A Macroscopic Pedestrian Model with Variable Maximal Density

The paper’s 1D steady-state analysis in the congested regime (ρ > σ), with u ≡ 0 and θ − β∂xθ = 0, assumes a constant flux f(ρ, τ) = d fmax and uses the triangular fundamental diagram to get (ρ − τ) = d(σ − τ), hence ρ = dσ + (1 − d)τ. Combined with θ = ρ − (τave − ν) and the choice c = 0 (i.e., θ ≡ 0), d = 0, and τ(x) = ax + b, averaging τ over [x, x + δ] yields τave = τ + (aδ)/2, so τ = τave − ν forces a = 2ν/δ. This is exactly the argument and result presented in the paper (Eqns. (4), (6)–(7), (10a–c), (11)–(14) in the 1D steady-state discussion), and the candidate solution follows the same steps to reach the same conclusion .