A probabilistic model for the variance to mean power law in ecology

The spatial distribution of plants and animals generally obeys a power law relationship between the variance and the mean of the numbers of individuals per unit area. The mechanistic basis for this relationship remains controversial. Here a probabilistic model is proposed, analogous to diffusion-limited aggregation, from whence the power law follows as an exact and unconstrained mathematical result. The model possesses a symmetry, self affinity, which suggests that aggregation of individuals follows a fractal pattern.