Abstract
Mimura and one of the authors (1996) proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis. For this model, Tello and Winkler (2007) [22] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. (2008) numerically showed several spatio-temporal patterns in a rectangle. Motivated by their work, we consider some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints in the present paper. First we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity. Next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, we exhibit several numerical results for the stationary and oscillating patterns.
| Original language | English |
|---|---|
| Pages (from-to) | 1629-1639 |
| Number of pages | 11 |
| Journal | Physica D: Nonlinear Phenomena |
| Volume | 241 |
| Issue number | 19 |
| DOIs | |
| Publication status | Published - 2012 Oct 1 |
| Externally published | Yes |
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Keywords
- Bifurcation
- Chemotaxis
- Pattern formation
ASJC Scopus subject areas
- Condensed Matter Physics
- Statistical and Nonlinear Physics
Cite this
Spatial pattern formation in a chemotaxis-diffusion-growth model. / Kuto, Kousuke; Osaki, Koichi; Sakurai, Tatsunari; Tsujikawa, Tohru.
In: Physica D: Nonlinear Phenomena, Vol. 241, No. 19, 01.10.2012, p. 1629-1639.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Spatial pattern formation in a chemotaxis-diffusion-growth model
AU - Kuto, Kousuke
AU - Osaki, Koichi
AU - Sakurai, Tatsunari
AU - Tsujikawa, Tohru
PY - 2012/10/1
Y1 - 2012/10/1
N2 - Mimura and one of the authors (1996) proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis. For this model, Tello and Winkler (2007) [22] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. (2008) numerically showed several spatio-temporal patterns in a rectangle. Motivated by their work, we consider some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints in the present paper. First we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity. Next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, we exhibit several numerical results for the stationary and oscillating patterns.
AB - Mimura and one of the authors (1996) proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis. For this model, Tello and Winkler (2007) [22] obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution, while Kurata et al. (2008) numerically showed several spatio-temporal patterns in a rectangle. Motivated by their work, we consider some qualitative behaviors of stationary solutions from global and local (bifurcation) viewpoints in the present paper. First we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity. Next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle. For this case, the directions of the branches near the bifurcation points are also obtained. Finally, we exhibit several numerical results for the stationary and oscillating patterns.
KW - Bifurcation
KW - Chemotaxis
KW - Pattern formation
UR - http://www.scopus.com/inward/record.url?scp=84864987912&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84864987912&partnerID=8YFLogxK
U2 - 10.1016/j.physd.2012.06.009
DO - 10.1016/j.physd.2012.06.009
M3 - Article
VL - 241
SP - 1629
EP - 1639
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
IS - 19
ER -