Kit Yates from Oxford's Mathematical Institute recently published in PNAS on his work into the modelling of locust swarms: I asked him about how seemingly 'random' changes in direction by individuals can add up to better collective choreography...
OxSciBlog: Why do animals such as locusts and starlings make these sudden changes in direction?
Kit Yates: Directional switching is an intrinsic property of the motion of many animal groups. This behaviour is not just limited to locusts and starlings, but extends to shoals of fish, flocks of various different bird types and groups of herding/swarming animals.
An important concept which unifies our understanding of these groups is that of transfer of directional information. It is clearly to the advantage of all the members of a group if they can rapidly change direction away from an oncoming predator or towards a food source which has been detected by only a few members of the group.
Individuals which change their direction of travel in response to the direction taken by their near neighbours can quickly transfer information about the presence of a predatory threat or food source across the whole group.
It is thought that locusts sense their near neighbours using their eyes for the neighbours in front and to the sides their hind legs and abdomen for their neighbours behind. It has been found that locusts without the sensory perception in their abdomen (but still retaining the mechanical ability to move their legs) will often remain motionless while other locusts eat them alive! It is believed that this stimulus encourages the onward march of the locusts. A slightly weaker 'pull' towards the locusts in front is also thought to encourage onwards marching.
OSB: What are the challenges involved in modelling this kind of collective motion?
KY: One of the main challenges in modelling this type of behaviour is obtaining sufficient data to justify the model. Tracking animals in the wild over large distances or over long periods of time can be extremely difficult especially if we are trying to track large enough numbers of individuals to say something useful about the collective dynamics of the group.
For example, a typical locust swarm can contain as many as a quarter of a billion locusts. Fortunately a well thought out laboratory experiment was devised which allowed us to track up to 100 individual locusts for periods of up to 8 hours. Once the data was obtained finding a model which was well suited to the data was quite simple.
There are well known models which display directional switching behaviour like that found in the actual locust data. One of these models which we decided to use is commonly known as the Czirók model or One-Dimensional Vicsek model. In this model, known as a 'self-propelled particle' model, individual locusts are modelled as computer generated particles.
These particles are initialised with a position, direction and a set of rules for interaction with the other particles. The individuals are given a common interaction radius inside which they can sample the velocities of neighbouring individuals. They use the average of these sampled velocities to update their velocity and position at regular time intervals. Some random noise is added to their updated velocities to incorporate the fact that in reality the locusts will not be able to sample the velocities of their neighbours perfectly.
The choice of the magnitude of this noise term is something which previously has not been given much thought but when the computationally generated data is compared to the experimental data it turns out to be very important.
OSB: How did you attempt to create more accurate mathematical models of this behaviour?
KY: We compared certain characteristics of the model to those of the experimental data in order to ascertain how well the model replicated the experimental findings. One of these characteristics was the magnitude of the noise term applied to the locusts' velocity update which models the the randomness in the locusts' new velocities.
We found that the model replicated the data more realistically when the magnitude of the noise term given to individual locusts varied with how aligned that locust was to its near neighbours. We altered the noise term in the model so that the locusts become worse judges of the velocities of their near neighbours when they were not aligned to them. This allowed us to replicate qualitatively the behaviour we saw in the experimental locust data.
OSB: What did you discover about the seemingly 'random' direction changes of individuals?
KY: There are several reasons why a locust might increase the noise term when it finds itself to be unaligned. For example, when unaligned, a locust might want to try several random directions in quick succession in order to reorientate itself with the other locusts.
When a locust swarm changes direction most of the individual locusts will find themselves unaligned at some point. Increased individual noise allows individuals to find alignment with their neighbours (and consequently reorientate and realign the swarm) quickly in order to avoid predation or head towards a food source.
Although the advantage at the group level is palpable (and what is best for the swarm is best for the individual in the tasks of avoiding predation and finding food sources), what makes the individual want to reorientate itself so rapidly?
Previous studies have found that locusts are cannibalistic and, if allowed, will eat each other in their desperate search for salt and protein. It has also been found in studies on immobilised locusts that locusts which are most likely to be eaten are those which are side-on to the swarm, rather than aligned with the swarm.
It is thought that the reason for this is that even when immobilised a locust's hind legs, which it uses to sense other locusts, can be used to kick out at locusts trying to take a quick nibble whereas locusts which are sideways-on to the swarm have no such defence. It could be that an attempt to not be cannibalised is a motivating force for realignment of individual locusts to the rest of the swarm.
Another contributing factor could be that when a locust is not aligned it becomes more difficult for it to measure its neighbours' velocities accurately as its usual sensing mechanisms (eyes, hind legs and abdomen) are disorientated. In short our modelling approach to the data allowed us to propose the idea that an individual-level response to local lack of alignment allows locust swarms to turn more efficiently.
OSB: How might your research help us understand other examples of collective motion - such as traffic jams?
KY: Self-propelled particle models have been used to model collective motion in animals from fish to birds. It would be interesting to compare our findings to data from other animal groups and to see whether this 'increased individual noise in response to a loss of alignment' is a ubiquitous phenomenon.
Noise in response to lost alignment may be an example of a general property of organization of collective motion. Another example is found in traffic-jam models where one way of avoiding “phantom traffic jams” is to introduce additional noise to traffic motion (although there is no direct link between this research and traffic calming measures).
The paper 'Inherent noise can facilitate coherence in collective swarm motion' is published online in PNAS.