Territorial behaviour of buzzards versus random matrix spacing distributions

Highlights

• We propose a first large-scale analysis of the two-dimensional spatial statistics that quantifies the repulsive behaviour of a bird of prey, the Common buzzard, as a function of its time dependent population.

• The wealth of 20 years of data allows us to compare the spacings between next and next-to-nearest nests of buzzards using methods from spatial statistics, that have so far seen little application to biological systems.

• The two-dimensional spacing distributions derived from non-Hermitian random matrices are tailored to give a universal description of the repulsive behaviour amongst particles. Its generalisation to a Coulomb gas system, that includes the uncorrelated Poisson case, allows us to perform a one-parameter fit to the spacings between nests.

• This quantifies the effect of the observed population increase on the correlation length between these very territorial birds.

Abstract

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.

Introduction

It is one of the major goals of population ecology, and indeed one of the oldest goals of ecology as a whole, to understand the processes that govern the distribution of animals in space and time (Lack, 1954). This is not only an important academic question, but has crucial implications for conservation planning and management (Krüger et al., 2012, O’Bryan et al., 2019). Birds of prey, or raptors, are often apex predators, are among the most-threatened groups of birds globally (Krüger and Radford, 2008), and their abundance and diversity is a key indicator of the state of the ecosystem as a whole (Sergio et al., 2005). Birds of prey have been used as model systems in other such contexts, for example vultures, including the interaction with humans (Gangoso et al., 2013), or even as an indicator for the well-being of entire ecosystems (Krüger et al., 2012, O’Bryan et al., 2019). Their charismatic nature, conspicuousness and relatively high vulnerability due to their trophic position increases their conservation value (Sergio et al., 2005) and hence the recording of their population dynamics has attracted a lot of attention for centuries (Newton, 1979). In this article, we focus on the behaviour of such a territorial bird of prey, the common buzzard, and model a change in its population density over 20 years. As we shall see, their territoriality, measured by the repulsion among their nests, increases significantly with population size.

In many animal and plant populations, reproductive success decreases with increasing population density. This density dependence of reproduction has been known since the awn of modern animal ecology (Lack, 1954). Although density dependence has been examined in different taxa (Sibly et al., 2005), it is rather easily studied in large, territorial species. One particularly prominent group used for disentangling hypotheses about density-dependent processes have been birds of prey (Krüger et al., 2012, Krüger and Lindström, 2001). Their predatory habit could amplify the occurrence of strong density dependence and make them especially suited for studies of the underlying mechanisms (De Roos and Person, 2002). Territoriality allows density effects to be examined in detail, while this possibility could be impaired in classically colonial species. Birds of prey also offer the advantage of being very site-faithful: once they have occupied a territory, they rarely move and they are highly aggressive against intruders and hence territorial aggression can be fatal. The study of mechanisms that could explain the spatial clustering of bird of prey territories is therefore of theoretical as well as applied value.

Here, the locations of buzzard nests collected over the 20 years 2000–2019 in the Teutoburger Wald around Bielefeld, Germany, are investigated. We are in the comfortable position to have about $100-200$ nests per year available in this approximately two dimensional (2D) landscape, over a period of 20 years; see Section 2 for details. Given such a data set, it is tempting to model the emergence and behaviour in space and time from an ecological angle. However, this would require to make many assumptions and to introduce even more parameters, which bears the danger of overfitting. Indeed, a less biased approach would begin by extracting structural properties from the data, such as distribution patterns, distance preferences, or any kinds of correlations in space and time. Although the inference point of view from point process theory would be natural, compare (Karr, 1991), the data set does not seem to be large enough for such an endeavour. Consequently, the goal of this initial approach is rather modest in the sense that we primarily look at the spacing distribution between neighbouring points, that is, the nest locations, in order to test whether a model with one parameter and no a priori biological assumptions could still explain the change in territorial spacing over time.

A popular approach to spacing distributions is based on random matrix ensembles. They allow to analytically quantify the repulsion among data points, without making further assumptions. In particular, the spacing distributions of the classical random matrix ensembles in one (1D) and two dimensions (2D) are parameter free. For some details on random matrices, we refer to Appendix A. Their history started independently in multivariate statistics (Wishart, 1928), inspired from agriculture by Wishart, and in a statistical theory of energy levels of complex quantum systems by Dyson (1962), motivated by Wigner and ideas of Bohr about the compound nucleus; see Guhr et al. (1998) for a historic account and various modern applications. Further examples without quantum mechanical background repeatedly show the signature of random matrix statistics, such as the spacing between subsequent buses in Cuernavaca (Mexico) (Krbálek and Šeba, 2000), and parked cars (Rawal and Rodgers, 2004, Abul-Magd, 2006) or birds on a power line (Šeba, 2009. While most comparisons are made for data in 1D, comparing with the statistics of real eigenvalues of symmetric or Hermitian random matrices, few examples exist in 2D. Here, one is comparing with complex eigenvalues of random matrices without symmetries, with applications ranging from quantum chaotic systems with dissipation (Grobe et al., 1988, Akemann et al., 2019) over quantum field theory with chemical potential (Markum et al., 1999) to the spacing between chief towns of departments or districts (Caër et al., 1993) and Swedish pine trees (Caër, 1990). Data sets similar to the latter two have been modelled by so-called determinantal point processes (Lavancier et al., 2015), for which random matrices provide a particular example. A novel feature we propose here is to track the time evolution of the repulsion strength, in this case as a function of the population density. This allows us to draw biologically more relevant conclusions, compared to the static distribution of birds on a power line.

In order to apply such ideas to the distribution of nests, we employ recent progress on the universality of certain distributions in random matrix theory. In Section 3, we start with a comparison to the distribution from a uniform Poisson process in 2D, which describes the distribution of uncorrelated points, and to the distribution of the complex Ginibre ensemble of random matrices (Ginibre, 1965), which displays a rather strong repulsion. The repulsive nature is also easily detectable from the diffuse scattering components in the diffraction image of random point sets in 2D (Baake and Kösters, 2011). We stress again that both distributions are parameter free, after fixing the normalisation and first moment to unity. As we shall see, the nest locations are indeed not adequately described by a Poisson process, in line with the known and frequently observed territorial repulsion of the buzzards.

In this first step, it also becomes clear that the repulsion in the Ginibre process is too strong, which is perhaps not too surprising either, as the ecological system should show some repulsion on a shorter scale (visual range), but not a long-range one. To deal with this situation, we embark on a simple one-parameter interpolation between Poisson and Ginibre statistics, which we derive from a known 2D Coulomb gas ensemble at variable temperature (being a non-determinantal, general Gibbs point process (Forrester, 2010, Serfaty, 2019)). Since the underlying model has no direct meaning in the biological system, the parameter $\beta$ (proportional to the inverse temperature) is just taken as an effective phenomenological quantity and then determined by a simple fitting procedure. It directly measures the power of local repulsion of two points at distance s, which is proportional to $s^{\beta +1}$.

It turns out that the employed one-parameter family of spacing distributions works well for the data. Moreover, our effective parameter proves sensitive to population density dependent properties in time, which indicates its suitability for our initial step in the data analysis. In Section 3, we explain how we analyse our data in moving time averages, with further details provided in Appendix C. Our conclusions and open questions are summarised in Section 4.