Self-organized digging activity in ant colonies

Abstract

Many ant species adjust the volume of their underground nest to the colony size. We studied whether the regulation of the volume of excavated sand could result from an interplay between recruitment processes and ant density. Experiments were performed with different group sizes of workers in the ant Messor sancta. When presented with a thin homogeneous sand disk, these groups excavated networks of galleries in less than 3 days. The excavation dynamics were logistic shaped, which suggests the existence of a double feedback system: a positive one resulting in an initial exponential growth phase, and a negative one leading the dynamics to a saturation phase. The total volume of excavated sand was almost proportional to the number of workers. We then developed a model in which we incorporated the quantitative behavioral rules of the workers’ digging activity. A positive feedback was introduced in the form of a recruitment process mediated by pheromones. The model predicts that the excavation dynamics should be logistic shaped and the excavation should almost stop despite the absence of any explicit negative feedback. Moreover, the model was able to reproduce the positive linear relationship between nest volume and colony size.

Appendix: description of the model implementation

At the beginning of a simulation, ants are randomly placed on the surface around the sand disk. All ants start in the free state. During a cycle, each ant is randomly chosen and performs a single action per time step.

Digging and carrying states

If an ant decides to dig, she comes into the digging state for T_d time steps; during this period the ant will not perform any other action. When this period of time ends, the ant enters into the transporting state. There exists another way for an ant to come into this state: each time an ant in a free state comes into contact with a pellet, she can pick it up with a constant probability P_p. When an ant is transporting a pellet, she has a constant probability P_d to spontaneously drop the pellet she carries and then enters back into the free state (Fig. 4b).

Ant movements

At the end of a simulation cycle, each ant that has not performed an action (digging, picking or dropping a sand pellet) moves randomly over a distance of one cell, in one of the six possible directions (0°, +/−45°,+/−90° and 180°; see also Fig. 4a) according to a weighted matrix (Md) that favors front directions. The weights applied to each direction are the following: 100 for 0°, 46 for +/−45°, 6 for +/−90°, and 1 for 180°.

The thigmotactic behavior of ants is implemented by weighting the cells, in the matrix Md, that are adjacent to the direction blocked by sand so that the probability to leave the sand wall is equal to the experimental probability (P_l). The movement matrix Md is modified by trail pheromones by adding the pheromone units present in each direction.

An ant cannot move on a cell already occupied by two other individuals. The simultaneous occupation of one cell by two ants was allowed since, in the experiments, we observed that one ant would frequently start to walk over another one when tunnels were crowded. Finally, an ant cannot move on a cell that contains sand and her displacement is restricted by the outside arena.

Evaporation and diffusion of pheromones

At the end of a cycle, evaporation and diffusion processes occur and are applied separately on the two kinds of pheromone. Evaporation is expressed by the following function:

$$Q_{{t + 1}} = Q_{t} - {\left( {{Q_{t} } \mathord{\left/ {\vphantom {{Q_{t} } \mu }} \right. \kern-\nulldelimiterspace} \mu } \right)} $$

(A1)

Q_t+1is the amount of pheromone in a cell at t+1 after it has lost a fraction of its previous amount Q_t; μ is a constant representing the half-life of the pheromone.

Diffusion is implemented in the following way: a fraction of the amount of pheromone Q_c on a cell i is diffused to all neighboring cells n,i that have a smaller amount of pheromone Q_n,i according to the function:

$$ \begin{aligned} & If\;Q_{c} > Q_{{n,i}} \;then\;Q_{c} = Q_{c} - {\left( {Q_{c} \times D} \right)}\;and\;Q_{{n,i}} = Q_{{n,i}} \; + {\left( {Q_{c} \times D} \right)} \\ & for\;each\;neighbouring\;cell\;i \\ \end{aligned} $$

(A2)

where D is a constant that represents the diffusion coefficient of the pheromone.