Adding a fecundity-survival trade-off to a discrete population model with maturation delay

For the BH-Constant model Nt+1 = Nt/(1+α+βNt) + m(τ)Nt−τ with 0<m(τ)<1, the paper proves the global dichotomy: (i) if (1+α)m(τ) < α then 0 is the only nonnegative equilibrium and is globally asymptotically stable; (ii) if (1+α)m(τ) > α then a unique positive equilibrium N* = ((1+α)m(τ) − α)/(β(1−m(τ))) exists and is globally asymptotically stable (see the model specification (18), the equilibrium formula (19), and Theorems 3.11 and 3.13) . The candidate solution derives exactly the same threshold and equilibrium and gives a direct min–max block argument. Its key min/max inequalities are stated too strongly in full generality but are used only in the regimes where they do hold, so its conclusions match the paper’s.