The niche model of food webs

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The niche model of food webs in R, compared to the cascade and random models: one feeding axis reproduces real structure, a random graph does not.

Author

Tidy Ecology

Published

2026-07-21

The previous post measured a food web that was written down by hand. This one asks where such structure comes from. Given only the number of species and the connectance, can a simple rule generate a web that looks like the real thing: the right mix of plants, herbivores and predators, chains of a believable length, a scatter of omnivores? The niche model of Williams and Martinez says yes, and it needs a single idea: lay every species on one feeding axis, and let each one eat a contiguous slice of it. This post builds the model in base R, draws a web from it, and pits it against two older models to show what “realistic” actually buys you.

One axis, one rule

Give every species a niche value between 0 and 1, a position on an abstract feeding axis (loosely, body size or some other ordering of who can eat whom). Each species eats every species whose niche value falls inside a feeding range: an interval placed at or below its own value. Two parameters control a range: its width, drawn so that the average width delivers the connectance you asked for, and its centre, placed below the species’ niche value so that a consumer mostly eats smaller things. One species, the one with the lowest niche value, is forced to eat nothing. That single rule makes it a basal resource and roots the whole web.

niche_web <- function(S, C) {
  n <- runif(S)                     # niche value: a position on the feeding axis
  beta <- 1 / (2 * C) - 1           # sets mean range width to hit connectance C
  r <- n * rbeta(S, 1, beta)        # feeding range, always at most the niche value
  r[which.min(n)] <- 0              # smallest niche eats nothing: a basal resource
  centre <- runif(S, r / 2, n)      # range centre, placed below the niche value
  A <- matrix(0L, S, S)
  for (i in seq_len(S)) {
    lo <- centre[i] - r[i] / 2
    hi <- centre[i] + r[i] / 2
    A[i, n >= lo & n <= hi] <- 1L   # eat everything whose niche is inside the range
  }
  A
}

The beta line is the only piece of arithmetic that matters. The feeding range is the niche value times a Beta random number with mean 1 / (1 + beta); setting beta = 1 / (2C) - 1 makes that mean equal to 2C, which works out so the expected number of links is C times the number of possible links. You ask for a connectance and the model delivers it on average, without any tuning.

set.seed(43)
S <- 30
A <- niche_web(S, 0.15)

gen <- rowSums(A); vul <- colSums(A)
c(species      = S,
  links        = sum(A),
  connectance  = round(sum(A) / S^2, 3),
  basal        = sum(gen == 0),
  intermediate = sum(gen > 0 & vul > 0),
  top          = sum(gen > 0 & vul == 0),
  cannibals    = sum(diag(A)))

     species        links  connectance        basal intermediate          top 
      30.000      137.000        0.152        7.000       18.000        5.000 
   cannibals 
       5.000

This draw has 7 basal species, 18 intermediate and 5 top, at a connectance of 0.152, close to the 0.15 requested. It also has 5 cannibals: species that eat their own kind, which the model allows whenever a feeding range covers a species’ own niche value. Real webs have some cannibalism, though rarely this much, and we will come back to that.

library(ggplot2)

# recover the ranges for the drawn web by rerunning with the same seed
set.seed(43)
n <- runif(S); beta <- 1 / (2 * 0.15) - 1; r <- n * rbeta(S, 1, beta)
r[which.min(n)] <- 0; centre <- runif(S, r / 2, n)

ord <- order(n)
df <- data.frame(niche = n[ord], lo = (centre - r / 2)[ord],
                 hi = (centre + r / 2)[ord], rank = seq_len(S))
df$role <- ifelse(rowSums(A)[ord] == 0, "basal",
           ifelse(colSums(A)[ord] == 0, "top", "intermediate"))

ggplot(df, aes(y = rank)) +
  geom_segment(aes(x = lo, xend = hi, yend = rank, colour = role), linewidth = 1.6,
               alpha = 0.5) +
  geom_point(aes(x = niche, colour = role), size = 2) +
  scale_colour_manual(values = c(basal = "#4c9f70", intermediate = "#e0c341",
                                 top = "#c1523f")) +
  labs(x = "Niche value", y = "Species (ordered by niche value)", colour = NULL) +
  theme_minimal(base_size = 12) +
  theme(panel.grid.minor = element_blank(),
        legend.position = "top",
        plot.background = element_rect(fill = "white", colour = NA),
        panel.background = element_rect(fill = "white", colour = NA))

