Facchini G, Lazarescu A, Perna A, Douady S
Life Sciences Department, University of Roehampton, London, UK.
Institut de Recherche en Mathématique et Physique, UCLouvain, Louvain-la-Neuve, Belgium.
J R Soc Interface. 2020 Jul;17(168):20200093. doi: 10.1098/rsif.2020.0093. Epub 2020 Jul 22.
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 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 . When 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 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.
我们提出了一个简单的三维模型来描述一种基质的自主扩张，该基质的生长由其表面的局部平均曲率驱动。该模型旨在重现树栖白蚁的巢穴建造过程，其合作可能同样由它们行走的结构形状介导，例如将白蚁的建造活动集中在局部平均曲率高的地方。我们采用了一个相场模型，其中巢穴由一个连续的标量场描述，其生长由一个带有一个可调参数的非线性方程控制。当该参数足够大时，方程线性不稳定，并相当逼真地再现了一个生长过程，即初始壁扩张、分支和合并，同时逐渐侵入所有可用空间，这与真实巢穴的复杂结构一致。有趣的是，与我们的生长方程相关的线性问题类似于薄弹性板在对称平面内压缩下的屈曲，已知这种屈曲也会通过非线性和二次失稳产生丰富的图案。我们通过从野外收集两种树栖白蚁的巢穴并用微型计算机断层扫描仪对其结构进行成像，验证了我们的模型。我们发现真实巢穴与模拟巢穴非常相似，其特征是出现了一个特征长度尺度以及大量具有零平均曲率的鞍形表面，这验证了我们生长模型驱动机制的选择。
