1. MMC
  2. All Volumes and Issues
  3. Volume 10, Number 3, 2023
  4. Stability analysis and Hopf bifurcation of a delayed prey–predator model with Hattaf–Yousfi functional response and Allee effect

Stability analysis and Hopf bifurcation of a delayed prey–predator model with Hattaf–Yousfi functional response and Allee effect

MMC.
2023;
Volume 10, Number 3
: pp. 668–673
https://doi.org/10.23939/mmc2023.03.668
Received: February 16, 2023
Accepted: July 06, 2023

Mathematical Modeling and Computing, Vol. 10, No. 3, pp. 668–673 (2023)

Authors:
  1. S. Bouziane
  2. E. M. Lotfi
  3. K. Hattaf
  4. N. Yousfi
1
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca
2
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca
3
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'sik, Hassan II University of Casablanca, Casablanca, Morocco; Centre Régional des Métiers de l'Education et de la Formation (CRMEF), Casablanca, Morocco
4
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca

The Allee effect is an important phenomena in the context of ecology characterized by a correlation between population density and the mean individual fitness of a population.  In this work, we examine the influences of Allee effect on the dynamics of a delayed prey–predator model with Hattaf–Yousfi functional response.  We first prove that the proposed model with Allee effect is mathematically and ecologically well-posed.  Moreover, we study the stability of equilibriums and discuss the local existence of Hopf bifurcation.

ecology
Allee effect
Hattaf–Yousfi functional response
stability
Hopf bifurcation
  1. Lotka A. J.  Elements of physical biology.  Williams and Wilkins (1925).
  2. Volterra V.  Fluctuations in the abundance of a species considered mathematically.  Nature.  118, 558–560 (1926).
  3. Bouziane S., Lotfi E., Hattaf K., Yousfi N.  Dynamics of a delayed prey–predator model with Hattaf–Yousfi functional response.  Communications in Mathematical Biology and Neuroscience.  2022, 104 (2022).
  4. Louartassi Y., Alla A., Hattaf K.,  Nabil A.  Dynamics of a predator–prey model with harvesting and reserve area for prey in the presence of competition and toxicity.  Journal of Applied Mathematics and Computing.  59, 305–321 (2019).
  5. Allee W. C.  Animal aggregations: A study in general sociology.  Chicago, The University of Chicago Press (1931).
  6. Pal P. J., Saha T., Sen M., Banerjee M.  A delayed predator–prey model with strong Allee effect in prey population growth.  Nonlinear Dynamics.  68, 23–42 (2012).
  7. Ye Y., Liu H., Wei Y., Zhang K., Ma M., Ye J.  Dynamic study of a predator-prey model with Allee effect and Holling type-I functional response.  Advances in Difference Equations.  2019, 369  (2019).
  8. Holling C. S.  The components of predation as revealed by a study of small mammal predation of the European pine sawfly.  The Canadian Entomologist.  91 (5), 293–320 (1959).
  9. Garain K., Mandal P. S.  Bubbling and hydra effect in a population system with Allee effect.  Ecological Complexity.  47, 100939 (2021).
  10. Hattaf K., Yousfi N.  A class of delayed viral infection models with general incidence rate and adaptive immune response.  International Journal of Dynamics and Control.  4, 254–265 (2016).
  11. Hattaf K.  A new generalized definition of fractional derivative with non-singular kernel.  Computation.  8 (2), 49 (2020).
  12. Hattaf K.  On the stability and numerical scheme of fractional differential equations with application to biology.  Computation.  10 (6), 97 (2022).
  13. Berec L., Angulo E., Courchamp F.  Multiple Allee effects and population management.  Trends in Ecology and Evolution.  22 (4), 185–191 (2007).
  14. Angulo E., Roemer G. W., Berec L., Gascoigne J., Courchamp F.  Double Allee effects and extinction in the island fox.  Conservation Biology.  21 (4), 1082–1091 (2007).