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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 and 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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

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