The Model

The self-organization of patterns of flow in social insect swarms is a beautiful example of how intelligent and efficient behavior of the whole can be achieved even in the absence of any particular intelligence or forethought of the individuals.[1] Indeed, such patterns can have functionality even without the awareness of the individual entities themselves. A study of the essential elements of swarm dynamics provides an understanding of such behaviors, and is our principal goal in this letter. A secondary, broader goal is to study the generic behavior of a kind of stochastic particle-field system in which the motion of the ``particles" both changes the field and is affected by the field. There are a number of systems like this in the physical sciences. We particularly have in mind systems where the relaxation time scales of the field are slow in comparison to the motion of the particles and this is what makes the present work unique and interesting.

Consider the following scenario. Suppose particles move in a field that field corresponds to an energy so that each particle experiences a force due to the field of . Suppose also that the particles are subject to random perturbations so that their motion is only statistically in the direction of . We could describe their motion by the stochastic (Langevin) type equation

where is Gaussian white noise with and . The parameter is a friction coefficient. Obviously for social insects this term is not to be taken literally. The ``friction" in such cases merely parameterizes the tendency of the organism-particles to continue in a given direction. A smaller ``friction" thus means that the organism's velocity vector is correlated over longer times. In physical systems this parameter may have a more literal meaning.

The noise strength is . Thus , which we call the osmotropotaxic sensitivity, determines the degree of determinacy with which the organism-particle follow the gradient of the field . If is large the predominant force felt by the particles is , and they follow more accurately. If is small the noise becomes more important and the organism-particles follow the field gradient with less certainty. It is necessary to assume probabilistic laws in order to reproduce the behaviors observed experimentally, and the osmotropotaxic sensitivity can be, and has been, observed experimentally.[5,6,7] As we will see, the degree of randomness plays a very important role in the functionality of the swarm. Most probably it is hardwired into the sensory apparatus of the organism.

Up to this point we have simply described the noisy motion of particles in some ``energy landscape". The principal element of interest here is that the energy landscape will be allowed to evolve in response to the noisy motion of the particles. We will choose one of the simplest possible models for how this occurs.[9,10,11] The presence of a particle at will ``deposit" an amount of field per unit time. For insect swarms is to be interpreted as the pheromonal field density[12] at . An amount of field will ``decay" per unit time. The field might also diffuse in space with diffusion constant D such that the field will evolve according to

where is the particle density at . When the particles are social insects this equation describes the evolution of pheromonal field laid down by the organisms as they walk, and the parameter is an evaporation rate.

The only thing which has been left unspecified here is . In this letter we use

The use of this ``field energy function" has already been justified elsewhere,[9,10,11] where it has been explained at length how this function is derived from the experimentally observed behavior of real ants. Other energy functions can be used for different physical situations, and the general framework of this model is capable of supporting many different types of systems.

Here we call the capacity. This models the fact that an actual ant's response to additional concentration of pheromone decreases somewhat at high concentrations. This gives rise to a peaked function for the average time an ant will stay on a line of ant traffic as the concentration of pheromone is varied -- a fact which has been repeatedly observed experimentally[13,14,15]. This is perhaps significant in light of one of the results presented here: trails and networks do not spontaneously form in the absence of this saturation effect. One reasonable physiological explanation for this effect is that the ant's sensing organs become saturated: since each antenna has only a finite number of pheromone receptor sites, the antennae are effectively jammed at high concentration, and the response to gradients can be expected to grow less pronounced.

The model described above was constructed precisely because it approximately reproduces the ``microscopic" behavior of individual ants as observed in the laboratory.[8,3,4,7] Notice that we have resisted the temptation to write a Fokker-Planck equation for from Eq. 1. This is to emphasize that we do not want to consider the continous limit, but rather the case where internal fluctuations are significant. Since the field corresponds to the pheromone which, unlike the ant density , is composed of a macroscopically large number of particles, we can use the continuous Equation 2 to describe it. We have expressed the model in this particularly suggestive form to emphasize its physical aspects, and its generic relationship to other physical systems. We will not expand on this point here, but will merely indicate the generic resemblence of this model to problems such as anomalous ionic diffusion in polymeric materials, stochastic growth processes, the evolution of river basins, and other types of complex physical systems that incorporate mobile elements and an evolving substrate of some kind.[16]