Dynamic transition and pattern formation for chemotactic systems

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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

Abstract
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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.

Keywords: general three-component Keller-Segel with moderated stimulant supplies, rich stimulant two-component system, Keller-Segel model, Hemotaxis, steady state patterns, spatiotemporal oscillatory patterns..

Mathematics Subject Classification: Primary: 92C17; Secondary: 35Q9.

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.
[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.
[3]	__________, Dynamics of formation of symmetric patterns of chemotactic bacteria,, Nature, 376 (1995), 49.
[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.
[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.
[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.
[7]	T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, (2013). doi: 10.1007/978-1-4614-8963-4.
[8]	_________, Dynamic transition theory for thermohaline circulation,, Physica D, 239 (2010), 167.
[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.
[10]	J. Murray, Mathematical Biology, II,, 3rd Ed. Springer-Verlag, (2002).
[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.
[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.
[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.

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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.
[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.
[3]	__________, Dynamics of formation of symmetric patterns of chemotactic bacteria,, Nature, 376 (1995), 49.
[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.
[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.
[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.
[7]	T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, (2013). doi: 10.1007/978-1-4614-8963-4.
[8]	_________, Dynamic transition theory for thermohaline circulation,, Physica D, 239 (2010), 167.
[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.
[10]	J. Murray, Mathematical Biology, II,, 3rd Ed. Springer-Verlag, (2002).
[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.
[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.
[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.

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