In this paper, using the localization method of compact invariant sets, we examine the ultimate dynamics of the 3D prey–predator model containing two subpopulations of susceptible and infected predators. Our attention is focused to finding ultimate sizes of interacting populations, and, in addition, we show the existence of a global attracting set. Then, we derive various global conditions of ultimate extinction of at least one of the predators subpopulations and describe conditions under which all types of internal bounded dynamics are ruled out. In particular, we describe convergence conditions to omega‐limit sets located (1) in the intersection of the prey‐free plane with the infected predators‐free plane and (2) in the infected predators‐free plane. Based on the dynamical analysis of the 2D infection‐free subsystem, we obtain conditions of global attraction to (i) the prey‐only disease‐free equilibrium point, (ii) the disease‐free prey‐predator equilibrium point (self‐healing of the predator population), and (iii) the omega‐limit set containing an equilibrium point or a periodic orbit. Main theoretical results are illustrated by numerical simulation. Tools and techniques developed in this work can be appropriated in the studies within predictive population ecology of more complex eco‐epidemiological models.

中文翻译：

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3D生态流行病学模型中的稳定化：从捕食者种群的彻底灭绝到他们的自我修复

在本文中，我们使用紧凑不变集的定位方法，研究了3D食饵-捕食者模型的最终动力学，该模型包含两个易感和感染的捕食者亚群。我们的注意力集中在寻找相互作用的群体的最终规模上，此外，我们还展示了存在全球吸引力的群体。然后，我们得出了至少一个捕食者亚群最终灭绝的各种全局条件，并描述了排除所有类型的内部有界动力学的条件。特别地，我们描述了ω极限集的收敛条件，该极限极限集位于（1）无猎物平面与被感染的无掠食性飞机的相交处和（2）在受感染的无掠食性飞机的相交处。基于2D无感染子系统的动态分析，我们获得了对（i）无猎物无疾病的平衡点，（ii）无病猎物-捕食者的平衡点（捕食者种群的自我修复）和（iii）欧米茄限制的全球吸引力的条件包含平衡点或周期轨道的集合。数值模拟说明了主要的理论结果。这项工作中开发的工具和技术可以在更复杂的生态流行病学模型的预测种群生态学中进行研究。