  This paper is devoted to investigate the pattern formation of a
volume-filling chemotaxis model with logistic cell growth. We first apply the
local stability analysis to establish sufficient conditions of destabilization
for uniform steady-state solution. Then, weakly nonlinear analysis with
multi-scales is used to deal with the emerging process of patterns near the
bifurcation point. For the single unstable mode case, we derive the
Stuart-Landau equations describing the evolution of the amplitude, and thus the
asymptotic expressions of patterns are obtained in both supercritical case and
subcritical case. While for the case of multiple unstable modes, we also derive
coupled amplitude equations to study the competitive behavior between two
unstable modes through the phase plane analysis. In particular, we find that
the initial data play a dominant role in the competition. All the theoretical
and numerical results are in excellently qualitative agreement and better
quantitative agreement than that in [1]. Moreover, in the subcritical case, we
confirm the existence of stationary patterns with larger amplitudes when the
bifurcation parameter is less than the first bifurcation point, which gives an
positive answer to the open problem proposed in [2].
