Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/6035
Title: Stationary solutions of a volume-filling chemotaxis model with logistic growth and their stability
Authors: Ma, M
Ou, C
Wang, Z 
Keywords: Chemotaxis
Volume-filling effect
Global-in-time existence
Stationary solutions
Pattern formation
Bifurcation
Stability
Issue Date: 2012
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on applied mathematics, v. 72, no. 3, p. 740–766 How to cite?
Journal: SIAM Journal on applied mathematics 
Abstract: In this paper, we derive the conditions for the existence of stationary solutions (i.e., nonconstant steady states) of a volume-filling chemotaxis model with logistic growth over a bounded domain subject to homogeneous Neumann boundary conditions. At the same time, we show that the same system without the chemotaxis term does not admit pattern formations. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, we establish the stability criteria and find a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue. We show that all bifurcations except the one at the first location of the bifurcation parameter are unstable, and if the pattern is stable, then its principal wave mode must be a positive integer which minimizes the bifurcation parameter. For a special case where the carrying capacity is one half, we find a necessary and sufficient condition for the stability of pattern solutions. Numerical simulations are presented, on the one hand, to illustrate and fit our analytical results and, on the other hand, to demonstrate a variety of interesting spatio-temporal patterns, such as chaotic dynamics and the merging process, which motivate an interesting direction to pursue in the future.
URI: http://hdl.handle.net/10397/6035
ISSN: 0036-1399 (print)
1095-712X (online)
DOI: 10.1137/110843964
Rights: © 2012 Society for Industrial and Applied Mathematics
Appears in Collections:Journal/Magazine Article

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