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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 and 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), 1677-1693. 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), 630-633. doi: 10.1038/349630a0. [3] __________, Dynamics of formation of symmetric patterns of chemotactic bacteria, Nature, 376 (1995), 49-53. [4] Y. Guo and H. J. Hwang, Pattern formation (I): the Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530. 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), 399-415. 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), 1-31. [7] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, 555+XXII pp, 2013. doi: 10.1007/978-1-4614-8963-4. [8] _________, Dynamic transition theory for thermohaline circulation, Physica D, 239 (2010), 167-189. [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), vi+71. 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), 517-538. 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), 2319-2335. 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), 1253-1270. doi: 10.1088/0951-7715/24/4/012.

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), 1677-1693. 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), 630-633. doi: 10.1038/349630a0. [3] __________, Dynamics of formation of symmetric patterns of chemotactic bacteria, Nature, 376 (1995), 49-53. [4] Y. Guo and H. J. Hwang, Pattern formation (I): the Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530. 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), 399-415. 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), 1-31. [7] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, 555+XXII pp, 2013. doi: 10.1007/978-1-4614-8963-4. [8] _________, Dynamic transition theory for thermohaline circulation, Physica D, 239 (2010), 167-189. [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), vi+71. 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), 517-538. 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), 2319-2335. 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), 1253-1270. doi: 10.1088/0951-7715/24/4/012.
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