A mathematical model for invasion range of population dispersion through a patchy environment

We focus on the question of how the dispersion of an invading population is affected by the spatial distribution of patches that have resource available for the population's settlement and reproduction. We have developed and analyzed a mathematical model with a simple stochastic process. The patches are grouped into three classes - free, occupied and abandoned - depending on the state of the patch used by the population. We especially consider the range expanded by invaded patches, the invaded range R, assuming a certain generalized relation between R and the total number of invaded patches k, making use of an index, a sort of fractal dimension, to characterize the spatial distribution of invaded patches. We show that the expected velocity is significantly affected by the nature of spatial distribution of resource patches, and is temporally variable. When the invading population finally becomes extinct at a certain moment, the terminal size of the invaded range at that the moment is closely related to the nature of the spatial distribution of resource patches, which is explicitly demonstrated by our analysis.