Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 807486, 16 pages
http://dx.doi.org/10.5402/2011/807486
Research Article

A Mathematical Study of a Predator-Prey Dynamics with Disease in Predator

Department of Mathematics, Mahadevananda Mahavidyalaya, Monirampore, Barrackpore PO, Kol-120, India

Received 24 March 2011; Accepted 19 April 2011

Academic Editor: M. Langthjem

Copyright © 2011 Krishna pada Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. O. Kermack and A. G. Mc Kendrick, “Contributions to the mathematical theory of epidemics, part 1,” Proceedings of the Royal Society Series A, vol. 115, pp. 700–721, 1927. View at Google Scholar
  2. R. M. Anderson and R. M. May, “Regulation and stability of host-parasite population interactions,” Journal of Animal Ecology, vol. 47, pp. 219–249, 1978. View at Google Scholar
  3. J. Chattopadhyay and O. Arino, “A predator-prey model with disease in the prey,” Nonlinear Analysis: Theory, Methods & Applications, vol. 36, no. 6, pp. 747–766, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  4. H. I. Freedman, “A model of predator-prey dynamics as modified by the action of a parasite,” Mathematical Biosciences, vol. 99, no. 2, pp. 143–155, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. W. Hethcote, W. Wang, L. Han, and Z. Ma, “A predator—prey model with infected prey,” Theoretical Population Biology, vol. 66, no. 3, pp. 259–268, 2004. View at Publisher · View at Google Scholar · View at PubMed
  6. E. Venturino, “Epidemics in predator-prey models: disease in the predators,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 19, no. 3, pp. 185–205, 2002. View at Google Scholar
  7. Y. Xiao and L. Chen, “Modeling and analysis of a predator-prey model with disease in the prey,” Mathematical Biosciences, vol. 171, no. 1, pp. 59–82, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. M. Anderson and R. M. May, “Population biology of infectious diseases: part I,” Nature, vol. 280, no. 5721, pp. 361–367, 1979. View at Google Scholar
  9. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1998.
  10. H. Curtis, Invitation to Biology, Worth Publishers, New York, NY, USA, 1972.
  11. R. P. Jaques, J. M. Hardman, J. E. Laing, R. F. Smith, and E. Bent, “Orchard trials in Canada on control of Cydia pomonella (Lep: Tortricidae) by granulosis virus,” Entomophaga, vol. 39, no. 3-4, pp. 281–292, 1994. View at Publisher · View at Google Scholar
  12. P. Caballero, E. Vargas-Osuna, and C. Santiago-Alvarez, “Efficacy of a spanish strain of Agrotis sehetum Granulosis virus (Baculoviridae) against Agrotis segetum Schiff. (Lep., Noctuidae) on corn,” Journal of Applied Entomology, vol. 112, pp. 59–64, 1991. View at Google Scholar
  13. A. Laarif, A. Ben Ammar, M. Trabelsi, and M. H. Ben Hamouda, “Histopathology and morphogenesis of the Granulovirus of the potato tuber moth Phthorimaea operculella,” Tunisian Journal of Plant Protection, vol. 1, pp. 115–124, 2006. View at Google Scholar
  14. R. M. Anderson and R. M. May, “The invasion, persistence and spread of infectious diseases within animal and plant communities,” Philosophical Transactions of the Royal Society of London Series B, vol. 314, no. 1167, pp. 533–570, 1986. View at Google Scholar
  15. K. P. Hadeler and H. I. Freedman, “Predator-prey populations with parasitic infection,” Journal of Mathematical Biology, vol. 27, no. 6, pp. 609–631, 1989. View at Publisher · View at Google Scholar
  16. M. E. Hochberg, “The potential role of pathogens in biological control,” Nature, vol. 337, no. 6204, pp. 262–265, 1989. View at Google Scholar
  17. E. Venturino, “The influence of disease on Lotka-Volterra systems,” Rocky Mountain Journal of Mathematics, vol. 24, pp. 381–402, 1994. View at Google Scholar
  18. L. Han, Z. Ma, and H. W. Hethcote, “Four predator prey models with infectious diseases,” Mathematical and Computer Modelling, vol. 34, no. 7-8, pp. 849–858, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. D. Greenhalgh and M. Haque, “A predator-prey model with disease in the prey species only,” Mathematical Methods in the Applied Sciences, vol. 30, no. 8, pp. 911–929, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Haque and E. Venturino, “Increase of the prey may decrease the healthy predator population in presence of a disease in the predator,” Hermis, vol. 7, pp. 38–59, 2006. View at Google Scholar
