Levy Flight Searches in the Foraging Behaviour of Ants

The foraging movements of Iridomyrmex species of ant are observed to follow a Levy flight distribution. The movement step size is shown to be characterised by a power law distribution. This makes the movement fractal in nature, which will produce an efficient search pattern. Introduction: Levy flights are a category of random walks first devised by Paul Levy in 1937 by generalizing Brownian motion to include non-Gaussian randomly distributed step sizes for the distance moved. Such a distribution has a non-finite variance, leading to much longer tails than a Gaussian distribution. The probability distribution used in a Levy Flight drops off as a power law given by 1/x for large values of x, where α is between 0 and 2. For the case α = 2 the behaviour is similar to a Gaussian, but for lower values of α, the behaviour becomes more dominated by the large jumps in step size allowed by the wide tails of the distribution. This also leads to the motion having no characteristic length scale, which characterises the motion as fractal. There are many very diverse natural and artificial phenomenons that can be described accurately by Levy flights. Examples are present in fluid dynamics, financial mathematics, earthquake analysis and even in the trajectories that abstract painter Jackson Pollock used to create his drip paintings [1]. Levy flights have also been proposed to describe the paths followed by animals engaged in foraging behaviour. This was first proposed for ants in 1986 [2], and it has since been demonstrated that the movement of Drosophila fruit flies [3], Honeybees [4] and Wandering Albatrosses [5] also follow Levy flights. Ants have been also been shown to follow Levy Flights, although there is limited quantitative data concerning the foraging paths of single ants, with most studies investigating the amount of time an individual ant is active or resting [6] (which is also fractal), or the behaviour of the colony as a whole [7]. To further investigate the movements of ants, the study described below was conducted to examine the foraging behaviour of an Iridomyrmex species of ants (commonly referred to as Meat Eater Ants, or Meat Ants). The aim will be to establish whether the search movement pattern follows a power law behaviour, or some other distribution for the step sizes. Method: The ants to be investigated were of an Iridomyrmex species studied at two locations, one at ANU in Canberra, and the other near Captains Flat in NSW. The species may have been different variants, however there was no easy way to distinguish them so they were assumed to be the same. An ant was chosen at a distance of at least 2 metres from the nearest nest and to try ensure that it was engaged in foraging behaviour it was observed for around a minute. Ants following pheromone trails tend to move in a general direction at roughly a constant speed, and could usually be distinguished after being watched for a short time. Once the ant was determined to be most likely foraging, the data recording was commenced by tracking its movement. The movement of the ant was tracked by placing numbered markers at the location of the ant at 5-second time intervals. The time interval was selected as it was the shortest that still allowed time to drop the counter, pick up the next one and keep track of the ant. The ants would often change direction within the 5-second time interval, so a shorter one would have been better. To attempt to minimise any disturbance the Figure 1: Plot of step size for distance (Figure 1 (a), top) and angle (Figure 2 (b), bottom) for the raw data of all 11 runs combined. dropping of the marker may cause to the ant, the markers were actually placed at the locations once the ant had moved away a few cm. The ant’s location was tracked over a total time length of 245s, which corresponded to 50 data points. When all of the 50 markers were in place, a co-ordinate origin was established and a protractor fixed there. A piece of string was attached at the origin, and this could be stretched to each point to allow the radial distance (r) and angle (θ) to be measured. Note that the overall precision of this method is quite low. The markers used to record ant position are ~1cm in width, and combined with the fact that they are only placed once the ant has moved on means that there would be at least a + 1cm error associated with the radial position measurements. There would also be an error of at Angular direction step size -4 -3 -2 -1 0 1 2 3 4 0 100 200 300 400 500 600 A n g le ( ra d ia n s ) Distance step size 0 200 400 600 80