Hopf Bifurcation Dynamics in a Delayed Turbidostat Model with Adaptive Proportional–Integral Control: Resource-Species Feedback and Stability Transitions

In ecosystems, the interplay between resource dynamics and species evolution drives complex adaptive behaviors of species. This study proposes a novel delayed turbidostat model that integrates resource consumption-induced delay and dynamic feedback between species and resources, aiming to elucidate oscillation patterns observed in microbial cultivation. We first establish the existence and stability criteria for the positive equilibrium, identifying conditions for steady-state coexistence without delay. Bifurcation and sensitivity analysis for the control parameters reveals that it governs Hopf bifurcation-induced periodic solutions, aligning with experimental observations of population oscillations. Furthermore, the delay parameter is shown to destabilize the equilibrium through Hopf bifurcation, while adaptive control strategies effectively increase the critical delay threshold for bifurcation, mitigating its destabilizing effects. Numerical simulations validate the theoretical framework and uncover various stability regimes under the coupled effects of delay and control scheme. Our findings provide a mechanistic understanding of how feedback control can counteract time-delay disturbances, offering practical insights for optimizing bioreactor operations and ecological management.