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Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Predator-Prey Model with Intraguild Predation in an Uncertain Environment

Journal of Vibration Testing and System Dynamics 7(4) (2023) 399--417 | DOI:10.5890/JVTSD.2023.12.001

Prabir Panja, Sailen Gayen, Dipak Kumar Jana

Department of Applied Science, Haldia Institute of Technology, Haldia-721657, West Bengal, India

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In this paper, a prey, intermediate predator and top predator interaction model has been developed. Here all parameters of the model have been considered as triangular fuzzy number. Positivity and boundedness of solutions of the proposed model have been investigated. Possible equilibrium points of the model are determined and also local stability of the model around these equilibrium points have been studied. Global stability of the model around the interior equilibrium point is also studied. Conditions for the existence of Hopf bifurcation have been investigated with respect to $\alpha$ (degree of uncertainty). It is found that the uncertain values of the parameters have a great influence in the solution of the proposed model. From the analysis of the model, it is observed that intra-species competition rate of prey as well as intermediate predator can be stabilized the system. It is also observed that the harvesting rate of intermediate predator has the ability to stabilize the system. Some complex behaviour of the system have been seen due to the increase of death rate of top predator species. Finally some numerical simulation results have been presented to verify the analytical findings.


  1. [1]  Lotka, A.J. (1925), Elements of Physical Biology, Williams and Wilkins.
  2. [2]  Volterra, V. (1926), Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Societá Anonima Tipografica Leonardo da Vinci, 2, 31-113.
  3. [3]  Panja, P. and Mondal, S.K. (2015), Stability analysis of coexistence of three species prey-predator model, Nonlinear Dynamics, 81, 373-382.
  4. [4]  Panja, P., Mondal, S.K. and Jana, D.K. (2017), Effects of toxicants on Phytoplankton-Zooplankton-Fish dynamics and harvesting, Chaos, Solitons $\&$ Fractals, 104, 389-399.
  5. [5]  Panja, P., Poria, S., and Mondal, S.K. (2018), Analysis of a harvested tritrophic food chain model in the presence of additional food for top predator, International Journal of Biomathematics, 11(04), p.1850059.
  6. [6]  Mortoja, S.G., Panja, P., and Mondal, S.K. (2018), Dynamics of a predator-prey model with stage-structure on both species and anti-predator behavior, Informatics in Medicine Unlocked, 10, 50-57.
  7. [7]  Mortoja, S.G., Panja, P., and Mondal, S.K. (2019), Dynamics of a predator-prey model with nonlinear incidence rate, Crowley-Martin type functional response and disease in prey population, Ecological Genetics and Genomics, 10, 100035.
  8. [8]  Sen, D., Ghorai, S., and Banerjee, M. (2018), Complex dynamics of a three species prey-predator model with intraguild predation, Ecological Complexity, 34, 9-22.
  9. [9]  Zadeh, L.A. (1965), Fuzzy sets, Information and Control, 8, 338-353.
  10. [10]  Zimmerman, H.J. (1985), Fuzzy Set Theory and its Applications, Kluwer-Nijhcff Publishing, USA.
  11. [11]  Seikkala, S. (1987), On the fuzzy initial value problem, Fuzzy Sets and Systems, 24, 319-330.
  12. [12]  Hullermeier, E. (1997), An approach to modelling and simulation of uncertain dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5, 117-137.
  13. [13]  Barros, L.C. and Pedro, F.S. (2017), Fuzzy differential equations with interactive derivative, Fuzzy Sets and System, 309, 64-80.
  14. [14]  Bassanezi, R.C., Barros, L.C. and Tonelli, A. (2000), Attractors and asymptotic stability for fuzzy dynamical systems, Fuzzy Sets and System, 113, 473-483.
  15. [15]  Bede, B. and Gal, S.G. (2005), Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151, 581-599.
  16. [16]  Bede, B., Rudas, I.J., and Bencsik, A.L. (2007), First order linear fuzzy differential equations under generalized differentiability, Information Science, 177, 1648-1662.
  17. [17]  Tuyako, M.M., Barros, L.C., and Bassanezi, R.C. (2009), Stability of fuzzy dynamic systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17, 69-83.
  18. [18]  Villamizar-Roa, E.J., Angulo-Castillo, V., and Chalco-Cano, Y. (2015), Existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles, Fuzzy Sets and System, 265, 24-38.
  19. [19]  Iwasa, Y., Hakoyama, H., Nakamaru, M., and Nakanishi, J. (2000), Estimate of population extinction risk and its application to ecological risk management, Population Ecology, 42, 73-80.
  20. [20]  Barros, L.C., Bassanezi, R.C., and Tonelli, P.A. (2000), Fuzzy modelling in population dynamics, Ecological Modelling, 128, 27-33.
  21. [21]  Samanta, G.P. and Maiti, A. (2003), Stochastic Gomatam model of interacting species: non-equilibrium fluctuation and stability, Systems Analysis Modelling Simulation, 43, 683-692.
  22. [22]  Pal, D. and Mahapatra, G.S. (2014), A bioeconomic modeling of two-prey and one-predator fishery model with optimal harvesting policy through hybridization approach, Applied Mathematics and Computation, 242, 748-763.
  23. [23]  Pal, D., Mahapatra, G.S. and Samanta, G.P. (2016), Stability and bionomic analysis of fuzzy prey-predator harvesting model in presence of toxicity: a dynamic approach, Bulletin of Mathematical Biology, 78, 1493-1519.
  24. [24]  Panja, P., Mondal, S.K. and Chattopadhyay, J. (2017), Dynamical study in fuzzy threshold dynamics of a cholera epidemic model, Fuzzy Information and Engineering, 9, 381-401.
  25. [25]  Mahata, A., Roy, B., Mondal, S.P., and Alam, S. (2017), Application of ordinary differential equation in glucose-insulin regulatory system modeling in fuzzy environment, Ecological Genetics and Genomics, 3-5, 60-66.
  26. [26]  Panja, P. (2018), Fuzzy parameter based mathematical model on forest biomass, Biophysical Reviews and Letters, 13, 179-193.
  27. [27]  Birkhoff, G. and Rota, G.C. (1982), Ordinray differential equations, Ginn Boston.
  28. [28]  Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981), Theory and Application of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press.