Hopf bifurcation analysis in a predator–prey model with time delay and food subsidies

We analyze the stability of positive equilibrium in a predator-prey model with time delay τ and subsidies. The sufficient conditions of the local Hopf bifurcations at the positive equilibrium are obtained. By center manifold theorem and normal form theory, we analyze the direction of Hopf bifurcations and stability of the bifurcating periodic solution. Using the global Hopf bifurcation theorem, we find that each connected component is unbounded. High-dimensional Bendixson theorem is used to prove that the system has no nonconstant periodic solutions of τ-period, then we obtain the global existence of periodic solutions. Finally, a numerical example is performed to support the theoretical results, and the effect of the food subsidy is discussed. We find that the food subsidy will make the stable interval [0 , τ) of positive equilibrium larger with τ the first Hopf bifurcation value.