Random perturbations of an eco-epidemiological model

Lopo F. de Jesus; César M. Silva; Helder Vilarinho

doi:10.3934/dcdsb.2021040

Article Contents

2022, Volume 27, Issue 1: 257-275. Doi: 10.3934/dcdsb.2021040

This issue Previous Article Phase portraits of the Higgins–Selkov system Next Article Strong solutions to a fluid-particle interaction model with magnetic field in

$ \mathbb{R}^2 $

Random perturbations of an eco-epidemiological model

1.
Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, and Instituto Superior de Ciências da Educação, Rua Sarmento Rodrigues, Lubango, Angola
2.
Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, 6201-001, Covilhã, Portugal

^* Corresponding author: Helder Vilarinho
^* Corresponding author: Helder Vilarinho

Received: August 31, 2020

Revised: November 30, 2020

Early access: January 2021

Published: January 2022

L. F. de Jesus, C. M. Silva and H. Vilarinho were partially supported by FCT through CMA-UBI (project UIDB/MAT/00212/2020). L. F. de Jesus was also supported by INAGBE

Abstract Full Text(HTML) Related Papers Cited by

Abstract

We consider random perturbations of a general eco-epidemiological model. We prove the existence of a global random attractor, the persistence of susceptibles preys and provide conditions for the simultaneous extinction of infectives and predators. We also discuss the dynamics of the corresponding random epidemiological $ SI $ and predator-prey models. We obtain for this cases a global random attractor, prove the prevalence of susceptibles/preys and provide conditions for the extinctions of infectives/predators.

Keywords:

Mathematics Subject Classification: Primary: 34F05, 60H25, 37A50.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	Y. Asai and P. E. Kloeden, Numerical schemes for random ODEs via stochastic differential equations, Commun. Appl. Anal., 17 (2013), 511–528.
[3]	T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynam., 84 (2016), 115-126. doi: 10.1007/s11071-015-2238-3.
[4]	T. Caraballo, R. Colucci, J. López-de-la-Cruz and A. Rapaport, A way to model stochastic perturbations in population dynamics models with bounded realizations, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 239-257. doi: 10.1016/j.cnsns.2019.04.019.
[5]	T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems, SpringerBriefs in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6.
[6]	T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162. doi: 10.3934/cpaa.2017007.
[7]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
[8]	I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Math, Vol. 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.
[9]	H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., 176 (1999), 57-72. doi: 10.1007/BF02505989.
[10]	H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc., 63 (2001), 413-427. doi: 10.1017/S0024610700001915.
[11]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[12]	H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0.
[13]	H. Crauel and M. Scheutzow, Minimal random attractors, J. Differential Equations, 265 (2018), 702-718. doi: 10.1016/j.jde.2018.03.011.
[14]	H. Crauel, Random Probability Measures on Polish Spaces Stochastics Monographs, V.11, London, 2002.
[15]	J. W. Cholewa and T. Dloko, Global Attractors in the Abstract Parabolics Problems, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.
[16]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[17]	M. Garrione and C. Rebelo, Persistence in seasonally varying predator-prey systems via the basic reproduction number, Nonlinear Anal. Real World Appl., 30 (2016), 73-98. doi: 10.1016/j.nonrwa.2015.11.007.
[18]	X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Probability Theory and Stochastic Modelling, 85, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.
[19]	L. F. de Jesus, C. M. Silva and H. Vilarinho, An Eco-epidemiological model with general functional response of predator to prey, preprint.
[20]	L. F. de Jesus, C. M. Silva and H. Vilarinho, Periodic orbits for periodic eco-epidemiological systems with infected prey, Electron. J. Qual. Theory Differ. Equ., 54 (2020), 1-20. doi: 10.14232/ejqtde.2020.1.54.
[21]	Y. Lu, X. Wang and S. Liu, A non-autonomous predator-prey model with infected prey, Discrete Contin. Dyn. Syst. B, 23 (2018), 3817-3836. doi: 10.3934/dcdsb.2018082.
[22]	C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64 (2012), 933-949. doi: 10.1007/s00285-011-0440-6.
[23]	C. M. Silva, Existence of Periodic Solutions for Eco-Epidemic Model with Disease in the Prey, J. Math. Anal. Appl., 453 (2017), 383-397. doi: 10.1016/j.jmaa.2017.03.074.
[24]	W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.
[25]	X. Niu, T. Zhang and Z. Teng, The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey, Appl. Math. Model., 35 (2011), 457-470. doi: 10.1016/j.apm.2010.07.010.