Dynamics of a discretized Nicholson's blowfly model with Dirichlet boundary condition

The dynamics of the classical diffusive Nicholson's blowfly model with zero Dirichlet boundary condition is less understood. A challenging question is that how the stability of the unique and implicit positive steady state changes in parameters. Numerics suggests that it could be stable or unstable, and a time periodic solution may appear in the latter case. In this paper, we make an effort toward this question by considering a discretized version of the model. To overcome the difficulty caused by the implicit steady state, we first establish some monotonicity properties, which are then used to find the parameter regions such that the positive steady state is locally stable, or unstable due to the occurrence of Hopf bifurcations. A set of sufficient conditions for the global convergence of the positive equilibrium are also obtained. Finally, the global existence of periodic solution is studied by using the global Hopf bifurcation theorem and high-dimensional Bendixson's criterion.