Horizontal axis from zero to one showing thirty species ordered by niche value, each with a point and a feeding-range bar; low-niche species have short or absent bars, high-niche species have wide bars reaching down the axis. — Figure 1: The feeding axis that generated the web. Each species sits at its niche value (point); the bar is its feeding range. A species eats every other species whose niche value falls under its bar.

Three models, same connectance

The niche model is one of three classic generators, and the differences between them are the whole point. The random model (an Erdos-Renyi directed graph) fills the whole matrix with links at probability C, ignoring any ordering. The cascade model gives species a rank and lets a consumer eat only lower-ranked species, so the web is strictly hierarchical with no loops. The niche model keeps the ordering of the cascade but replaces “eat anything below you” with “eat a contiguous interval”, which is what lets it produce omnivores, cannibals and the occasional loop.

cascade_web <- function(S, C) {
  p <- 2 * C * S / (S - 1)          # link probability that matches connectance C
  A <- matrix(0L, S, S)
  for (i in 2:S) for (j in 1:(i - 1)) if (runif(1) < p) A[i, j] <- 1L
  A
}
random_web <- function(S, C) matrix(rbinom(S * S, 1, C), S, S)

troph <- function(A) {              # prey-averaged trophic level, or NA if undefined
  S <- nrow(A); W <- A / pmax(rowSums(A), 1)
  tryCatch(solve(diag(S) - W, rep(1, S)), error = function(e) rep(NA_real_, S))
}

props <- function(f, S = 30, C = 0.15, reps = 1000) {
  z <- t(replicate(reps, {
    A <- f(S, C); g <- rowSums(A); v <- colSums(A); tl <- troph(A)
    c(connectance = sum(A) / S^2,
      no_basal    = sum(g == 0) == 0,
      basal_frac  = mean(g == 0),
      tl_undef    = any(!is.finite(tl)) | any(tl > 100))
  }))
  colMeans(z)
}

set.seed(2024)
summ <- rbind(niche   = props(niche_web),
              cascade = props(cascade_web),
              random  = props(random_web))
round(summ, 3)

        connectance no_basal basal_frac tl_undef
niche         0.151    0.000      0.207    0.354
cascade       0.150    0.000      0.105    0.000
random        0.150    0.786      0.008    0.807

Every model hits the target connectance: the realised values are all near 0.15. What separates them is whether the web they build is a web at all. Look at the no_basal column, the fraction of generated webs with no basal species. For the niche and cascade models it is 0: every web has a producer. For the random model it is 0.786. At a connectance typical of real food webs, a random graph of 30 species almost always leaves every single species eating something, so there is no bottom to the web and no energy source. It is not a worse food web; most of the time it is not a food web.

nb <- data.frame(model = rownames(summ), no_basal = summ[, "no_basal"])
nb$model <- factor(nb$model, levels = c("niche", "cascade", "random"))

ggplot(nb, aes(model, no_basal, fill = model)) +
  geom_col(width = 0.6) +
  scale_fill_manual(values = c(niche = "#4c9f70", cascade = "#3d6cb0",
                               random = "#c1523f"), guide = "none") +
  labs(x = NULL, y = "Fraction of webs with no basal species") +
  ylim(0, 1) +
  theme_minimal(base_size = 12) +
  theme(panel.grid.minor = element_blank(),
        panel.grid.major.x = element_blank(),
        plot.background = element_rect(fill = "white", colour = NA),
        panel.background = element_rect(fill = "white", colour = NA))

Bar chart with three bars: niche and cascade at zero, random near 0.79, showing that only the random model regularly produces webs with no basal species. — Figure 2: Fraction of 1000 generated webs that lack any basal species, by model. The random graph fails to root the web most of the time; the niche and cascade models always leave a producer.

