This paper is devoted to investigate the pattern formation of avolume-filling chemotaxis model with logistic cell growth. We first apply thelocal stability analysis to establish sufficient conditions of destabilizationfor uniform steady-state solution. Then, weakly nonlinear analysis withmulti-scales is used to deal with the emerging process of patterns near thebifurcation point. For the single unstable mode case, we derive theStuart-Landau equations describing the evolution of the amplitude, and thus theasymptotic expressions of patterns are obtained in both supercritical case andsubcritical case. While for the case of multiple unstable modes, we also derivecoupled amplitude equations to study the competitive behavior between twounstable modes through the phase plane analysis. In particular, we find thatthe initial data play a dominant role in the competition. All the theoreticaland numerical results are in excellently qualitative agreement and betterquantitative agreement than that in [1]. Moreover, in the subcritical case, weconfirm the existence of stationary patterns with larger amplitudes when thebifurcation parameter is less than the first bifurcation point, which gives anpositive answer to the open problem proposed in [2].
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