One of the most fundamental ecological relationships is that as the area of a region increases, so does the number of different species encountered. While this makes sense and may even seem obvious, this observation seems to have first occurred late in the eighteenth century and slowly taken hold in the nineteenth century. During this period, naturalists such as Alfred Wallace and Charles Darwin accompanied sailing expeditions to islands around the globe. In the process of recording and collecting what were new and exotic species to Europeans, certain ecological patterns and trends slowly became apparent. Johann Rheinhold Forster, the naturalist on Captain Cook's second voyage to the South Pacific (1772), seems to be the first to have noticed this particular point [Quamen 1996].

Islands only produce a greater or less number of species as their circumference is more or less extensive.Simply put, the number of species increases with area. A less obvious insight would occur later to others making careful collections of data: the increase in species occurs at a decreasing rate. This species-area relation may be the oldest ecological pattern to be recognized; H. C. Watson described a species-area curve for plant species in Britain in 1859 and deCandolle produced a similar study in 1855 [Williams 1964 and Rosenzweig 1995]. Beginning with a small plot in county Surrey, Watson identified the plant species present in ever increasing areas of Great Britain. The general pattern is one of increasing species diversity with increasing area sampled. The first mathematical description of the species-area relationship was proposed by Arrhenius in 1920 and modified by Gleason in 1922.

The association of increased area with an increasing number of species at a declining rate has been tested numerous times. It persists over areas both small and large and with animals as well as plants.

**Example 1** Load the data file `Darlington.dat` into the
ScatterPlot applet to the right and click the "Plot the Data" button.
Here and in subsequent examples, **A** is the
area of the region and **S** is the number of species present in the
corresponding region.

Darlington [1957] found this pattern with
amphibian and reptile species in the Greater and Lesser Antilles
The areas, **A,** of these islands range from
about 1 mi^{2} to nearly 40000 mi^{2}. The number of
species, **S,** present on the islands ranged from 3 on the smallest island
Redonda to 84 on Hispaniola. Note that Hispaniola is the second largest island
in this group with Cuba being the largest. Consequently, the number of species
does not always increase with island size, though that is the general
trend.

**Example 2** The same pattern persists when small areas are surveyed.
Arrhenius [1921] observed this effect when counting plant
species in a variety of ecological communities.
His plots ranged from 1 to 100 square decimeters. You can load `Arrhenius.dat`
into the scatterplot at the right and see the same pattern.

The simple species-area pattern elucidated by Watson and deCandolle has, of course, been shown to depend on a number of other variables besides area. For example, elevation and latitude may change the shape of the species-area curve. So does isolation (mainland versus island). This may be one reason for the large number of amphibian and reptile species that were found on Hispaniola. Likewise, habitat heterogeneity contributes to the rate at which new species are added as area increases. For example, whether you sampled a square meter or a square kilometer at the North Pole, you'd probably find few if any species. However, a similar study in an area with several different types of habitats would yield many new species with each increase in plot area. In fact, some studies suggest that the best explanation for the species-area relationship is increasing habitat diversity [Johnson and Simberloff 1974].

Ecologists have produced hundreds of examples of increasing species diversity with increasing area (see [Rosenzweig 1995]). Yet despite numerous empirical examples, debate continues to exist over the cause(s) of the species-area relation [McGuinness 1984]. The debate has become even more contentious in recent years as species-area curves have been used to address important questions such as:

- What is the minimum protected area that will sustain a particular endangered species?
- Should we protect species or ecosystems?
- What management practices will result in species extinction and at what rate?

Reconsider the data on amphibians and reptiles in the Antilles. Darlington [1957] noticed that there was a pattern in the data when he excluded a few of the islands from consideration.

**Table 1.** Island size versus number of amphibian and
reptile species in the Antilles.
Based on [Darlington 1957, Table 17; also see Quamen 1996, 388].

