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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 & 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. doi: 10.1112/S0024610706023015. Google Scholar

[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. doi: 10.1007/s00285-010-0369-1. Google Scholar

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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. doi: 10.1016/j.physa.2011.12.054. Google Scholar

[4]

D. Barkley, Simplifying the complexity of pipe flow,, Physical Review E, 84 (2011). 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]

J. T. Bonner, The Social Amoebae: The Biology of Cellular Slime Molds,, Princeton University Press, (2008). Google Scholar

[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. doi: 10.1038/349630a0. Google Scholar

[8]

G. de Vries, Bursting as an emergent phenomenon in coupled chaotic maps,, Phys. Rev. E, (0519). doi: 10.1103/PhysRevE.64.051914. Google Scholar

[9]

Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 46 (2003), 153. doi: 10.1007/s00285-002-0173-7. Google Scholar

[10]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[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. doi: 10.1142/S0218202512500480. Google Scholar

[12]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jahresberichte der DMV, 105 (2003), 103. Google Scholar

[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. doi: 10.1093/imamat/hxl028. Google Scholar

[14]

J. P. Keener, Chaotic Behavior in Piecewise Continuous Difference Equations,, Trans. AMS, 261 (1980), 589. doi: 10.1090/S0002-9947-1980-0580905-3. Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theo. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[16]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[17]

T. Kolokolnikov and J. Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth,, submitted, (2013). Google Scholar

[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. doi: 10.1016/j.physd.2012.06.009. Google Scholar

[19]

H. G. Othmer and J. Dallon, Models of Dictyostelium aggregation,, In W. Alt, (1996). Google Scholar

[20]

H. G. Othmer and C. Xue, Multiscale models of taxis-driven patterning in bacterial populations,, SIAM Appl. Math., 70 (2009), 133. doi: 10.1137/070711505. Google Scholar

[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. doi: 10.1006/jtbi.1997.0494. Google Scholar

[22]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011. Google Scholar

[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. doi: 10.1007/s00285-007-0088-4. Google Scholar

[24]

R. Šimkus and R. Baronas, Metabolic self-organization of bioluminescent escherichia coli,, Luminescence, 26 (2011), 716. Google Scholar

[25]

Z. A. Wang and T. Hillen, Pattern formation for a chemotaxis model with volume filling,, Chaos, 17 (2007). doi: 10.1063/1.2766864. Google Scholar

[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. Google Scholar

show all references

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. doi: 10.1112/S0024610706023015. Google Scholar

[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. doi: 10.1007/s00285-010-0369-1. Google Scholar

[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. doi: 10.1016/j.physa.2011.12.054. Google Scholar

[4]

D. Barkley, Simplifying the complexity of pipe flow,, Physical Review E, 84 (2011). 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]

J. T. Bonner, The Social Amoebae: The Biology of Cellular Slime Molds,, Princeton University Press, (2008). Google Scholar

[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. doi: 10.1038/349630a0. Google Scholar

[8]

G. de Vries, Bursting as an emergent phenomenon in coupled chaotic maps,, Phys. Rev. E, (0519). doi: 10.1103/PhysRevE.64.051914. Google Scholar

[9]

Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 46 (2003), 153. doi: 10.1007/s00285-002-0173-7. Google Scholar

[10]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[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. doi: 10.1142/S0218202512500480. Google Scholar

[12]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jahresberichte der DMV, 105 (2003), 103. Google Scholar

[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. doi: 10.1093/imamat/hxl028. Google Scholar

[14]

J. P. Keener, Chaotic Behavior in Piecewise Continuous Difference Equations,, Trans. AMS, 261 (1980), 589. doi: 10.1090/S0002-9947-1980-0580905-3. Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theo. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[16]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[17]

T. Kolokolnikov and J. Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth,, submitted, (2013). Google Scholar

[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. doi: 10.1016/j.physd.2012.06.009. Google Scholar

[19]

H. G. Othmer and J. Dallon, Models of Dictyostelium aggregation,, In W. Alt, (1996). Google Scholar

[20]

H. G. Othmer and C. Xue, Multiscale models of taxis-driven patterning in bacterial populations,, SIAM Appl. Math., 70 (2009), 133. doi: 10.1137/070711505. Google Scholar

[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. doi: 10.1006/jtbi.1997.0494. Google Scholar

[22]

K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011. Google Scholar

[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. doi: 10.1007/s00285-007-0088-4. Google Scholar

[24]

R. Šimkus and R. Baronas, Metabolic self-organization of bioluminescent escherichia coli,, Luminescence, 26 (2011), 716. Google Scholar

[25]

Z. A. Wang and T. Hillen, Pattern formation for a chemotaxis model with volume filling,, Chaos, 17 (2007). doi: 10.1063/1.2766864. Google Scholar

[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. Google Scholar

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