Species-Area Relationship (Power Law) Equation, Derivation & Rearrangements

Core idea

Overview

The Species-Area Relationship (SAR), often expressed as a power law, describes the empirical relationship between the area of a habitat or region and the number of species found within it. This fundamental ecological pattern, S = cA^z, shows that larger areas generally contain more species. The constant 'c' represents the species richness of a unit area, while the exponent 'z' indicates how rapidly species richness increases with area, typically ranging from 0.1 to 0.5 for continental areas and 0.25 to 0.55 for islands.

When to use: Use this equation to estimate species richness in different-sized habitats, predict biodiversity loss due to habitat destruction, or compare biodiversity patterns across different regions or taxa. It's particularly useful in conservation biology for setting reserve sizes or understanding island biogeography.

Why it matters: The SAR is a cornerstone of ecology and conservation, providing a quantitative framework for understanding biodiversity distribution. It informs conservation strategies, helps predict the impact of habitat fragmentation, and is crucial for designing protected areas to maximize species preservation.

Symbols

Variables

c = Constant, A = Area, z = Exponent, S = Species Richness

c

Constant

d im e n s i o n l ess

A

Area

k m^{2}

z

Exponent

d im e n s i o n l ess

S

Species Richness

s p ec i es

Walkthrough

Derivation

Formula: Species-Area Relationship (Power Law)

The Species-Area Relationship (SAR) describes how the number of species (S) increases with the area (A) of a habitat, often modeled by a power law.

The relationship between species richness and area follows a power law, which is an empirical observation rather than a universal law.
The constants 'c' and 'z' are specific to the taxonomic group and geographic region being studied and are derived from empirical data.
The habitat within the area is relatively homogeneous or the heterogeneity is accounted for in the constants.

1

Empirical Observation:

Early ecologists observed that larger areas tend to contain more species. This suggests a direct, though not necessarily linear, relationship between species richness (S) and area (A).

S \propto A

2

Introducing the Power Law Model:

To quantify this relationship, the power law model was proposed. Here, 'c' is a constant representing the number of species in a unit area, and 'z' is the exponent, indicating the rate at which species richness increases with area. Both 'c' and 'z' are empirically determined parameters.

S = c A^{z}

3

Linearization for Estimation (Optional):

Taking the logarithm of both sides transforms the power law into a linear equation (y = mx + b), where log(S) is y, log(A) is x, z is the slope (m), and log(c) is the y-intercept (b). This linearization allows for easier estimation of 'c' and 'z' using linear regression on empirical data.

lo g (S) = lo g (c) + z lo g (A)

Note: The choice of base for the logarithm (natural log or base 10) does not affect the value of 'z', but it will affect the value of 'c'.

Result

lo g (S) = lo g (c) + z lo g (A)

Source: MacArthur, R. H., & Wilson, E. O. (1967). The Theory of Island Biogeography. Princeton University Press.

Free formulas

Rearrangements

Solve for $c$

Species-Area Relationship: Make c the subject

c = \frac{S}{A ^{z}}

To make 'c' (constant) the subject of the Species-Area Relationship formula, divide both sides by A^z.

Difficulty: 2/5

Solve for $A$

Species-Area Relationship: Make A the subject

A = (\frac{S}{c})^{\frac{1}{z}}

To make 'A' (Area) the subject of the Species-Area Relationship formula, first divide by 'c', then raise both sides to the power of 1/z.

Difficulty: 3/5

Solve for $z$

Species-Area Relationship: Make z the subject

z = \frac{lo g ( \frac{S}{c} )}{lo g ( A )}

To make 'z' (exponent) the subject of the Species-Area Relationship formula, first isolate A^z, then take the logarithm of both sides.

Difficulty: 4/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a power law curve where species richness increases as area grows, starting from the origin and becoming increasingly steep. For a biology student, this shape shows that small areas contain few species, while large areas support a rapidly expanding number of species as more habitats become available. The most important feature is that the curve never levels off, meaning that adding more area always results in a higher predicted number of species.

