Population Persistence in Stream Networks: Growth Rate and Biomass

The paper’s main claims are correct and rigorously justified: (a) the biomass upper bound K ≤ K_env ∑_i (1+q/d)^{f(i)} with equality when all growth is allocated to the most upstream nodes (Theorem 5.6), and (b) for sufficiently small diffusion d, the network growth rate is maximized by concentrating all resources at exactly one downstream end node (Theorem 5.5(i)); the uniform-baseline perturbation result also prioritizes the most downstream nodes (Theorem 3.3) . By contrast, the candidate uses an incorrect symmetrization (they take V=diag(v) instead of the standard W=diag(√v)) and then relies on Rayleigh–Ritz with a matrix that is not symmetric under that transformation; they also assert a zero first-order effect at the uniform baseline, which contradicts the paper’s w^T E v calculation that shows a strictly positive first-order gain when increasing resources at downstream nodes (Theorem 3.3) . While their biomass bound argument can be repaired (a maximum principle works for the Laplacian-form operator without requiring symmetry), the spectral optimization and sensitivity parts contain substantive technical errors.