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Impacts of Incubation Delay on the Dynamics of an Eco-Epidemiological System—A Theoretical Study

  • N. BairagiEmail author
  • R. R. Sarkar
  • J. Chattopadhyay
Original Article

Abstract

Parasite and predator play significant role in trophic interaction, productivity and stability of an ecosystem. In this paper, we have studied a host-parasite-predator interaction that incorporates incubation delay. How the qualitative and quantitative behaviors of the system alter with the incubation delay have been discussed both from mathematical and biological point of views. It is observed that for a lower infection rate, the system is stable for all delays; but for a higher infection rate, there exists a threshold value of the delay above which the system is unstable and below which the system is stable leading to the persistence of all the species. Also, the instability arising from the incubation delay may be controlled if somehow the growth rate of predator population is increased. Numerical studies have also been performed to illustrate different analytical findings.

Keywords

Susceptible population Infected population Predator Linear mass action Holling type II Delay Local stability Hopf-bifurcation 

References

  1. Adams, V.D., De Angelis, D.L., Goldstein, R.A., 1980. Stability analysis of the time delay in a host parasitoid model. J. Theor. Biol. 83, 43–2. CrossRefGoogle Scholar
  2. Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans—Dynamics and Control. Oxford University Press, London. Google Scholar
  3. Arriola, L., Hyman, J., 2005. Lecture Notes, Forward and Adjoint Sensitivity Analysis: with Applications in Dynamical Systems, Linear Algebra and Optimization. Mathematical and Theoretical Biology Institute. Google Scholar
  4. Chattopadhyay, J., Arino, O., 1999. A predator-prey model with disease in the prey. Nonlinear Anal. 36, 747–66. CrossRefMathSciNetGoogle Scholar
  5. Chattopadhyay, J., Bairagi, N., 2001. Pelicans at risk in Salton Sea—an eco-epidemiological study. Ecol. Model. 136, 103–12. CrossRefGoogle Scholar
  6. Cohn, J.P., 2000. Saving the Salton Sea. Bioscience 50(4), 295–01. CrossRefGoogle Scholar
  7. Culshaw, R.V., Ruan, S., 2000. A Delay-differential equation model of HIV infection of CD4+ T-cells. Math. Biosci. 165, 27–9. zbMATHCrossRefGoogle Scholar
  8. Culshaw, R.V., Ruan, S., Webb, G., 2003. A mathematical model of cell to cell spread of HIV-1 that includes a time delay. J. Math. Biol. 46, 425–44. zbMATHCrossRefMathSciNetGoogle Scholar
  9. Curio, E., 1988. Behavior and parasitism. In: Mehlhorn, H. (Ed.), Parasitology in Focus. Springer, New York. Google Scholar
  10. Dieudonne’ J., 1960. Foundations of Modern Analysis. Academic, New York. Google Scholar
  11. Dobson, A.P., Hudson, P.J., 1992. Regulation and stability of a free-leaving host-parasite system, Trichostrongylus tenuis in red grouse. II. Population models. J. Anim. Ecol 61, 487–00. CrossRefGoogle Scholar
  12. Dwyer, G., Dushoff, J., Yee, S.H., 2004. Generalist predators, specialist pathogens and insect outbreaks. Nature 43, 341–45. CrossRefGoogle Scholar
  13. Fenton, A., Rands, S.A., 2006. The impact of parasite manipulation and predator foraging behavior on predator-prey communities. Ecology 87(11), 2832–841. CrossRefGoogle Scholar
  14. Freedman, H.I., Rao, V.S.H., 1983. The trade-off between mutual interference and time-lags in predator-prey systems. Bull. Math. Biol. 45, 991. zbMATHMathSciNetGoogle Scholar
  15. Freedman, H.I., Addicot, J.F., Rai, B., 1987. Obligate mutualism with a predator: stability and persistence of three species models. Theor. Popul. Biol. 32, 157–75. zbMATHCrossRefGoogle Scholar
  16. Freeland, W.J., 1983. Parasites and coexistence of animal host species. Am. Nat. 121, 223–36. CrossRefGoogle Scholar
  17. Friend, M., 2002. Avian disease in the Salton Sea. Hydrobiologia 473, 293–06. CrossRefGoogle Scholar
  18. Gonzalez, M.R., Hart, C.M., Verfaillie, J.R., Hurlbert, S.H., 1998. Salinity and fish effects on Salton Sea microecosystems: water chemistry and nutrient cycling. Hydrobiologia 381, 105–28. CrossRefGoogle Scholar
  19. Gopalsamy, K., 1992. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht. zbMATHGoogle Scholar
