American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$
  • DCDS-B Home
  • This Issue
  • Next Article
    Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring
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
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.  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), 141-171. 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), 4061-4062. doi: 10.1016/j.physa.2011.12.054.  Google Scholar

[4]

D. Barkley, Simplifying the complexity of pipe flow, 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]

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), 630-633. doi: 10.1038/349630a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 165-198. 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), 103-165.  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), 140-162. doi: 10.1093/imamat/hxl028.  Google Scholar

[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.  Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415. 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), 225-234. 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), 1629-1639. doi: 10.1016/j.physd.2012.06.009.  Google Scholar

[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. Google Scholar

[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.  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), 63-80. 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), 363-375. 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), 365-388. 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), 716-721. Google Scholar

[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.  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), 2181-2189. 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), 453-474. 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), 141-171. 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), 4061-4062. doi: 10.1016/j.physa.2011.12.054.  Google Scholar

[4]

D. Barkley, Simplifying the complexity of pipe flow, 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]

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), 630-633. doi: 10.1038/349630a0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 165-198. 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), 103-165.  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), 140-162. doi: 10.1093/imamat/hxl028.  Google Scholar

[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.  Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415. 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), 225-234. 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), 1629-1639. doi: 10.1016/j.physd.2012.06.009.  Google Scholar

[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. Google Scholar

[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.  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), 63-80. 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), 363-375. 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), 365-388. 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), 716-721. Google Scholar

[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.  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), 2181-2189. Google Scholar

[1]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91
[2]

Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012
[3]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[4]

Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4665-4681. doi: 10.3934/dcds.2015.35.4665
[5]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129
[6]

B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537
[7]

Jean-Luc Chabert, Ai-Hua Fan, Youssef Fares. Minimal dynamical systems on a discrete valuation domain. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 777-795. doi: 10.3934/dcds.2009.25.777
[8]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187
[9]

Mostafa Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont, Olivier Goubet. Discrete Schrödinger equations and dissipative dynamical systems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 211-227. doi: 10.3934/cpaa.2008.7.211
[10]

Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential trichotomy of dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2929-2962. doi: 10.3934/dcds.2014.34.2929
[11]

Jacobo Pejsachowicz, Robert Skiba. Topology and homoclinic trajectories of discrete dynamical systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1077-1094. doi: 10.3934/dcdss.2013.6.1077
[12]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[13]

Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243
[14]

Simone Fiori. Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2785-2808. doi: 10.3934/dcdsb.2014.19.2785
[15]

Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124
[16]

Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807
[17]

Adina Luminiţa Sasu, Bogdan Sasu. Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3199-3220. doi: 10.3934/dcdsb.2017170
[18]

Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287
[19]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[20]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences