Island biogeography and the species-area relationship

Intuition Beginner

Picture a bathtub with the tap running and the plug partly out. The water climbs until the outflow matches the inflow, then holds level. An island's species count behaves the same way. Fresh species arrive like water from a tap — that is immigration. Resident species die out like water down the drain — that is extinction. The level where the two balance is the equilibrium species number.

Two knobs move that level. A bigger island shrinks the drain: larger populations persist longer, so extinction runs slower. A nearer island opens the tap wider: colonizers cross a short sea gap more easily, so immigration runs faster. A large island close to a mainland sits high on the curve; a small, remote island sits low. The number $S^{\ast }$ stays steady, but the identities holding each slot keep swapping — this is species turnover.

The same logic now rules parks and forest fragments ringed by farmland. When continuous habitat is chopped into pieces, each piece "relaxes" toward its own smaller $S^{\ast }$, shedding species for decades. That raises the central question of reserve design: is one large reserve better than several small ones of equal total area? This debate — SLOSS, Single Large or Several Small — is island biogeography turned into policy.

Visual Beginner

The equilibrium model as two crossing rate curves. Immigration falls as the island fills toward the mainland pool $P$; extinction rises as crowding grows. Where they cross sits $S^{\ast }$.

Rate
 ^
 | I_0  \                         immigration  I(S) = I_0 (1 - S/P)
 |       \                          (high on empty island -> 0 at full pool)
 |        \
 |         \                      extinction  E(S) = (e/A) * S
 |          \                       (rises with crowding; flatter on big islands)
 |           \      /
 |            \    /
 |             \  /
 |              \/  <-  equilibrium S*   (I = E here)
 |              /\
 |             /  \
 |            /    \   turnover T* = I(S*) = E(S*)
 |           /      \   (rate at which identities swap while S* holds)
 |          /        \
 |_________/__________\____________________>  S  (species on island)
 0        S*          P                     (mainland species pool)

How area and isolation move the crossing point:

Island type	Immigration curve	Extinction curve	Equilibrium $S^{\ast }$
Large and near	high (starts at large $I_0$)	flat (small slope $e/A$)	highest
Small and near	high	steep (large $e/A$)	intermediate
Large and far	low (small $I_0$)	flat	intermediate
Small and far	low	steep	lowest

The SLOSS choice, drawn at equal total area:

Option A — Single Large         Option B — Several Small
area A, one block               n blocks each of area A/n

+------------------------+      +------+ +------+ +------+
|                        |      |      | |      | |      |
|   S = c * A^z          |      |c(A/n)^z| |...  | |c(A/n)^z|
|                        |      |      | |      | |      |
+------------------------+      +------+ +------+ +------+

because z < 1, species            each block holds fewer species;
richness rises sublinearly        the UNION can exceed option A
with area, so concentrating      only if blocks differ in species
area usually wins                 (high beta diversity)

Worked example Beginner

A conservation agency can buy land for forest birds in one of two configurations with the same total area. For this region the species-area rule is $S=10A^{0.30}$, with $A$ in km${}^2$.

Option A — one large reserve of 100 km${}^2$:

$$S_A=10\times 100^{0.30}=10\times 3.98\approx 40\text{ species.}$$

Option B — four small reserves of 25 km${}^2$ each:

$$S_{\text{each}}=10\times 25^{0.30}=10\times 2.63\approx 26\text{ species per block.}$$

If the four blocks sit in the same forest and share most of their birds, the union is close to 26 species — far below the 40 of the single large block. If the blocks are spread across different habitats with little overlap, their union could approach $4\times 26=104$, but real reserves always share species, so the truth lies between.

What this tells us. Because $z$ is less than 1, doubling area does not double species: richness rises sublinearly with area, so concentrating habitat usually packs in more species than splitting it. That is the engine behind the Single Large side of SLOSS. The exception — when several small blocks actually win — is the subject of the deeper analysis below.

