Abstract This paper proposes a multiple-delayed predator–prey system with habitat complexity and harvesting effort, and investigates the dynamical behavior including stability properties and Hopf bifurcation. Firstly, stability of equilibrium points and the existence of Hopf bifurcation are investigated and some critical conditions which guarantee the corresponding results are obtained based on mathematical view. Secondly, the explicit formulae for determining the direction, stability and period of the bifurcating periodic solutions are derived by using the center manifold theory and the normal form theory. Finally, in order to verify the theoretical results, some numerical simulations are done to illustrate the results. It is observed that the level of abundance of prey and predator populations depends on the gestation delay if the gestation delay exceeds some critical values.

中文翻译：

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具有栖息地复杂性的多延迟捕食者-猎物系统的 Hopf 分叉

摘要 本文提出了一种具有栖息地复杂性和收获努力的多延迟捕食者-猎物系统，并研究了包括稳定性特性和 Hopf 分叉在内的动力学行为。首先研究了平衡点的稳定性和Hopf分岔的存在性，并基于数学观点得到了保证相应结果的一些临界条件。其次，利用中心流形理论和范式理论推导出确定分岔周期解的方向、稳定性和周期的显式公式。最后，为了验证理论结果，进行了一些数值模拟来说明结果。