Spreading Speeds And Traveling Waves For A Single Species Model In 2-Dimension Lattice

In this paper, we derive a lattice model for a single species in a two-dimensional patch environment with infinite number of patches connected locally by diffusion. New lattice differential equation models with delayed global interaction are developed in two-dimensional patch domain. The important feature of the models is the reflection of the joint effect of the diffusion dynamics, the nonlocal delayed effect and the direction of propagation. We study the well-posedness of the initial-value problem and obtain the existence of monotone traveling waves for wave speed c > c_*(θ), whereθis any fixed direction of propagation. In particular, we show that the minimal wave speed c_*(θ) coincides with the asymptotic speed of spread for any fixed directionθ. Our main finding here is that the asymptotic speed of spread depends on not only the maturation period and the diffusion rate of mature population mono-tonically but also the direction of propagation, which is different from the case that the spatial variable is continuous.