Check your understanding Beginner

Formal definition Intermediate+

Let an island lie at distance $D$ from a species-source mainland whose pool contains $P$ species. Define two rate functions of the number of species $S$ currently on the island ^{[MacArthur & Wilson 1967]}:

• the immigration rate $I(S)$, the number of new species arriving per unit time when $S$ species are already present;
• the extinction rate $E(S)$, the number of resident species lost per unit time.

In the linear model the two rates are straight lines:

$$I(S)=I_0\left(1-\frac{S}{P}\right),E(S)=\frac{e}{A}S,$$

where $I_0=I_0(D)$ is the maximum immigration rate (a decreasing function of isolation $D$), $e$ is a per-species extinction coefficient, and $A$ is island area. Two derived quantities carry the model's predictive content:

• the equilibrium species number $S^{\ast }$, defined by $I(S^{\ast })=E(S^{\ast })$;
• the turnover rate at equilibrium $T^{\ast }=I(S^{\ast })=E(S^{\ast })$, the rate at which species identities are exchanged while $S^{\ast }$ holds.

The species-area relationship (SAR) is the power law

$$S=c{A}^z⟺\log S=\log c+z\log A,$$

where $c$ is a taxon- and region-dependent constant and $z$ is the slope of the line on a log-log plot. For isolated islands and other true isolates, $z\in [0.25,0.35]$; for nested subareas censused within a single continuous mainland, $z\in [0.10,0.20]$ ^{[Rosenzweig 1995]}. The two ranges reflect different processes, examined in the Advanced results.

Core model [Intermediate+] {#core-model}

Setting $I(S^{\ast })=E(S^{\ast })$ and solving gives the equilibrium count and the turnover rate ^{[MacArthur & Wilson 1963]}:

$$S^{\ast }=\frac{P{I}_{0}A}{I_{0}A+eP},T^{\ast }=\frac{eP{I}_0}{I_{0}A+eP}.$$

The formula encodes four qualitative claims, each readable as a statement about which way the two rate lines move. Increasing area $A$ flattens the extinction line (its slope $e/A$ shrinks); the crossing point slides right and down, so $S^{\ast }$ rises while $T^{\ast }$ falls. Large islands therefore hold more species and turn over more slowly. Raising isolation depresses $I_0$; the immigration line pivots downward, the crossing slides left, and both $S^{\ast }$ and $T^{\ast }$ fall. Remote islands are species-poor and sluggish. Combining the two, the four island types rank as $S_{\text{large, near}}^{\ast }>S_{\text{large, far}}^{\ast }\approx S_{\text{small, near}}^{\ast }>S_{\text{small, far}}^{\ast }$, with the middle ordering parameter-dependent but the extremes robust.

The rescue effect: a correction to the basic model

Brown and Kodric-Brown (1977) observed that immigration does more than add new species — incoming individuals also prop up sinking populations of species already present, depressing their extinction rate ^{[Brown & Kodric-Brown 1977]}. The corrected extinction rate acquires an isolation dependence:

$$E(S,D)=\frac{e}{A}(1-\mu I_0(D))S,0\le \mu <1,$$

so near islands (large $I_0$) suffer lower extinction than far islands of identical size. The rescue effect amplifies the area-by-isolation interaction: it steepens the contrast between near and far islands beyond what the original model allows, and it explains why metapopulations of poor dispersers persist only within well-connected archipelagos.

Evidence: fumigation, volcanoes, and turnover

The most direct test is the Simberloff-Wilson mangrove experiment. Small mangrove islands in Florida Bay were censused for arthropods, encased in tents, and fumigated with methyl bromide to eliminate the fauna; recolonization was then tracked ^{[Simberloff & Wilson 1969]}. Three predictions were confirmed. Species numbers returned to approximately pre-defaunation levels, supporting an equilibrium. The particular species differed from the original community, confirming turnover. Larger and nearer islands regained species faster and supported higher equilibria, matching the area and isolation gradients. Natural experiments agree: Surtsey, which emerged from the sea south of Iceland in 1963, accumulated vascular plants along a decelerating curve toward an equilibrium consistent with its area and isolation; Krakatau, sterilized by eruption in 1883, recolonized along the same predicted trajectory.

