Basins of attraction in a modified ratio-dependent predator-prey model with prey refugee

Khairul Saleh*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this paper, we analyze a modified ratio-dependent predator-prey model with a strong Allee effect and linear prey refugee. The model exhibits rich dynamics with the existence of separatrices in the phase plane in-between basins of attraction associated with oscillation, coexistence, and extinction of the interacting populations. We prove that if the initial values are positive, all solutions are bounded and stay in the interior of the first quadrant. We show that the system undergoes several bifurcations such as transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Consequently, a homoclinic bifurcation curve exists generating an unstable periodic orbit. Moreover, we find that the Bogdanov-Takens bifurcation acts as an organizing center for the scenario of surviving or extinction of both interacting species. Topologically different phase portraits with all possible trajectories and equilibria are depicted illustrating the behavior of the system.

Original languageEnglish
Pages (from-to)14875-14894
Number of pages20
JournalAIMS Mathematics
Issue number8
StatePublished - 2022

Bibliographical note

Funding Information:
The author would like to thank King Fahd University of Petroleum and Minerals (KFUPM), Saudi Arabia, for supporting this research.

Publisher Copyright:
© 2022 the Author(s), licensee AIMS Press.


  • basin of attraction
  • bifurcation
  • predator-prey
  • refugee
  • stability

ASJC Scopus subject areas

  • Mathematics (all)


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