Asymptotic behavior of solutions for time-delayed nonlocal dispersion equations with Dirichlet boundary

This paper is concerned with the asymptotic behavior of the solutions for Nicholson's blowflies equation with nonlocal dispersion subjected to Dirichlet boundary condition. We first prove the existence and uniqueness of the solution for the initial boundary value problem and its non-trivial steady state. Then we give a threshold result on global stability of equilibria: when D lambda(1) + delta > p, the solution time-exponentially converges to the constant equilibrium 0 for any large initial data; when D lambda(1) + d < p, the solution time-asymptotically converges to its positive steady-state f(x) for any large initial data, once 1 < p d = e, where D> 0 is the diffusion coefficient, d > 0 is the death rate, p> 0 is the birth rate and.1 > 0 is the principal eigenvalue for the nonlocal characteristic equation. The adopted approach is the energy method and the monotonic technique.