Ricerche di Matematica

, Volume 68, Issue 2, pp 435–452 | Cite as

Asymptotic stability of one prey and two predators model with two functional responses

  • Harsha KharbandaEmail author
  • Sachin Kumar


We formulate a mathematical model to study the complex dynamical behavior of a three dimensional model consisting of one prey and two predators involving Beddington–DeAngelis and Crowley–Martin functional responses. The existence and stability conditions of the equilibrium points are analyzed. The global asymptotic stability of the interior equilibrium point, if exists, is proved by considering Lyapunov function. Several numerical simulations are performed to illustrate the theoretical analysis. The multiple states of stability are observed in one example whereas another example exhibits the global stability of interior equilibrium point.


Prey–predator model Functional response Equilibrium points Stability Lyapunov function 

Mathematics Subject Classification

Primary 92D40 Secondary 34D20 49K15 70K05 



The authors would like to thank the anonymous referees for their extensive comments on the revision of the manuscript which really improved the quality of the paper. This research work is supported by University Grant Commission (UGC), Government of India to the author, Harsha Kharbanda (Sr. No. 2121440663).


  1. 1.
    Alebraheem, J., Hasan, Y.A.: Dynamics of a two predator-one prey system. Comput. Appl. Math. 33(3), 767–780 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)CrossRefGoogle Scholar
  3. 3.
    Cantrell, R.S., Cosner, C.: On the dynamics of predator–prey models with the Beddington–DeAngelis functional response. J. Math. Anal. Appl. 257(1), 206–222 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Capone, F., et al.: On the dynamics of an intraguild predator–prey model. Math. Comput. Simul. 149, 17–31 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Crowley, P.H., Martin, E.K.: Functional response and interference within and between year classes of a dragonfly population. J. North Am. Benthol. Soc. 8, 211–221 (1989)CrossRefGoogle Scholar
  6. 6.
    DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for trophic interaction. Ecology 56, 881–892 (1975)CrossRefGoogle Scholar
  7. 7.
    Du, Z., Feng, Z.: Periodic solutions of a neutral impulsive predator–prey model with Beddington–DeAngelis functional response with delays. J. Comput. Appl. Math. 258, 87–98 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dubey, B., Upadhyay, R.K.: Persistence and extinction of one-prey and two-predators system. Nonlinear Anal. Model. Control 9(4), 307–329 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gakkhar, S., Singh, B., Naji, R.K.: Dynamical behavior of two predators competing over a single prey. BioSystems 90, 808–817 (2007)CrossRefGoogle Scholar
  10. 10.
    Haque, M., Venturino, E.: The role of transmissible diseases in Holling–Tanner predator–prey model. Theor. Popul. Biol. 70(3), 273–288 (2006)CrossRefGoogle Scholar
  11. 11.
    Haque, M.: A detailed study of the Beddington–DeAngelis predator–prey model. Math. Biosci. 234(1), 1–16 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Ent. 91, 385–395 (1959)CrossRefGoogle Scholar
  13. 13.
    Huang, J., et al.: Complex dynamics in predator–prey models with nonmonotonic functional response and harvesting. Math. Model. Nat. Phenom. 8(5), 95–118 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jazar, N.A.M.: Global dynamics of a modified Leslie–Gower predator–prey model with Crowley–Martin functional responses. J. Appl. Math. Comput. 43, 271–293 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Koch, A.L.: Competitive coexistence of two predators utilizing the same prey under constant environmental conditions. J. Theor. Biol. 44, 387–395 (1974)CrossRefGoogle Scholar
  16. 16.
    Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  17. 17.
    Lee, J., Baek, H.: Dynamics of a Beddington–DeAngelis-type predator-prey system with constant rate harvesting. Electron. J. Qual. Theory Differ. Equ. 2017(1), 1–20 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lotka, A.J.: Elements of Mathematical Biology (formerly published under the title Elements of Physical Biology). Dover Publications, Inc., New York (1958)Google Scholar
  19. 19.
    Malthus, T.R.: An Essay on the Principle of Population. J. Johnson in St. Paul’s Churchyard, London (1798)Google Scholar
  20. 20.
    Meng, X.-Y., et al.: Stability in a predator–prey model with Crowley–Martin function and stage structure for prey. Appl. Math. Comput. 232, 810–819 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Misra, O.P., Sinha, P., Singh, C.: Dynamics of one-prey two-predator system with square root functional response and time lag. Int. J. Biomath. 8(3), 1550029 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Naji, R.K., Balasim, A.T.: Dynamical behavior of a three species food chain model with Beddington–DeAngelis functional response. Chaos Solitons Fractals 32(5), 1853–1866 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol. 7, 3rd edn. Springer, New York (2001)CrossRefGoogle Scholar
  24. 24.
    Petraitis, P.: Multiple Stable States in Natural Ecosystems. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar
  25. 25.
    Rionero, S.: Stability of ternary reaction-diffusion dynamical systems. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22(3), 245–268 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Shi, X., Zhou, X., Song, X.: Analysis of a stage-structured predator–prey model with Crowley–Martin function. J. Appl. Math. Comput. 36(1–2), 459–472 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tian, X., Xu, R.: Global dynamics of a predator–prey system with Holling type II functional response. Nonlinear Anal. Model. Control 16(2), 242–253 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tripathi, J.P., Tyagi, S., Abbas, S.: Global analysis of a delayed density dependent predator-prey model with Crowley–Martin functional response. Commun. Nonlinear Sci. Numer. Simul. 30(1–3), 45–69 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Upadhyay, R.K., Naji, R.K.: Dynamics of a three species food chain model with Crowley–Martin type functional response. Chaos Solitons Fractals 42(3), 1337–1346 (2009)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)CrossRefGoogle Scholar
  31. 31.
    Zhang, T., et al.: Geometric analysis of a pest management model with Holling’s type III functional response and nonlinear state feedback control. Nonlinear Dyn. 84(3), 1529–1539 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of DelhiNew DelhiIndia

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