Bulletin of Mathematical Biology

, Volume 80, Issue 3, pp 626–656 | Cite as

Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

  • Maitri Verma
  • A. K. Misra
Original Article


The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.


Predator–prey system Allee effect Ratio dependent Prey refuge Bifurcation 

Mathematics Subject Classification

34C23 34C25 34D20 92B05 92D25 


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematics, School of Physical and Decision SciencesBabasaheb Bhimrao Ambedkar UniversityLucknowIndia
  2. 2.Department of Mathematics, Institute of ScienceBanaras Hindu UniversityVaranasiIndia

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