Bifurcation analysis of an ecological model with nonlinear state–dependent feedback control by Poincaré map defined in phase set

Ecosystems often need to integrate instantaneous interventions such as biological resource management or integrated pest control. The implementation of interventions, in turn, depends on the state and its change rate of the system, which brings great challenges for the stability and bifurcation analyses. To develop qualitative analysis techniques, we propose a predator–prey model with state-dependent pulse interventions, including spraying insecticides and releasing natural enemies, in which the implementation of control measures depends on whether the weighted value of pest density and its change rate reaches the action threshold. We first address the threshold condition for the existence and stability of the boundary periodic solutions and then define a one-parameter family of discrete (Poincaré) maps. By analyzing the properties of those discrete maps, we conclude that if there is no interior equilibrium for the system without control measures and the releasing amount of natural enemies is greater than zero, then there is at least one positive periodic solution. While if the system without control measures has an internal equilibrium and only the chemical tactic is applied, then there could be an unstable positive periodic solution near the boundary periodic solution. Consequently, backward bifurcation and bi-stability occur. The analytical techniques developed here could be applied to analyze more generalized models and other fields, including infectious disease control.