Stability and Hopf bifurcation in a delayed competition system

In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given.