/s00285-002-0173-7. [10] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [11] T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480. [12] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV, 105 (2003), 103-165. [13] K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162. doi: 10.1093/imamat/hxl028. [14] J. P. Keener, Chaotic Behavior in Piecewise Continuous Difference Equations, Trans. AMS, 261 (1980), 589-604. doi: 10.1090/S0002-9947-1980-0580905-3. [15] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [16] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [17] T. Kolokolnikov and J. Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, submitted, 2013. [18] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009. [19] H. G. Othmer and J. Dallon, Models of Dictyostelium aggregation, In W. Alt, A. Deutsch, and G. Dunn, editors, Dynamics of Cell and Tissue Motion, Birkhäuser, 1996. [20] H. G. Othmer and C. Xue, Multiscale models of taxis-driven patterning in bacterial populations, SIAM Appl. Math., 70 (2009), 133-167. doi: 10.1137/070711505. [21] M. R. Owen and J. A. Sherratt, Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions, J. Theor. Biol., 189 (1997), 63-80. doi: 10.1006/jtbi.1997.0494. [22] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011. [23] I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson and S. F. Hubbard, Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388. doi: 10.1007/s00285-007-0088-4. [24] R. Šimkus and R. Baronas, Metabolic self-organization of bioluminescent escherichia coli, Luminescence, 26 (2011), 716-721. [25] Z. A. Wang and T. Hillen, Pattern formation for a chemotaxis model with volume filling, Chaos, 17 (2007), 037108, 13 pp. doi: 10.1063/1.2766864. [26] D. D. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. Budrene and H. C. Berg, Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189.

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##### References:
 [1] M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453-474. doi: 10.1112/S0024610706023015. [2] V. Andasari, A. Gerisch, G. Lolas, A. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1. [3] S. Banerjee, A. P. Misra and L. Rondoni, Spatiotemporal evolution in a (2+ 1)-dimensional chemotaxis model, Physica A: Statistical Mechanics and its Applications, 391 (2012), 4061-4062. doi: 10.1016/j.physa.2011.12.054. [4] D. Barkley, Simplifying the complexity of pipe flow, Physical Review E, 84 (2011), 016309 [8 pages]. doi: 10.1103/PhysRevE.84.016309. [5] R. Baronas and R. Šimkus, Modelling the bacterial self-organization in circular container along the contact line as detected by bioluminescence imaging,, Nonlinear Anal. Model. Control, 16 (): 270. [6] J. T. Bonner, The Social Amoebae: The Biology of Cellular Slime Molds, Princeton University Press, 2008. [7] E. O. Budrene and H. C. Berg, et al, Complex patterns formed by motile cells of Escherichia coli, letters to Nature, 349 (1991), 630-633. doi: 10.1038/349630a0. [8] G. de Vries, Bursting as an emergent phenomenon in coupled chaotic maps, Phys. Rev. E, 64, 051914, 2001. doi: 10.1103/PhysRevE.64.051914. [9] Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170. doi: 10.1007/s00285-002-0173-7. [10] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [11] T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480. [12] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV, 105 (2003), 103-165. [13] K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162. doi: 10.1093/imamat/hxl028. [14] J. P. Keener, Chaotic Behavior in Piecewise Continuous Difference Equations, Trans. AMS, 261 (1980), 589-604. doi: 10.1090/S0002-9947-1980-0580905-3. [15] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [16] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [17] T. Kolokolnikov and J. Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, submitted, 2013. [18] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009. [19] H. G. Othmer and J. Dallon, Models of Dictyostelium aggregation, In W. Alt, A. Deutsch, and G. Dunn, editors, Dynamics of Cell and Tissue Motion, Birkhäuser, 1996. [20] H. G. Othmer and C. Xue, Multiscale models of taxis-driven patterning in bacterial populations, SIAM Appl. Math., 70 (2009), 133-167. doi: 10.1137/070711505. [21] M. R. Owen and J. A. Sherratt, Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions, J. Theor. Biol., 189 (1997), 63-80. doi: 10.1006/jtbi.1997.0494. [22] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011. [23] I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson and S. F. Hubbard, Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388. doi: 10.1007/s00285-007-0088-4. [24] R. Šimkus and R. Baronas, Metabolic self-organization of bioluminescent escherichia coli, Luminescence, 26 (2011), 716-721. [25] Z. A. Wang and T. Hillen, Pattern formation for a chemotaxis model with volume filling, Chaos, 17 (2007), 037108, 13 pp. doi: 10.1063/1.2766864. [26] D. D. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. Budrene and H. C. Berg, Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189.
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