# Uniform persistence in a prey–predator model with a diseased predator

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## Abstract

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.

## Keywords

Uniform persistence Infectious disease Basic reproduction number Prey–predator model## Mathematics Subject Classification

37C70 92D30## Notes

### Acknowledgements

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”.

## References

- Bacaër N (2011) A short history of mathematical population dynamics. Springer, LondonCrossRefGoogle Scholar
- Bacaër N, Ait Dads EH (2012) On the biological interpretation of a definition for the parameter \(R_0\) in periodic population models. J Math Biol 65(4):601–621. https://doi.org/10.1007/s00285-011-0479-4 MathSciNetCrossRefzbMATHGoogle Scholar
- Bacaër N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco. J Math Biol 53(3):421–436. https://doi.org/10.1007/s00285-006-0015-0 MathSciNetCrossRefzbMATHGoogle Scholar
- Bate AM, Hilker FM (2013) Predator–prey oscillations can shift when diseases become endemic. J Theoret Biol 316:1–8. https://doi.org/10.1016/j.jtbi.2012.09.013 MathSciNetCrossRefzbMATHGoogle Scholar
- Butler G, Freedman HI, Waltman P (1986) Uniformly persistent systems. Proc Am Math Soc 96(3):425–430. https://doi.org/10.2307/2046588 MathSciNetCrossRefzbMATHGoogle Scholar
- Cheng KS (1981) Uniqueness of a limit cycle for a predator–prey system. SIAM J Math Anal 12(4):541–548. https://doi.org/10.1137/0512047 MathSciNetCrossRefzbMATHGoogle Scholar
- Conley C (1978) Isolated invariant sets and the Morse index, CBMS regional conference series in mathematics, vol 38. American Mathematical Society, Providence, R.I.CrossRefGoogle Scholar
- Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J Math Biol 28(4):365–382. https://doi.org/10.1007/BF00178324 MathSciNetCrossRefzbMATHGoogle Scholar
- Fonda A (1988) Uniformly persistent semidynamical systems. Proc Am Math Soc 104(1):111–116. https://doi.org/10.2307/2047471 MathSciNetCrossRefzbMATHGoogle Scholar
- Fonda A, Gidoni P (2015) A permanence theorem for local dynamical systems. Nonlinear Anal 121:73–81. https://doi.org/10.1016/j.na.2014.10.011 MathSciNetCrossRefzbMATHGoogle Scholar
- Garrione M, Rebelo C (2016) Persistence in seasonally varying predator–prey systems via the basic reproduction number. Nonlinear Anal Real World Appl 30:73–98. https://doi.org/10.1016/j.nonrwa.2015.11.007 MathSciNetCrossRefzbMATHGoogle Scholar
- Hofbauer J (1989) A unified approach to persistence. Acta Appl Math 14(1–2):11–22. https://doi.org/10.1007/BF00046670 MathSciNetCrossRefzbMATHGoogle Scholar
- Hsu SB, Hubbell SP, Waltman P (1978) A contribution to the theory of competing predators. Ecol Monogr 48(3):337–349CrossRefGoogle Scholar
- Hutson V (1984) A theorem on average Liapunov functions. Monatsh Math 98(4):267–275. https://doi.org/10.1007/BF01540776 MathSciNetCrossRefzbMATHGoogle Scholar
- Lakshmikantham V, Leela S (1969) Differential and integral inequalities: theory and applications. Vol. I: Ordinary differential equations. Academic Press, New York, Mathematics in Science and Engineering, Vol 55-IGoogle Scholar
- LaSalle JP (1976) The stability of dynamical systems. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
- Lefschetz S (1963) Differential equations: geometric theory. 2nd edn. Pure and applied mathematics, Vol. VI, Interscience Publishers, a division of Wiley, New YorkGoogle Scholar
- Rebelo C, Margheri A, Bacaër N (2012) Persistence in seasonally forced epidemiological models. J Math Biol 64(6):933–949. https://doi.org/10.1007/s00285-011-0440-6 MathSciNetCrossRefzbMATHGoogle Scholar
- Rebelo C, Margheri A, Bacaër N (2014) Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete Contin Dyn Syst Ser B 19(4):1155–1170. https://doi.org/10.3934/dcdsb.2014.19.1155 MathSciNetCrossRefzbMATHGoogle Scholar
- Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 97(895):209–223CrossRefGoogle Scholar
- Waltman P (1991) A brief survey of persistence in dynamical systems. In: Delay differential equations and dynamical systems (Claremont, CA, 1990), Lecture Notes in Math., vol 1475, Springer, Berlin, pp 31–40. https://doi.org/10.1007/BFb0083477 zbMATHGoogle Scholar
- Wang W, Zhao XQ (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dynam Differ Equ 20(3):699–717. https://doi.org/10.1007/s10884-008-9111-8 MathSciNetCrossRefzbMATHGoogle Scholar
- Yuan Y, Chen H, Du C, Yuan Y (2012) The limit cycles of a general Kolmogorov system. J Math Anal Appl 392(2):225–237. https://doi.org/10.1016/j.jmaa.2012.02.065 MathSciNetCrossRefzbMATHGoogle Scholar