Approx.Area | Species | Species | Index No. |

mi^{2} | (Approximate) | (Actual) | k |

4 | 5 | 5 | 0 |

40 | 10 | 9 | 1 |

(400) | (20) | -- | (2) |

4000 | 40 | 39--40 | 3 |

40000 | 80 | 76--84 | 4 |

These data may be loaded into the ScatterPlot by using the file

Though there was no 400 mi^{2} island in the Antilles,
Darlington still included that size in his table and filled in an
expected number of 20 amphibian and
reptile species for such an island.
He did so because he noticed that for each tenfold increase in area,
the number of species doubled.
Mathematically this means that both
the area and the number of species are geometric series.
If **A** denotes
the area and **S** the number of species, then
the area of the islands can be written as
**A = 4*10 ^{k},**
where

The goal is to "predict" the number of species based on the area as Darlington
did. In other words,
**S** should be expressed as a function of **A,** not as a function of some
arbitrary index **k.** To do this, solve for **k** in terms of **A** by using logarithms
and then substitute this into the expression for **S.**

If we rewrite

= 5(10

= 5(10

Again, using basic logarithm properties, this simplifies to

Since

Notice how the exponent **log 2** of **A** ensures that a tenfold
increase in area produces a doubling of the number of species. If we start with an area of
size **A** and then evaluate **S** for an area of size **10A** we
see that the function predicts there to be twice as many species as in an area of size
**A**:

Of course, we can evaluate the log of 2:

So, in this particular case, we are able to express

because we spotted a pattern in (an approximation to) the data. Such patterns are seldom so obvious. Nonetheless, the expectation is that species and area are related in the general way that we have found in this example,

where

- Suppose Darlington had found
that a tenfold increase in area produced a tripling
of species. What would the exponent
**z**be in the equation**S=cA**?^{z} - Suppose that a ninefold increase in area produced a doubling
of species, what would the exponent
**z**be? - For those familiar with natural logarithms, recall that
**log 2=ln 2/ ln 10.**Re-express your answers to parts (a) and (b) using natural logs. - More generally, if an
**n**-fold increase in area produces an**m**-fold increase in the number of species, express the corresponding exponent**z**using natural logs.

The power curve description of the species-area relation,

Because **c** and **z** are fitted to the data, some have criticized power curve
models because they don't explain anything about the system, they merely describe
it. Biologically, why should species-area curves be described by power curves
at all? Taking up this criticism, many others (see [McGuinness 1984] for an account)
have modified, adapted or even scrapped this basic
model in an attempt to build a model with explanatory power. Nonetheless,
McGuinness [1984] notes that the
basic relation is often viewed as "one of community ecology's few genuine laws."
For beginning students, the model provides an introduction to the
process of transforming data in order to determine the relationship that
would seem to exist between two variables that can be measured relatively easily
in the field.

Reconsider Darlington's original data set for Antillean amphibians and reptiles. How does one find the power curve

Again, the key is logarithms. Assuming a power model

so

and finally

This linearizes the original relation, that is, this new equation has the form of a line: the constant

So if a power model describes the data, then when we
graph **log A** on the horizontal axis and **log S** on the vertical
the points should lie nearly along a line. This line has slope **z** and
intercept **log c.**

- Reload
`Darlington.dat`into the ScatterPlot Applet. Carry out a transformation of Darlington's data. Take the logs of both variables and plot the result. Do this by- typing
`log(A)`into the box labelled "Plot" - typing
`log(S)`into the box labelled "versus:" - and then clicking the "Plot the Data" button.

- typing
- Are the data reasonably linear? Remember that the point representing Hispaniola will probably not be very close to the regression line. Why?
- What are the slope and intercept for the regression line? (Read these above the scatterplot.)
- What is the correlation coefficient (also called
**r**)? It measures the strength of the linear relationshipd between the two variables and ranges from -1 to +1. When the value is close to -1 or +1, there is a strong correlation between the two variables and when the value is close to 0 there is a little or no correlation between the two variables. Negative correlation coefficients arise when the regression line has a negative slope.

or, using the formulation of the power model,

This means that

Thus, the power model for these data would be

where

- Many values of
**z**fall within a narrow range between 0.15 and 0.39 [Preston 1962; MacArthur and Wilson 1967].- Suppose that
**z=0.15.**Using the general power-curve model**S = cA**what is the effect on the number of species^{z},**S**if the area is increased from**A**to**10A**? - Suppose that
**z=0.39.**What is the effect on the number of species**S**if the area is increased from**A**to**10A**? - Darlington's observation that for every tenfold increase in
area there is a doubling of the number of species often is given as
a rule of thumb.
Do your two calculations support this claim? Explain.

