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January  2015, 20(1): 231-248. doi: 10.3934/dcdsb.2015.20.231

The stability of bifurcating steady states of several classes of chemotaxis systems

Qian Xu 1,

1. 

Department of Basic Courses, Beijing Union University, Beijing 100101

Received  October 2013 Revised  July 2014 Published  November 2014

This paper concerns with the stability of bifurcating steady states obtained in [13] of several chemotaxis systems. By spectral analysis and the principle of the linearized stability, we prove that the bifurcating steady states are stable when the parameters satisfy some certain conditions.
Keywords: Stability, spectral analysis, bifurcating steady states, chemotaxis systems., expansion.
Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 92C1.
Citation: Qian Xu. The stability of bifurcating steady states of several classes of chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 231-248. doi: 10.3934/dcdsb.2015.20.231
References:
[1]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of the Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134. doi: 10.1016/j.jde.2014.06.008.  Google Scholar

[2]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[4]

M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch.Rational Mech.Anal, 52 (1973), 161-180.  Google Scholar

[5]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[6]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165.  Google Scholar

[7]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69.  Google Scholar

[8]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

X. Lai, X. Chen, C. Qin and Y. Zhang, Existence, uniqueness, and stability of bubble solutions of a chemotaxis model,, preprint., ().   Google Scholar

[10]

A. B. Potapov and T. Hillen, Metastability in chemotaxis models, J. of Dynamics and Diff. Eqs., 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3.  Google Scholar

[11]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[12]

B. Sleeman, M. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117.  Google Scholar

[13]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.  Google Scholar

show all references

References:
[1]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of the Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134. doi: 10.1016/j.jde.2014.06.008.  Google Scholar

[2]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[4]

M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch.Rational Mech.Anal, 52 (1973), 161-180.  Google Scholar

[5]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[6]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165.  Google Scholar

[7]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69.  Google Scholar

[8]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

X. Lai, X. Chen, C. Qin and Y. Zhang, Existence, uniqueness, and stability of bubble solutions of a chemotaxis model,, preprint., ().   Google Scholar

[10]

A. B. Potapov and T. Hillen, Metastability in chemotaxis models, J. of Dynamics and Diff. Eqs., 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3.  Google Scholar

[11]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[12]

B. Sleeman, M. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117.  Google Scholar

[13]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.  Google Scholar

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