Stability and bifurcation analysis for a discrete-time model of Lotka-Volterra type with delay

A discrete model of Lotka-Volterra type with delay is considered, and a bifurcation analysis is undertaken for the model. We derive the precise conditions ensuring the asymptotic stability of the positive equilibrium, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, fold or Neimark-Sacker bifurcations occur, but codimension 2 (fold-Neimark-Sacker, double Neimark-Sacker and resonance 1:1) bifurcations may also be present. The direction and the stability of the Neimark-Sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory.