×

Role of environmental disturbance in an eco-epidemiological model with disease from external source. (English) Zbl 1236.92059

Summary: An eco-epidemiological model with random environmental disturbance is proposed and analyzed. We assume that the susceptible prey population can acquire infection both from external sources and from internal transmission of the disease. It is also assumed that there is no recovery of the disease, and the consumption of diseased prey has a deleterious effect on the predator population. Conditions for the extinction of the predator and the prey populations are worked out. The most important observation of the present investigation is that oscillatory behavior of the populations observed in deterministic framework undergoes stable coexistence in the stochastic framework.

MSC:

92D40 Ecology
92D30 Epidemiology
34C60 Qualitative investigation and simulation of ordinary differential equation models
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C60 Computational problems in statistics (MSC2010)
37N25 Dynamical systems in biology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Freedman, A model of predator-prey dynamics as modified by the action of a parasite, Mathematical Biosciences 99 pp 143– (1990) · Zbl 0698.92024
[2] Chattopadhyay, A predator-prey model with disease in the prey, Nonlinear Analysis 36 pp 747– (1999) · Zbl 0922.34036
[3] Xiao, Modelling and analysis of a predator-prey model with disease in the prey, Mathematical Biosciences 171 pp 59– (2001) · Zbl 0978.92031
[4] Venturino, Epidemics in predator-prey models: disease in the predators, IMA Journal of Mathematics Applied in Medicine and Biology 19 pp 185– (2002) · Zbl 1014.92036
[5] Hethcote, A predator-prey model with infected prey, Theoretical Population Biology 66 pp 259– (2004)
[6] Das, Effect of disease-selective predation on prey infected by contact and external sources, BioSystems 95 (3) pp 188– (2009)
[7] Holmes, Behavioural Aspects of Parasites Transmission pp 123– (1972)
[8] Lafferty, Altered behaviour of parasitized killifish increases susceptibility to predation by bird final hosts, Ecology 77 pp 1390– (1996)
[9] Chattopadhyay, Pelicans at risk in Salton Sea-an eco-epidemiological model-II, Ecological Modelling 167 pp 199– (2003)
[10] Blanchong, Persistence of Pasteurella multocida in wetlands following avian cholera outbreaks, Journal of Wildlife Diseases 42 (1) pp 33– (2006)
[11] Botzler, Epizootiology of avian cholera in wildfowl, Journal of Wildlife Diseases 27 pp 367– (1991)
[12] Wobeser, Avian cholera and waterfowl biology, Journal of Wildlife Diseases 28 pp 674– (1992)
[13] Rupier, Prevalence of Trichomonas gallinae in central California mourning doves, California Fish and Game 74 (4) pp 471– (1988)
[14] Anderson DP Hanson RP Influence of environment on virus diseases of poultry Symposium on Poultry Viruses. American Veterinary Medical Association 1964
[15] Isham, Assessing the variability of stochastic epidemics, Mathematical Biosciences 107 pp 209– (1991) · Zbl 0739.92015
[16] van Herwaarden, Stochastic epidemics: major outbreaks and the duration of the endemic period, Journal of Mathematical Biology 33 pp 581– (1995) · Zbl 0830.92024
[17] Garmendia, Recovery and identification of West Nile virus from a hawk in winter, Journal of Clinical Microbiology 38 (8) pp 3110– (2000)
[18] Venturino, The effect of diseases on competing species, Mathematical Biosciences 174 pp 111– (2001) · Zbl 0986.92025
[19] Pielou, Introduction to Mathematical Ecology (1969)
[20] Hutson, Permanent coexistence in general models of three interacting species, Journal of Mathematical Biology 21 pp 289– (1985) · Zbl 0579.92023
[21] Butler, Uniformly persistent systems, Proceedings of the American Mathematical Society 96 pp 425– (1986) · Zbl 0603.34043
[22] Kumar, A mathematical model of facultative mutualism with populations interacting in a food chain, Mathematical Biosciences 97 pp 235– (1989) · Zbl 0695.92016
[23] Hofbauer, Mathematical Ecology, Proceedings of Trieste (1986)
[24] Beretta, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation 45 pp 269– (1998) · Zbl 1017.92504
[25] Gikhman, The Theory of Stochastic Process-I (1979) · Zbl 0066.37901
[26] Adomian, Mathematics in Science and Engineering 169 (1983)
[27] Afanas’ev, Mathematical Theory of Control Systems Design (1996)
[28] Packer, Keeping the herds healthy and alert: implications of predator control for infectious disease, Ecology Letters 6 pp 797– (2003)
[29] Chattopadhyay, Classical predator-prey system with infection of prey population-a mathematical model, Mathematical Methods in the Applied Sciences 26 pp 1211– (2003) · Zbl 1044.34001
[30] Anderson, The invasion, persistence and spread of infectious diseases within animal and plant communities, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 314 pp 533– (1986)
[31] Hadeler, Predator-prey populations with parasitic infection, Journal of Mathematical Biology 27 pp 609– (1989) · Zbl 0716.92021
[32] Beltrami, Modelling the role of viral disease in recurrent phytoplankton blooms, Journal of Mathematical Biology 32 pp 857– (1994) · Zbl 0825.92122
[33] Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences 175 pp 117– (2002) · Zbl 0987.92027
[34] Sih, Predation, competition and prey communities: a review of field experiments, Annual Review of Ecology and Systematics 16 pp 269– (1985)
[35] Hudson, Grouse in Space and Time (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.