Bifurcation dynamics and chaos control in a discrete stock-dependent resource-extraction model

This study investigates a discrete nonlinear resource-effort system with a state-dependent carrying capacity, where the maximum exploitation capacity adapts to resource availability. Unlike classical fixed-capacity models, the proposed framework incorporates adaptive feedback between resource and exploitation variables, providing a generalized approach for analyzing complex dynamics in nonlinear discrete systems. Using the method of piecewise constant argument, the continuous model is transformed into a nonlinear discrete map. Fundamental properties, including positivity, boundedness, equilibrium points and local stability, are analyzed. The loss of stability at the interior equilibrium is explored through flip and Neimark-Sacker bifurcations, leading to period-doubling cascades, quasiperiodic oscillations and chaos. Normal form theory is applied to determine bifurcation direction and stability. Numerical simulations using bifurcation diagrams, phase portraits, time series and maximum Lyapunov exponents confirm complex transitions. A feedback control strategy is further proposed to stabilize unstable dynamics and enlarge the stability region.