Dynamics of a delay-diffusion prey-predator model with disease in the prey



A mathematical model dealing with a prey-predator system with disease in the prey is considered. The functional response of the predator is governed by a Hoilling type-II function. Mathematical analysis of the model regarding stability and persistence has been performed. The effect of delay and diffusion on the above system is studied. The role of diffusivity on stability and persistence criteria of the system has also been discussed.

AMS Mathematics Subject Classification

Primary 92D30 

Keywords and phrases

Prey-predator system persistence impermanence discrete time delay diffusivity 


  1. 1.
    R. M. Anderson and R. M. May,The population dynamics of macroparasites and their invertebrate hosts, Phil. Trans. Roy. Soc. LondonB291 (1981), 451–524.CrossRefGoogle Scholar
  2. 2.
    R. M. Anderson and R. M. May,Directly transmitted infectious diseases: control by vaccination, Science,215 (1982), 1053–1060.CrossRefMathSciNetGoogle Scholar
  3. 3.
    R. M. Anderson and R. M. May,The invasion and spread of infectious diseases within animal and plant communities, Philos. Trans. R. Soc. Lond.B314 (1986), 533–570.CrossRefGoogle Scholar
  4. 4.
    R. Bhattacharyya, M. Bandyopadhyay and S. Banerjee,Stability and bifurcation in a diffusive prey-predator system: non-linear bifurcation analysis, J. Appl. Math. & Computing10 (2002), 17–26.MathSciNetCrossRefGoogle Scholar
  5. 5.
    R. Bhattacharyya, B. Mukhopadhyay and M. Bandyopadhyay,Diffusion driven stability analysis of a prey-predator system with Holling type-IV functional response, System Analysis Modelling Simulation,43(8) (2003), 1085–1093.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. Birkhoff and G. C. Rota,Ordinary Differential Equations, Ginn. and Co., 1982.Google Scholar
  7. 7.
    G. J. Bulter, H. I. Freedman and P. Waltman,Uniformly persistent system, Proc. Am. Math. Soc.96 (1986), 425–430.CrossRefGoogle Scholar
  8. 8.
    J. Chattopadhyay, G. Ghosal and K. S. Chaudhuri,Nonselective harvesting of a preypredator community with infected prey, Korean J. Comput. & Appl. Math.6(3) (1999), 601–616.MATHMathSciNetGoogle Scholar
  9. 9.
    K. Das and A. K. Sarkar,Effect of time delay in an autotroph-herbivore system with nutrient recycling, Korean J. Comput. & Appl. Math.5(3) (1998), 507–516.MATHMathSciNetGoogle Scholar
  10. 10.
    A. P. Dobson,The population biology of parasite induced changes in host behaviour, Q. Rev. Biol.63 (1988), 139–165.CrossRefGoogle Scholar
  11. 11.
    P. C. Fife,Mathematical Aspects of Reacting and Diffusing Systems, Lect. Notes in Biomathematics,28, Springer, Berlin, Heidelberg, New York, 1979.MATHGoogle Scholar
  12. 12.
    H. I. Freedman and P. Waltman,Persistence in models of three interacting predator-prey populations, Math. Biosci.68 (1984), 213–231.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. I. Freedman and P. Waltman,Persistence in a model of three competitive populations, Math. Biosci.73 (1985), 89–101.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    H. I. Freedman,A model of predator-prey dynamics as modified by the action of a parasite, Math. Biosci.99 (1990), 143–155.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    T. C. Gard,Persistence in food chains general interactions, Math. Biosci.51 (1980), 165–174.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. K. Ghosh, J. Chattopadhyay and P.K. Tapaswi, An SIRS epidemic model on a dispersive population, Korean J. Comput. & Appl. Math.,7(3) (2000), 693-MATHMathSciNetGoogle Scholar
  17. 17.
    K. P Hadeler and H. I. Freedman,Predator-prey populations with parasite infection, J. Math. Biol.27 (1989), 609–631.MATHMathSciNetGoogle Scholar
  18. 18.
    H. W. Hethcote,A thousand and one epidemic models, InFrontiers in Mathematical Biology, (Ed.) Levin, S.A., Lecture Notes in Biomathematics100, Springer, Berlin, 1994.Google Scholar
  19. 19.
    J. Hofbauer,General co-operation theorem for hypercycles, Monatsh. Math.91 (1981), 233–240.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    J. C. Holmes and W. M. Bethel,Modification of intermediate host behaviour by parasite InBehavioral Aspects of Parasite Transmission, No.1 to the Zool. J. Linnean. Soc., (Eds.) Cunning, E. V. and Wright, C. A.51 (1972), 123–149.Google Scholar
  21. 21.
    V. Hutson and G. T. Vickers,A criterion for permanent co-existence of species with an application to a two prey one predator system, Math. Biosci.63 (1983), 253–269.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    S. Kováis,Spatial inhomogenity due to Turing bifurcation in a system of Gierer-Meinhardt type, J. Appl. Math. & Computing11(1–2) (2003), 125–142.Google Scholar
  23. 23.
    W. O. Kermack and A. G. Mckendrick,Contributions to the mathematical theory of epidemics, Proc. Roy. Soc.A115 (1927), 700.CrossRefGoogle Scholar
  24. 24.
    R. M. May,Population biology of microparasite infections InMathematical Ecology, (eds.) Hallam, T.G. and Levin, S. A., Biomathematics17, Springer, Berlin, 1986, 405–442.Google Scholar
  25. 25.
    D. Mukherjee,Uniform persistence in a generalised prey-predator system with parasitic infection, Biosystems,47 (1998), 149–155.CrossRefGoogle Scholar
  26. 26.
    S. Muratori and S. Rinaldi,Low and high frequency oscillations in three-dimensional food chain system, SIAM. J. Appl. Math.52(6) (1992), 1688–1706.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    A. Okubo,Diffusion and Ecological Problems: Mathematical Models, Biomathematics,10, Springer, Berlin, 1980.MATHGoogle Scholar
  28. 28.
    A.M. Turing,The chemical basis of morphogenesis, Philos. R. Sec. Ser.B237 (1952), 37–72.CrossRefGoogle Scholar
  29. 29.
    Y. Xiao and L. Chen,Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci.171 (2001), 59–82.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2005

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of CalcuttaKolkataIndia

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