Bifurcation analysis in a delayed diffusive Nicholson's blowflies equation

The dynamics of a diffusive Nicholson's blowflies equation with a finite delay and Dirichlet boundary condition have been investigated in this paper. The occurrence of steady state bifurcation with the changes of parameter is proved by applying phase plane ideas. The existence of Hopf bifurcation at the positive equilibrium with the changes of specify parameters is obtained, and the phenomenon that the unstable positive equilibrium state without dispersion may become stable with dispersion under certain conditions is found by analyzing the distribution of the eigenvalues. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived.