Dynamical complexities in a predator-prey system involving teams of two prey and one predator

  • P. Mishra
  • S. N. RawEmail author
Original Research


Empirical studies have shown that animals often focus on short-term benefits under conditions of predation risk, which increases the likelihood that they will co-operate with others. With this motivation, we propose and analyze a predator and two prey model with the assumption that during predation the members of both prey make a team to reduce risk of predation. We incorporate Monod–Haldane and Holling type II functional response to model the interaction with predator. Firstly, we discuss conditions which ensure that model system has a unique positive solution. We investigate stability and Hopf bifurcation conditions to explore dynamics of system around positive equilibrium. We also derive Kolmogorov conditions for the parametric restriction of the system. Secondly, we present numerical solution which substantiate our analytical results. In numerical simulation, we observe period doubling and period halving cascade which explore the dynamical complexity of predator-prey system. Finally, we conclude that partial co-operation and low defense may lead to extinction of prey species.


Predator-prey Team of prey Stability analysis Defense mechanism Bifurcation analysis 

Mathematics Subject Classification

37N25 92D25 34D20 34D45 



The work done in this paper is supported by a grant (File No. ECR/2017/000141) under Early Career Research (ECR) Award, Science and Engineering Research Board (SERB), Department of Science and Technology, New Delhi, to the corresponding author (SNR).


