Dynamics of numerical discretization in a delayed diffusive Nicholson's blowflies equation

In this article, we study the dynamics of a diffusive Nicholson's blowflies equation with a finite delay and Dirichlet boundary condition by the forward and backward Euler difference schemes and Crank-Nicolson scheme. The existence of Hopf bifurcation at the equilibrium is obtained. Through analyzing the distribution of the eigenvalues we find that the unstable equilibrium state without dispersion may become stable with dispersion under certain conditions. Our results show that Crank-Nicolson and the backward Euler difference schemes are superior to the forward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results.