Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 92B05, 93E15, 60H10; Secondary: 34E10.

 Citation:

•  [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.