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December  2013, 18(10): 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

Pattern formation of the attraction-repulsion Keller-Segel system

1. 

Y.Y. Tseng Functional Analysis Research Center and School of Mathematics Science, Harbin Normal University, Harbin, Heilongjiang, 150025

2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, United States

3. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  December 2012 Revised  June 2013 Published  October 2013

In this paper, the pattern formation of the attraction-repulsion Keller-Segel (ARKS) system is studied analytically and numerically. By the Hopf bifurcation theorem as well as the local and global bifurcation theorem, we rigorously establish the existence of time-periodic patterns and steady state patterns for the ARKS model in the full parameter regimes, which are identified by a linear stability analysis. We also show that when the chemotactic attraction is strong, a spiky steady state pattern can develop. Explicit time-periodic rippling wave patterns and spiky steady state patterns are obtained numerically by carefully selecting parameter values based on our theoretical results. The study in the paper asserts that chemotactic competitive interaction between attraction and repulsion can produce periodic patterns which are impossible for the chemotaxis model with a single chemical (either chemo-attractant or chemo-repellent).
Citation: Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597
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show all references

References:
[1]

Science, 153 (1966), 708-716. Google Scholar

[2]

Differential Integral Equations, 3 (1990), 13-75.  Google Scholar

[3]

in Delay differential equations and dynamical systems (Claremont, CA, 1990), vol. 1475 of Lecture Notes in Math., Springer, Berlin, 1991, 53-63. doi: 10.1007/BFb0083479.  Google Scholar

[4]

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

[5]

IMA J. Math. Appl. Med., 10 (1993), 149-168. Google Scholar

[6]

Dev. Biol., 296 (2006), 137-149. doi: 10.1016/j.ydbio.2006.04.451.  Google Scholar

[7]

J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[8]

Arch. Rational Mech. Anal., 67 (1977), 53-72. doi: 10.1007/BF00280827.  Google Scholar

[9]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 315-329.  Google Scholar

[10]

Nonlinear Anal., 13 (1989), 1091-1113. doi: 10.1016/0362-546X(89)90097-7.  Google Scholar

[11]

R. Firtel, Dictyostelium cinema,, http://people.biology.ucsd.edu/firtel/video.htm., ().   Google Scholar

[12]

Phys. Rev. Lett., 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101.  Google Scholar

[13]

Euro. J. Neuroscience, 19 (2004), 831-844. Google Scholar

[14]

Phys. Rev. Lett., 77 (1996), 775-778. doi: 10.1103/PhysRevLett.77.775.  Google Scholar

[15]

IMA J. Math. Appl. Med. Biol., 6 (1989), 69-79. doi: 10.1093/imammb/6.2.69.  Google Scholar

[16]

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[17]

Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.  Google Scholar

[18]

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[19]

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[21]

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[22]

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[23]

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[24]

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[25]

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in Stationary partial differential equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, 157-233. doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

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Can. Appl. Math. Q., 10 (2002), 501-543.  Google Scholar

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Birkhäuser Verlag, Basel, 2007.  Google Scholar

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Math. Biosci., 136 (2003), 35-63. Google Scholar

[41]

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[42]

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[43]

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