The additive partition of the net biodiversity effect is exact arithmetic, which makes it easy to trust more than it deserves. The identity always holds; whether the two numbers mean anything depends on the monocultures, the experimental design, and whether one species is quietly running the whole thing. Here are three checks, each a few lines of R, that a biodiversity-ecosystem function result should survive.
Every relative yield divides a mixture yield by a monoculture yield, so a biased monoculture biases the whole partition. The most common way to get this wrong is to grow the monocultures at a different density from the mixture components, so they underperform. Underperforming monocultures make every species look like an overachiever in the mixture.
Mtrue <- c(120, 300, 450, 220, 600); RYe <- rep(1/5, 5)
dRY <- 0.06 + 0.0002 * (Mtrue - mean(Mtrue))
Yobs <- (RYe + dRY) * Mtrue # observed mixture yields (fixed)
rbind(honest = round(lh(Mtrue, Yobs / Mtrue, RYe), 2),
biased_M = round(lh(0.8 * Mtrue, Yobs / (0.8 * Mtrue), RYe), 2)) dY comp sel
honest 130.22 101.4 28.82
biased_M 197.82 169.0 28.82With honest references the net effect is 130.2 g/m2. Measure the monocultures a fifth too low and the same mixture data return a net effect of 197.8, an inflation of 52%, almost all of it landing in the complementarity effect. Nothing about the mixture changed; only the denominator did. The check is to ask, before believing any complementarity number, whether the monocultures were grown at matched density and conditions, and to treat species that barely persist alone as a warning rather than a data point.
db <- data.frame(effect = rep(c("net","complementarity","selection"), 2),
ref = rep(c("honest","biased M"), each = 3),
val = c(part_true, part_bias))
db$effect <- factor(db$effect, levels = c("net","complementarity","selection"))
ggplot(db, aes(effect, val, fill = ref)) +
geom_col(position = "dodge", width = 0.7) +
scale_fill_manual(values = c("honest" = col_true, "biased M" = col_bias)) +
labs(x = NULL, y = expression(Effect~(g/m^2)), fill = NULL,
title = "A fifth-too-low monoculture inflates the net effect")
A substitutive (replacement) design holds total density constant: a four-species mixture puts each species at a quarter density. An additive design adds species at full density, so the mixture is four times as dense as each monoculture. That difference alone produces more yield, and if you feed an additive experiment into the partition as if it were substitutive, the extra density is misread as a diversity effect.
a <- c(1.8, 2.2, 2.0, 2.4); dfull <- 100; RYe4 <- rep(1/4, 4)
mono <- a * dfull # linear yield, NO interaction
sub_each <- a * (dfull / 4) # substitutive: quarter density each
add_each <- a * dfull # additive: full density each
rbind(substitutive = round(lh(mono, sub_each / mono, RYe4), 2),
additive = round(lh(mono, add_each / mono, RYe4), 2)) dY comp sel
substitutive 0 0 0
additive 630 630 0There is no interaction here at all: yield is a fixed multiple of density. The substitutive design correctly returns a net effect of 0. The additive design returns 630 g/m2 of pure phantom overyielding, all of it from the extra plants. The check is to know which design produced the data, and to compare a mixture only with monocultures at the matching total density.
Even substitutive designs carry a quieter assumption: that yield is proportional to density. When the yield-density curve saturates, a species grown at a quarter density returns more than a quarter of its full-density monoculture, and the partition reports complementarity that is really nonlinearity. In the saturating case here, with still no interaction, the net effect comes out at 66.7 g/m2, entirely spurious.
The selection effect is a covariance, and a covariance can be set by a single extreme point. If one species has a far larger monoculture yield than the rest, its term in the sum can swamp everyone else’s.
Mc <- c(90, 140, 180, 220, 900); RYe5 <- rep(1/5, 5)
dRYc <- c(0.14, 0.10, 0.06, 0.02, 0.20)
part_c <- lh(Mc, RYe5 + dRYc, RYe5)
cross <- (dRYc - mean(dRYc)) * (Mc - mean(Mc)) # per-species contribution to selection
round(rbind(partition = part_c), 2) dY comp sel
partition 221.8 159.12 62.68round(cross, 2)[1] -7.78 0.66 5.54 7.22 57.02The selection effect is 62.7 g/m2 and looks positive: the productive species overperform. But one species (monoculture yield 900) supplies 91% of it. Among the other four, the relationship between monoculture yield and relative-yield change is the opposite sign, so removing that one species turns the selection contribution to -8.6, a reversal. A selection effect that flips when you drop a single species is a description of that species, not of the community.
dc <- data.frame(M = Mc, dRY = dRYc, grp = c(rep("other four",4), "dominant"))
ggplot(dc, aes(M, dRY)) +
geom_smooth(data = dc, method = "lm", se = FALSE, colour = col_true, linewidth = 0.9) +
geom_smooth(data = subset(dc, grp == "other four"), method = "lm", se = FALSE,
colour = col_bias, linewidth = 0.9, linetype = "dashed") +
geom_point(aes(shape = grp), size = 3, colour = col_ref) +
scale_shape_manual(values = c("dominant" = 17, "other four" = 16)) +
labs(x = expression(Monoculture~yield~(g/m^2)), y = "Change in relative yield",
shape = NULL, title = "One extreme species sets the selection slope")
These checks bound a biodiversity-ecosystem function result; they do not validate the biology. A partition can survive all three, with matched monocultures, a clean substitutive design, and no single dominant species, and still only describe a pattern. It says the mixture used the site more completely than expected and tells you whether the gain was spread evenly or concentrated in the productive species. It does not tell you the mechanism, and its own interpretation is still debated (Hooper et al. 2005 Ecological Monographs 75(1):3-35; Pillai and Gouhier 2019 Ecology 100(7):e02645). A net effect that is corroborated only by the partition is a weaker claim than one backed by a measured process. Run the checks, report both effects, and keep the mechanism as a separate question.
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