Sharp Rate of the Accelerating Propagation for a Recursive System

How to characterize the rate of accelerating propagation in recursive systems is a challenging topic though it has attracted great attention of theoretical and empirical ecologists. In this paper, we determine the sharp rate of accelerating propagation for a unimodal recursive system with a heavy-tailed dispersal kernel $J$ through tracking of level sets of solutions with compactly supported initial data. It turns out that the solution level set $E_{\lambda }\left(n\right)$ satisfies $J\left(E_{\lambda }\left(n\right)\right)\sim e^{-{\rho }^{\ast }n}$ for large $n$, where $\lambda$ is the level and ${\rho }^{\ast }$ is determined by the linearized system at zero.