Following the well-extablished mathematical approach to persistence and its developments contained in Rebelo et al. (Discrete Contin Dyn Syst Ser B 19(4):1155-1170. https://doi.org/10.3934/dcdsb.2014.19.1155, 2014) we give a rigorous theoretical explanation to the numerical results obtained in Bate and Hilker (J Theoret Biol 316:1-8. https://doi.org/10.3934/dcdsb.2014.19.1155, 2013) on a prey-predator Rosenzweig-MacArthur model with functional response of Holling type II, resulting in a cyclic system that is locally unstable, equipped with an infectious disease in the predator population. The proof relies on some repelling conditions that can be applied in an iterative way on a suitable decomposition of the boundary. A full stability analysis is developed, showing how the "invasion condition" for the disease is derived. Some in-depth conclusions and possible further investigations are discussed.

中文翻译：

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具有疾病食肉动物的食肉动物-食肉动物模型的一致持久性。

遵循Rebelo等人中包含的关于持久性及其发展的数学方法。（Discrete Contin Dyn Syst Ser B 19（4）：1155-1170。https://doi.org/10.3934/dcdsb.2014.19.1155，2014）我们对在Bate和Hilker中获得的数值结果进行了严格的理论解释（ J Theoret Biol 316：1-8。https://doi.org/10.3934/dcdsb.2014.19.1155，2013）上具有Holling II型功能性反应的捕食者Rosenzweig-MacArthur模型，导致了一个循环系统是局部不稳定的，在食肉动物种群中患有传染病。该证明依赖于某些排斥条件，这些条件可以迭代方式应用于边界的适当分解。进行了全面的稳定性分析，显示了如何得出该疾病的“入侵条件”。