Cells move up gradients of self-secreted chemoattractant. When total mass exceeds the critical threshold (8π in 2D), the solution blows up — finite-time aggregation. Below threshold, diffusion wins and cells spread out. Toggle saturation to prevent blowup.
∂ₜρ = D_ρΔρ − χ∇·(ρ∇c)
∂ₜc = D_cΔc + ρ − c
Critical mass: M* = 8π ≈ 25.1
M > M* → finite-time blowup
M < M* → global existence