• Previous Article
    Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds
  • DCDS-B Home
  • This Issue
  • Next Article
    Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative
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.

[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.

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.

[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.

[1]

Qi Wang, Jingyue Yang, Lu Zhang. Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3547-3574. doi: 10.3934/dcdsb.2017179

[2]

Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012

[3]

Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81

[4]

Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061

[5]

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

[6]

Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042

[7]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231

[8]

Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078

[9]

Jaewook Ahn, Kyungkeun Kang. On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5165-5179. doi: 10.3934/dcds.2014.34.5165

[10]

Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks & Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181

[11]

Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317

[12]

Shangbing Ai, Zhian Wang. Traveling bands for the Keller-Segel model with population growth. Mathematical Biosciences & Engineering, 2015, 12 (4) : 717-737. doi: 10.3934/mbe.2015.12.717

[13]

Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic & Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035

[14]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046

[15]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

[16]

Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109

[17]

Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023

[18]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243

[19]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049

[20]

Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]