The model's limitations are the entry to its modern descendants. It treats the species pool $P$ as fixed and external, ignoring in-situ evolution; yet oceanic islands are evolutionary engines whose endemic radiations (Hawaiian honeycreepers, Galapagos finches, anoline lizards) change the pool itself over time. It is silent on habitat heterogeneity, trophic structure, and environmental stochasticity. These gaps motivate the incidence-function, neutral, and general-dynamic generalizations taken up at Master tier.

Exercises Intermediate+

Advanced results [Master] {#advanced-results}

Three threads extend the equilibrium model: a theory of the exponent $z$, a per-species stochastic reformulation, and the resolution of SLOSS in terms of beta diversity. Each dissolves a limitation of the original theory without abandoning its core.

Why $z\approx 0.25$ to $0.35$: Preston's canonical lognormal

Preston (1962) observed that the abundances of species in a large community typically follow a lognormal distribution. If that lognormal has its "veil line" (the abundance below which species go unsampled) fixed by the sample size in the canonical position, the species-area slope predicted for an independent sample is $z\approx {\log }_{10}2/{\log }_{10}e\approx 0.301$ — Preston's canonical value, close to the empirical isolate mean ^{[Rosenzweig 1995]}. May (1975) recovered the same result from the sampling statistics of a broken-stick abundance model. The deeper point is Rosenzweig's typology: the species-area relationship is not one phenomenon but four. A within-area curve (nested sub-transects of one census) gives shallow $z\approx 0.13$ because the sub-samples share species by construction. A continental curve (independent points within one biome) gives $z\approx 0.15$ to $0.20$. An inter-provincial curve (comparing separate biogeographic provinces) gives steeper $z\approx 0.6$ to $1.0$, and the global curve steeper still. Only the first two are "ecological"; the last two mix ecological with evolutionary time and confound ecological turnover with phylogenetic divergence. Treating all species-area data as interchangeable is the most common error in applied uses of the model.

Incidence functions: from $S^{\ast }$ to per-species probabilities

The deterministic $S^{\ast }$ is the mean of an underlying stochastic process. For each species $i$, model presence as a birth-death process with colonization rate $c_i$ and local extinction rate $e_i$ ^{[Whittaker & Fernandez-Palacios 2007]}. At stochastic equilibrium the probability that species $i$ occupies the island is

$$p_i^{\text{eq}}=\frac{c_i}{c_i+e_i},$$

and the expected richness is $S^{\ast }={\sum }_i{p}_i^{\text{eq}}$, recovering the aggregate model as a sum over species-specific incidences. This reformulation does three things at once. It makes the rescue effect automatic, since $e_i$ depends on the immigration it receives. It connects island biogeography to metapopulation theory, of which it is the multi-species limit. And it admits heterogeneity: area-sensitive and isolation-sensitive species acquire different $c_i,e_i$, so the community is no longer a homogeneous aggregate. The price is parameter explosion: a community of $P$ species carries $2P$ rates, which must be estimated or constrained by traits.

The SLOSS resolution

The worked example and Exercise 5 encode the formal content of the SLOSS debate. For one block of area $A$ against $n$ blocks each of area $A/n$, the single-large count is $c{A}^z$ and the each-small count is $c(A/n{)}^z$. If each small block contributes a fraction $f$ of unique species, the union is $c(A/n{)}^z[(1-f)+nf]$. Single Large wins when $f<f^{\ast }$, Several Small when $f>f^{\ast }$, where $f^{\ast }=\frac{(A/(A/n){)}^z-1}{n-1}=\frac{n^z-1}{n-1}$. The threshold is set entirely by $z$ and $n$, and the data question is whether real beta diversity clears it. Diamond (1975) argued from incidence functions that large, connected reserves minimize extinction for area-sensitive species; Simberloff and Abele (1976) countered that high beta diversity could make a network of small reserves superior. The synthesis, due to Haila, Margules, Higgs, and others, is that no universal ranking exists: Single Large minimizes extinction risk for populations requiring large contiguous habitat, Several Small maximizes sampled diversity when habitats differ, and the practical answer requires knowing both $z$ and the beta-diversity structure of the region ^{[Losos & Ricklefs 2009]}.

