[nlin.AO] What about the birds?

 By: Jeffrey Heninger    11:13 AM  nonlinear dynamics, self-organizing systems, swarming   2 comments

The Role of Projection in the Control of Bird Flocks
Pearce, Miller, Rowlands, and Turner

Many animals form swarms: cohesive and coherent groups that help individuals gain protection from predators. Some of the most dramatic of these swarms are the murmurations of starlings, shown here. These swarms are usually modeled by assuming that each bird interacts only with a couple of the birds that immediately surround it. These models can describe the shape and structure of large flocks.



Pearce et. al. focus on a particular aspect of the flocks, the opacity, that they believe are inadequately described by current models. Regardless of the size of the flock, the density of the birds tends to be such that a distant observer can see substantial fractions of both birds and sky through the flocks. The opacity is the percent of the sky visible through the birds. Measured flocks have opacity between 25% and 60%. The opacity also changes immediately before rapid acceleration of the flock, indicating that the opacity allows for long-range information transfer.

This level of opacity is evolutionarily favorable. Denser swarms force predators to face target degeneracy, in which it is hard to distinguish an individual target because so many of the birds move together. Sparser swarms allow more of the birds to be able to see an approaching predator, allowing them to aware sooner and evade the predator. Marginal opacity balances both of these benefits.

Current models fix the density of the birds by providing interaction models between the birds, creating a fictitious potential between them, or by constraining the volume the birds fly in. However, having a fixed density results in different opacity for different sized flocks. In order to maintain the opacity, these models would have to change their parameters depending on the number of bids in the flock.

This paper proposes a hybrid model in which the individual birds respond both to their nearest neighbors and to the density of birds in their line of sight. Birds tend to align themselves with their nearest neighbors: absent other factors, a bird will fly in the direction given by the average velocity of its nearest neighbors. This term leads to collective behavior of the birds. Each bird also responds to its view of the rest of the flock. An individual bird can see dark regions (where its line of sight is intercepted by another bird) and light regions (where it can see through the flock to the sky). Models in which a bird moves towards the lightest or darkest regions it can see result in flock expansion or flock collapse, respectively. Instead, the birds respond to the edges between light and dark regions. Absent other factors, a bird will orient itself towards the average angle of all of the edges it can see. A third term is also included to provide uncorrelated noise to the direction a bird flies.

The orientation of a bird depends on three terms: (1) the average orientation of its immediate neighbors, (2) the average direction of the edges of bird and non-bird regions it can see in the rest of the flock, and (3) a stochastic term. The authors numerically explore the parameter space of the relative weights of these three terms.

Four numbers are important for each combination of parameters: (1) the opacity of the flock; (2) the maximum distance between any two birds, which measures whether the flock has dispersed; (3) the center of mass velocity of the flock; and (4) the autocorrelation time of the flock, which measures how quickly birds respond to changes on the other edge of the flock. Each of these numbers in averaged over 400,000 time steps for a flock with 100 individuals.

If the opacity term is neglected, the flock disperses. The maximum distance between birds is large and the opacity drops to near zero. It is also difficult for the flock to transmit information. The average velocity is small and the autocorrelation time is large.

Once the opacity term is included, the flock stabilizes. For most parameter values, the maximum distance between the birds is small and constant. The opacity is consistent with experiments and increases as this term dominates. The average velocity of the flock increases as the correlation between neighboring birds increases relative to the noise term. The autocorrelation time of the flock increases with the opacity term as long as the neighbor interactions are sufficiently strong.

The authors also investigated the effects of swarm size. They found that the opacity remains approximately constant as the size of the swarm changes. Larger swarms are slightly more opaque, but the opacity remains less than 75% even for large swarms. This contrasts with other swarming models for which the swarms rapidly become opaque when there are large numbers of birds.

Modeling the behavior of large flocks of birds can be described by having each bird respond to a few external visual stimuli. The birds tend to fly in the same direction as their nearest neighbors. Pearce et. al. argue that the birds also respond to the rest of the birds by observing the opacity of the flock. This mechanism results in flocks which have both the target degeneracy to protect individuals from predators and the ability for all of its members to be aware of approaching threats.