Stability and metastability in a chemotaxis model. (English) Zbl 1518.35062
Summary: This work studies the stability and metastability of stationary patterns in a diffusion-chemotaxis model without cell proliferation. We first establish the interval of unstable wave modes of the homogeneous steady state, and show that the chemotactic flux is the key mechanism for pattern formation. Then, we treat the chemotaxis coefficient as a bifurcation parameter to obtain the asymptotic expressions of steady states. Based on this, we derive the sufficient conditions for the stability of one-step pattern, and prove the metastability of two or more step patterns. All the analytical results are demonstrated by numerical simulations.
MSC:
| 35B35 | Stability in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 92C15 | Developmental biology, pattern formation |
| 92C17 | Cell movement (chemotaxis, etc.) |
References:
| [1] | Adler, J., Chemotaxis in bacteria, Science153 (1966) 708-716. |
| [2] | Adler, J., Chemoreceptors in bacteria, Science166 (1969) 1588-1597. |
| [3] | Brenner, M. P., Levitor, L. S. and Budrene, E. O., Physical mechanisms for chemotactic pattern formation by bacterial, Biophys. J.74 (1988) 1677-1693. |
| [4] | Budrene, E. and Berg, H., Complex patterns formed by motile cells of Escherichia coli, Nature349 (1991) 630-633. |
| [5] | Budrene, E. and Berg, H., Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature376 (1995) 49-53. |
| [6] | Goldstein, R. E., Traveling-wave chemotaxis, Phys. Rev. Lett.77 (1996) 775-778. |
| [7] | Horstmann, D., From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. Verein.105(3) (2003) 103-165. · Zbl 1071.35001 |
| [8] | Horstmann, D. and Winkler, M., Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ.215 (2005) 52-107. · Zbl 1085.35065 |
| [9] | Keller, E. F. and Segel, L. A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol.26 (1970) 399-415. · Zbl 1170.92306 |
| [10] | Keller, E. F. and Segel, L. A., Model for chemotaxis, J. Theor. Biol.30 (1971) 225-234. · Zbl 1170.92307 |
| [11] | Lee, K. J., Cox, E. C. and Goldstein, R. E., Competing patterns of signaling activity in Dictyostelium Discoideum, Phys. Rev. Lett.76 (1996) 1174-1177. |
| [12] | Ma, M. and Wang, Z., Patterns in a generalized volume-filling chemotaxis model with cell proliferation, Anal. Appl.15(1) (2017) 83-106. · Zbl 1361.35074 |
| [13] | Ma, M. and Wang, Z., Global bifurcation and stability of steady states for a reaction-diffusion-chemotaxis model with volume-filling effect, Nonlinearity28 (2015) 2639-2660 · Zbl 1331.35185 |
| [14] | Ma, M., Ou, C. and Wang, Z., Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math.72 (2012) 740-766. · Zbl 1259.35031 |
| [15] | Patlak, C. S., Random walk with persistence and external bias, Bull. Math. Biophys.15 (1953) 311-338. · Zbl 1296.82044 |
| [16] | Painter, K. and Hillen, T., Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q.10(4) (2002) 501-543. · Zbl 1057.92013 |
| [17] | Potapov, A. and Hillen, T., Metastability in chemotaxis models, J. Dyn. Differ. Equ.17 (2005) 293-330. · Zbl 1170.35460 |
| [18] | Schaaf, R., Stationary solutions of chemotaxis systems, Trans. AMS292(2) (1985) 531-556. · Zbl 0637.35007 |
| [19] | Wang, Z. and Hillen, T., Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos17 (2007) 037108. · Zbl 1163.37383 |
| [20] | Wang, Z., Winkler, M. and Wrzosek, D., Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity24 (2011) 3279-3297. · Zbl 1255.35213 |
| [21] | Wang, Z., Winkler, M. and Wrzosek, D., Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume filling effect and degenerate diffusion, SIAM J. Math. Anal.44 (2012) 3502-3525. · Zbl 1277.35223 |
| [22] | Wrzosek, D., Model of chemotaxis with threshold density and singular diffusion, Nonlinear Anal.73 (2010) 338-349. · Zbl 1197.35068 |
| [23] | Xia, P., Han, Y., Tao, J. and Ma, M., Existence and metastability of non-constant steady states in a Keller-Segel model with density-suppressed motility, Math. Appl. Sci. Eng.1(1) (2020) 1-15. · Zbl 1498.92037 |
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