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January  2015, 20(1): 295-321. doi: 10.3934/dcdsb.2015.20.295

The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise

Qiumei Zhang 1, , Daqing Jiang 1, and  Li Zu 1,

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China, China

Received  August 2013 Revised  July 2014 Published  October 2014

In this paper, we have proposed and analyzed a perturbed eco-epidemiological model with Holling type II functional response by white noise. By Lyapunov analysis methods, we prove the stochastic stability, its long time behavior around the equilibrium of deterministic eco-epidemiological model and the stochastic asymptotic stability. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.
Keywords: stochastically asymptotically stable., Holling type II functional response, extinction, Eco-epidemiological model, asymptotically stable.
Mathematics Subject Classification: Primary: 92B05, 93E15, 60H10; Secondary: 34E1.
Citation: Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295
References:
[1]

E. Berettaa, V. Kolmanovskiib and L. Shaikhetc, Stability of epidemic model with time delays influenced by stochastic perturbations, Math Comput Simulat, 45 (1998), 269-277. doi: 10.1016/S0378-4754(97)00106-7.  Google Scholar

[2]

J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766. doi: 10.1016/S0362-546X(98)00126-6.  Google Scholar

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J. Chattopadhyay and N. Bairagi, Pelicans at risk in Salton Sea - an eco-epidemiological study, Ecol. Model., 136 (2001), 103-112. Google Scholar

[4]

J. Chattopadhyay, P. D. N. Srinivasu and N. Bairagi, Pelicans at risk in Salton Sea - an eco-epidemiological model II, Ecol. Model., 167 (2003), 199-211. doi: 10.1016/S0304-3800(03)00187-X.  Google Scholar

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J. Chattopadhyay, P. K. Roy and N. Bairagi, Role of infection on the stability of a predator-prey system with several response functions - A comparative study, Journal of Theoretical Biology., 248 (2007), 10-25. doi: 10.1016/j.jtbi.2007.05.005.  Google Scholar

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D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302.  Google Scholar

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H. W. Hethcote, W. Wang, L. Han and Z. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010.  Google Scholar

[8]

K. P. Hadeler and H. I. Freedman, Predator-prey population with parasite infection, J. Math. Biol., 27 (1989), 609-631. doi: 10.1007/BF00276947.  Google Scholar

[9]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. doi: 10.1016/j.automatica.2011.09.044.  Google Scholar

[10]

C. Y. Ji, D. Q. Jiang, Q. S. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.  Google Scholar

[11]

C. Y. Ji and D. Q. Jiang, Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation, Commun Nonlinear Sci Numer Simulat., 17 (2012), 2501-2516. doi: 10.1016/j.cnsns.2011.07.025.  Google Scholar

[12]

C. Y. Ji and D. Q. Jiang, Analysis of a predator-prey model with disease in the prey, Int. J. Biomath., 6 (2013), 1350012, 21 pp. doi: 10.1142/S1793524513500125.  Google Scholar

[13]

X. Y. Li and X. R. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete and Continuous Dynamical Systems A, 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.  Google Scholar

[14]

X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. doi: 10.1533/9780857099402.  Google Scholar

[15]

E. Venturino, Epidemics in predator-prey models: Disease in the prey, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Theory of EpidemicsWuerz, 1, Winnipeg, Canada, 1995, 381-393. Google Scholar

[16]

E. Venturino, Epidemics in predator-prey models: Disease in the predators, IMA J. Math. Appl. Med. Biol., 19 (2002), 185-205. doi: 10.1093/imammb19.3.185.  Google Scholar

[17]

Y. Xiao and L. Chen, Modelling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82. doi: 10.1016/S0025-5564(01)00049-9.  Google Scholar

show all references

References:
[1]

E. Berettaa, V. Kolmanovskiib and L. Shaikhetc, Stability of epidemic model with time delays influenced by stochastic perturbations, Math Comput Simulat, 45 (1998), 269-277. doi: 10.1016/S0378-4754(97)00106-7.  Google Scholar

[2]

J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766. doi: 10.1016/S0362-546X(98)00126-6.  Google Scholar

[3]

J. Chattopadhyay and N. Bairagi, Pelicans at risk in Salton Sea - an eco-epidemiological study, Ecol. Model., 136 (2001), 103-112. Google Scholar

[4]

J. Chattopadhyay, P. D. N. Srinivasu and N. Bairagi, Pelicans at risk in Salton Sea - an eco-epidemiological model II, Ecol. Model., 167 (2003), 199-211. doi: 10.1016/S0304-3800(03)00187-X.  Google Scholar

[5]

J. Chattopadhyay, P. K. Roy and N. Bairagi, Role of infection on the stability of a predator-prey system with several response functions - A comparative study, Journal of Theoretical Biology., 248 (2007), 10-25. doi: 10.1016/j.jtbi.2007.05.005.  Google Scholar

[6]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302.  Google Scholar

[7]

H. W. Hethcote, W. Wang, L. Han and Z. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010.  Google Scholar

[8]

K. P. Hadeler and H. I. Freedman, Predator-prey population with parasite infection, J. Math. Biol., 27 (1989), 609-631. doi: 10.1007/BF00276947.  Google Scholar

[9]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. doi: 10.1016/j.automatica.2011.09.044.  Google Scholar

[10]

C. Y. Ji, D. Q. Jiang, Q. S. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.  Google Scholar

[11]

C. Y. Ji and D. Q. Jiang, Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation, Commun Nonlinear Sci Numer Simulat., 17 (2012), 2501-2516. doi: 10.1016/j.cnsns.2011.07.025.  Google Scholar

[12]

C. Y. Ji and D. Q. Jiang, Analysis of a predator-prey model with disease in the prey, Int. J. Biomath., 6 (2013), 1350012, 21 pp. doi: 10.1142/S1793524513500125.  Google Scholar

[13]

X. Y. Li and X. R. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete and Continuous Dynamical Systems A, 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.  Google Scholar

[14]

X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. doi: 10.1533/9780857099402.  Google Scholar

[15]

E. Venturino, Epidemics in predator-prey models: Disease in the prey, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Theory of EpidemicsWuerz, 1, Winnipeg, Canada, 1995, 381-393. Google Scholar

[16]

E. Venturino, Epidemics in predator-prey models: Disease in the predators, IMA J. Math. Appl. Med. Biol., 19 (2002), 185-205. doi: 10.1093/imammb19.3.185.  Google Scholar

[17]

Y. Xiao and L. Chen, Modelling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82. doi: 10.1016/S0025-5564(01)00049-9.  Google Scholar

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