Holling-Tanner predator-prey model with state-dependent feedback control (Q1727066)

Summary: In this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré map are investigated in detail which play important roles in the proofs of the existence of limit cycles, and it is concluded that the definition domain of the Poincaré map has a complicated shape with discontinuity points under certain conditions. Subsequently, the existence of the boundary order-1 limit cycle is discussed and it is shown that this limit cycle is unstable. Furthermore, the conditions for the existence and stability of an order-1 limit cycle are provided, and the existence of order-\(k\) (\(k \geq 2\)) limit cycle is also studied. Moreover, numerical simulations are carried out to substantiate our results. Finally, biological implications related to the mathematical results which are beneficial for successful pest control are addressed in the Conclusions section.