Partitioning the union of disks in plant competition models

Abstract

The problem of partitioning the union of disks, which arises in theories of competition between plants, is analysed from a general point of view. Some simple and natural assumptions such as connectedness, invariance under translation, rotation and change of scale, and monotonicity and continuity properties of the partition sets are imposed. From these it is proved that the partition is specified by giving a boundary curve between two disks with the very explicit form: $R_{x}^{\alpha}-|r-x|^{\alpha}=R_{y}^{\alpha}-|r-y|^{\alpha},\alpha \geq 1$ where $R_{x},R_{y}$ are the radii of the disks, $x,y$ are their centres and $r$ is a point on the boundary. This includes the Johnson-Mehl construction $(\alpha =1)$, the common chord $(\alpha =2)$ and the perimeter of the larger disc $(\alpha \rightarrow \infty)$. Larger disks dominate smaller ones if and only if $\alpha >1$ so that $\eta =1-1/\alpha $, $0\leq \eta 0,\alpha \geq 1$ and $\beta >0$. The obvious special cases are $\eta =0$ (cones, for conifers), $\eta =\frac{1}{2}$ (ellipsoids of revolution, used by other authors for broadleaf trees) and $\eta =1$ (cylinders or umbrellas, perhaps for flat topped plants). The results apply equally to $N$-dimensional spheres, while some apply to more general shapes. The disk partitions also generate partitions of $\text{R}^{2}$, giving a natural prescription for 'growing space' which is simpler than those of some other authors.