Network growth dynamics of fire ant ({\em Solenopsis invicta}) nests

We study the construction dynamics and topology of fire ant ({\em Solenopsis invicta}) nests. Fire ants in colonies of hundreds to hundreds of thousands create subterranean tunnel networks through the excavation of soil. We observed the construction of nests in a laboratory experiment. Workers were isolated from focal colony and placed in a quasi 2D, vertically oriented arena with wetted soil. We monitored nest growth using time-lapse photography. We found that nests grew linearly in time through tunnel lengthening and branching. Tunnel path length followed an extended power law distribution, $P ~ (l - l_0)^\beta$. Average degree of tunnel nodes was $k = 2.17 \pm 0.40$ and networks were cyclical. In simulation we model the nest growth as a branching and annihilating levy-flight process. We study this as a function of dimensionality (2D and 3D space considered) and step length distribution function $P(l_s)$. We find that in two-dimensions path length distribution is exponential, independent of the functional form of $P(l_s)$ consistent with a poisson spatial process while in three-dimensions $P(l) = P(l_s)$. Comparing simulation and experiment we attribute the slower than exponential tail of $P(l)$ in experiment as a result of a behavioral component to the ant digging program.