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

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 witho...

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Bibliographic Details
Published inSIAM journal on applied mathematics Vol. 72; no. 3; pp. 740 - 766
Main Authors MA, MANJUN, OU, CHUNHUA, WANG, ZHI-AN
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Subjects
Aggregation
Applied mathematics
Asymptotic properties
Bifurcations
Boundary conditions
Cell aggregates
Chemotaxis
Eigenvalues
Eigenvectors
Estimating
Integers
Logistics
Mathematical models
Modeling
Morphogenesis
Parametric models
Position (location)
Simulation
Stability
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Summary: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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ISSN:0036-1399
1095-712X
DOI:10.1137/110843964