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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

Thomas Hillen 1, , Jeffery Zielinski 1, and  Kevin J. Painter 2,

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.
Keywords: Chemotaxis patterns, emerging, discrete dynamical systems, merging, set-valued dynamical systems..
Mathematics Subject Classification: Primary: 92C17, 35B36; Secondary: 37N2.
Citation: Thomas Hillen, Jeffery Zielinski, Kevin J. Painter. Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2513-2536. doi: 10.3934/dcdsb.2013.18.2513
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show all references

References:
[1]

J. London Math. Soc., 74 (2006), 453-474. doi: 10.1112/S0024610706023015.  Google Scholar

[2]

J. Math. Biol., 63 (2011), 141-171. doi: 10.1007/s00285-010-0369-1.  Google Scholar

[3]

Physica A: Statistical Mechanics and its Applications, 391 (2012), 4061-4062. doi: 10.1016/j.physa.2011.12.054.  Google Scholar

[4]

Physical Review E, 84 (2011), 016309 [8 pages]. doi: 10.1103/PhysRevE.84.016309.  Google Scholar

[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.   Google Scholar

[6]

Princeton University Press, 2008. Google Scholar

[7]

letters to Nature, 349 (1991), 630-633. doi: 10.1038/349630a0.  Google Scholar

[8]

Phys. Rev. E, 64, 051914, 2001. doi: 10.1103/PhysRevE.64.051914.  Google Scholar

[9]

J. Math. Biol., 46 (2003), 153-170. doi: 10.1007/s00285-002-0173-7.  Google Scholar

[10]

J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[11]

Math. Mod. Meth. Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480.  Google Scholar

[12]

Jahresberichte der DMV, 105 (2003), 103-165.  Google Scholar

[13]

IMA J. Appl. Math., 72 (2007), 140-162. doi: 10.1093/imamat/hxl028.  Google Scholar

[14]

Trans. AMS, 261 (1980), 589-604. doi: 10.1090/S0002-9947-1980-0580905-3.  Google Scholar

[15]

J. Theo. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[16]

J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[17]

submitted, 2013. Google Scholar

[18]

Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009.  Google Scholar

[19]

In W. Alt, A. Deutsch, and G. Dunn, editors, Dynamics of Cell and Tissue Motion, Birkhäuser, 1996. Google Scholar

[20]

SIAM Appl. Math., 70 (2009), 133-167. doi: 10.1137/070711505.  Google Scholar

[21]

J. Theor. Biol., 189 (1997), 63-80. doi: 10.1006/jtbi.1997.0494.  Google Scholar

[22]

Physica D: Nonlinear Phenomena, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[23]

J. Math. Biol., 55 (2007), 365-388. doi: 10.1007/s00285-007-0088-4.  Google Scholar

[24]

Luminescence, 26 (2011), 716-721. Google Scholar

[25]

Chaos, 17 (2007), 037108, 13 pp. doi: 10.1063/1.2766864.  Google Scholar

[26]

Biophys. J., 68 (1995), 2181-2189. Google Scholar

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