Lecture notes. Written from the material of MIT 8.591J 2014, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
18. Neutral Theory of Species Abundance¶
What this covers¶
This is the last lecture of the semester, and it is deliberately a change of scale: everywhere else the course worked from enzyme kinetics and molecular binding up to stochastic gene expression, all at the scale of one cell. Here the same master-equation machinery is pointed at a 50-hectare plot of rainforest, to ask why some tree species are common and some are rare. The chapter assumes the birth-death master equation and the trick of solving a one-step chain at steady state by balancing probability fluxes — both used earlier in the course for stochastic mRNA counts — and asks what happens when the birth rate, instead of being constant, is proportional to how many individuals are already there.
The question it answers is: what do two very different families of model — one that assumes species are ecologically different, one that assumes they are identical — actually predict about the pattern of relative species abundance (RSA), and how much can fitting that pattern to data tell you about which family is right?
The data: a fifty-hectare census¶
The paper under discussion analyses a single data set: Barro Colorado Island (BCI), a 50-hectare plot in the Panama Canal — an island only because the canal was built around it about a hundred years ago, and close enough to the mainland that cougars swim across. A hectare is \(10^4\,\text{m}^2\), so 50 hectares is a \(100\,\text{m}\times 100\,\text{m}\) square repeated fifty times, roughly half a square kilometre.
On this plot, biologists identified every tree with diameter greater than 10 cm at breast height (DBH) — the threshold you need so the counting is well defined — and assigned each one a species. That gave 21,457 trees sorted into 225 distinct species. The point of restricting to one trophic level (here, canopy trees, elsewhere beetles, or birds) is that the species in it are plausibly competing with each other for the same kind of resource, which is exactly the situation the models below are trying to describe.
Reading the abundance histogram¶
The paper has one figure, and it is worth being careful about what it actually shows, because most people misread it. The figure plots the number of species that have a given number of individuals, binned on a \(\log_2\) scale of individuals. Read naively, the tallest bar sits around 30 individuals, and it is tempting to call that the mode — the "typical" abundance of a species.
That is wrong, and the reason is the binning. On a log scale the bins grow geometrically: the first bar covers essentially one integer value, but a bar far to the right covers a whole range — everything from roughly 20-something individuals up to 50. So the height of a bar out there is already a sum over many true abundance values, and a tall bar there does not mean any single abundance value is common. To find the real mode you have to undo the binning.
The paper actually tells you how: the first bar is not the number of species with one individual, \(\phi_1\), but \(\phi_1/2\). Reading that bar as 9 gives \(\phi_1 = 18\). The next plotted bar is \((\phi_1+\phi_2)/2\), and unwinding the arithmetic bar by bar recovers the true, linearly-binned counts: 18 species with exactly one individual, 19 with two, 13 with three, 9 with four, 6 with five, and falling off from there — 5, then 4, then 3, and so on, becoming too sparse to pin down exactly. Nothing in that sequence looks anything like a bump at 30. The mode of the distribution is 1.
The mean, by contrast, is \(21457/225 \approx 95\), just under 100 — pulled far above the mode by a handful of extremely abundant species (a couple with well over a thousand individuals). The median sits close to where the log-binned figure's visually tallest bar actually is, which is presumably why that bar gets mistaken for the mode. So mean, median and mode are wildly different numbers for this distribution: roughly 100, 30, and 1. A distribution this skewed cannot even be plotted sensibly on a linear axis — which is the honest reason to use the log-binned figure — but plotting it that way leaves you with the false mental picture that most species have a moderate, similar-sized population. The correct one-line summary, worth holding onto: rare species are common, and common species are rare. This is not particular to BCI; it is what people have found wherever they have counted, for hundreds of years (Darwin himself remarked on it), regardless of exactly how fast the tail falls off or whether the plot is an island or a mainland.
The species-area relationship¶
A second robust empirical pattern: if you count species in a larger area, you find more of them, but sublinearly. If \(S\) is the number of species observed and \(A\) the area sampled, $$ S \propto A^{z}, \qquad z \approx \tfrac14 . $$ \(z<1\) has to hold — if it were 1, doubling the area would double the species count, meaning a second identical plot shares essentially no species with the first, which is not plausible. Both of the model families below reproduce a power law of roughly this shape from quite different microscopic pictures, which is a first hint of the chapter's real theme: a macroscopic pattern being explained by a model does not mean the model's assumptions are the reason nature looks that way. Many different microscopic processes generate power laws.
Niche models: breaking the resource axis¶
The niche-theory account starts from MacArthur's 1957 broken-stick model. Picture a single, homogeneous resource axis — some scarce thing that limits how many individuals a species can support — and imagine dividing it among \(N\) species, with each species' abundance proportional to the length of axis it captures.
