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Phase portraits of the Higgins–Selkov system
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 |
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.
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. Google Scholar |
[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. |
show all references
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. Google Scholar |
[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. |
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