A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

中文翻译：

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Bogdanov–在具有恒定努力量的收获的Holling IV捕食者－捕食者模型中分叉

本文研究了具有恒定工作量的猎物和食肉动物的食肉动物-食肉动物模型。首先，我们通过分析平衡方程和特征值的分布来讨论平衡的数量和类型。其次，以重新分配的收割努力作为分叉参数，通过分析模型的横向展开，在多焦点附近显示了次临界霍普夫分叉，并且在BT奇点附近也显示了Bogdanov-Takens分叉。随着分叉参数的变化，系统显示出多稳态结构，不同吸引子的吸引域由稳定和不稳定的鞍形流形和由Hopf和Bogdanov-Takens分叉产生的极限环组成。最后，在鞍节点分叉曲线和Hopf分叉曲线上分别找到一个尖点和两个广义Hopf点。通过广义Hopf分叉，可以显示出与广义Hopf点之一接近的参数的几个相位图。