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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| Published in | SIAM journal on applied mathematics Vol. 72; no. 3; pp. 740 - 766 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Philadelphia
Society for Industrial and Applied Mathematics
01.01.2012
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0036-1399 1095-712X |
| DOI | 10.1137/110843964 |
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| 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. |
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| AbstractList | 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. 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. [PUBLICATION ABSTRACT] |
| Author | MA, MANJUN WANG, ZHI-AN OU, CHUNHUA |
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| Cites_doi | 10.1038/376049a0 10.1063/1.2766864 10.3233/ASY-2009-0948 10.1016/S0362-546X(04)00327-X 10.1098/rspb.1995.0037 10.1016/0022-5193(71)90050-6 10.1016/j.physd.2010.09.011 10.1073/pnas.96.10.5549 10.1002/bit.260370708 10.1080/03605300903473426 10.1137/070684173 10.1007/s00285-008-0201-3 10.1007/s10884-005-2938-3 10.1016/0022-5193(70)90092-5 10.1016/S1534-5807(02)00256-3 10.1038/349630a0 10.1006/jmaa.1998.6167 10.1016/S0362-546X(01)00815-X 10.1006/bulm.1999.0166 10.1137/08072797X 10.1080/03605300701319003 10.1017/S0308210500004649 10.1137/1018114 10.1002/bit.260370707 |
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| SubjectTerms | 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 |
| Title | STATIONARY SOLUTIONS OF A VOLUME-FILLING CHEMOTAXIS MODEL WITH LOGISTIC GROWTH AND THEIR STABILITY |
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