Basic offspring number and robust feedback design for the biological control of vectors by sterile insect release technique

The paper states the key results (extinction under the unilateral lower bound u ≥ α|Ṁ|+ + ωαM + (δs−ω)Ms with α > αcrit, and finiteness of total releases under explicit upper bounds) and sketches the proofs, but repeatedly defers full details to a forthcoming extended paper, hence it is incomplete as written . The candidate solution reaches the same conclusions via a different route (a monotone inequality on D := γ(Ms − αM) implying limsup r(t) ≤ 1/(1+γα), then a linear comparison for E,F), and its treatment of finite releases matches the paper’s bounds . However, in Step 3 the candidate’s scalar Lyapunov estimate Ẇ ≤ μE(Nr−1)W is sign-incorrect when Nr−1 < 0; a correct argument needs a weighted functional Wc = E + cF with c in (βE/δF, μE/(λ2r*)) to obtain Ẇc ≤ −κE − εF ≤ −c0Wc once r(t) ≤ r* < 1/N. With that fix, the candidate’s proof aligns with the paper’s conclusions. Therefore, both are incomplete: the paper due to omitted details, and the model due to a fixable inequality gap.