The naive way to divide it is to drop \(N-1\) points uniformly at random on the stick and let the gaps between them be the species' shares. Class discussion talked through why this does not give a log-normal-shaped abundance distribution: uniform random points cannot produce either a very long gap or a very short one with any appreciable frequency, so the resulting shares are far less spread out than the real, heavily skewed data. A single random partition is the wrong generative picture.
What does work is sequential, hierarchical breaking — the same axis broken multiple times, each break acting only on the piece produced by the break before it. The motivating example given in class: a bird community first divides its foraging resource between ground foragers and tree foragers (say 30% versus 70%); within tree foraging, the resource divides again between trunk and branch feeders; within branch feeding, it divides again between surface grubs and grubs under the bark — and so on, at whatever level actually corresponds to real speciation events.
Repeated enough times, this sequential breaking is exactly the process that gives a log-normal distribution — and it has nothing special to do with biology. If you take a stone and crush it, weigh the resulting fragments, the mass distribution is log-normal, for the same structural reason: a piece breaks into smaller pieces, those break again, and a quantity built by successive multiplicative shrinkage is log-normal by the same logic that a sum of independent random quantities is normal (central limit theorem, but in log space, since multiplying corresponds to adding logs). So a log-normal-shaped RSA distribution is evidence for some multiplicative breaking process having gone into the abundances — but it is not, by itself, evidence for any particular biological story about what did the breaking, or how many times, or according to what probability rule. All of that is a separate, harder empirical question — and it was flagged in class that a fair amount of work goes into asking what distribution actually governs each break (uniform, or tilted, e.g. because whoever arrives at a resource first can monopolize a disproportionate share of it).
Neutral theory: an island fed by a metacommunity¶
The competing account, neutral theory, makes an almost absurdly minimal assumption: every individual is identical, regardless of species. Same birth rate, same death rate, no advantage or disadvantage from being a member of a large or small population (no Allee effect), no species-specific competition. The claim is not that this is biologically true — it obviously is not — but that it may already be sufficient to reproduce the observed pattern, which is itself an important thing to learn if true.
The setup: a large metacommunity and a small island of fixed size \(J\) individuals (the island stands in for the 50-hectare plot). Each cycle, the process is a Moran process: pick one random individual on the island and remove it. With probability \(m\), replace it with an individual drawn at random from the metacommunity (so migrants arrive in proportion to how abundant a species already is on the mainland); with probability \(1-m\), replace it by copying another individual already on the island (a birth event local to the island). The island's total size never changes. As \(m \to 0\) this is pure ecological drift and one species eventually takes over the island entirely, exactly as in genetic drift; as \(m\) grows large, the island's composition increasingly mirrors the metacommunity's.
Speciation — new species appearing — is assumed to happen only in the metacommunity, at some small constant rate, and to be negligible on the island itself, because the island is small.
The metacommunity distribution: Fisher's log series¶
The metacommunity's own steady-state abundance distribution can be solved exactly, by the same detailed-balance argument used earlier in the course for one-step birth-death chains: at equilibrium, the probability flux up from state \(n\) must equal the flux down into it, $$ P_n\, b_n = P_{n+1}\, d_{n+1}. $$ Here the birth and death rates are, respectively, $$ b_0 = \text{(constant speciation rate)}, \qquad b_n = b\,n \ (n\ge 1), \qquad d_n = d\,n, $$ i.e. a species with \(n\) individuals gains new members at a rate proportional to \(n\) and loses them at a rate proportional to \(n\) — every individual can give birth and every individual can die — plus a constant trickle of brand-new species entering at \(n=1\). Telescoping the flux balance, $$ P_n = P_0\left(\frac{b_0}{d}\right)\left(\frac{b}{d}\right)^{n-1}\prod_{k=1}^{n-1}\frac{k}{k+1} = P_0\,\frac{b_0}{b}\cdot\frac{x^n}{n}, \qquad x \equiv \frac{b}{d}, $$ using \(\prod_{k=1}^{n-1} k/(k+1) = 1/n\). This is Fisher's log series: the expected number of species with \(n\) individuals in the metacommunity is proportional to \(x^n/n\).
