# American Institute of Mathematical Sciences

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

 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.
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.  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.  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.  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.   Google Scholar [5] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  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.   Google Scholar [7] D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences II,, Jahresber. DMV, 106 (2004), 51.   Google Scholar [8] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret Biol., 26 (1970), 399.  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.  doi: 10.1007/s10884-005-2938-3.  Google Scholar [11] R. Schaaf, Stationary solutions of chemotaxis systems,, Trans. Amer. Math. Soc., 292 (1985), 531.  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.  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.  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.  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.  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.  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.   Google Scholar [5] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  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.   Google Scholar [7] D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences II,, Jahresber. DMV, 106 (2004), 51.   Google Scholar [8] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret Biol., 26 (1970), 399.  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.  doi: 10.1007/s10884-005-2938-3.  Google Scholar [11] R. Schaaf, Stationary solutions of chemotaxis systems,, Trans. Amer. Math. Soc., 292 (1985), 531.  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.  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.  doi: 10.1007/s00285-012-0533-x.  Google Scholar
 [1] Tian Xiang. A study on the positive nonconstant steady states of nonlocal chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2457-2485. doi: 10.3934/dcdsb.2013.18.2457 [2] Anne Nouri, Christian Schmeiser. Aggregated steady states of a kinetic model for chemotaxis. Kinetic & Related Models, 2017, 10 (1) : 313-327. doi: 10.3934/krm.2017013 [3] P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1 [4] Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206 [5] Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049 [6] Yongli Cai, Yun Kang, Weiming Wang. Global stability of the steady states of an epidemic model incorporating intervention strategies. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1071-1089. doi: 10.3934/mbe.2017056 [7] Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849 [8] Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541 [9] Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777 [10] Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063 [11] O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110 [12] Soohyun Bae. Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 823-837. doi: 10.3934/dcds.2010.26.823 [13] Wei-Ming Ni, Yaping Wu, Qian Xu. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271 [14] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [15] Francesca Romana Guarguaglini, Corrado Mascia, Roberto Natalini, Magali Ribot. Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 39-76. doi: 10.3934/dcdsb.2009.12.39 [16] Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067 [17] Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359 [18] Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1 [19] Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367 [20] Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317

2019 Impact Factor: 1.27

## Metrics

• PDF downloads (39)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]