Synthesis. Putting these together, the foundational reason the equilibrium model survives is that its two-rate structure appears again in every descendant: incidence functions are per-species rate equations, metapopulation theory is the rescue effect generalized, neutral theory is the same pool-colonization-extinction balance with demographic equivalence imposed, and the general dynamic model adds only an ontogenetic time axis to the same immigration-extinction bookkeeping. The central insight — that richness is a balance of input and loss rates, not a static property of a place — generalises from oceanic islands to forest fragments, mountaintops, lakes, and host-parasite systems alike, and this is exactly why habitat fragmentation behaves as a second-generation island biogeography. The bridge is that every patch in a fragmented landscape is an island whose $S^{\ast }$ has just been moved downward, and the debt it owes is the integral of the difference.

Full proof set Master

Proposition (equilibrium species number and turnover). In the linear MacArthur-Wilson model with $I(S)=I_0(1-S/P)$ and $E(S)=(e/A)S$, the equilibrium count and turnover rate are $S^{\ast }=\frac{P{I}_{0}A}{I_{0}A+eP}$ and $T^{\ast }=\frac{eP{I}_0}{I_{0}A+eP}$.

Proof. Equilibrium requires $I(S^{\ast })=E(S^{\ast })$, that is $I_0(1-S^{\ast }/P)=(e/A)S^{\ast }$. Expanding the left side gives $I_0-(I_0/P)S^{\ast }=(e/A)S^{\ast }$. Collecting the terms in $S^{\ast }$ onto the right yields $I_0=S^{\ast }(\frac{I_0}{P}+\frac{e}{A})$, and dividing through by the bracketed factor produces $S^{\ast }=I_0/(\frac{I_0}{P}+\frac{e}{A})$. Multiplying numerator and denominator by $PA$ gives $S^{\ast }=\frac{P{I}_{0}A}{I_{0}A+eP}$, the first claim. For the turnover, evaluate either rate at $S^{\ast }$: $E(S^{\ast })=(e/A)\cdot \frac{P{I}_{0}A}{I_{0}A+eP}=\frac{eP{I}_0}{I_{0}A+eP}$, where the factor $A$ cancels, and the same value is obtained from $I(S^{\ast })$ by direct substitution. This is $T^{\ast }$, the second claim. $\square$

Proposition (area loss and species loss). If habitat area is reduced to a fraction $\varphi \in (0,1]$ of its original value, the species-area law $S=c{A}^z$ predicts a retained fraction $S_{\text{new}}/S_{\text{old}}={\varphi }^z$ of species and a lost fraction $1-{\varphi }^z$.

Proof. Write $S_{\text{old}}=c{A}^z$ for the original area $A$ and $S_{\text{new}}=c(\varphi A{)}^z$ for the reduced area. Forming the ratio, the constant $c$ and the original area cancel: $S_{\text{new}}/S_{\text{old}}=c(\varphi A{)}^z/(c{A}^z)={\varphi }^z{A}^z/A^z={\varphi }^z$. Because $0<\varphi <1$ and $z>0$, the power ${\varphi }^z$ lies strictly between $\varphi$ and $1$, so the proportional species loss $1-{\varphi }^z$ is always smaller than the proportional area loss $1-\varphi$: halving habitat loses fewer than half the species, which is the asymmetry that makes the species-area curve a predictive tool for conservation. $\square$

Proposition (SLOSS threshold). Comparing one reserve of area $A$ to $n$ reserves each of area $A/n$ with per-block uniqueness fraction $f$, the two configurations protect equal species when $f=f^{\ast }=\frac{n^z-1}{n-1}$, with Several Small favored for $f>f^{\ast }$ and Single Large for $f<f^{\ast }$.