The mean basal fraction tells the same story from the other side. The niche model puts about 21% of species at the base, in the range real webs show. The cascade model roots the web too but keeps fewer species basal, around 11%. The random model averages under 1%: not just often unrooted, but structurally top-heavy in a way no real community is.

What the model gets wrong

None of this makes the niche model true. It is a null model for structure, not a mechanism: it says nothing about who actually eats whom, only that if you assume one feeding axis and interval diets, the coarse counts come out right. Two failures are worth stating plainly. First, cannibalism: the example web had 5 self-eaters, and across the ensemble the niche model produces far more cannibalism than real webs do, because nothing stops a feeding range from covering its owner’s niche value. Second, loops. The interval rule permits feeding cycles, and at this connectance about 35% of niche webs contain a cycle that leaves the prey-averaged trophic level undefined. The cascade model, being strictly hierarchical, never does: its tl_undef is 0. That is the trade. The cascade model is cleaner but rigid; the niche model is messier but reproduces the omnivory and looping that real webs actually contain.

So the niche model earns its place not by being right but by being the simplest rule that fails in interesting ways. A random graph fails boringly, by not making a web. The niche model makes a web and then makes the same kinds of mistakes a real web does. That is why it became the baseline against which structural claims are tested, and why the next post can ask a sharper question: given a web with this structure, is it stable?

Where to go next

The connectance and trophic level post defines the roles and measures used here. The next post, stability and complexity, takes a web of a given size and connectance and asks whether it can persist. For the general machinery of testing observed structure against a generated baseline, see null models for interaction networks.

References

Williams RJ, Martinez ND 2000 Nature 404(6774):180-183 (10.1038/35004572)

Cohen JE, Newman CM 1985 Proceedings of the Royal Society of London B 224(1237):421-448 (10.1098/rspb.1985.0042)

Williams RJ, Martinez ND 2004 American Naturalist 163(3):458-468 (10.1086/381964)

Dunne JA, Williams RJ, Martinez ND 2002 Proceedings of the National Academy of Sciences 99(20):12917-12922 (10.1073/pnas.192407699)

