Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice

Abstract

Recently, we derived a lattice model for a single species with stagestructure in a two-dimensional patchy environment with infinitenumber of patches connected locally by diffusion and globalinteraction by delay (IMA J. Appl. Math., 73 (2008), 592-618.). Theimportant feature of the model is the reflection of the joint effectof the diffusion dynamics, the nonlocal delayed effect and thedirection of propagation. In this paper we study the asymptoticstability of traveling wavefronts of this model when the immaturepopulation is not mobile. Under the assumption that the birthfunction satisfies monostable condition, we prove that the travelingwavefront is exponentially stable by means of weighted energymethod, when the initial perturbation around the wave is suitablysmall in a weighted norm. The exponential convergent rate is alsoobtained.