Bifurcation thresholds in a bi-trophic turbidostat system: Refuge-mediated critical transitions and delay-induced...

Interspecies interactions within ecosystems generate intricate ecological networks and spatial structures. To mitigate predation risks during ecological engagement, species frequently adopt adaptive survival strategies such as refuge concealment. This study develops a bi-trophic food-chain turbidostat model incorporating multiple time delays and refuge protection mechanisms to systematically investigate how critical parameters influence population dynamics and evolutionary patterns. Through rigorous stability analysis of system equilibria, we establish sufficient conditions for equilibrium stability and characterize parameter perturbation effects on system dynamics. Our bifurcation analysis reveals that both transcritical and Hopf bifurcations emerge when refuge parameters approach critical thresholds, demonstrating how parameter variations can transition population growth patterns from stable equilibrium to sustained oscillations. Notably, our refuge parameter analysis demonstrates the dual-edged nature of protective strategies: both excessive and insufficient refuge utilization destabilize population equilibrium. By employing center manifold and normal form theory, we quantitatively assess the nonlinear dynamics near bifurcation points and derive stability criteria for emergent periodic solutions. The temporal analysis further uncovers that time delays induce Hopf bifurcations when surpassing critical values, generating persistent population oscillations that endanger ecological stability. Numerical simulations across multiple parameter regimes consistently validate our theoretical predictions.