Pattern formation in Keller-Segel chemotaxis models with logistic growth. (English) Zbl 1334.35003
Summary: 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 \(\chi\) increases. Then using Crandall-Rabinowitz local theory with \(\chi\) 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.
MSC:
| 35B36 | Pattern formations in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 35B32 | Bifurcations in context of PDEs |
| 35B35 | Stability in context of PDEs |
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