Delayed weak-Allee harvesting model reveals collapse thresholds and feedback stabilisation limits

A new mathematical study dissects when a harvested population model collapses, when it recovers, and how feedback control can — and cannot — rescue it. The work, published in Advances in Continuous and Discrete Models, analyses a two-dimensional discrete-time population model built from one-step sampling of delayed weak-Allee recruitment under constant harvesting, with the map interpreted for populations that have non-overlapping census intervals.

The analysis reduces positive equilibria to an explicit scalar harvesting function of the model. This reduction makes the equilibrium structure transparent: the authors derive a nondegenerate equilibrium fold, the mathematical signature of a tipping point where two equilibria merge and disappear, and they resolve numerically a harvest-induced basin boundary that separates basins of attraction leading to persistence or extinction. Crossing such a boundary, even with an equilibrium still present, can send trajectories to collapse.

Stability is handled with standard discrete-time tools. Jury conditions supply complete local stability tests for the equilibria, while closed-form period-doubling boundaries and analytical transversality conditions identify exactly when the upper equilibrium loses stability. Because the delay-shift structure forces the Jacobian determinant to be negative, the model excludes a fixed-point Neimark–Sacker bifurcation, narrowing the possible routes to instability to fold and flip mechanisms.

Extinction dynamics are examined through a separately analysed projected map. There, extinction behaves as a locally finite-time attracting absorbing state: once a trajectory enters the relevant region, it reaches extinction in finite time and cannot leave. The authors run over-harvest simulations that quantify collapse transients, and they are explicit that the maximum harvest level used in these simulations is not treated as a global persistence theorem — a caution against over-reading the numerical results as a universal guarantee.

To characterise complicated dynamics, the study uses bifurcation diagrams and Lyapunov exponents, which diagnose bounded sensitive trajectories — chaotic-type behaviour confined to bounded regions of state space. This diagnostic combination distinguishes stable equilibria, periodic regimes and sensitive dynamics across the harvesting parameter range.

Finally, the paper tests three idealised feedback laws designed to stabilise the system locally. The authors emphasise that these laws illustrate local stabilisation only, subject to explicit operational limitations, so the results should not be read as a prescription for real-world harvest management. The study is a theoretical analysis of an idealised model, not an empirical assessment of any specific population.

By making the harvesting function explicit and the bifurcation boundaries closed-form, the model offers a compact setting in which collapse thresholds, delay effects and feedback control can be analysed together.

Subject of Research: Mathematical analysis of collapse thresholds, bifurcations and feedback stabilisation in a delayed weak-Allee harvesting difference model.

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

Article References: Tassaddiq, A., Naz, S., Tanveer, M., Alharbi, R., Almutairi, D. K., & Md Kasmani, R. (2026). Collapse thresholds, bifurcation, hysteresis, and feedback stabilisation in a delayed weak-Allee harvesting difference model. Advances in Continuous and Discrete Models. https://doi.org/10.1186/s13662-026-04141-9

Image Credits: AI Generated

DOI: 10.1186/s13662-026-04141-9

Keywords: difference equations, weak Allee effect, harvesting, collapse thresholds, bifurcation, hysteresis, period-doubling, Neimark-Sacker, Lyapunov exponents, feedback stabilisation, extinction, discrete-time population model