Quantifying Fish School Fragmentation under Predation Using Stochastic Differential Equations

The paper cleanly defines the graph-based metrics (r* = 1.2, first split time, final component count) and the two predator strategies (nearest vs center), and reports simulation trends: increasing α delays splitting and reduces the final component count; overly large β leads to complete disintegration (C_final = N); center attack fragments more than nearest attack at relatively high α; and a sharp prey-noise threshold appears near σ_i ≈ 0.4. These statements are documented in the text and figures, but are empirical rather than proved (e.g., r* and the β-claim are justified narratively rather than analytically) . The candidate solution offers partial analysis: a plausible virial/second-moment route to α-monotonicity (under explicit structural assumptions) and a mechanism for why center attack can be more disruptive at high α, plus a back-of-the-envelope rationale for a noise threshold around 0.4. However, its blanket refutation of the paper’s large-β fragmentation phenomenon leans on classical Cucker–Smale flocking without carefully incorporating the paper’s predator forcing and fast-decaying communication weights, which can invalidate unconditional flocking. Net: the paper’s claims are supported numerically but under-justified theoretically; the model adds helpful structure but leaves important steps open and overreaches on the β point.