A Stochastic Capital-Labour Model with Logistic Growth Function

Both the paper and the candidate solution aim to prove persistence-in-mean for v(t) in the stochastic logistic capital–labour SDE du = [r u(1 - u/K) - m u v] dt - σ u v dB, dv = [m u v - d v] dt + σ u v dB, and both conclude liminf ⟨v⟩ ≥ d(Rs0-1)/(m+d) under Rs0 = r/d - σ^2 K^2/(2d) > 1 and m > r/K. The paper’s main steps are: derive time-averaged identities for v, ln u, ln v, combine them so the martingale terms vanish using boundedness from Theorem 1, then obtain (d+m)⟨v⟩ ≥ r - d - (σ^2/2)K^2 + o(1), and finally the claimed bound (Theorem 3) . However, in the proof the paper (i) rewrites quadratic terms with an algebraic slip, using (m - σ^2)⟨uv⟩ - (σ^2/2)⟨(u+v)^2⟩ in place of the correct identity m⟨uv⟩ - (σ^2/2)(⟨u^2⟩ + ⟨v^2⟩) = - (σ^2/2)⟨(u+v)^2⟩ + (m + σ^2)⟨uv⟩, and (ii) crucially bounds −(σ^2/2)⟨(u+v)^2⟩ below by −(σ^2/2)K^2 without having established an a.s. upper bound (uniform-in-time) of u+v by K; the only bound available is lim sup(u+v) ≤ rK/μ from Theorem 1, which is not the same and does not justify replacing the time-average of (u+v)^2 by K^2 . The candidate’s argument has the same structural goal but similarly invokes an unproven uniform bound on u (or on (u+v)) by K to replace quadratic averages by K^2 in the final step. Hence, while the headline result matches the paper’s Theorem 3, both proofs have gaps at the precise step that introduces the “−(σ^2/2)K^2” term, and neither currently closes that step rigorously (despite otherwise correct SDE manipulations, SLLN applications justified by eventual boundedness, and the use of the noise cancellation in u+v) .