Nonlinear Dynamics

, Volume 92, Issue 2, pp 627–643 | Cite as

Dynamical behaviour of a delayed three species predator–prey model with cooperation among the prey species

  • Soumen Kundu
  • Sarit Maitra
Original Paper


In this paper we have discussed about the dynamics of three species (two preys and one predator) delayed predator–prey model with cooperation among the preys against predation. We accept that the rate of change of density of population relies on the growth, death and in addition intra-species competition for the predators. The growth rate for preys is thought to be logistic. Delays are taken just in the growth components for each of the species. With this model we have demonstrated that the system has permanence. Taking the delays as the bifurcation parameter, the stability of the interior equilibrium point has been discussed analytically and numerically. Critical value of the delay is obtained where the Hopf-bifurcation happens. In presence of delay by constructing a Lyapunov function local asymptotic stability of the positive equilibrium point is discussed.


Predator–prey logistic model Delay Mutualism Permanence Bifurcation 



We thank the anonymous referee for valuable suggestions. The first author is thankful to DST, New Delhi, India, for its financial support under INSPIRE fellowship, without which this research would not have been possible.


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Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology, DurgapurDurgapurIndia

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