Hopf Bifurcation and Stability Analysis of a Delayed Logistic Harvesting Model Applied to the Koran Fish Population

This paper investigates the stability and bifurcation behavior of a delayed logistic population model with constant harvesting, motivated by the management of the endemic Koran fish (Salmo letnica) in Lake Ohrid. Classical logistic models typically assume instantaneous population responses and neglect biological time delays associated with maturation and recruitment, which may significantly affect population dynamics. To address this limitation, a discrete maturation delay is introduced to represent time-lag effects in reproduction and recruitment. By linearizing the system around the positive equilibrium and analyzing the corresponding transcendental characteristic equation, explicit criteria for local asymptotic stability are established. It is shown that the maturation delay acts as a bifurcation parameter: when it exceeds a critical threshold, a Hopf bifurcation occurs, giving rise to sustained periodic oscillations. Closed-form expressions for the critical delay and bifurcation frequency are derived, allowing precise identification of stability switching conditions. Numerical simulations and sensitivity analysis with respect to key ecological parameters confirm the theoretical results and demonstrate the robustness of delay-induced oscillations. The findings underline the destabilizing role of maturation delay and highlight its interaction with harvesting pressure. From a practical perspective, the results suggest that neglecting delay effects may lead to inaccurate predictions in fisheries management. Overall, this study provides a mathematically rigorous and biologically relevant framework that contributes to a deeper understanding of delayed population dynamics and supports more sustainable management strategies.