• Received : 2018.04.24
  • Accepted : 2018.09.07
  • Published : 2018.09.25


The paper explores a tri-trophic food chain model with density dependent mortality of intermediate predator. To analyze this aspect, we have worked out the local stability of different equilibrium points. We have also derived the conditions for global stability of interior equilibrium point and conditions for persistence of model system. To observe the global behaviour of the system, we performed extensive numerical simulations. Our simulation results reveal that chaotic dynamics is produced for increasing value of half-saturation constant. We have also observed trajectory motions around different equilibrium points. It is noticed that chaotic dynamics has been controlled by increasing value of density dependent mortality parameter. So, we conclude that the density dependent mortality parameter can be used to control chaotic dynamics. We also applied basic tools of nonlinear dynamics such as Poincare section and Lyapunov exponent to investigate chaotic behaviour of the system.


  1. Anderson R. M. & May R. M. (1980). Infectious diseases and population cycles of forest insects. Science 210, 658-61
  2. Anderson, R.M. & May, R.M., Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, New York,(1991).
  3. Aziz-Alaoui MA. Study of a Leslieower-type tritrophic population model. Chaos Solitons Fractals 2002;14:1275-93.
  4. Butler, G. J., Freedman, H., Waltman, P., Uniformly persistent systems. in: Proc. Am. Math. Soc., 96, 425-429, (1986).
  5. M. Bandyopadhyay, S. Chatterjee, S. Chakraborty, J. Chattopadhyay; Density Dependent Predator Death Prevalence Chaos In A Tri-Trophic Food Chain Model. Nonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 3, 305-324
  6. Bairagi,N., Chaudhuri,S., Chattopadhyay,J.,2009. Harvesting as a disease control measure in an ecoepidemiological system - A theoretical study. Mathematical Biosciences 217 (2009) 134-144
  7. Chattopadhyay, J.,Sarkar,R.R.,2003. Chaos to order: Preliminary experiments with a population dynamics models of three trophic levels. Ecological Modelling 163:45-50
  8. Dwyer, G., Dushoff, Yess, S.H., Generalist predators, specialist pathogens and insect outbreaks. Nature, 43, 341-345, (2004).
  9. Diekmann, O.,Heesterbeek, J.A.P., Metz, J.A.J., On the definition and the computation of the basic reproductive ratio $R_0$ in models for infectious diseases in heterogeneous populations.J.Math.Biol., 28,365-382, (1990).
  10. Younghae Do, Hunki Baek, Yongdo Lim,Dongkyu Lim,2011. A Three-Species Food Chain System with Two Types of Functional Responses. Abstract and Applied Analysis Volume 2011 (2011), Article ID 934569
  11. Eisenberg, J.N., Maszle, D.R., 1995. The structural stability of a three-species food chain model. J. Theo. Biol. 176, 501-510.
  12. Gilpin, M. E. (1979). Spiral chaos in a predator-prey model. Am. Natural. 113, 306-308. doi: 10.1086/283389
  13. Greenhalgh,D.,Khan,Q.J.A.,Pettigrew,J.S.,2016. An eco-epidemiological predatorlrey model where predators distinguish between susceptible and infected prey. Mathematical methods in the applied sciences.Vol 40 ,Issue 1
  14. Hastings, A., Powell, T., 1991. Chaos in three-species food chain. Ecology 72 (3), 896-903.
  15. Hadeler, K.P., Freedman, H.I., 1989. Predator-prey populations with parasitic infection. J. Math. Biol. 27, 609-631.
  16. Hassell, M., J. Lawton and R. M. May, 1976. Pattern of dynamical behavior in single-species populations. Journal of Animal Ecology 45: 471-486
  17. Hallegraeff, G.M., A review of harmful algae blooms and the apparent global increase. Phycologia, 32, 79-99, (1993).
  18. Han L, Ma Z, Hethcote HW, Four predator prey models with infectious diseases, Math Comp Model 34:849-858, 2001.
  19. Hilker,F.M.,Malchow,H.,2006. Strange periodic attractors in a prey-predator system with infected prey. Math.Popul.Stud.13,119-134.
