Population Extinction on a Random Fitness Seascape

The candidate solution reproduces the paper’s mean-field self-consistency for seascape noise, ȳ = (cD/ca)[Kβ(2x)/Kβ+1(2x)]^2 with β = cD − cµ and x = √(cD ca ȳ, and then uses the same small-x expansion of the Bessel K-ratio as the paper to obtain the two regimes: (i) β>1 gives ȳ ≈ (µ/a)(1−σ^2/(2D)); (ii) 0<β<1 gives ȳ ∝ µ^{σ^2/(2D)} with the same prefactor structure, matching Eqs. (23)–(28) and Appendix A in the paper . The identity a⟨y^2⟩ = µ⟨y⟩ is also used consistently to obtain the moment ratios as in the text . For demographic (square-root) noise, the candidate derives the same stationary density and self-consistency, identifies the critical µc via c′D∫0∞ e^{py−qy^2}dy=1, and obtains the small- and large-σd asymptotics µc ∝ σd^2 and µc ∝ σd√lnσd exactly as in the paper’s Appendix C; the near-critical scaling ȳ ∝ (µ−µc) and ⟨y^2⟩/⟨y⟩^2 ∝ 1/(µ−µc) also agree with the main text and appendices . For mixed noise, the candidate’s argument that √y fluctuations dominate near y=0 (hence DP mean-field scaling) matches the paper’s conclusion and Appendix D . Minor differences are cosmetic (e.g., retaining an additive D in the large-σd asymptotic for µc that is subleading in the paper’s limit), and a likely small typo in the paper’s Appendix A equation (A4) dimensional form does not affect the results. Net: both are correct and follow substantially the same proof structure.