Journal
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
Volume 26, Issue 2, Pages -Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218127416500334
Keywords
Pattern formation; steady state; bifurcation; chemotaxis model; logistic growth
Funding
- SRF for ROCS, SEM
- Department of Education, Sichuan China [15ZA0382]
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In this paper, we investigate pattern formation in Keller-Segel chemotaxis models over a multidimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate. increases. Then using Crandall-Rabinowitz local theory with. being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations. Our results provide a selection mechanism of stable wavemode which states that the only stable bifurcation branch must have a wavemode number that minimizes the bifurcation value. Finally, we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary spikes, interior spikes, stripes, etc. These nontrivial patterns can model cellular aggregation that develop through chemotactic movements in biological systems.
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