Stability and Hopf bifurcation for a competition and cooperation model of two enterprises with delay

• We model a competition and cooperation model of two enterprises with delay.
• We prove existence of the unique positive equilibrium.
• We study the existence of Hopf bifurcation at the positive equilibrium according to three cases and property of Hopf bifurcation.
• The interesting dynamical behaviors of the model by two numerical examples are shown.
• We show some economic meanings.

In this paper, a retarded competition and cooperation model of two enterprises is investigated. We first prove that the existence of the unique positive equilibrium for the mentioned model. By choosing the delay ττ as a bifurcation parameter, we show that Hopf bifurcation at the positive equilibrium of the system can occur as ττ crosses some critical values. Further, using the normal form theory and center manifold theorem, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, some numerical simulations supporting our theoretical results and the economic meaning of model are also included.