Dispersal-induced growth in a time-periodic environment

The paper proves Theorem 1 (sink–sink case) via a reduction to a one-dimensional Riccati equation for z=x2/x1, deriving (i) Λ(m,ω) ≤ χ (hence χ<0 ⇒ decay) and (ii) asymptotics Λ(m,ω)=Λ0(m)+O(ω) as ω→0 and Λ(m,ω)=Λ∞(m)+O(1/ω) as ω→∞, with Λ0 and Λ∞ given explicitly; from these, a unique threshold m* is characterized and high-frequency decay is established . The candidate solution reaches the same conclusions: (I) Λ ≤ χ, so χ<0 ⇒ Λ<0; (II.A) an adiabatic eigenbasis argument yields the same Λ0(m) and uniqueness of m*; (II.B) high-frequency averaging gives the averaged matrix’s top eigenvalue, ensuring decay for sinks. The model’s proof is different in method (instantaneous diagonalization of a symmetric 2×2 family vs. ratio ODE) but aligns with the paper’s results and formulas. Minor gaps in the model’s write-up (e.g., using y2/y1 without fully justifying non-vanishing of y1) are fixable and don’t affect the conclusions. Overall, both are correct, with the paper’s proof complete and rigorous.