4.5 Article

STATIONARY SOLUTIONS OF A VOLUME-FILLING CHEMOTAXIS MODEL WITH LOGISTIC GROWTH AND THEIR STABILITY

期刊

SIAM JOURNAL ON APPLIED MATHEMATICS
卷 72, 期 3, 页码 740-766

出版社

SIAM PUBLICATIONS
DOI: 10.1137/110843964

关键词

chemotaxis; volume-filling effect; global-in-time existence; stationary solutions; pattern formation; bifurcation; stability

资金

  1. Zhejiang province [NSF-Y6100166]
  2. NSERC of Canada
  3. Mathematical Biosciences Institute (MBI) at The Ohio State University through the National Science Foundation [DMS 0931642]
  4. Hong Kong RGC General Research Fund [502711]

向作者/读者索取更多资源

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.

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