Biodiversity and ecosystem function in R

The biodiversity-ecosystem function relationship in R: relative yields, overyielding, and why beating the average monoculture differs from beating the best.

A recurring result in community ecology is that a mixture of species often produces more (more biomass, more yield, more of whatever function you measure) than you would guess from growing each species on its own. This is the biodiversity-ecosystem function relationship, and it sits behind a large experimental literature (Tilman et al. 2001 Science 294(5543):843-845; Cardinale et al. 2012 Nature 486(7401):59-67).

The idea is simple, but two versions of it get mixed up, and they are not the same claim. The first: does a mixture beat the average of its component monocultures? The second: does a mixture beat the best single species grown alone? The first is the everyday biodiversity-ecosystem function signal. The second, called transgressive overyielding, is a much stronger statement, and it is far rarer. This post builds both in R, and the follow-up posts take apart why a mixture overyields.

One mixture, worked by hand

Take four grassland species with monoculture yields of 100, 250, 400 and 600 g/m². We sow them in an equal, replacement (substitutive) mixture: each species gets a quarter of the plot, so its expected relative yield is 0.25. With no interaction at all, each species should return a quarter of its monoculture, and the mixture should yield the average monoculture.

M   <- c(100, 250, 400, 600)
RYe <- rep(1/4, 4)
YE  <- sum(RYe * M)          # expected: no interaction
RYo <- RYe + 0.10           # observed relative yields: a complementarity boost
YO  <- sum(RYo * M)         # observed mixture yield
c(expected = YE, observed = YO, net = YO - YE, RYT = sum(RYo))

expected observed      net      RYT 
   337.5    472.5    135.0      1.4 

The expected yield is 337.5 g/m², exactly the mean of the four monocultures. The observed mixture returns 472.5, so the net biodiversity effect is 135.0 g/m²: the mixture overyields its average monoculture by 40%.

Now the second question. The best single species yields 600 g/m² on its own. The mixture returns 472.5, which falls 21% short of that. So the same mixture that clearly beats the average monoculture still loses to the best one. Both statements are true at once, and reporting only the first while implying the second is one of the most common overstatements in the field.

d <- data.frame(sp = factor(c("A","B","C","D","mixture")),
                y  = c(M, YO),
                kind = c(rep("monoculture",4), "observed mixture"))
ggplot(d, aes(sp, y, fill = kind)) +
  geom_col(width = 0.7) +
  geom_hline(yintercept = YE, colour = col_exp, linewidth = 0.9, linetype = "dashed") +
  geom_hline(yintercept = max(M), colour = col_ref, linewidth = 0.7, linetype = "dotted") +
  annotate("text", x = 1.1, y = YE + 18, label = "expected (average monoculture)",
           colour = col_exp, hjust = 0, size = 3.4) +
  annotate("text", x = 1.1, y = max(M) + 18, label = "best monoculture",
           colour = col_ref, hjust = 0, size = 3.4) +
  scale_fill_manual(values = c("monoculture" = col_mono, "observed mixture" = col_mix)) +
  labs(x = NULL, y = expression(Yield~(g/m^2)), fill = NULL,
       title = "Overyielding is measured against the average, not the best")

Figure 1: Four monoculture yields (grey), the expected mixture (no interaction, ochre) and the observed mixture (green). The mixture beats the average monoculture but not the tallest bar.

Relative yields and the total

The quantity doing the work above is the relative yield: a species’ yield in the mixture divided by its yield in monoculture. Summed over the species, it gives the relative yield total (RYT). If every species returns exactly its sown fraction, RYT is one and there is no net effect. Here RYT is 1.40, above one, which is the signature of complementarity: together the species use the plot more completely than the replacement expectation (Loreau 1998 Oikos 82(3):600-602). RYT below one would mean the species interfere with each other.

Relative yield is why monoculture references matter so much. Every number in this literature is anchored to how each species performs alone, measured in the same conditions. Get the monocultures wrong and the whole partition drifts; the checking post returns to this.

The relationship across a richness gradient

One mixture is an anecdote. The biodiversity-ecosystem function relationship is a curve: function plotted against the number of species. Here is a synthetic experiment with a pool of 12 species, drawing many random mixtures at each richness level from one to eight and giving each mixture a modest complementarity boost.

ggplot(res, aes(k, mixY)) +
  geom_jitter(width = 0.15, height = 0, alpha = 0.15, colour = col_mix) +
  geom_line(data = agg, aes(k, mixY), colour = col_mix, linewidth = 1) +
  geom_line(data = agg, aes(k, meanMono), colour = col_mono, linewidth = 1) +
  geom_line(data = agg, aes(k, maxMono), colour = col_ref, linewidth = 0.8, linetype = "dotted") +
  scale_x_continuous(breaks = 1:8) +
  labs(x = "Species richness", y = expression(Yield~(g/m^2)),
       title = "The biodiversity-ecosystem function curve")

Figure 2: Mixture yield rising with species richness (green points, jittered; line is the mean). The average monoculture (grey) stays flat; the best monoculture (dotted) climbs because richer draws are more likely to include a productive species.

Mean mixture yield climbs from 355 g/m² in monoculture to 591 at eight species, a log-linear rise of about 107 g/m² per unit of log richness. The net biodiversity effect at eight species averages 229 g/m². The average monoculture line, by contrast, is flat: adding species to a plot does not change what the typical species does alone.

The dotted line is the quiet complication. As richness rises, the best monoculture in the draw also rises, because a larger random sample is more likely to contain a highly productive species. That is the sampling effect (Huston’s concern, formalised later): part of what looks like a diversity benefit is just a better chance of including a winner. Across all mixtures of two or more species here, only 19% actually beat their best monoculture. The average is easy to beat; the best is not.

What the curve does and does not say

A rising biodiversity-ecosystem function curve is real and repeatable, and it is a genuine argument that species number matters for function (Hector et al. 1999 Science 286(5442):1123-1127; Isbell et al. 2011 Nature 477(7363):199-202). But on its own it does not tell you why. A positive net effect can come from species partitioning resources, from one species subsidising the others, or simply from a productive species being present and dominating. RYT above one narrows it to “the mixture uses the site more completely than expected”, and no further.

Separating those causes is the job of the additive partition, which splits the net effect into a complementarity part and a selection part. That is the next post. For now the honest summary is two sentences: mixtures here beat their average monoculture almost always, and beat their best monoculture about one time in five. Keep those two claims apart and most of the confusion in this literature disappears.

Related tutorials

• Partitioning the net biodiversity effect
• Functional diversity from traits
• Alpha diversity indices in R
• Additive diversity partitioning

References

• Cardinale BJ et al. 2012. Nature 486(7401):59-67 (10.1038/nature11148).
• Hector A et al. 1999. Science 286(5442):1123-1127 (10.1126/science.286.5442.1123).
• Isbell F et al. 2011. Nature 477(7363):199-202 (10.1038/nature10282).
• Loreau M 1998. Oikos 82(3):600-602 (10.2307/3546381).
• Tilman D et al. 2001. Science 294(5543):843-845 (10.1126/science.1060391).