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IMA Journal of Applied Mathematics Advance Access published online on February 27, 2008
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxn003
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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

Cui-Ping Cheng, Wan-Tong Li{dagger} and Zhi-Cheng Wang

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People's Republic of China

{dagger}Corresponding author. Email: wtli{at}lzu.edu.cn

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.

Keywords: lattice differential equation; travelling wave; spreading speeds; minimal wave speed; global interaction.


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