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


Dynamical Complexity in a Tritrophic Food Chain Model with Prey Harvesting

Discontinuity, Nonlinearity, and Complexity 10(4) (2021) 705--722 | DOI:10.5890/DNC.2021.12.010

Krishnendu Sarkar$^1$, Nijamuddin Ali$^2$, Lakshmi Narayan Guin$^1$

$^1$ Department of Mathematics, Visva-Bharati, Santiniketan, West Bengal, India, 731235

$^2$ Department of Mathematics, Vivekananda Mahavidyalaya, Purba Bardhaman, West Bengal, India, 713103

Download Full Text PDF



The present investigation deals with a tritrophic food web model with Holling-Tanner type II functional response to clarify the dynamical complexity of the eco-systems in the natural environment. The objective of this study is to explore the harvesting mechanism scenario in a three-dimensional interacting species system such as one prey and two specialist predators. Attention has been given to demonstrate the system characteristics near the biologically feasible equilibria. Specifically, stability, Hopf-Andronov bifurcation for the respective system parameters and dissipativeness has been performed in order to scrutinize the system behaviour. Lyapunov exponents are worked out numerically and an unstable scenario for significant parameters of the model system has been executed to characterize the complex dynamics. In addition to, we put forward a detailed numerical simulation to justify the chaotic dynamics of the present system. We conclude that chaotic dynamics can be executed by the prey harvesting parameters.


The present form of the paper owes much to the useful suggestions of the referees, whose careful study we are pleased to acknowledge. The first and third authors gratefully acknowledge the financial support in part from Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.510 / 3 / DRS-III / 2015 (SAP-I)).


  1. [1]  Atangana, A. and Qureshi, S. (2019), Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Soliton Fract, 123(5), 320-337.
  2. [2]  Ajraldi, V., Pittavino, M., and Venturino, E. (2011), Modeling herd behavior in population systems, Nonlinear Analysis: Real World Applications, 12(4), 2319-2338.
  3. [3]  Cramer, N.F. and May, R.M. (1972), Interspecific competition, predation and species diversity: a comment, J. Theor. Biol., 34(2), 289-293.
  4. [4]  Djilali, S. (2019), Impact of prey herd shape on the predator-prey interaction, Chaos Soliton Fract, 120, 139-148.
  5. [5]  Feng, W. (1993), Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179(2), 592-609.
  6. [6]  Fujii, K. (1977), Complexity-stability relationship of two-prey- one-predator species system model: local and global stability, J. Theor. Biol., 69(4), 613-623.
  7. [7]  Ghanbari, B. and Djilali, S. (2020), Modeling and analysis of a predator-prey model with disease in the prey, Math. Method Appl. Sci., 43(4), 1736-1752.
  8. [8]  Hale, J.K. (1976), Theory of Functional Differential Equations, Springer-Verlag, New York.
  9. [9]  Han, R., Guin, L.N., and Dai, B. (2020), Cross-diffusion-driven pattern formation and selection in a modified Leslie-Gower predator-prey model with fear effect, J. Biol. Syst., 28(1), 1-38.
  10. [10]  Hastings, A. and Powell, T. (1991), Chaos in three-species food chain, Ecology, 72, 896-903.
  11. [11]  Holt, R.D. and Polis, G.A. (1997), A theoretical framework for intraguild predation, Am. Nat., 149, 745-764.
  12. [12]  Hutson, V. and Vickers, G.T. (1983), A criterion for permanent coexistence of species, with an application to a two-prey one predator system, Math. Biosci., 63(2), 253-269.
  13. [13]  Kar, T.K. and Ghorai, A. (2011), Dynamic behaviour of a delayed predator-prey model with harvesting, Appl. Math. Comput., 217(22), 9085-9104.
  14. [14]  Kar, T.K., Chattopadhyay, S.K. and Pati, C.K. (2009), A bio-economic model of two-prey one-predator system, J. Appl. Math. Inform., 27(5-6), 1411-1427.
  15. [15]  Klebanoff, A. and Hastings, A. (1994), Chaos in one-predator, two prey models: general results from bifurcation theory, Math. Biosci., 122(2), 221-233.
  16. [16]  Koster, F.W., M\"{o}llmann, C., Neuenfeldt, S.P., John, M.A., and Voss, R. (2001), Developing Baltic cod recruitment models. I. Resolving spatial and temporal dynamics of spawning stock and recruitment for cod, herring, and sprat, Can. J. Fish. Aquat. Sci., 58, 1516-1533.
  17. [17]  Kot, M. (2001), Elements of Mathematical Ecology, {Cambridge University Press.}
  18. [18]  Liu, Z. and Yuan, R. (2006), Stability and bifurcation in a harvested one-predator-two-prey model with delays, Chaos Soliton Fract, 27(5), 1395-1407.
  19. [19]  Lotka, A.J. (1956), Elements of Mathematical Biology, Dover Publications, New York.
  20. [20]  Lv, S. and Zhao, M. (2008), The dynamic complexity of a three species food chain model, Chaos Soliton Fract, 37(5), 1469-1480.
  21. [21]  May, R.M. (2001), Stability and complexity in model ecosystems, Princeton University Press.
  22. [22]  Nomdedeu, M.M., Willen, C., Schieffer, A. and Arndt, H. (2012), Temperature-dependent ranges of coexistence in a model of a two-prey-one-predator microbial food web, Marine Biol., 159(11), 2423-2430.
  23. [23]  Paine, R.T. (1966), The Pisaster-Tegula interaction: prey patches, predator food preference, and intertidal community structure, Am. Nat., 100, 65-75.
  24. [24]  Pandey, P., Pal, N., Samanta, S. and Chattopadhyay, J. (2018), Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Fear, Int. J. Bifurcation and Chaos, 28(1), 1850009.
  25. [25]  Parrish, J.D. and Saila, S.B. (1970), Interspecific competition, predation and species diversity, J. Theor. Biol., 27(2), 207-220.
  26. [26]  Polis, G.A. and Holt, R.D. (1992), Intraguild predation: The dynamics of complex trophic interactions, Trends Ecol. Evol., 7, 151-154.
  27. [27]  Qureshi, S., Atangana, A., and Shaikh, A.A. (2019), Strange chaotic attractors under fractal-fractional operators using newly proposed numerical methods, Eur. Phys. J. Plus., 134(10), 523.
  28. [28]  Rosenstein, M., Collins, J.J., and De Luca, C.J. (1993), A practical method for calculating largest Lyapunov exponents from small data sets, Physica D: Nonlinear Phenomena, 65(1-2), 117-134.
  29. [29]  Sayekti, I.M., Malik, M., and Aldila, D. (2017), One-prey two-predator model with prey harvesting in a food chain interaction, AIP Conference Proceedings, 1862(1), 030124.
  30. [30]  Sooknanan, J., Bhatt, B., and Comissiong, D.M.G. (2012), Criminals treated as predators to be harvested: a two prey one predator model with group defense, prey migration and switching, J. Math. Res., 4(4), 92-106.
  31. [31]  Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York.
  32. [32]  Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A. (1985), Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16, 285-317.
  33. [33]  Xiao, Y. and Chen, L. (2001), Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171, 59-82.
  34. [34]  Younghae, D., Hunki, B., Yongdo, L., and Dongkyu, L. (2011), A three-Species Food Chain system with Two Types of Functional Responses, Abstr. Appl. Anal., 2011, 934569.