Qualitative Analysis of a Generalized Planar Impulsive Predator-Prey System with Nonlinear Threshold Level

In this paper, we aim to construct a general research framework for the generalized planar impulsive semidynamic system (ISDS) and evaluate the effectiveness of control strategies by exploring its dynamics. To do this, we propose a Kolmogorov-type predator-prey system with nonlinear state-dependent feedback control, which means that when the combination of the number of prey species and its growth rates is below the action threshold, no control measures are implemented; once the action threshold is reached, control measures are immediately implemented. We define two one-dimensional discrete Poincaré maps, which are determined by the difference equation iterated by the pulse point sequence. Based on those, the existence and stability of order-1 periodic solutions, the existence of order-[Formula: see text] periodic solutions, and the sufficient conditions for the ISDS to undergo transcritical bifurcation with respect to the key parameters are identified. Finally, to illustrate the applications of the main results, an example in pest control and its numerical analyses are provided.