In this article, a delayed phytoplankton-zooplankton system with Allee effect and linear harvesting is proposed, where phytoplankton species protects themselves from zooplankton by producing toxin and taking shelter. First, the existence and stability of the possible equilibria of system are explored. Next, the existence of Hopf bifurcation is investigated when the system has no time delay. What's more, the stability of limit cycle is demonstrated by calculating the first Lyapunov number. Then, the condition that Hopf bifurcation occurs is obtained by taking the time delay describing the maturation period of zooplankton species as a bifurcation parameter. Furthermore, based on the normal form theory and the central manifold theorem, we derive the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. In addition, by regarding the harvesting effort as control variable and employing the Pontryagin's Maximum Principle, the optimal harvesting strategy of the system is obtained. Finally, in order to verify the validity of the theoretical results, some numerical simulations are carried out.

中文翻译：

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具有Allee效应和线性收获的延迟浮游植物-浮游动物模型的稳定性和Hopf分叉分析。

在本文中，提出了具有Allee效应和线性收获的延迟浮游植物-浮游植物系统，其中浮游植物通过产生毒素和躲藏来保护自己免受浮游动物的侵害。首先，探讨了系统可能平衡的存在性和稳定性。接下来，研究了系统没有时间延迟时Hopf分叉的存在。此外，极限环的稳定性通过计算第一个李雅普诺夫数来证明。然后，通过将描述浮游动物物种成熟期的时间延迟作为分叉参数来获得发生霍普夫分叉的条件。此外，基于正规形式理论和中心流形定理，我们推导了Hopf分支的方向和分支周期解的稳定性。此外，通过将收割工作量作为控制变量并采用庞特里亚金的最大原理，可以获得系统的最佳收割策略。最后，为了验证理论结果的有效性，进行了一些数值模拟。