Bacterial aggregation and blowup: ∂ρ/∂t = D∇²ρ − χ∇·(ρ∇c)
t=0 | max ρ=— | max c=— | Status: —
Keller-Segel (1970): bacteria move up chemical gradients they themselves produce. ∂ρ/∂t = Dρ∇²ρ − χ∇·(ρ∇c), ∂c/∂t = Dc∇²c + αρ − βc. When χ exceeds a critical value (χ_c = 4πDρ/α for 2D), the uniform state is unstable and bacteria collapse to finite-time singularities (blowup). Below threshold, stable patterns form. Left: bacterial density ρ (orange=high). Right: chemoattractant concentration c (green=high).