Graph type: power_law

Why it behaves this way

Intuition

Imagine progressively expanding a sampling quadrat in a landscape; initially, many new species are added, but as the area grows, the rate of discovering *new* species slows down, even though the total number continues to

S

Number of species (species richness)

The total count of distinct species found within a given area, representing the biodiversity level.

c

Species richness of a unit area (intercept constant)

A baseline value indicating the expected number of species in a hypothetical area of one unit, reflecting regional biodiversity.

A

Area of the habitat or region

The physical extent of the space being studied; larger areas generally offer more diverse habitats and resources.

z

Exponent defining the rate at which species richness increases with area

Indicates how rapidly species richness accumulates with increasing area; a higher 'z' means species richness grows more steeply with area.

Signs and relationships

exponent z: A positive exponent 'z' ensures that species richness (S) increases as area (A) increases. Its value dictates the curve's shape, reflecting how quickly new species are encountered with increasing area

Free study cues

Insight

Canonical usage

The primary unit consideration is ensuring consistency between the chosen unit for area (A) and the empirically derived constant 'c', while 'S' represents a dimensionless count of species and 'z' is a dimensionless

Common confusion

Students often forget that the constant 'c' is not universally fixed but depends on the specific units chosen for area 'A'. Using 'c' derived from km^2 with 'A' in m^2 will lead to incorrect results.

Dimension note

The number of species (S) is a dimensionless count, and the exponent (z) is also dimensionless. The constant 'c' has units that depend on the units chosen for area 'A' and the value of 'z'.

Unit systems

$S$ species (count) · Represents the number of species, a dimensionless count.

$A$ m^2 | km^2 | ha | acres · Area of the habitat or region. The specific unit chosen for A must be consistent with the unit implicit in the constant 'c'.

$c$ species / (unit of A)^z · A constant representing species richness per unit area. Its numerical value and implicit unit depend on the chosen unit for 'A' and the value of 'z'.

$z$ dimensionless · The exponent indicating the rate at which species richness increases with area. It is always dimensionless.

Ballpark figures

Quantity:
Quantity:

One free problem

Practice Problem

A study in a tropical rainforest found that for a specific group of insects, the constant 'c' is 3.5 and the exponent 'z' is 0.28. If a conservation area is established with an area of 1500 km², how many species of this insect group would be predicted to be present according to the power law species-area relationship?

Constant3.5 dimensionless

Area1500 km²

Exponent0.28 dimensionless

Solve for: S

Hint: Remember to calculate A^z first before multiplying by c.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating the number of plant species expected in a newly designated national park of a certain size.

Study smarter

Tips

Ensure 'A' (Area) is in consistent units (e.g., km² or m²).
The constants 'c' and 'z' are empirically derived and vary depending on the taxonomic group, geographic region, and habitat type.
The power law model is often linearized by taking the logarithm of both sides: log(S) = log(c) + z log(A), making it easier to estimate 'c' and 'z' from empirical data using linear regression.
Be aware that the SAR can be influenced by sampling effort and habitat heterogeneity.

Avoid these traps

Common Mistakes

Using inappropriate 'c' and 'z' values for the specific ecosystem or taxonomic group being studied.
Extrapolating the relationship far beyond the range of areas for which 'c' and 'z' were derived.
Confusing the power law model with other SAR models (e.g., exponential, logarithmic).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Species-Area Relationship (SAR) describes how the number of species (S) increases with the area (A) of a habitat, often modeled by a power law.

Use this equation to estimate species richness in different-sized habitats, predict biodiversity loss due to habitat destruction, or compare biodiversity patterns across different regions or taxa. It's particularly useful in conservation biology for setting reserve sizes or understanding island biogeography.

The SAR is a cornerstone of ecology and conservation, providing a quantitative framework for understanding biodiversity distribution. It informs conservation strategies, helps predict the impact of habitat fragmentation, and is crucial for designing protected areas to maximize species preservation.

Using inappropriate 'c' and 'z' values for the specific ecosystem or taxonomic group being studied. Extrapolating the relationship far beyond the range of areas for which 'c' and 'z' were derived. Confusing the power law model with other SAR models (e.g., exponential, logarithmic).