Termite Mound Building Model

Abstract

In 1977, Deneubourg introduced a diffusion–advection model for describing the initial stage that termites build spontaneously their mound. The model consisting of three variables is written in the form

$$\begin{cases}\frac{\partial u}{\partial t}=a\varDelta u-\mu\nabla\cdot[u\nabla\rho]-cu+f&\text{in}\ \varOmega\times(0,\infty), \\\noalign{\vspace{3pt}}\frac{\partial v}{\partial t}=-dv+\nu(1-\frac{v}{K})u&\text{in}\ \varOmega\times(0,\infty), \\\noalign{\vspace{3pt}}\frac{\partial\rho}{\partial t}=b\varDelta\rho-g\rho+\zeta v&\text{in}\ \varOmega\times(0,\infty).\end{cases}$$

Here, Ω⊂ℝ³ is a domain where termites walk around. The function u=u(x,t) denotes the density of termites loading soils in their mouthes in Ω at time t, v=v(x,t) is the density of deposited material still active in Ω, and ρ=ρ(x,t) is the concentration of pheromone. Termites deposit the loading soils at rate $\nu(1-\frac{v}{K})$ , where K is the capacity. Pheromone is emitted from the active deposited material at rate ζ. Pheromone acts as chemoattractant for laden termites. The attraction process is described by Keller–Segel model as in Chap. 12 with sensitivity function χ(ρ)=ρ. This model was brought up by Nicolis–Prigogine in their book Self-Organization in Nonequilibrium System.