- Suppose that
- The nonvolant (flightless) mammal fauna for the
Channel Islands was surveyed and displayed the general
trend of increased area with an increased number of species
at a declining rate. Load the ChannelData into the ScatterPlot Applet and
find the power curve that best fits these data.
**Table 2:**Total numbers of nonvolant mammal species versus area for the islands of the British Channel [Wright 1981; adapted from Table A2].Island Area (km ^{2})Species Jersey 116.3 9 Guernsey 63.5 5 Alderney 7.9 3 Sark 5.2 2 Herm 1.3 2 - The data below give the number of endemic vascular
plant species in mainland coastal areas
(mi
^{2}) of California at or above 33 degrees latitude.**Table 3:**Data from [Johnson, Mason, and Raven 1968].Location Area Species Tiburon Peninsula 5.9 370 San Francisco 45 640 Santa Barbara area 110 680 Santa Monica Mountains 320 640 Marin County 529 1060 Santa Cruz Mountains 1386 1200 Monterey County 3324 1400 San Diego County 4260 1450 California Coast 24520 2525 - Find the power curve function that
best fits the endemic (native) plant species data for coastal areas of California
at or above 33 degrees latitude given above.
- Now predict the number of species present in a
24210 mi
^{2}region. - Johnson, Mason, and Raven [1968] were interested in the effects of
latitude and elevation as well as area on the number of species present.
The Baja region of California is considerably south (28 degrees latitude)
of the regions in Table 3. Its area is 24210
^{2}and the number of endemic plant species is 1450. How does this compare to your prediction? Does latitude seem to be an important factor?

- Find the power curve function that
best fits the endemic (native) plant species data for coastal areas of California
at or above 33 degrees latitude given above.
- Islands occur not just in
oceans. There are also "virtual" islands such as mountain tops where
the surrounding lowland region represents a physical barrier to montane species.
Other islands can be lakes or ponds or even wooded areas surrounded by open tracts of
land. In this view, a nature preserve or wildlife refuge acts like an
"island" to many of the species which inhabit it.
- Power curve models can be used to estimate the effect on the
number of species present when natural areas are encroached upon
by human activity (e.g., clear cutting of rain forest). Suppose that
50% of an existing area is deforested. Does the power curve model predict that
50% of the species will be lost? Use a value of
**z**of 0.25 to make your estimate. - Suppose that
90% of an existing area is deforested. What proportion of species does the power
curve model predict will remain?
- How many Amazonian plant and animal species ultimately can
be preserved if only 1% of the Amazonian rain forest is maintained in
a ``natural" state?
(Diamond and May [1981] comment, `"Such relations are admittedly
crude and neglectful of detail, but they provide an informed first guess
at the relation between the area of a reserve and the number of species
which are eventually likely to be preserved in it.")

- Power curve models can be used to estimate the effect on the
number of species present when natural areas are encroached upon
by human activity (e.g., clear cutting of rain forest). Suppose that
50% of an existing area is deforested. Does the power curve model predict that
50% of the species will be lost? Use a value of

Suppose two researchers (perhaps two students in class that should have talked to each other before starting the project!) in the same location are counting plant species in relatively small areaa. One measures area in square meters and the other in square feet. How will their species-area curves differ? Consider their two species-area equations for the same location with area measured using two different scales,

- Use a
**log A**versus**log S**transformation to find the species-area curve. How good is the fit? (Use the coefficient of determination.) - What are the slope and intercept of the regression line?
- What is the equation of the species-area curve with area measured in meters?
- Now let's graph the data as if area had been measured in square feet instead.
We will need to use the scaling factor on the areas only: Plot
**10.50*A**verus**S**. The graph should be curved again, like the original plot. - Now comes the tricky part: We want to straighten out this new curve. Do so by taking
logs of both variables. Plot
**log(10.50*A)**versus**log S**. What is the slope of the regression line? What is its intercept? - What is the equation of the species-area curve with area measured in feet?
- How were the two species-area curves similar? How were they different?

It is important to note that there are competing models of the species-area relation being used by biologists and ecologists. One whose history is nearly as long as the power function model is the so-called ``exponential model" proposed by Gleason [1922]. In this model the number of species is a linear function of the logarithm of the area:

Even when the power function model **S=cA ^{z}** is used, there are different methods for finding
the constants

Mitchell, Kevin and James Ryan. The species-area relation.

Quamen, David. 1996. *Song of the Dodo*. Touchstone, New York.

Rosenzweig, M. 1995. *Species Diversity in Space and Time*. Cambridge University Press.

Williams, C. 1964. *Patterns in the Balance of Nature*. Academic Press, London.

Kevin Mitchell mitchell@hws.edu Hobart and William Smith Colleges Copyright © 1997-2001 Last updated: 17 July 2001