The FRAC-MN usually describes the complexity and fragmentation of patch dynamics; and it approaches the value of one for shapes with very simple perimeters such as circles or squares; it approaches two for shapes with highly convoluted and plane-filling perimeters (McGarigal et al. 2002).
In contrast, Oryza longiglumis and the two non-Oryza species (P. parviflora and M. stipoides) had simple continuous perimeters without lobes (Figs. 1c and d).
Let (K_{k}) ((k=0,1)) be two domains of areas (A_{k}) with simple boundaries of perimeters (P_{k}) in (mathbb{R}^{2}).
Theorem 1 Let K k ( k = 0, 1 ) be two domains of areas A k with simple boundaries of perimeters P k in R 2. Let K 1 be convex.
However, if the predator attacks from outside the group's perimeter, these simple movement rules might not lead to aggregation.
The trenches were part of a simple, but effective makeshift perimeter defense: A low wall of dirt was thrown up behind the trench, then topped with tall stakes, to create a defensive barrier almost 3 meters wide and 3 meters high.
Proposition 2 Let K be a domain of area A, bounded by a simple closed curve of perimeter P in the Euclidean plane R 2. Let r and R be, respectively, the maximum inscribed radius and minimum circumscribed radius of K. Then we have the following Bonnesen's isoperimetric inequality: P 2 − 4 π A ≥ π 2 ( R − r ) 2, (4).
That is: Let Γ be a simple closed curve of perimeter P in the Euclidean plane (mathbb{R}^{2}), and A be the area of the domain K enclosed by Γ; then P^{2}-4pi Age0, (1.9) where the equality holds if and only if Γ is a circle.
Let K be a plane domain of area A and bounded by a simple closed curve of perimeter P. Denote by R and r, respectively, the radius of the minimum circumscribed disc and radius of the maximum inscribed disc of K. Then we have begin{aligned}& pi t^{2}-Pt+Aleq0;quad rleq tleq R, & frac{P-sqrt{P^{2}-4pi A}{2pi}leqq rleq Rleq frac{P+sqrt{P^{2}-4pi A}{2pi}}, & P^{2}-4pi Agepi^{2}(R- r)^{2}.
The survivability of the species is higher for those inhabiting patches with large perimeters and simple shapes than for those inhabiting patches of equivalent area but complex shape.