  20. Kevin McCann, Alan Hastings 1997. Re evaluating the omnivorytability relationship in food webs. Proceedings of the Royal Society B Biological Sciences
  21. Kooi, B. W., L. D. J. Kuijper, M. P. Boer and S. A. L. M. Kooijman (2002a). Numerical bifurcation analysis of a tri-trophic food web with omnivory. Mathematical Biosciences 177: 201-228.
  22. Kumar, R., Freedman, H., A mathematical model of facultative mutualism with populations interacting in a food chain. Math. Biosci., 97, 235-261, (1989).
  23. Lonngren, K.E., Bai, E.W., Ucar, A., 2004. Dynamics and synchronization of the Hastingsowell model of the food chain. Chaos Solitons Fract. 20, 387-393
  24. Lotka, A. (1925), n the True Rate of Natural Increase? Journal of the American Statistical Association, 20(151):305-339.
  25. Maionchi, D.O., Reis, S.F.d., Aguiar, M.A.M.d., 2005. Chaos and pattern formation in a spatial tritrophic food chain. Ecol. Model., doi:10.1016/j.ecol.model.2005.04.028.
  26. MAY, R. M. 1973. Itability and Complexity in Model Ecosystems,?Princeton Univ. Press, Princeton, N. J.
  27. May R.M.,Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science. 1974 Nov 15;186(4164):645-7
  28. May,M.R., Leonard J.W. 1975. Nonlinear Aspects of Competition Between Three Species. SIAM Journal on Applied Mathematics,Vol 29,Issue-2 243-253
  29. Olsen, L.F., Truty G.L. and Schaffer W.M., Oscillations and chaos in epidemics: a non-linear dynamic study of six childhood diseases in Copenhagen, Denmark. Theoretical Population Biology, 33, 344-370, (1988).
  30. Thomas Powell,Peter J Richerson, 1985. Temporal Variation, Spatial Heterogeneity, and Competition for Resources in Plankton Systems: A Theoretical Model. The American Naturalist 125(3) ?
  31. Rai, V., Sreenivasan, R., 1993. Period-doubling bifurcations leading to chaos in a model food chain. Ecol. Model. 69 (1-2), 63-77.
  32. Rosenzweig, M.L., Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science, 171, 385-387,(1971).
  33. Ruxton, G.D., 1994. Low levels of immigration between chaotic populations can reduce system extinctions by inducing asynchronous cycles Proc. R. Soc. Lond. B 256, 189-193.
  34. Ruxton, G.D., 1996. Chaos in a three-species food chain with a l ower bound on the bottom population. Ecology 77 (1), 317-319.
  35. Mada Sanjaya Waryano Sunaryo,Zabidin Salleh,Mustafa Mamat,2013.Mathematical model of three species food chain with Holling Type-III functional response. International Journal of Pure and Applied Mathematics Volume 89 No. 5 2013, 647-657
  36. Schaffer, W. M., and M. Kot. 1986a. Chaos in ecological systems: the coals that Newcastle forgot. Trends in Ecology and Evolution 1:58-63.
  37. Schaffer, W. M., and M. Kot. 1986b. Differential systems in ecology and epidemiology. Pages 158-178 in A. V. Holden, editor. Chaos: an introduction. University of Manchester Press, Manchester, England.
  38. Schaffer, W. M., and M. Kot. 1985a. Nearly one dimensional dynamics in an epidemic. Journal of Theoretical Biology 112:403-427.
  39. Turchin,P., Complex Population Dynamics .A Theoretical / empirical Synthesis. Princeton University Press, Princeton, NJ, (2003).
  40. Upadhyay,R.K.,Raw, S.N.,2011. Complex dynamics of a three species food-chain model with Holling type IV functional response. Nonlinear Analysis: Modelling and Control, 2011, Vol. 16, No. 3, 353-374
  41. Vandermeer, J. 2006. Omnivory and the stability of food webs. Journal of Theoretical Biology 238: 497-504.
  42. Varriale,M.C., Gomes, A.A. 1998. A Study of a three species food chain, Ecol. Model.110, 119-133
  43. Xu, C., Li, Z., 2002. Influence of intraspecific density dependence on a three-species food chain with and without external stochastic disturbances. Ecol. Model. 155, 71-83
  44. Zhang P, Sun J, Chen J, Wei J, Zhao W, Liu Q, Sun H.2013. Effect of feeding selectivity on the transfer of methylmercury through experimental marine food chains. Mar Environ Res 89:39-44