Uniform persistence in a prey–predator model with a diseased predator
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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.
KeywordsUniform persistence Infectious disease Basic reproduction number Prey–predator model
Mathematics Subject Classification37C70 92D30
The author is truly grateful to prof. Fabio Zanolin without whose supervision this work wouldn’t have been accomplished. The author wishes to thank also prof. Carlota Rebelo and prof. Alessandro Margheri for the helpful remarks and even more for suggesting the paper (Bate and Hilker 2013) which introduced our main subject of investigation. Work partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2017: “Problemi differenziali con peso indefinito: tra metodi topologici e aspetti dinamici”.
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