Proof. The single large block holds $S_{\text{L}}=c{A}^z$. Each small block holds $S_{\text{s}}=c(A/n{)}^z=c{A}^z{n}^{-z}=S_{\text{L}}{n}^{-z}$. Under the overlap model the union over $n$ small blocks is $S_{\text{union}}=S_{\text{s}}[(1-f)+nf]=S_{\text{L}}{n}^{-z}[1+(n-1)f]$. Setting $S_{\text{union}}=S_{\text{L}}$ and cancelling $S_{\text{L}}$ gives $n^{-z}[1+(n-1)f]=1$, that is $1+(n-1)f=n^z$. Solving for $f$ yields $f=(n^z-1)/(n-1)=f^{\ast }$. Because $S_{\text{union}}$ is strictly increasing in $f$ while $S_{\text{L}}$ is independent of it, the inequality reverses across the threshold: Several Small wins above $f^{\ast }$ and Single Large wins below it. Since $0<z<1$ implies $1<n^z<n$, the threshold $f^{\ast }$ lies in $(0,1)$, so a decisive answer always exists once beta diversity is measured. $\square$

Connections Master

• Biogeography (general) 19.12.01. This unit deepens the island subtheory sketched in the parent unit, which named the equilibrium idea but left its rate-curve mechanics, turnover algebra, rescue effect, and SLOS consequences for the present treatment. The species-area exponent $z$ studied here is the same quantity the parent used heuristically; here it receives its Prestonian and Rosenzweigian interpretation, and the four-type typology distinguishes the isolate $z$ from the mainland $z$ that the parent conflated.

• Community ecology 19.10.01. Island biogeography is at base a community-assembly model: it predicts how many species coexist given a pool and a set of arrival and loss rates. The parent community-ecology unit's treatment of interactions (competition, predation, mutualism) supplies the mechanism by which the extinction rate $E(S)$ steepens as the island fills, since biotic resistance and competitive displacement raise local extinction. Rescue-effect and incidence-function models are most realistic when $E(S)$ embeds those interaction terms rather than treating them as a single slope.

• Conservation biology 19.14.01. The SLOSS threshold, the extinction-debt calculation, and the species-area prediction of biodiversity loss from habitat reduction are direct inputs to reserve design and red-list forecasting. The Full proof set's area-loss proposition is the equation behind estimates of future species commitment under deforestation scenarios, and the SLOSS proposition converts directly into the decision rule a reserve planner applies when allocating a fixed conservation budget across one large or several small parcels.

• Metapopulation dynamics 19.09.02 pending. The incidence-function reformulation $p_i^{\text{eq}}=c_i/(c_i+e_i)$ is, species-by-species, the Levins metapopulation model. The rescue effect is the colonization-rescue term of metapopulation persistence, and an archipelago of habitat fragments is precisely the spatial setting in which the single-island $S^{\ast }$ must be replaced by a coupled patch network whose colonization rates depend on inter-patch distances.

• Natural selection and speciation 19.03.01. The equilibrium model is ecological: it takes the mainland pool $P$ as fixed. Over evolutionary time the pool itself is not fixed, and adaptive radiation on isolated archipelagos generates the endemic species that constitute an island's deviation above its immigration-only equilibrium. The selective pressures and founder-event speciation treated in the evolution units are what turn an island from a passive sampler of the mainland into a generator of novelties, which is the engine MacArthur and Wilson's purely ecological model does not capture.

Historical & philosophical context Master

Island biogeography before 1963 was a descriptive enterprise. Wallace, Darlington, and Mayr had catalogued the regularities — more species on larger islands, fewer on remote ones, impoverished but endemic biotas on oceanic islands — without a predictive mechanism. Robert MacArthur and E. O. Wilson changed this with a 1963 paper in Evolution and the 1967 monograph The Theory of Island Biogeography ^{[MacArthur & Wilson 1963]}, casting species number as the equilibrium of two measurable rate processes. The move was programmatic: ecology, they argued, could aspire to the predictive, model-based form of physics, with immigration and extinction playing the roles of birth and death rates in a demographic balance.

