This paper proposes a mathematical model describing the dynamics of both Antarctic krill and Blue whale population where Antarctic krill species (prey) has group defense when they aggregate in herds in order to provide a self-defense from Blue whale species (predator). Firstly, a sufficient condition for the global asymptotical stability of the positive equilibrium is given using Poincaré–Bendixson theorem excluding periodic solution. Furthermore, we discuss the existence of Hopf-bifurcation and give criteria for stability switches through the study of a first-order exponential polynomial characteristic equation. Finally, some numerical simulations are given to illustrate our results.

中文翻译：

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具有延迟的捕食者-猎物系统中群体防御的社会行为

本文提出了一个描述南极磷虾和蓝鲸种群动态的数学模型，其中南极磷虾物种（猎物）在成群聚集时具有群体防御能力，以提供针对蓝鲸物种（捕食者）的自卫。首先，利用不包括周期解的庞加莱-本迪克森定理，给出了正平衡全局渐近稳定的充分条件。此外，我们通过对一阶指数多项式特征方程的研究，讨论了Hopf分岔的存在，并给出了稳定性开关的判据。最后，给出了一些数值模拟来说明我们的结果。