Spatio-Temporal Patterns and Finite-Dimensional Dynamical System generated by a Reaction-Diffusion Chemotaxis System

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We examined the Mimura-Tsujikawa system, associated with advection due to chemotaxis, and global existence, blow-up and the pattern formation of solutions. The global existence of solutions was demonstrated by newly introducing a secretion term in a two-dimensional domain. We proved the blow-up of solutions only for the parabolic-elliptic approximated system by using the method of Jager-Luckhaus (Jager-Luckhaus, Trans. Amer. Math. Soc. 329(1992)). The pattern formation of the solutions was investigated by the reduction employing the center manifold theory. In the resulting solution, a constant solution was transversally bifurcated to an unstable hexagonal pattern solution, after which the bifurcated branches were inverted and become stable and simultaneously the stripe pattern solution caused an unstable pitch fork bifurcation.