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

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

MMC.
2023;
Volume 10, Number 1
: pp. 66–79
https://doi.org/10.23939/mmc2023.01.066
Received: April 14, 2022
Revised: September 24, 2022
Accepted: October 31, 2022

Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 66–79 (2023)

Authors:
  1. M. Hafdane
  2. I. Agmour
  3. Y. El Foutayeni
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.

 

predator–prey
stability analysis
Hopf bifurcation
discrete delay
fishing effort
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