Collapse thresholds, bifurcation, hysteresis, and feedback stabilisation in a delayed weak-Allee harvesting difference model

We analyse a two-dimensional discrete-time population model motivated by one-step sampling of delayed weak-Allee recruitment under constant harvesting. The map is interpreted independently for populations with non-overlapping census intervals. Positive equilibria reduce to an explicit scalar harvesting function. This yields a nondegenerate equilibrium fold and a numerically resolved harvest-induced basin boundary. Jury conditions give the complete local stability tests, while closed-form period-doubling boundaries and analytical transversality conditions identify stability loss of the upper equilibrium. The delay-shift structure makes the Jacobian determinant negative, excluding a fixed-point Neimark–Sacker bifurcation. A projected map is analysed separately: extinction is a locally finite-time attracting absorbing state, and over-harvest simulations quantify collapse transients without treating \(hₘₐₓ\) as a global persistence theorem. Bifurcation diagrams and Lyapunov exponents diagnose bounded sensitive trajectories, and three idealised feedback laws illustrate local stabilisation subject to explicit operational limitations.