Species-Area Relationship (S = cA^z)

Count the species in a plot, then count them in a plot ten times bigger, and you do not find ten times as many. You find roughly ten to the power z as many, where z is usually between 0.20 and 0.35 for nested plots inside one region and between 0.25 and 0.45 for genuine islands. Olof Arrhenius wrote the power law down in 1921; Frank Preston connected it to the lognormal distribution of abundances in 1962; MacArthur and Wilson built the theory of island biogeography on it in 1967. It is the closest thing community ecology has to a law, and it is the arithmetic behind every argument about how much habitat a reserve needs.

The consequence people underestimate is the cost of the last few species. At z = 0.25, doubling the species list from 50 to 100 needs sixteen times the area, because you must invert a fourth root. Run the same sum backwards and it becomes the extinction arithmetic that drives conservation policy: destroy 90% of a habitat, leaving a tenth of the area, and 0.1 to the 0.25 is 0.56 — you keep a bit over half the species, and lose nearly half. That single line is why habitat loss estimates and species loss estimates look so different from each other, and why they are consistent.

Now the units, because c is where this page can bite you. S is a count and z is a bare exponent, so for the equation to balance, c must carry the units — specifically species per (area unit) to the power z. Its dimensions therefore change with z, which means a c fitted with areas in hectares is not the same number as a c fitted with the same data in square metres. This calculator works internally in square metres, so the c you enter must be the c that fits square metres. If your source quotes c against hectares, refit it, or enter your area in hectares when you solve for c here and keep the pair together. This is the reason the page tags c with a display unit but deliberately declines to give it a fixed SI dimension: there isn't one, because the exponent moves.

Two more cautions. The power law is a fit, not a mechanism, and at very small and very large areas the curve genuinely bends away from a straight line on log-log axes. And z is not a property of a place; it is a property of a place, a taxon and a sampling design together. Birds and beetles on the same islands give different z. Quote all three or the number is decoration.

It is also worth knowing why nested plots give a shallower z than islands do. A plot inside a continuous habitat shares its species pool with everything around it, so enlarging it mostly adds more of the same; a true island has to have received each species across water and hold it there, so island lists turn over far more between islands and the curve steepens. That difference is the whole reason island biogeography became a theory rather than a curve fit, and it is why applying an island z to a set of survey quadrats overstates what a bigger quadrat would give you.