A growth model driven by curvature reproduces geometric features of arboreal termite nests

We present a simple three-dimensional model to describe the autonomous expansion of a substrate whose growth is 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 in which 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 with 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 patterns through nonlinear 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-computed tomography scanner. We found a strong resemblance between real and simulated nests, characterized 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.