Nonlinear stability of wavefronts for a delayed stage-structured population model on a 2-D lattice

Abstract

In this paper, we study the nonlinear stability of wavefronts in a delayed stage-structured population model on a 2-D spatial lattice. For all wavefronts with the speed \begin{equation*} c>\max\{c(\eta_{0}),c_{*}(\theta)\}, \end{equation*} where $\eta_{0}$ is some positive constant, $c_{*}(\theta)>0$ is the critical wave speed and $\theta$ is the direction of propagation, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the wavefronts decays exponentially as $i\cos\theta+j\sin\theta \to -\infty$, but it can be arbitrary large in other locations. This essentially improves the previous work with more strongly restricted wave speed and the small initial perturbation. Our approach adopted in this paper is the weighted energy method and the squeezing technique.