A New Perspective on Determining Disease Invasion and Population Persistence in Heterogeneous Environments

The paper establishes the asymptotic expansion r(µ)=s(Q−µL)=∑i qiθi + (1/µ)∑i∑j qi ℓ#_{ij} qj θj + o(1/µ), proves monotone-decreasing and convex-in-µ behavior, and gives the sharp bounds ∑i qiθi ≤ r(µ) ≤ max_i qi, with the scalar case r(µ)≡q, all under the standard assumptions on an irreducible Laplacian L and diagonal Q. These appear in Theorem 3.2, Corollary 3.2.1, and Theorem 3.4, with supporting preliminaries on L# and θ (e.g., 1^⊤L=0, Lθ=0, LL#=I−θ1^⊤) . The candidate solution reproduces the same main expansion using a Lyapunov–Schmidt/group-inverse reduction around the simple zero eigenvalue of −L and derives the same scalar case and bounds. Its approach is essentially the same as the paper’s perturbative derivation with L#, albeit via an explicit projector notation P=θ1^⊤ and N=I−P. Two minor issues: (i) a typographical slip omits an ε factor in the identity relating λ and 1^⊤Qx, though the subsequent formulas use the correct scaling; and (ii) the stated O(1/µ^2) remainder is stronger than the paper’s o(1/µ) and is not justified in the write-up. The lower bound A ≤ r(µ) is obtained in the paper via monotonicity in µ, while the model argues via convexity in q; the latter is plausible (and numerically consistent) but needs a brief justification. Overall, both are correct and closely aligned in proof strategy, with the paper slightly more careful about remainder order and µ-monotonicity .