Table 2.

Predictions of the piecewise linear approximation. The species-area relationship (SAR) and the two invariability–area relationships [IAR with decorrelation by species turnover (IAR-DST) and IAR with decorrelation by distance (IAR-DD)] are approximated by three line segments. The first one (‘First phase’ column) is linear in the area A, the second one (‘Second phase’ column) is constant, and the third one (‘Third phase’ column) is again linear in the area A. The downward inflection point is approximated as the intersection of the first and second line segment; the corresponding area is given in the ‘Downward’ column. The upward inflection point is approximated as the intersection of the second and third line segment; the corresponding area is given in the ‘Upward’ column. Note that both IAR-DST and IAR-DD are proportional to the productivity invariability of a single plant, $I_1=m_{\rho }^2/{\sigma }_{\rho }^2$.

	First phase	Downward	Second phase	Upward	Third phase
SAR	λIλSQ×A	1/λI	λSQ	Q	λS×A
IAR-DST	I1λIλSQ×A	1/λI	I1λSQ	Q	I1λS×A
IAR-DD	I1λIλSQ×A	1/ρoλIλSQ	I1/ρo	2Do	I1/2ρoDo×A