  We present a simple three-dimensional model to describe the autonomous
expansion of a substrate which grows driven by the local mean curvature of its
surface. The model aims to reproduce the nest construction process in arboreal
Nasutitermes termites, whose cooperation may similarly be mediated by the shape
of the structure they are walking on, for example focusing the building
activity of termites where local mean curvature is high. We adopt a phase-field
model where the nest is described by one continuous scalar field and its growth
is governed by a single nonlinear equation with one adjustable parameter d.
When d is large enough the equation is linearly unstable and fairly reproduces
a growth process where the initial walls expand, branch and merge, while
progressively invading all the available space, which is consistent with the
intricate structures of real nests. Interestingly, the linear problem
associated to our growth equation is analogous to the buckling of a thin
elastic plate under symmetric in-plane compression which is also known to
produce rich pattern through non linear and secondary instabilities. We
validated our model by collecting nests of two species of arboreal Nasutitermes
from the field and imaging their structure with a micro-CT scanner. We found a
strong resemblance between real and simulated nests, characterised by the
emergence of a characteristic length-scale and by the abundance of
saddle-shaped surfaces with zero-mean curvature which validates the choice of
the driving mechanism of our growth model.

Adaptation and Self-Organizing Systems