Local Stability and Bifurcation Analysis of a Mangrove Detritus Small-Fish Model with Dynamic Harvesting Effort under a Beddington-DeAngelis Functional Response

This study develops a three-dimensional mangrove detritus small fish model with a dynamics harvesting effort governed by a threshold rule and a Beddington DeAngelis functional response. The main objective is to understand how adaptive harvesting and consumer interference shape long-term dynamics and stability. Equilibrium points are derived and their local stability is examined using the Jacobian matrix and eigenvalue-based criteria, supported by numerical simulations. Bifurcation analysis with respect to the threshold parameter is performed using numerical continuation to detect qualitative transitions in system behavior. The results show the existence of biologically feasible coexistence equilibria with active harvesting, including a stable branch and an unstable branch. A Hopf bifurcation is detected at approximately a = 6.6159, confirming the emergence of sustained oscillations limit cycles in fish density and harvesting effort, while a fold limit point occurs near a = 22.001, indicating changes in equilibrium structure. Moreover, simulation scenarios demonstrate that reducing environmental capacity and increasing fish mortality can drive the system toward a no-effort equilibrium where harvesting collapses. These findings highlight the threshold parameter as a key control factor that can switch the system between stable harvesting, effort extinction, and oscillatory harvesting regimes.