Agent-based modelling of sports riots

The paper derives, for the same ABM (two species with exclusion on a square lattice, von Neumann neighborhood, fast motility m ≫ rates), the cross-diffusion PDE system ∂r/∂t = D ∇·[(1−b)∇r + r∇b] + ρ(r,b), ∂b/∂t = D ∇·[(1−r)∇b + b∇r] − ρ(r,b), with D = mΔ²/4 and ρ(r,b) = b∑ λrn Bn,4(r) − r∑ λdn Bn,4(b), and shows that T := r+b satisfies the linear heat equation ∂T/∂t = D∇²T (Eqs. (33)–(36) in the paper) . In the spatially uniform limit, the paper presents the ODE dr/dt = b∑ λrn (4 choose n) r^n(1−r)^{4−n} − r∑ λdn (4 choose n) b^n(1−b)^{4−n} (Eq. (8)) and frames an inverse problem using the degree-4 Bernstein basis with the change-of-basis x^m = ∑_{n=m}^4 [(n choose m)/(4 choose m)] Bn,4(x) and its upper-triangular matrix M (Eqs. (14)–(16)) . The candidate solution reproduces these results: it sketches the same continuum-limit derivation yielding identical fluxes and D, proves that T obeys the heat equation, and uses the same Bernstein–monomial transformation to give the explicit upper-triangular identification λ_{rn} = ∑_{m=0}^n [(n choose m)/(4 choose m)] α_m (and analogously for λ_{dn}). A very minor discrepancy is a sentence that informally describes move acceptance as “no opposite species present,” whereas the ABM uses single-occupancy (empty destination) exclusion; however, the discrete balance the candidate writes matches the paper’s continuum limit and is algebraically identical to the correct empty-destination update after cancellation. The paper also notes a non-identifiability caveat if only net growth is known (e.g., logistic example with free parameters), which the candidate does not rely on and does not contradict . Overall, both are correct and follow substantially the same argument.