Intransitivity Index Formula or Derivation Variables or Terms Original Source
i Minimum number of competitive reversals separating the competitive system from a pure hierarchy Competitive reversals (see text) Slater, 1961
d d = \(\left(\frac{n}{3}\right)-\sum_{i=1}^{n}a_{i}(\frac{a_{i}-1}{2})\) n = the number of species in the system ai = row sum of species i in a competitive outcome matrix Kendall and Babington Smith, 1940
ν v = \(\frac{1}{2}||out\left(R\right)-\ <0,1,2,\ldots n-1>||_{1}\) Where [\(||out\left(R\right)-\ <0,1,2,\ldots n-1>||_{1}\)] is defined as the sum of absolute differences between row sums of the n species system and an n-species hierarchy after row sums of both lists are arranged in ascending order Monsuur and Storcken, 1997
δ’ δ′ = δn\(\frac{(n-1)}{2}\) Where δ is the proportion of competitive outcomes between a pair of species within a n species intransitive relationship Laird and Schamp, 2018 derived from Bezembinder, 1981
u or a u = unbeatability a = always-beatability A species is unbeatable if it outcompetes every other species in the system (u = 1) and is always-beatable if it never wins a competitive interactions (a = 1) Laird and Schamp, 2018
Δri \(r_{i}=\ \frac{\sum_{j\neq i}^{S}{r_{i}-r_{i,-j}}}{S-1}\) S = the number of species in the system ri = the growth rate of species i during an invasion of all S resident species ri,-j = the growth rate of species i during an invasion of the community after the removal of species j Gallien et al, 2017