  21. M. Haque and E. Venturino, “An ecoepidemiological model with disease in predator: the ratio-dependent case,” Mathematical Methods in the Applied Sciences, vol. 30, no. 14, pp. 1791–1809, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. Y. Xiao and F. Van Den Bosch, “The dynamics of an eco-epidemic model with biological control,” Ecological Modelling, vol. 168, no. 1-2, pp. 203–214, 2003. View at Publisher · View at Google Scholar
  23. A. P. Dobson, “The population biology of parasite-induced changes in host behavior,” Quarterly Review of Biology, vol. 63, no. 2, pp. 139–165, 1988. View at Google Scholar
  24. A. Fenton and S. A. Rands, “The impact of parasite manipulation and predator foraging behavior on predator-prey communities,” Ecology, vol. 87, no. 11, pp. 2832–2841, 2006. View at Publisher · View at Google Scholar
  25. A. P. Dobson and A. E. Keymer, “Life history models,” in Biology of the Acanthocephala, D. W. T. Crompton and B. B. Nickol, Eds., pp. 347–384, Cambridge University Press, Cambridge, UK, 1985. View at Google Scholar
  26. E. C. Pielou, Introduction to Mathematical Ecology, Wiley-Interscience, New York, NY, USA, 1969.
  27. Y. H. Hsieh and C. K. Hsiao, “Predator-prey model with disease infection in both populations,” Mathematical Medicine and Biology, vol. 25, no. 3, pp. 247–266, 2008. View at Publisher · View at Google Scholar · View at PubMed
  28. O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990. View at Google Scholar
  29. G. J. Butler, H. Freedman, and P. Waltman, “Uniformly persistent systems,” Proceedings of the American Mathematical Society, vol. 96, no. 3, pp. 425–429, 1986. View at Google Scholar
  30. R. Kumar and H. I. Freedman, “A mathematical model of facultative mutualism with populations interacting in a food chain,” Mathematical Biosciences, vol. 97, no. 2, pp. 235–261, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  31. V. Hutson and R. Law, “Permanent coexistence in general models of three interacting species,” Journal of Mathematical Biology, vol. 21, no. 3, pp. 285–298, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  32. J. Hofbauer, “Saturated equilibria, permanence and stability for ecological systems,” in Proceedings of the 2nd Autumn Course on Mathematical Ecology, L. Groos, T. Hallam, and S. Levin, Eds., World Scientific, Trieste, Italy, 1986.
  33. C. Packer, R. D. Holt, P. J. Hudson, K. D. Lafferty, and A. P. Dobson, “Keeping the herds healthy and alert: implications of predator control for infectious disease,” Ecology Letters, vol. 6, no. 9, pp. 797–802, 2003. View at Publisher · View at Google Scholar
  34. E. Beltrami and T. O. Carroll, “Modelling the role of viral disease in recurrent phytoplankton blooms,” Journal of Mathematical Biology, vol. 32, no. 8, pp. 857–863, 1994. View at Publisher · View at Google Scholar
  35. S. R. Hall, M. A. Duffy, and C. E. Cáceres, “Selective predation and productivity jointly drive complex behavior in host-parasite systems,” American Naturalist, vol. 165, no. 1, pp. 70–81, 2005. View at Publisher · View at Google Scholar · View at PubMed
  36. F. De Castro and B. M. Bolker, “Parasite establishment and host extinction in model communities,” Oikos, vol. 111, no. 3, pp. 501–513, 2005. View at Publisher · View at Google Scholar
  37. F. Courchamp and G. Sugihara, “Modeling the biological control of an alien predator to protect island species from extinction,” Ecological Applications, vol. 9, no. 1, pp. 112–123, 1999. View at Google Scholar
  38. F. Courchamp, J. L. Chapuis, and M. Pascal, “Mammal invaders on islands: impact, control and control impact,” Biological Reviews of the Cambridge Philosophical Society, vol. 78, no. 3, pp. 347–383, 2003. View at Publisher · View at Google Scholar
  39. M. Nogales, A. Martín, B. R. Tershy et al., “A review of feral cat eradication on islands,” Conservation Biology, vol. 18, no. 2, pp. 310–319, 2004. View at Publisher · View at Google Scholar
  40. S. Roy and J. Chattopadhyay, “Enrichment and ecosystem stability: effect of toxic food,” BioSystems, vol. 90, no. 1, pp. 151–160, 2007. View at Publisher · View at Google Scholar · View at PubMed
  41. K. D. Lafferty, S. Allesina, M. Arim et al., “Parasites in food webs: the ultimate missing links,” Ecology Letters, vol. 11, no. 6, pp. 533–546, 2008. View at Publisher · View at Google Scholar · View at PubMed