Pattern Formation in Keller-Segel Chemotaxis Models with Logistic Growth

In this paper, we investigate the pattern formation in Keller–Segel chemotaxis over multi– dimensional bounded domains subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate χ increases. Then, using Crandall–Rabinowitz local theory with χ being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcates from a homogeneous steady state. The stability of the bifurcating solutions is also established. Our results indicate that L–norm of the k–th eigenfunction of −∆ under NBC determines the stability, together with system parameters. Finally, we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary, interior spikes, stripes, etc. These spiky solutions can model cellular aggregations that develop through chemotactic movements in biological systems.