Stability and bifurcation of mutual system with time delay

In this paper, we study the stability and bifurcation in a mutual model with a delay τ, where τ is regarded as a parameter. It is found that there are stability switches, and Hopf bifurcation occur when the delay τ passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions in the first bifurcation value is given using the normal form method and center manifold theorem