Spreading speeds and travelling waves in a delayed population model with stage structure on a 2D spatial lattice

In this paper, we derive a lattice model for a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. The important feature of the model is the reflection of the joint effect of the diffusion dynamics, the non-local delayed effect and the direction of propagation. We study the well-posedness of the initial-value problem and establish the existence of monotone travelling waves for wave speed c >= c(*)(theta) > 0, where theta is any fixed direction of propagation. In particular, we show that the minimal wave speed c(*)(theta) coincides with the asymptotic speed of spread for any fixed direction theta. Moreover, we find that the asymptotic speed of spread depends on not only the maturation period and the diffusion rate of mature population monotonically but also the direction of propagation, which is different from the case when the spatial variable is continuous.