It is worth pausing on why this differs from the mRNA steady state solved earlier in the course, which came out Poisson from the same kind of balance argument. There, the birth rate (transcription) was a constant, independent of how much mRNA was already present, while the death rate (degradation) was proportional to the amount present. Here it is the opposite: the birth rate is proportional to how many individuals already exist — a species with twice as many members gains new members twice as fast — while the death rate is also proportional. Constant-birth against linear-death gives Poisson; linear-birth against linear-death gives the much heavier-tailed \(x^n/n\) instead. Convergence requires \(x = b/d < 1\), i.e. death rates exceed birth rates for every species individually — which sounds like it should drive everything extinct, and would, if there were no constant speciation term \(b_0\) continually replenishing the pool of species at \(n=1\). Every individual species does go extinct eventually; the metacommunity distribution is stationary only because new ones are constantly created to replace them.
Metacommunity versus island¶
The metacommunity's Fisher log series and the island's realized abundance distribution are not the same object, and it is worth asking which falls off faster in the tail. Because of the extra \(1/n\) factor, \(x^n/n\) already decays faster than a plain geometric series would. But the island distribution, built from the metacommunity by migration plus local drift, turns out to be more heavy-tailed — more skewed toward abundant species — than the metacommunity itself. The mechanism: migration onto the island samples the metacommunity in proportion to abundance there, so abundant metacommunity species keep getting reinforced on the island, while rare species on the island are constantly at risk of drifting to local extinction and are replaced disproportionately often by migrants from the metacommunity's already-abundant species. The net effect is that the island ends up with relatively more frequent, abundant species than the mainland it is fed from — the migration process itself pushes the island's distribution toward the abundant end.
Can the shape of the data tell the models apart?¶
Both families — the niche hierarchy (log-normal) and the neutral island-metacommunity model — give curves that fit the BCI abundance data about equally well; nothing resembling a decisive chi-squared verdict is available, partly because with counts this small the sampling noise on each bin (of order \(\sqrt{n}\)) is already comparable to the difference between the two fitted curves.
Nor does a naive count of free parameters settle it. Both papers claim to have fewer free parameters than their competitor, which cannot both be right, and the disagreement comes down to bookkeeping: the neutral model's authors do not count the community size \(J\) as free, because it is externally measured from the data (the number of individuals sampled) rather than fit. But by the same logic, the log-normal fit's overall amplitude is also fixed by the measured number of individuals rather than truly free — so if \(J\) doesn't count as a parameter, the amplitude shouldn't either, and the two models end up with the same effective parameter count.
The lesson, stated plainly in class: writing down a model that is consistent with an observation does not prove that model's assumptions are correct. When two models built on essentially opposite assumptions — species are ecologically different, versus species are identical — both reproduce the same static abundance pattern and the same species-area scaling, that pattern is telling you less about the underlying microscale process than it might seem to.
What does discriminate between them is dynamics, not the static snapshot. Neutral theory predicts that today's abundant species are only transiently abundant — since no species is intrinsically better suited than any other, dominance should drift and turn over. Niche-based models predict that an abundant species is abundant because it is actually well-matched to its niche, and so should stay abundant. The observation reported in class is that, empirically, abundant species tend to persist for longer than the neutral model predicts — evidence against strict neutrality. That does not make the neutral model useless; on the contrary, the fact that a model with no species-level differences at all can nonetheless reproduce the RSA pattern and the species-area law is itself the valuable, surprising result. It shows how little of the macroscopic pattern actually depends on species being different — and, as with the physical crushed-stone analogy, is a caution about reading biological content into a shape a model produces for purely mathematical reasons.
Sources¶
- MIT 8.591j Systems Biology (Fall 2014), final lecture, transcript
recordings/m41dwardioc.md(converted fromm41dwardioc-captions.srt). This chapter draws on the whole transcript; timestamps for the main threads: BCI data description [04:45–11:49]; reading the abundance figure and reconstructing the true histogram [12:55–30:57]; species-area relationship [47:09–51:44]; niche/broken-stick models [33:25–47:09]; neutral theory setup [51:44–59:23]; Fisher log series derivation [59:23–1:09:22]; metacommunity-versus-island comparison [1:09:22–1:11:40]; model comparison and free parameters [1:11:40–1:17:07]. - Not supplied, referred to only: the assigned paper the lecture discusses — described as giving a closed-form (semi-analytic) alternative to simulating the neutral model, and containing the single figure being reconstructed — is not named in the transcript and is not in this source set. Its Figure 1, and the professor's board redrawing of it, are likewise not captured here; the histogram values in this chapter are the professor's own verbal reconstruction of the underlying linear counts, not a reproduction of the figure. An email Q&A sent to students before class about how to read the figure is mentioned but not reproduced. MacArthur's 1957 broken-stick paper is referred to by author and year only. The classic RSA data sets used as points of comparison (beetles on the Thames, Fisher's original data) and comparative mainland-versus-island abundance measurements are mentioned as existing but not detailed.
- No slides, written notes, or exercises were supplied for this lecture (transcript-only source).
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