Bistability and tristability in a deterministic prey–predator model: Transitions and emergent patterns in its stochastic counterpart

Nazmul Sk, Bapin Mondal*, Ashraf Adnan Thirthar, Manar A. Alqudah, Thabet Abdeljawad

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


The paper focuses on studying the global dynamics of a simple prey–predator model. The model incorporates cooperative hunting behavior among predators and takes into account the effect of harvesting on the predator population. The interaction between the prey and predator follows the Crowley–Martin functional response. The study involves analyzing the equilibrium points of the system and investigating their stability properties through mathematical analysis. Various types of bifurcations, including Hopf bifurcation, saddle–node bifurcation, and transcritical bifurcation, are numerically demonstrated in the figures, highlighting the dynamic behavior of the model. One of the intriguing findings of the study is the occurrence of bistability and tristability in the model. To incorporate stochasticity into the model, white noise is added to the deterministic system. This allows for the examination of transitions between different steady states in the stochastic system. We have conducted an analysis of species persistence and extinction in relation to the presence of noise. The paper presents the stochastic sensitivity function (SSF) technique and the use of confidence ellipses to assess the likelihood of such transitions occurring in the system. Overall, the study provides insights into the complex dynamics of prey–predator interactions, considering factors such as cooperation, harvesting, and stochasticity. The results contribute to our understanding of population dynamics and the potential effects of environmental and human-induced perturbations on ecosystem stability.

Original languageEnglish
Article number114073
JournalChaos, Solitons and Fractals
Publication statusPublished - Nov 2023
Externally publishedYes


  • Confidence ellipse
  • Cooperation
  • Harvesting
  • Noise
  • Transition
  • Tristability


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