Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


A Mathematical Study of a Two Species Eco-Epidemiological Model with Different Predation Principles

Discontinuity, Nonlinearity, and Complexity 9(2) (2020) 309--325 | DOI:10.5890/DNC.2020.06.011

Aktar Saikh, Nurul Huda Gazi

Department of Mathematics and Statistics, Aliah University, IIA/27, New Town, Kolkata-700160, India

Download Full Text PDF



This paper formulates and analyzes a predator-prey model with disease in the prey. Mathematical analysis of the model system concerns the existence, uniqueness and uniform boundedness of solutions in the positive octant. The threshold condition for epidemic and the conditions for persistence are obtained. Moreover, the system is analyzed for local stability, global stability around several equilibria. Hopf-bifurcation with its nature and the stability of the bifurcating limit cycle are studied around the disease free equilibrium point. Numerical simulations are performed to justify the analytical findings. Eco-epidemilogical significance and implications of the concluded results are discussed as well.


  1. [1]  Kot, M. (2001), Elements of Mathematical Ecology, Cambridge University Press: Cambridge.
  2. [2]  Brauer, F. and Chávez, C.C. (2001), Mathematical Models in population Biology and Epidemiology, Springer-Verlag: New York.
  3. [3]  Hethcote, H.W. (2000), The mathematics of infectious diseases, SIAM review, 42(4), 599-653.
  4. [4]  Cash, J.R. and Mazzia, F. (2011), Efficient Global Methods for the Numerical Solution of Nonlinear Systems of Two Point Boundary Value Problems, In: Simos T. (eds) Recent Advances in Computational and Applied Mathematics, Springer, Dordrecht, pp. 23-39.
  5. [5]  Hadeler, K.P. and Freedman, H.I. (1989), Predator-Prey population with parasite infection, J. Math Biol., 27, 609-631.
  6. [6]  Freedman, H.I. (1990), A model of predator-prey dynamics as modified by the action of a parasite, Math. Biosci., 99, 143-155.
  7. [7]  Beltrami, E. and Carrol, T.O. (1994), Modelling the role of viral disease in recurrent phytoplankton blooms, J. Math. Biol., 32, 857-863.
  8. [8]  Chattopadhyay, J. and Arino, O. (1999), A predator-preymodel with disease in the prey, Nonlinear Anal., 36, 747-766.
  9. [9]  Xiao, Y. and Chen, L. (2001),Modelling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171, 59-82.
  10. [10]  Venturino, E. (2002), Epidemics in predator-prey models: disease in the predators, IMA J. Math. Appl. Med. Biol., 19, 185-205.
  11. [11]  Hethcote, H.W., Wang, W., Han, L., and Ma, Z. (2004), A predator-prey model with infected prey, Theor. Popul. Biol., 66, 259-268.
  12. [12]  Anderson, R.M. and May, R.M. (1981), The population dynamics of microparasites and their invertebrates hosts, Proc R Soc London, 291, 451-463.
  13. [13]  Beretta, E. and Kuang, Y. (1998), Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal., 32, 381-408.
  14. [14]  Venturino, E. (1995), Epidemics in prey-predator models: Disease in the prey, Mathematical Population Dynamics: Analysis of Heterogeneity: Theory of Epidemics, 1, 381-393.
  15. [15]  Cosner, C., Angelis, D.L., Ault, J.S., and Olson, D.B. (1999), Effects of spatial grouping on functional response of predators, Theor Popul Biol, 56, 65-75.
  16. [16]  Hwang, T.W. (2003), Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J Math Anal Appl, 281, 395-401.
  17. [17]  Malthus, T.R. (1798), An Essay on the Principle of Population, J. Johnson in St. Paul’s Churchyard: London.
  18. [18]  Verhulst, P.F. (1838), Notice sur la loi que la population suit dans son accroissement, Correspondence Mathématique et Physique Publiée par A. Quételet, 10, 113-121.
  19. [19]  Lotka, A.J. (1956), Elements of Mathematical Biology, Dover, New York, .
  20. [20]  Volterra, V. and D’Ancona, U. (1931), La concorrenza vitale tra le specie dell’ambiente marino, In: Vlle Congr. Int. acquicult et de pêche, Paris, pp.1-14.
  21. [21]  Han, L., Ma, Z., and Hethcote, H.W. (2001), Four predator prey models with infectious diseases, Math. Comput. Modelling, 34, 849-858.
  22. [22]  Hilker, F.M. and Malchow, H. (2006), Strange periodic attractors in a prey-predator system with infected prey, Math. Popul. Stud., 13, 119-134.
  23. [23]  Temple, S.A. (1987), Do predators always capture substandard individuals disproportionately from prey population?, Ecology, 68, 669-674.
  24. [24]  Mech, L.D. (1970), The Wolf, Natural History Press: New York.
  25. [25]  Schaller, G.B. (1972), The Serengeti Lion: A Study of Predator Prey Relations, University of Chicago Press: Chicago.
  26. [26]  Saikh, A. and Gazi, N.H. (2017), Mathematical analysis of a predator–prey eco-epidemiological system under the reproduction of infected prey, Journal of Applied Mathematics and Computing, 58(1-2), 621-646.
  27. [27]  Kulma, K., Low, M., Bensch, S., and Qvarnström, A. (2014), Malaria-infected female collared flycatchers (Ficedula albicollis) do not pay the cost of late breeding, PLoS ONE, 9(1), e85822.
  28. [28]  Birkhoff, G. and Rota, G.C. (1982), Ordinary Differential Equation, Ginn. and Co., Boston.
  29. [29]  Gard, T.C. and Hallam, T.G. (1979), Persistence in Food web-1, Lotaka-Volterra food chains, Bull Math Biol, 41, 877-891.
  30. [30]  Anderson, R.M. and May, R.M. (1978), Regulation and stability of host-parasite population interactions. I. Regulatory processes, J. Anim. Ecol., 47, 219-247.
  31. [31]  Diekmann, O., Heesterbeek, J.A.P., and Metz, J.A.J. (1990), On the definition and the computation of the basic reproduction ratio R0 in models for infectiuos diseases in heterogeneous populations, J. Math. Biol., 28, 365-382.
  32. [32]  May, R.M. (2001), Stability and Complexity in Model Ecosystems, Princeton University Press: New Jersey.
  33. [33]  Murray, J.D. (2002),Mathematical Biology, Springer-Verlag: New York.
  34. [34]  LaSalle, J. and Lefschetz, S. (1961), Stability by Liapunov’s Direct Moethod, Academic Press: New York.
  35. [35]  Liu, W. (1994), Criterion of Hofp Bifurcation without Using Eigenvalues, Journal of Mathematical Analysis and Application, 182, 250-256. doi:10.1006/jmaa.1994.1079.
  36. [36]  Haque, M. and Venturino, E. (2006), The role of transmissible diseases in the Holling-Tanner predator-prey model, Theoretical Population Biology, 70, 273-288.
  37. [37]  Wiggins, S. (2003), Introduction to applied nonlinear dynamical systems and Chaos, 2nd edn., Springer: New York.
  38. [38]  Carr, J. (1981), Application of centre manifold theory, Springer: New York.
  39. [39]  Kar, T.K., Gorai, A., and Jana, S. (2012), Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide, J Theor Biol, 310, 187-198.
  40. [40]  Perko, L. (2000), Differential Equations and Dynamical Systems, Springer-Verlag: Heidelberg.
  41. [41]  Dawes, J.H.P. and Souza,M.O. (2013). A derivation of Holling’s type I, II and III functional responses in predator-prey systems, Journal of theoretical biology, 327, 11-22.