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November  2014, 19(9): 2809-2835. doi: 10.3934/dcdsb.2014.19.2809

Dynamic transition and pattern formation for chemotactic systems

1. 

Department of Mathematics, Sichuan University, Chengdu

2. 

Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  June 2012 Revised  June 2014 Published  September 2014

The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$, which is derived by incorporating the nonlinear interactions of both stable and unstable modes. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.
Citation: Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809
References:
[1]

M. P. Brenner, L. S. Levitov and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacteria,, Biophysical Journal, 74 (1998), 1677.  doi: 10.1016/S0006-3495(98)77880-4.  Google Scholar

[2]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630.  doi: 10.1038/349630a0.  Google Scholar

[3]

__________, Dynamics of formation of symmetric patterns of chemotactic bacteria,, Nature, 376 (1995), 49.   Google Scholar

[4]

Y. Guo and H. J. Hwang, Pattern formation (I): the Keller-Segel model,, J. Differential Equations, 249 (2010), 1519.  doi: 10.1016/j.jde.2010.07.025.  Google Scholar

[5]

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

[6]

H. Liu, T. Sengul and S. Wang, Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility,, Journal of Mathematical Physics, 53 (2012), 1.   Google Scholar

[7]

T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, (2013).  doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[8]

_________, Dynamic transition theory for thermohaline circulation,, Physica D, 239 (2010), 167.   Google Scholar

[9]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models,, Mem. Amer. Math. Soc., 202 (2009).  doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[10]

J. Murray, Mathematical Biology, II,, 3rd Ed. Springer-Verlag, (2002).   Google Scholar

[11]

G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms,, Interfaces Free Bound., 10 (2008), 517.  doi: 10.4171/IFB/200.  Google Scholar

[12]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model,, Trans. Amer. Math. Soc., 361 (2009), 2319.  doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[13]

B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: Existence and branching instabilities,, Nonlinearity, 24 (2011), 1253.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

show all references

References:
[1]

M. P. Brenner, L. S. Levitov and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacteria,, Biophysical Journal, 74 (1998), 1677.  doi: 10.1016/S0006-3495(98)77880-4.  Google Scholar

[2]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630.  doi: 10.1038/349630a0.  Google Scholar

[3]

__________, Dynamics of formation of symmetric patterns of chemotactic bacteria,, Nature, 376 (1995), 49.   Google Scholar

[4]

Y. Guo and H. J. Hwang, Pattern formation (I): the Keller-Segel model,, J. Differential Equations, 249 (2010), 1519.  doi: 10.1016/j.jde.2010.07.025.  Google Scholar

[5]

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

[6]

H. Liu, T. Sengul and S. Wang, Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility,, Journal of Mathematical Physics, 53 (2012), 1.   Google Scholar

[7]

T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, (2013).  doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[8]

_________, Dynamic transition theory for thermohaline circulation,, Physica D, 239 (2010), 167.   Google Scholar

[9]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models,, Mem. Amer. Math. Soc., 202 (2009).  doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[10]

J. Murray, Mathematical Biology, II,, 3rd Ed. Springer-Verlag, (2002).   Google Scholar

[11]

G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms,, Interfaces Free Bound., 10 (2008), 517.  doi: 10.4171/IFB/200.  Google Scholar

[12]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model,, Trans. Amer. Math. Soc., 361 (2009), 2319.  doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[13]

B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: Existence and branching instabilities,, Nonlinearity, 24 (2011), 1253.  doi: 10.1088/0951-7715/24/4/012.  Google Scholar

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