Study of Hopf bifurcation of delayed tritrophic system: dinoflagellates, mussels, and crabs

2023;
: pp. 66–79
https://doi.org/10.23939/mmc2023.01.066
Received: April 14, 2022
Revised: September 24, 2022
Accepted: October 31, 2022
1
Analysis, Modeling and Simulation Laboratory, Hassan II University, Morocco
2
Analysis, Modeling and Simulation Laboratory, Hassan II University, Casablanca, Morocco
3
Analysis, Modeling and Simulation Laboratory, Hassan II University, Morocco; Unit for Mathematical and Computer Modeling of Complex Systems, IRD, France

In this paper, we have a discrete delayed dynamic system of three marine species: prey, predator, and superpredator.  In addition to the effect of prey toxicity, we consider the negative fishing effect of these species.  The study of this model consists of the search for equilibria with eigenvalue analysis, the existence of Hopf bifurcations at interior equilibria, and the determination of direction and stability analysis of Hopf bifurcation using the theory of normal form and center manifold.  Some examples are given with numerical simulations to illustrate the results in different cases of delay.

 

  1. El Foutayeni Y., Bentounsi M., Agmour I., Achtaich N.  Bioeconomic model of zooplankton–phytoplankton in the central area of Morocco.  Modeling Earth Systems and Environment. 6, 461–469 (2020).
  2. El Foutayeni Y., Khaladi M., Zegzouti A.  Profit maximization of fishermen exploiting two fish species in competition.  Amer. J. Comput. Appl. Math. to appear (2023).
  3. El Foutayeni Y., Khaladi M.  A generalized bio-economic model for competing multiple-fish populations where prices depend on harvest.  Journal of Advanced Modeling and Optimization.  14 (3), 531–542 (2012).
  4. El Foutayeni Y., Khaladi M.  A bio-economic model of fishery where prices depend harvest.  Journal of Advanced Modeling and Optimization.  14 (3), 543–555 (2012).
  5. Bentounsi M., Agmour I., Achtaich N., El Foutayeni Y.  The impact of price on the profits of fishermen exploiting tritrophic prey–predator fish populations.  International Journal of Differential Equations.  13, 2381483 (2018).
  6. Bentounsi M., Agmour I., Achtaich N., El Foutayeni Y.  Intrinsic growth rates influence on the net economic rents of fishermen.  International Journal of Dynamical Systems and Differential Equations.  9 (4), 362–379 (2019).
  7. Agmour I., Achtaich N., El Foutayeni Y.  Carrying capacity influence on the incomes of seiners exploiting marine species in the Atlantic coast of Morocco.  Mathematical Biosciences. 305, 10–17 (2018).
  8. Agmour I., Achtaich N., El Foutayeni Y.  Stability analysis of a competing fish populations model with the presence of a predator.  International Journal of Nonlinear Science.  26 (2), 108–121 (2018).
  9. Bentounsi M., Agmour I., Achtaich N., El Foutayeni Y.  The Hopf bifurcation and stability of delayed predator–prey system.  Computational and Applied Mathematics.  37 (5), 5702–5714 (2018).
  10. Song Y., Wei J.  Bifurcation analysis for Chen’s System with delayed feedback and its application to Control of chaos.  Chaos, Solitons & Fractals.  22 (1), 75–91 (2004).
  11. Ruan S., Wei J.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays.  Dynamics of Continuous, Discrete & Impulsive Systems A.  10 (6), 863–874 (2003).
  12. Hale J. K.  Theory of Functional Differential Equations. Springer, New York (1977).
  13. Hassard B. D., Kazarinoff N. D., Wan Y.-H.  Theory and applications of hopf bifurcation. Cambridge University Press, Cambridge (1981).
Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 66–79 (2023)