The 1969 Simberloff-Wilson mangrove fumigation experiment was the field's first manipulative test of a population-level theory and remains its most cited ^{[Simberloff & Wilson 1969]}. It also seeded the discipline's longest-running policy argument. Jared Diamond's 1975 design principles for nature reserves, derived from incidence functions, asserted that large, connected reserves invariably outperform small, isolated ones. Daniel Simberloff and Lawrence Abele's 1976 rebuttal in Science argued from the same theory that a network of small reserves could hold more species when beta diversity was high. The SLOSS debate, sharpened through the 1980s by Higgs, Usher, Margules, and Haila, was eventually resolved not by victory for either side but by the recognition that the answer is conditional on measurable quantities — the area exponent $z$ and the between-site complementarity $f$ — rather than on a universal rule ^{[Losos & Ricklefs 2009]}.

The 1977 rescue effect of Brown and Kodric-Brown and the 2001 neutral theory of Hubbell each challenged a simplifying assumption of the original model. The rescue effect showed that immigration and extinction are not independent, since the former depresses the latter. Neutral theory showed that the species-area exponent $z\approx 0.25$ could be recovered from a model with no niches and no equilibrium at all, raising the question of whether the regularities MacArthur and Wilson explained were evidence for their mechanism or merely compatible with it. The current view holds both: equilibrium and neutral processes operate at different scales and time horizons, and their signatures are distinguishable in turnover and abundance-distribution data.

The most recent extension is Whittaker, Triantis, and Ladle's general dynamic model, which adds an ontogenetic axis: real islands are not static targets but pass through youth, maturity, and old age, with area, elevation, and habitat diversity changing over geological time. The immigration-extinction equilibrium still holds, but its parameters become functions of island age. This recovers the original model as the snapshot at a fixed ontogenetic stage and reconciles the equilibrium view with the evolutionary reality that MacArthur and Wilson bracketed away — that islands are, over sufficient time, generators of species, not merely collectors of them.

Bibliography Master

1. MacArthur, R. H. & Wilson, E. O. "An equilibrium theory of insular zoogeography." Evolution 17 (1963) 373-387.

2. MacArthur, R. H. & Wilson, E. O. The Theory of Island Biogeography. Princeton University Press, Princeton, 1967.

3. Preston, F. W. "The canonical distribution of commonness and rarity." Ecology 43 (1962) 185-215, 410-432.

4. Simberloff, D. S. & Wilson, E. O. "Experimental zoogeography of islands: the colonization of empty islands." Ecology 50 (1969) 278-296.

5. Diamond, J. M. "The island dilemma: lessons of modern biogeographic studies for the design of natural reserves." Biological Conservation 7 (1975) 129-146.

6. Simberloff, D. S. & Abele, L. G. "Island biogeography and conservation: strategy and tactics." Science 193 (1976) 893.

7. Brown, J. H. & Kodric-Brown, A. "Turnover rates in insular biogeography: effect of immigration on extinction." Ecology 58 (1977) 445-449.

8. May, R. M. "Patterns of species abundance and diversity." In Cody, M. L. & Diamond, J. M. (eds.) Ecology and Evolution of Communities, Harvard University Press (1975) 81-120.

9. Higgs, A. J. & Usher, M. B. "Should nature reserves be large or small?" Nature 285 (1980) 568-569.

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

11. Whittaker, R. J. & Fernandez-Palacios, J. M. Island Biogeography: Ecology, Evolution, and Conservation, 2nd ed. Oxford University Press, Oxford, 2007.

12. Hubbell, S. P. The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, Princeton, 2001.

13. Whittaker, R. J., Triantis, K. A. & Ladle, R. J. "A general dynamic theory of oceanic island biogeography." Journal of Biogeography 35 (2008) 977-994.

14. Losos, J. B. & Ricklefs, R. E. "Adaptation and diversification on islands." Nature 457 (2009) 830-836.