The Species-Area Relationship
The most robust empirical pattern in biogeography is the species-area relationship (SAR):
\[S = c A^z\]or equivalently, $\log S = \log c + z \log A$, where $S$ is species richness, $A$ is island area, $c$ is a taxon- and region-specific constant, and $z$ is the slope on a log-log plot.
Typical values of $z$:
| Context | $z$ range |
|---|---|
| True oceanic islands | $0.25$–$0.35$ |
| Habitat islands (fragments) | $0.20$–$0.35$ |
| Within continuous areas (species-area curves) | $0.10$–$0.18$ |
| Across continents (inter-provincial) | $\approx 0.65$ |
The steeper slope for true islands reflects the additional filtering imposed by overwater dispersal. The mechanistic explanation from MacArthur-Wilson is that both immigration and extinction rates depend on $A$ through their effects on population sizes.
Equilibrium Theory: Immigration and Extinction Rates
MacArthur and Wilson modeled species richness as a dynamic equilibrium. Let $P$ be the size of the mainland species pool and $S$ the number of species currently on the island.
Immigration rate $I(S)$: the rate at which new species (not yet on the island) arrive. When $S = 0$, all mainland species can arrive ($I$ is maximal). When $S = P$, no new species are possible ($I = 0$). A linear approximation:
\[I(S) = \lambda(P - S)\]where $\lambda$ is the per-species immigration rate. Distance reduces $\lambda$: $\lambda \propto e^{-\alpha d}$ for distance $d$.
Extinction rate $E(S)$: the rate at which resident species go locally extinct. When $S = 0$, extinction is impossible. As $S$ increases, each species occupies smaller average populations, increasing extinction risk:
\[E(S) = \mu S\]where $\mu$ is the per-species extinction rate. Larger islands have smaller $\mu$ because they support larger populations: $\mu \propto A^{-\gamma}$.
Equilibrium species richness: setting $I(S^) = E(S^)$:
\[S^* = \frac{\lambda P}{\lambda + \mu}\]Species turnover at equilibrium: $T^* = I(S^) = E(S^) = \frac{\lambda\mu P}{\lambda + \mu}$. This predicted turnover—species replacement without change in $S$—was confirmed by Wilson and Simberloff’s mangrove island defaunation experiments in the Florida Keys (1969).
Distance and Area Effects
Both effects enter naturally:
- Distance effect: Far islands have lower $\lambda$ (fewer propagules arrive), so $S^_\text{far} < S^_\text{near}$ at the same area.
- Area effect: Large islands have lower $\mu$, so $S^_\text{large} > S^_\text{small}$ at the same distance.
These can be combined. With $\lambda = \lambda_0 e^{-\alpha d}$ and $\mu = \mu_0 / A^\gamma$:
\[S^* = \frac{\lambda_0 e^{-\alpha d}\,P}{\lambda_0 e^{-\alpha d} + \mu_0 A^{-\gamma}}\]This recovers the qualitative predictions of both the SAR and the distance effect, and unifies them in a single mechanistic expression.
Applications to Conservation: The SLOSS Debate
The theory sparked the SLOSS debate (Single Large Or Several Small): does one large reserve hold more species than several small reserves of the same total area? The SAR suggests the large reserve wins if $z > 0$, since:
\[S_\text{large}(nA) = c(nA)^z = c n^z A^z > n \cdot c A^z = n \cdot S_\text{small}(A)\]because $n^z < n$ for $z < 1$. However, the debate is more nuanced in practice: several small reserves may capture more habitat heterogeneity, reduce disease transmission, and hedge against catastrophe. Modern reserve design incorporates connectivity (dispersal corridors) alongside size, extending the MacArthur-Wilson framework into the metapopulation paradigm.