Open Quasispecies Systems: New Approach to Evolutionary Adaptation

The paper’s Theorem 2.2 proves that any steady-state ū of D^{-1}Q(τ)M(τ)ū = e^{γS(ū)}ū lies in the convex set U_τ with S(ū) ≤ (1/γ) ln(K/ď), by combining the steady-state identity f̄(τ) = e^{γS(ū)} with f̄ = (m,ū)/(Dū,1) and the bounds Σ_i m_i ≤ K and Σ_i d_i ū_i ≥ ď Σ_i ū_i, yielding inequality (15) . The candidate solution reaches the same bound via a different, valid route: a column-sum/eigenvalue argument for A = D^{-1}Q(τ)M(τ) shows e^{γS(ū)} ≤ max_j c_j ≤ K/ď, and hence the same Ŝ, with the convexity of U_τ immediate. Therefore, both are correct, using different proofs.