Causes of variation of darkness in flocks of starlings, a computational model

The coordinated motion of large flocks of starlings is fascinating for both laymen and scientists. During their aerial displays, the darkness of flocks often changes, for instance dark bands propagate through the flock (so-called agitation waves) and small or large parts of the flock darken. The causes of dark bands in agitation waves have recently been shown to depend on changes in orientation of birds relative to the observer rather than changes in density of the flock, but what causes other changes in darkness need to be studied still and this is the aim of the present investigation. Because we cannot empirically relate changes in darkness in flocks to quantities, such as position and orientation of the flock and of its members relative to the observer, we study this in a computational model. We use StarDisplay, a model of collective motion of starlings, because its flocks resemble empirical data in many properties, such as their three-dimensional shape, their manner of turning, the correlation of heading of its group-members, and its internal structure regarding density and stability of neighbors. We show that the change in darkness in the flocks perceived by an observer on the ground mostly depends on the observer’s distance to the flock and on the degree of exposure of the wing surface of flock members to the observer, and that darkness appears to decrease when birds roll during sharp turns. Remarkably, the darkness of the flock perceived by the observer was neither affected by the orientation of the flock relative to the observer nor by the density of the flock. Further studies are needed to investigate changes in darkness for flocks under predation

Damping of waves of agitation in starling flocks

When a predator attacks a flock of starlings (Sturnus vulgaris), involving thousands of individuals, a typical collective escape response is the so-called agitation wave, consisting of one or more dark bands (pulses) propagating through the flock and moving away from the predator (usually a Peregrine falcon, Falco peregrinus). The mechanism underlying this collective behavior remains debated. A theoretical study has suggested that the individual motion underlying a pulse could be a skitter (in the form of a zigzag), that is copied by nearby neighbors, and causes us to temporarily observe a larger surface of the wing because the bird is banking during turning while zigzagging. It is not known, however, whether pulses during a wave event weaken over time. This is of interest, because whereas during the usual turning by an undisturbed flock the motion is copied completely without weakening, we may expect that pulses dampen during a wave event because individuals that are further away from a predator react less because of reduced fear. In the present paper, we show in empirical data that pulses during a wave event weaken over time. Our computational model, StarDisplay, reveals that this is most likely a consequence of a reduction of the maximum banking angle during the zigzag escape maneuver rather than by a reduced tendency to copy this maneuver with time. The response seems adaptive because of lowered danger at a larger distance to the location of attack

Some Causes of the Variable Shape of Flocks of Birds

Flocks of birds are highly variable in shape in all contexts (while travelling, avoiding predation, wheeling above the roost). Particularly amazing in this respect are the aerial displays of huge flocks of starlings (Sturnus vulgaris) above the sleeping site at dawn. The causes of this variability are hardly known, however. Here we hypothesise that variability of shape increases when there are larger local differences in movement behaviour in the flock. We investigate this hypothesis with the help of a model of the self-organisation of travelling groups, called StarDisplay, since such a model has also increased our understanding of what causes the oblong shape of schools of fish. The flocking patterns in the model prove to resemble those of real birds, in particular of starlings and rock doves. As to shape, we measure the relative proportions of the flock in several ways, which either depend on the direction of movement or do not. We confirm that flock shape is usually more variable when local differences in movement in the flock are larger. This happens when a) flock size is larger, b) interacting partners are fewer, c) the flock turnings are stronger, and d) individuals roll into the turn. In contrast to our expectations, when variability of speed in the flock is higher, flock shape and the positions of members in the flock are more static. We explain this and indicate the adaptive value of low variability of speed and spatial restriction of interaction and develop testable hypotheses

Fluctuation-Driven Flocking Movement in Three Dimensions and Scale-Free Correlation

Recent advances in the study of flocking behavior have permitted more sophisticated analyses than previously possible. The concepts of “topological distances” and “scale-free correlations” are important developments that have contributed to this improvement. These concepts require us to reconsider the notion of a neighborhood when applied to theoretical models. Previous work has assumed that individuals interact with neighbors within a certain radius (called the “metric distance”). However, other work has shown that, assuming topological interactions, starlings interact on average with the six or seven nearest neighbors within a flock. Accounting for this observation, we previously proposed a metric-topological interaction model in two dimensions. The goal of our model was to unite these two interaction components, the metric distance and the topological distance, into one rule. In our previous study, we demonstrated that the metric-topological interaction model could explain a real bird flocking phenomenon called scale-free correlation, which was first reported by Cavagna et al. In this study, we extended our model to three dimensions while also accounting for variations in speed. This three-dimensional metric-topological interaction model displayed scale-free correlation for velocity and orientation. Finally, we introduced an additional new feature of the model, namely, that a flock can store and release its fluctuations

Non-local kinetic and macroscopic models for self-organised animal aggregations

The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient ϵ is varied from ϵ=1 (for 1D transport models) to ϵ=0 (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit ϵ→0, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams

A new approach to spatially explicit modelling of forest dynamics: spacing, ageing and neighbourhood competition of mangrove trees

This paper presents a new approach to spatially explicit modelling that enables the influence of neighbourhood effects on the dynamics of forests and plant communities to be analysed. We refer to this approach as 'field of neighbourhood' (FON). It combines the 'neighbourhood philosophy' of grid-based models with the description of individual spacing in the 'zone of influence' (ZOI) approach. The novel feature of FON is that modelling of local competition between neighbouring trees is based on the notion of a field of neighbourhood exerted by each tree. This field is defined only on the ZOI of a tree and depends on the distance to the stemming point. For the demonstration of FON's power, a simulation model (KiWi) was implemented that focuses on the dynamic of mangrove forests. The realistic self-thinning behaviour of modelled stands of Avicennia germinans and Rhizophora mangle confirms the suitability of the FON approach for the description of intra- and inter-specific competition. In KiWi, mortality is modelled in terms of a 'memory function', i.e. the yearly stem increment of each tree is stored in its 'memory' over a certain time period and determines - as a sign of vitality - tree mortality. The results of KiWi demonstrate that this description is sufficient to keep the maximum age of the trees within a reasonable limit. The model thus manages without a description of individual tree age. This is an important feature considering the fact that a direct relationship between tree age and mortality is questioned and there is no established method as yet for determining the age of mangrove trees. (C) 2000 Elsevier Science B.V. All rights reserve