Equivalence in the dynamic behavior of ecosystems with switching, impulsive and adaptive control strategies: Insights into equilibrium and bifurcation

While continuous, switching, and impulsive controls are widely applied in integrated pest management and disease control, few studies have provided a theoretical demonstration of their equivalence. To address this gap, we propose a modeling framework for adaptive dynamic threshold control based on Kolmogorov's predator-prey model, incorporating a threshold policy-guided continuous adaptive strategy. We first prove the equivalence between a simplified two-dimensional model with adaptive control and a one-dimensional model with switching control in terms of global dynamics. Additionally, we show that the local dynamics of equilibria in the full (three-dimensional) system with adaptive control are equivalent to a two-dimensional system with switching control. Moreover, the adaptive model can exhibit Hopf bifurcation while the switching model cannot, but the dynamics is consistent to the model with impulsive control. This research demonstrates the feasibility of transforming between non-smooth/discontinuous and smooth control systems, providing a theoretical foundation for selecting the optimal control strategy.