  A deeper understanding of the processes underlying the distribution of
animals in space is crucial for both basic and applied ecology. The Common
buzzard (Buteo buteo) is a highly aggressive, territorial bird of prey that
interacts strongly with its intra- and interspecific competitors. We propose
and use random matrix theory to quantify the strength and range of repulsion as
a function of the buzzard population density, thus providing a novel approach
to model density dependence. As an indicator of territorial behaviour, we
perform a large-scale analysis of the distribution of buzzard nests in an area
of $300$ square kilometres around the Teutoburger Wald, Germany, as gathered
over a period of $20$ years. The nearest and next-to-nearest neighbour spacing
distribution between nests is compared to the two-dimensional Poisson
distribution, originating from uncorrelated random variables, to the complex
eigenvalues of random matrices, which are strongly correlated, and to a
two-dimensional Coulomb gas interpolating between these two. A one-parameter
fit to a time-moving average reveals a significant increase of repulsion
between neighbouring nests, as a function of the observed increase in absolute
population density over the monitored period of time, thereby proving an
unexpected yet simple model for density-dependent spacing of predator
territories. A similar effect is obtained for next-to-nearest neighbours,
albeit with weaker repulsion, indicating a short-range interaction. Our results
show that random matrix theory might be useful in the context of population
ecology.
