In this paper we study the phenomena of the extinction and persistence of predator populations of the three-dimensional Kooi et al. model in the global formulation of the problem. This model contains three populations: prey, susceptible predators and infected predators. We compute ultimate sizes of interacting populations and establish that all biologically feasible trajectories eventually enter in some bounded domain and remain there. We derive analytical conditions for the extinction of the infected predator population in cases of different/equal mortality rates of predators. In particular, we find conditions under which 1) the population of prey persists, while both of predator populations die out, 2) the populations of prey and susceptible predators persist, while the population of infected predators dies out. Besides, we describe the case when at least one periodic orbit exists in the disease-free invariant plane. Our analysis is based on using the localization method of compact invariant sets and the theorem of LaSalle. Main theoretical results are illustrated by numerical simulation.

在本文中，我们研究了三维Kooi等人的捕食者种群的灭绝和持续现象。模型在全球范围内提出问题。该模型包含三个种群：猎物，易感性掠食者和感染性掠食者。我们计算相互作用种群的最终大小，并确定所有生物学上可行的轨道最终都进入某个有界域并保持在那里。我们得出了在掠夺者死亡率不同/相等的情况下被感染的掠食者种群灭绝的分析条件。特别是，我们找到了以下条件：1）猎物种群持续存在，而两个捕食者种群都灭绝了； 2）猎物种群和易感捕食者种群持续了，而被感染的捕食者种群灭绝了。除了，我们描述了在无病不变平面中至少存在一个周期性轨道的情况。我们的分析是基于使用紧不变集的定位方法和LaSalle定理的。数值模拟说明了主要的理论结果。