Hopf bifurcation and chaos in a delayed Nicholson's blowflies equation with nonlinear density-dependent mortality rate

The stability and bifurcation of the Nicholson's blowflies equation with nonlinear density-dependent mortality rate and delay feedback are investigated. First, we study the existence of positive equilibria and the stability of positive equilibria in the absence of delay. Then, how delay influences the stability of positive equilibria is studied. According to our analysis, as delay increases, there are a sequence of critical values where the system undergoes Hopf bifurcations. Using the theory of normal form and center manifold reduction, the direction and stability of Hopf bifurcations can be determined according to the parameters of the system. Finally, numerical simulations are applied to illustrate and expend our theoretical analysis, and delay-induced chaos has been found.