Revisiting Theron’s hypothesis on the origin of fairy circles after four decades: Euphorbias are not the cause

Table 1 Nearest-neighbor based statistics and pattern analysis of fairy circles and Euphorbias in the three regions Giribes, Brandberg and Garub

Region	Data name	Number of points	Mean NN distance (m)	CV of NN distance	Clark-Evans index R	p-value	General type of pattern
Giribes	Gir-1 Fairy Circles	277	22.6	22.4	1.51	0.002	Regular
Giribes	Gir-1 Euphorbias	130	19.8	60.7	0.90	0.006	Clustered
Giribes	Gir-3 Fairy Circles	1143	12.0	18.8	1.62	0.002	Regular
Brandberg	Bra-1 Fairy Circles	284	17.4	42.7	1.17	0.002	Regular
Brandberg	Bra-1 Euphorbias	58	30.1	56.4	0.92	0.066	Random
Brandberg	Bra-2 Fairy Circles	138	24.2	51.4	1.14	0.036	Regular
Brandberg	Bra-2 Euphorbias	87	24.2	65.2	0.90	0.024	Clustered
Brandberg	Bra-3 Fairy Circles	437	18.0	21.1	1.50	0.002	Regular
Garub	Gar-1 Fairy Circles	113	26.6	38.9	1.10	0.32	Random
Garub	Gar-1 Euphorbias	90	21.6	95.3	0.80	0.002	Clustered
Garub	Gar-2 Fairy Circles	45	38.5	55.5	0.91	0.074	Random
Garub	Gar-2 Euphorbias	358	14.5	52.0	0.97	0.082	Random

NN = nearest neighbor, CV = coefficient of variation. Assessment of the spatial pattern was based on the Clark-Evans index R
Significant deviations from random distributions towards clustered or regular patterns are indicated by p-values < 0.05. Note that there were no Euphorbias in Gir-3 and only nine Euphorbias were in Bra-3, hence these plots have not been analyzed for Euphorbias. The specific type of pattern of the Gir-3 plot was spatially periodic (cf. g-function in Fig. 5), which is a special form of a regular pattern with an extraordinary degree of ordering

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