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The authors have declared that no competing interests exist.

Analyzed the data: FL. Wrote the paper: FL AVC RK.

The movement of organisms is subject to a multitude of influences of widely varying character: from the bio-mechanics of the individual, over the interaction with the complex environment many animals live in, to evolutionary pressure and energy constraints. As the number of factors is large, it is very hard to build comprehensive movement models. Even when movement patterns in simple environments are analysed, the organisms can display very complex behaviours. While for largely undirected motion or long observation times the dynamics can sometimes be described by isotropic random walks, usually the directional persistence due to a preference to move forward has to be accounted for, e.g., by a correlated random walk. In this paper we generalise these descriptions to a model in terms of stochastic differential equations of Langevin type, which we use to analyse experimental search flight data of foraging bumblebees. Using parameter estimates we discuss the differences and similarities to correlated random walks. From simulations we generate artificial bumblebee trajectories which we use as a validation by comparing the generated ones to the experimental data.

The characteristics of the movement of animals play a key role in a variety of ecologically relevant processes, from foraging and group behaviour of animals

The planar horizontal movement of an animal is often approximated by a sequence of steps: an angle

This class of models can generate a variety of different dynamics. Two special cases with a uniform distribution for

As Lévy-type models show anomalous diffusive behaviour, in contrast to models with a finite variance of the step length distribution and a fixed time step

The estimation of the tortuosity of a trajectory is intimately connected to the distributions of the turning angle and speed

In the following we will present a generalisation of the CRW above, which we then use to analyse bumblebee flight data. Given movement data with a constant time step

where we distinguish between the deterministic parts

Analysing measured movement data of animals in their natural habitat is intricate due to a variety of factors which may influence the animal's behaviour, ranging from heterogeneous food source distributions

The bumblebees forage on a grid of artificial flowers on one wall of the box. While being on the landing platforms, the bumblebees have access to food supply. Of interest in this paper is the movement when the bumblebee is

Given the experimental data, we start determining the unknown parameters in our model by first estimating the deterministic parts

where and

The regular structure shows the quick relaxation to small angles, and the absence of strong cross-dependencies in the drift, i.e., the

Examining the drift

The deterministic drift

While this reduction of the turning angle dynamics from

The speed drift

The experimental deterministic drift coefficient

where

What we did not specify before was that the turning angle distribution may depend on the speed of the bumblebees. Given that the force a bumblebee can use to change directions is finite, the largest turning angles have to be smaller when flying with high speeds (see

Assuming a constant maximal force (circle) available to the bumblebee to accelerate during a time step, the distribution of the turning angle

The standard-deviation

Instead

For the two stochastic parts of the Langevin equations, we estimated the normalised auto-correlation functions from the data. The turning angle auto-correlation is approximated by a steep power-law as seen in

The experimental data (black crosses) together with an exponential (magenta) and a power-law (blue) fit is shown with the large-lag standard error (grey). The green circles show the auto-correlation extracted from the simulated data.

The auto-correlation function of

As the observed anti-correlation between delays of

Given all the parameters of the full model (see

The complete model (Eqs. (5,6)) is simulated for 200 s (

The green (dashed) line shows the probability density

The green (dashed) line shows the auto-correlation extracted from the simulated data, the black (solid) line from the experimental data with two times the large-lag standard error (grey).

We generalised a reorientation model which is often used to describe the correlated random walk of animals by explicitly modelling accelerations via Langevin equations. Analysing movement data from bumblebees, we extracted information on the deterministic and stochastic terms of Eqs. (1,2). Simulations of our model and comparison to the data have shown that the resulting model agrees very well with the experimental data despite the approximations we made for the model. With the estimation of the turning angle drift

Given that the experiment which yielded our data is rather small and provided the bumblebees with an artificial environment, it would be interesting to apply our new model to free-flying bumblebees to reveal how much the results depend on the specific set-up. This would clarify whether the flight behaviour seen in the laboratory experiment survives as a flight mode for foraging in a patch of flowers in an intermittent model, with an additional flight mode for long flights between flower patches. The analysis of data from other flying insects and birds by using our model could be interesting in order to examine whether the piecewise linear nature of the speed drift and the trivial drift of the turning angle are a common feature. In view of understanding the small-scale bio-mechanical origin of flight dynamics, our model might serve as a reference point for any more detailed dynamical modelling. That is, we would expect that any more microscopic model should reproduce our dynamics after a suitable coarse graining over relevant degrees of freedom.

In this experiment 30 bumblebees (

For calculating auto-correlations small gaps in the time series have been interpolated linearly. As the number of gaps was small the correlations for short times were not affected, however, the interpolation increased the usable data for long time delays. Trajectories were split at larger gaps, e.g., when entering a flower zone, to exclude correlations induced by flower visits.

For a discussion of the influence of the boundedness of the flight arena and for the analysis of the foraging dynamics under varying environmental conditions see

The full set of parameters estimated from the data set which was used for the simulation is given here. For the deterministic drift of the speed the change of slope is at

A simple model showing a dependence of the turning angles on the speed (see

for

We thank Thomas C. Ings and Lars Chittka for providing us with their experimental data and for their helpful comments.