The main concern of this paper is to discuss stability and bifurcation analysis for a class of discrete predator-prey interaction with Holling type II functional response and harvesting effort. Firstly, we establish a discrete singular bioeconomic system, which is based on the discretization of a system of differential algebraic equations. It is shown that the discretized system exhibits much richer dynamical behaviors than its corresponding continuous counterpart. Our investigation reveals that, in the discretized system, two types of bifurcations (i.e., period-doubling and Neimark–Sacker bifurcations) can be studied; however, the dynamics of the continuous model includes only Hopf bifurcation. Moreover, the state delayed feedback control method is implemented for controlling the chaotic behavior of the bioeconomic model. Numerical simulations are presented to illustrate the theoretical analysis. The maximal Lyapunov exponents (MLE) are computed numerically to ensure further dynamical behaviors and complexity of the model.

中文翻译：

---

离散单一生物经济系统的稳定性与分岔分析

本文的主要关注点是讨论一类具有 Holling II 型功能反应和收获努力的离散捕食者-猎物相互作用的稳定性和分叉分析。首先，我们建立了一个基于微分代数方程组离散化的离散奇异生物经济系统。结果表明，离散系统比其相应的连续系统表现出更丰富的动态行为。我们的研究表明，在离散化系统中，可以研究两种类型的分岔（即周期倍增和 Neimark-Sacker 分岔）；然而，连续模型的动力学仅包括 Hopf 分岔。此外，采用状态延迟反馈控制方法来控制生物经济模型的混沌行为。给出了数值模拟来说明理论分析。以数值方式计算最大李雅普诺夫指数 (MLE) 以确保模型的进一步动力学行为和复杂性。