Pattern formation in the Keller-Segel model and Turing instability

We investigate nonlinear dynamics near an unstable constant equilibrium in the two classical models. Given any general perturbation of magnitude $\delta$, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of $ln(1/\delta)$. Our result can be interpreted as a rigourous mathematical characterization for early pattern formation in the Keller-Segel model and Turing instability.