Analysis of a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response

Advances in Difference Equations

Volumn 2015, Issue 1, 2015, Pages

  Li, Shuang ^a   Wang, Xiaopan ^a  

^a HENAN NORMAL UNIVERSITY   (China)

Author keywords

Beddington DeAngelis functional response; extinction; persistence; predator prey; stability; stationary distribution; stochastic

Indexed keywords

EID: 84938887383     PISSN: 16871839     EISSN: 16871847     Source Type: Journal    
DOI: 10.1186/s13662-015-0448-0     Document Type: Article

References (46)

1. Kermack, WO, Mckendrick, AG: A contribution to the mathematical theory of epidemics. Proc. R. Soc. A 115, 700-721 (1927)
2. Kermack, WO, Mckendrick, AG: Contributions to the mathematical theory of epidemics II. Proc. R. Soc. A 138, 55-83 (1932)
3. Raggett, GF: Modeling the Eyam plague. IMA J. 18, 221-226 (1982)
4. Liu, WM, Levin, SA, Iwasa, Y: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187-204 (1986)
5. Liu, WM, Hethcote, HW, Levin, SA: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359-380 (1987)
6. Hethcote, HW: The mathematics of infectious diseases. SIAM Rev. 42, 599-653 (2000)
7. Busenderg, S, Driessche, PV: Analysis of a disease transmission model in a population with varying size. J. Math. Biol. 28, 257-270 (1990)
8. Hethcote, HW: An immunization model for a heterogeneous population. Theor. Popul. Biol. 14, 338-349 (1978)
9. Anderson, RM, May, RM: The invasion, persistence and spread of infectious diseases within animal and plant communities. Philos. Trans. R. Soc. Lond. B 314, 533-570 (1986)
10. Bowers, RG, Begon, M: A host-host-pathogen model with free living infective stages, applicable to microbial pest control. J. Theor. Biol. 148, 305-329 (1991)
11. Begon, M, Bowers, RG, Kadianakis, N, Hodgkinson, DE: Disease and community structure: the importance of host-regulation in a host-host-pathogen model. Am. Nat. 139, 1131-1150 (1992)
12. Begon, M, Bowers, RG: Host-host-pathogen models and microbial pest control: the effect of host self-regulation. J. Theor. Biol. 169, 275-287 (1995)
13. Hadeler, KP, Freedman, HI: Predator-prey populations with parasitic infection. J. Math. Biol. 27, 609-631 (1989)
14. Han, LT, Ma, ZE, Hethcote, HW: Four predator-prey models with infectious diseases. Math. Comput. Model. 34, 849-858 (2001)
15. Han, LT, Ma, ZE, Tan, S: An SIRS epidemic model of two competitive species. Math. Comput. Model. 37, 87-108 (2003)
16. Xiao, YN, Chen, LS: Modeling and analysis of a predator-prey model with disease in the prey. Math. Biosci. 171, 59-82 (2001)
17. Xiao, YN, Chen, LS: A ratio-dependent predator-prey model with disease in the prey. Appl. Math. Comput. 131, 397-414 (2002)
18. Chattopadhyay, J, Arino, O: A predator-prey model with disease in the prey. Nonlinear Anal. 36, 747-766 (1999)
19. Hassell, MP, Varley, CC: New inductive population model for insect parasites and its bearing on biological control. Nature 223, 1133-1137 (1969)
20. Beddington, JR: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331-340 (1975)
21. DeAngelis, DL, Goldstein, AH, O’Neill, RV: A model for trophic interaction. Ecology 56, 881-892 (1975)
22. Hwang, TW: Global analysis of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 281, 395-401 (2003)
23. Hwang, TW: Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 290, 113-122 (2004)
24. Crowley, PH, Martin, EK: Functional response and interference within and between year classes of a dragonfly population. J. North Am. Benthol. Soc. 8, 211-221 (1989)
25. Li, HY, Takeuchi, Y: Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 374, 644-654 (2011)
26. Mao, XR, Marion, G, Renshaw, E: Environmental Brownian noise suppresses explosions in population dynamics. Stoch. Process. Appl. 97, 95-110 (2002)
27. Mao, XR, Sabanis, S, Renshaw, E: Asymptotic behavior of stochastic Lotka-Volterra model. J. Math. Anal. Appl. 287, 141-156 (2003)
28. Mao, XR, Yuan, C, Zou, J: Stochastic differential delay equations of population dynamics. J. Math. Anal. Appl. 304, 296-320 (2005)
29. Liu, M, Wang, K: Survival analysis of stochastic single-species population models in polluted environments. Ecol. Model. 9, 1347-1357 (2009)
30. Li, XY, Mao, XR: Population dynamical behavior of nonautonomous Lotka-Volterra competitive system with random perturbation. Discrete Contin. Dyn. Syst. 24, 523-545 (2009)
31. Jiang, DQ, Shi, NZ, Zhao, YN: Existence, uniqueness and global stability of positive solutions to the food-limited population model with random perturbation. Math. Comput. Model. 42, 651-658 (2005)
32. Ji, CY, Jiang, DQ, Shi, NZ: Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 359, 482-498 (2009)
33. Ji, CY, Jiang, DQ, Shi, NZ: A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 377, 435-440 (2011)
34. Ji, CY, Jiang, DQ: Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 381, 441-453 (2011)
35. Jiang, DQ, Ji, CY, Shi, NZ, Yu, JJ: The long time behavior of DI SIR epidemic model with stochastic perturbation. J. Math. Anal. Appl. 372, 162-180 (2010)
36. Liu, M, Wang, K: Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 1114-1121 (2011)
37. Mao, XR: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)
38. Jiang, DQ, Shi, NZ: A note on nonautonomous logistic equation with random perturbation. J. Math. Anal. Appl. 303, 164-172 (2005)
39. Zhu, C, Yin, G: On competitive Lotka-Volterra model in random environments. J. Math. Anal. Appl. 357, 154-170 (2009)
40. Friedman, A: Stochastic Differential Equations and Applications. Academic Press, New York (1975)
41. Liu, M, Wang, K, Wu, Q: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. Bull. Math. Biol. 73, 1969-2012 (2011)
42. Liu, M, Wang, K: Persistence and extinction in stochastic non-autonomous logistic systems. J. Math. Anal. Appl. 375, 443-457 (2011)
43. Khasminskii, R: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn (1980)
44. Gard, TC: Introduction to Stochastic Differential Equations. Dekker, New York (1988)
45. Zhu, C, Yin, G: Asymptotic properties of hybrid diffusion systems. SIAM J. Control Optim. 46, 1155-1179 (2007)
46. Higham, DJ: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525-546 (2001)