Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters

In this paper we consider a delayed population model with delay-dependent parameters. Its dynamics are studied in terms of stability and Hopf bifurcations. We prove analytically that the positive equilibrium switches from being stable to unstable and then back to stable as the delay ττ increases, and Hopf bifurcations occur finite times between the two critical values of stability changes. Moreover, the critical values for stability switches and Hopf bifurcations can be analytically determined. Using the perturbation approach and Floquet technique, we also obtain an approximation to the bifurcating periodic solution and derive the formulas for determining the direction and stability of the Hopf bifurcations. Finally, we illustrate our results with some numerical examples.