  1. 1.
    Agrawal, T., Saleem, M.: Complex dynamics in a ratio-dependent two-predator one-prey model. Comput. Appl. Math. 34(1), 265–274 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, J.F.: A mathematical model for the continuous culture of micro-organisms utilizing inhibitory substrate. Biotecnhnol. Bioeng. 10, 700–723 (1968)Google Scholar
  3. 3.
    Ali, N., Jafar, M.: Global dynamics of a modified Leslie–Gower predator-prey model with Crowley–Martin functional responses. J. Appl. Math. Comput. 13, 271–293 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agrawal, R., Jana, D., Upadhayay, R., Rao, V.S.H.: Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos. J. Appl. Math. Comput. 55, 513–547 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)CrossRefGoogle Scholar
  6. 6.
    Cramer, N., May, R.: Interspecific competition, predation and species diversity: a comment. J. Theor. Biol. 34, 280–292 (1972)CrossRefGoogle Scholar
  7. 7.
    De Angelis, D.L., Goldstein, R.A., ONeill, R.V.: A model for tropic interaction. Ecology 56, 881–892 (1975)CrossRefGoogle Scholar
  8. 8.
    Dubey, B., Upadhyay, R.K.: Persistence and extinction of one-prey and two- predators system. Nonlinear Anal. 9, 307–329 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dugatkin, L.A.: Co-operation Among Animals: A Evolutionary Prospective. Oxford University Press, New York (1997)Google Scholar
  10. 10.
    Freedman, H.I.: Hopf bifurcation in three species food chain models with group defense. Math. Biosci. 111, 73–87 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Freedman, H.I., Hongshun, Q.: Interaction leading to persistence in predator-prey systems with group defense. Bull. Math. Biol. 50, 517–530 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Freedman, H.I.: Deterministic Mathematical Models in Population Ecology. M. Dekker, New York (1980)zbMATHGoogle Scholar
  13. 13.
    Gause, G.F.: Struggle for Existence. Williams and Wilkins, Baltimore (1934)CrossRefGoogle Scholar
  14. 14.
    Hasting, A., Powell, T.: Chaos in a three-species food chain. Ecology 72, 896–903 (1991)CrossRefGoogle Scholar
  15. 15.
    Holmes, J.C., Bethel, W.M.: Modification of intermediate host behavior parasites. Zool. J. Linn. Soc. 51, 123–49 (1972)Google Scholar
  16. 16.
    Holling, C.: The functional response of predators to prey density and its role in mimicry and population regulation. Memo. Entom. Soc. Can. 97, 5–60 (1965)CrossRefGoogle Scholar
  17. 17.
    Hwang, Z.W.: Global analysis of the predator-prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 281, 395–401 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Kuang, Y., Freedman, H.I.: Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math. Biol. 88, 67–84 (1988)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Klebanoff, A., Hastings, A.: Chaos in one-predator, two-prey models: general results from bifurcation theory. Math. Biol. 122, 221–233 (1994)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kolmogorov, A.N.: Sulla Teoria di Voltera della Lotta per IEsisttenza. Giorn. Instituto Ital. Attuari. 7, 74–80 (1936)Google Scholar
  22. 22.
    Lotka, A.: Elements of Mathematical Biology. Dover Publications, New York (1956)zbMATHGoogle Scholar
  23. 23.
    Martin, M.M., Mitani, J.C.: Conflict and co-operation in wild life chimpanzees. Adv. Study Behav. 35, 275–331 (2005)CrossRefGoogle Scholar
  24. 24.
    Mischaikow, K., Wolkowicz, G.S.: A predator-prey system involving group defense: a connection matrix approach. Nonlinear. Anal. 14, 955–969 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    May, R.M.: Limit cycles in predator-prey communities. Science 177, 900–902 (1972)CrossRefGoogle Scholar
  26. 26.
    Murray, J.D.: Mathematical Biology I: An Introduction. Springer, New Delhi (2002)zbMATHGoogle Scholar
  27. 27.
    Pal, R., Basu, D., Banerjee, M.: Modelling of phytoplankton allelopathy with Monod–Haldane type functional response–a mathematical study. Biosystems 95, 243–253 (2009)CrossRefGoogle Scholar
  28. 28.
    Pasquet, A., Krafft, B.: Cooperation and prey capture efficiency in a social spider, Anelosimus eximius (Araneae, Theridiidae). Ethology 90, 121–133 (1992)CrossRefGoogle Scholar
  29. 29.
    Raw, S.N., Mishra, P., Kumar, R., Thakur, S.: Complex behavior of prey-predator system exhibiting group defense: a mathematical modeling study. Chaos. Soli. Frac. 100, 74–90 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sokol, J., Howell, J.A.: Kinetics of phenol oxidation by washed cell. Biotecnhnol. Bioeng. 23, 203–249 (1980)Google Scholar
  31. 31.
    Shen, C.: Permanence and global attractivity of the food-chain system with Holling IV type functional response. Appl. Math. Comput. 194(1), 179–185 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Strogatz, S.H.: Non-linear Dyanmics and Chaos with Applications to Physics, Bilogy, Chemistry, and Engineering. Westview Press, Colorado (2001)Google Scholar
  33. 33.
    Tripathi, J.P., Abbas, S., Thankur, M.: Local and global stability analysis of a two prey one predator model with help. Commun. Nonlinear Sci. Numer. Simul. 19, 3284–3297 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tener, J.S.: Muskoxen. Queens Printer, Ottawa (1965)Google Scholar
  35. 35.
    Upadhyay, R.K., Raw, S.N.: Complex dynamics of a three species food-chain model with Holling type IV functional response. Nonlinear Anal. Model Control 16, 353–374 (2011)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Upadhyay, R.K., Naji, R.K., Raw, S.N., Dubey, B.: The role of top predator interference on the dynamics of a food chain model. Commun. Nonlinear. Sci. Numer. Simul. 18, 757–768 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)CrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, W., Wang, H., Li, Z.: The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy. Chaos. Soli. Frac. 32, 1772–1785 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical System and Chaos, 2nd edn. Springer, New York (2003)zbMATHGoogle Scholar
  40. 40.
    Wolkowicz, G.S.K.: Bifurcation analysis of a predator-prey system involving group defense. SIAM. J. Appl. Math. 48, 592–606 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhao, M., Songjuan, L.V.: Chaos in a three-species food chain model with a Beddington–DeAngelis functional response. Chaos. Soli. Frac. 40, 2305–2316 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhang, S.W., Tan, D.J., Chen, L.S.: Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations. Chaos. Soli. Frac. 27, 980–90 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology, RaipurRaipurIndia

Personalised recommendations