  20. Hadeler, K.P., Freedman, H.I., 1989. Predator–prey population with parasite infection. J. Math. Biol. 27, 609–31. zbMATHMathSciNetGoogle Scholar
  21. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H., 1981. Theory and Application of Hopf Bifurcation. Cambridge University Press, Cambridge. Google Scholar
  22. Hethcote, H.W., Levin, S.A., 1989. Periodicity in epidemiological models. In: Gross, L., Levin, S.A. (Eds.), Applied Mathematical Ecology, pp. 193–11. Springer, New York. Google Scholar
  23. Holling, C.S., 1965. The functional response…population regulation. Memories of the entomological society of Canada, 45, pp. 1–0. Google Scholar
  24. Holmes, J.C., Bethel, W.M., 1972. Modification of intermediate host behavior by parasites. In: Canning, E.V., Wright, C.A. (Eds.), Behavioral Aspects of Parasite Transmission. Suppl. I to Zool. f. Linnean Soc., vol. 51, pp. 123–49. Google Scholar
  25. Hudson, P.J., Dobson, A.P., Newborn, D., 1992. Do parasites make pray vulnerable to predation? Red grouse and parasites. J. Anim. Ecol. 61, 681–92. CrossRefGoogle Scholar
  26. Hudson, P.J., Newborn, D., Dobson, A.P., 1992. Regulation and stability of a free-leaving host-parasite system, Trichostrongylus tenuis in red grouse. I. Monitoring and parasite reduction experiment. J. Anim. Ecol. 61, 477–86. CrossRefGoogle Scholar
  27. Hudson, P.J., Dobson, A.P., Newborn, D., 1998. Prevention of population cycles by parasite removal. Science 282, 2256–258. CrossRefGoogle Scholar
  28. Ives, A.R., Murray, D.L., 1997. Can sublethal parasitism destabilize predator-prey population dynamics? a model of snowshoe hares predators, and parasites. J. Anim. Ecol. 66, 265–78. CrossRefGoogle Scholar
  29. Jehl, J.R., 1996. Mass mortality events of eared grebes in North America. Am. J. Field Ornithol. 67(3), 471–76. Google Scholar
  30. Kaiser, J., 1999. Salton Sea: Battle over a dying sea. Science 284(5411), 28–0. CrossRefGoogle Scholar
  31. Kuang, Y., 1993. Delay Differential Equations with Applications in Population Dynamics. Academic, New York. zbMATHGoogle Scholar
  32. Lafferty, K.D., 1992. Foraging on prey that are modified by parasites. Am. Nat. 140, 854–67. CrossRefGoogle Scholar
  33. Lafferty, K.D., Morris, A.K., 1996. Altered behaviour of parasitized killifish increases susceptibility to predation by bird final hosts. Ecology 77, 1390–397. CrossRefGoogle Scholar
  34. MacDonald, M., 1978. Time Delays in Biological Models. Springer, Heidelberg. Google Scholar
  35. MacDonald, M., 1989. Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge. zbMATHGoogle Scholar
  36. McCallum, H., Harvell, D., Dobson, A., 2003. Rates of spread of marine pathogen. Ecol. Lett. 6, 1062–067. CrossRefGoogle Scholar
  37. McCallum, H.I. et al., 2004. Does terrestrial epidemiology apply to marine systems? Trends Ecol. Evol. 19(11), 585–91. CrossRefGoogle Scholar
  38. Minchella, D., Scott, M.E., 1991. Parasitism: A cryptic determinant of animal community structure. Trends Ecol. Evol. 6, 250–54. CrossRefGoogle Scholar
  39. Moore, J., 2002. Parasites and the Behaviour of Animals. Oxford University Press, Oxford. Google Scholar
  40. Packer, C., Holt, R.D., Hudson, P.J., Lafferty, K.D., Dobson, A.P., 2003. Keeping the herds healthy and alert: implications of predator control for infectious disease. Ecol. Lett. 6, 792–02. CrossRefGoogle Scholar
  41. Slack, G., 1997. Salton Sea Sickness. Pacific Discovery, Winter. Google Scholar
  42. Tompkins, D.N., et al., 2002. Parasites and host population dynamics. In: Hudson, P.J., Rizzoli, A., Grenfell, B.T., Heesterbeek, H., Dobson, A.P. (Eds.), The Ecology of Wildlife Disease, pp. 45–2. Oxford University Press, Oxford Google Scholar
  43. Thomas, F., Cezilly, F., Meeus, T.D., Crivelli, A., Renaud, F., 1997. Parasitism and ecology of wetlands: a review. Estuaries 20(3), 646–54. CrossRefGoogle Scholar
  44. Venturino, E., 2002. Epidemics in predator-prey models: disease in the predators. IMA J. Math. Appl. Med. Biol. 19, 185–05. zbMATHCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  1. 1.Centre for Mathematical Biology and Ecology, Department of MathematicsJadavpur UniversityKolkataIndia
  2. 2.Centre for Cellular and Molecular Biology500 007India
  3. 3.Agricultural and Ecological Research UnitIndian Statistical InstituteKolkataIndia

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