--- title: "The niche model of food webs" description: "The niche model of food webs in R, compared to the cascade and random models: one feeding axis reproduces real structure, a random graph does not." date: "2026-07-21 14:00" categories: [R, food webs, ecological networks, community ecology, ecology tutorial] image: thumbnail.png image-alt: "A niche-model food web drawn from basal species up to top predators, beside the one-dimensional feeding axis that generated it." --- The previous post measured a food web that was written down by hand. This one asks where such structure comes from. Given only the number of species and the connectance, can a simple rule generate a web that looks like the real thing: the right mix of plants, herbivores and predators, chains of a believable length, a scatter of omnivores? The niche model of Williams and Martinez says yes, and it needs a single idea: lay every species on one feeding axis, and let each one eat a contiguous slice of it. This post builds the model in base R, draws a web from it, and pits it against two older models to show what "realistic" actually buys you. ## One axis, one rule Give every species a niche value between 0 and 1, a position on an abstract feeding axis (loosely, body size or some other ordering of who can eat whom). Each species eats every species whose niche value falls inside a feeding range: an interval placed at or below its own value. Two parameters control a range: its width, drawn so that the average width delivers the connectance you asked for, and its centre, placed below the species' niche value so that a consumer mostly eats smaller things. One species, the one with the lowest niche value, is forced to eat nothing. That single rule makes it a basal resource and roots the whole web. ```{r} #| label: niche-model niche_web <- function(S, C) { n <- runif(S) # niche value: a position on the feeding axis beta <- 1 / (2 * C) - 1 # sets mean range width to hit connectance C r <- n * rbeta(S, 1, beta) # feeding range, always at most the niche value r[which.min(n)] <- 0 # smallest niche eats nothing: a basal resource centre <- runif(S, r / 2, n) # range centre, placed below the niche value A <- matrix(0L, S, S) for (i in seq_len(S)) { lo <- centre[i] - r[i] / 2 hi <- centre[i] + r[i] / 2 A[i, n >= lo & n <= hi] <- 1L # eat everything whose niche is inside the range } A } ``` The `beta` line is the only piece of arithmetic that matters. The feeding range is the niche value times a Beta random number with mean `1 / (1 + beta)`; setting `beta = 1 / (2C) - 1` makes that mean equal to `2C`, which works out so the expected number of links is `C` times the number of possible links. You ask for a connectance and the model delivers it on average, without any tuning. ```{r} #| label: one-web set.seed(43) S <- 30 A <- niche_web(S, 0.15) gen <- rowSums(A); vul <- colSums(A) c(species = S, links = sum(A), connectance = round(sum(A) / S^2, 3), basal = sum(gen == 0), intermediate = sum(gen > 0 & vul > 0), top = sum(gen > 0 & vul == 0), cannibals = sum(diag(A))) ``` This draw has `r sum(rowSums(A) == 0)` basal species, `r sum(rowSums(A) > 0 & colSums(A) > 0)` intermediate and `r sum(rowSums(A) > 0 & colSums(A) == 0)` top, at a connectance of `r round(sum(A)/S^2, 3)`, close to the 0.15 requested. It also has `r sum(diag(A))` cannibals: species that eat their own kind, which the model allows whenever a feeding range covers a species' own niche value. Real webs have some cannibalism, though rarely this much, and we will come back to that. ```{r} #| label: fig-niche-axis #| fig-cap: "The feeding axis that generated the web. Each species sits at its niche value (point); the bar is its feeding range. A species eats every other species whose niche value falls under its bar." #| fig-alt: "Horizontal axis from zero to one showing thirty species ordered by niche value, each with a point and a feeding-range bar; low-niche species have short or absent bars, high-niche species have wide bars reaching down the axis." #| fig-width: 7 #| fig-height: 5.2 library(ggplot2) # recover the ranges for the drawn web by rerunning with the same seed set.seed(43) n <- runif(S); beta <- 1 / (2 * 0.15) - 1; r <- n * rbeta(S, 1, beta) r[which.min(n)] <- 0; centre <- runif(S, r / 2, n) ord <- order(n) df <- data.frame(niche = n[ord], lo = (centre - r / 2)[ord], hi = (centre + r / 2)[ord], rank = seq_len(S)) df$role <- ifelse(rowSums(A)[ord] == 0, "basal", ifelse(colSums(A)[ord] == 0, "top", "intermediate")) ggplot(df, aes(y = rank)) + geom_segment(aes(x = lo, xend = hi, yend = rank, colour = role), linewidth = 1.6, alpha = 0.5) + geom_point(aes(x = niche, colour = role), size = 2) + scale_colour_manual(values = c(basal = "#4c9f70", intermediate = "#e0c341", top = "#c1523f")) + labs(x = "Niche value", y = "Species (ordered by niche value)", colour = NULL) + theme_minimal(base_size = 12) + theme(panel.grid.minor = element_blank(), legend.position = "top", plot.background = element_rect(fill = "white", colour = NA), panel.background = element_rect(fill = "white", colour = NA)) ``` ## Three models, same connectance The niche model is one of three classic generators, and the differences between them are the whole point. The **random** model (an Erdos-Renyi directed graph) fills the whole matrix with links at probability `C`, ignoring any ordering. The **cascade** model gives species a rank and lets a consumer eat only lower-ranked species, so the web is strictly hierarchical with no loops. The niche model keeps the ordering of the cascade but replaces "eat anything below you" with "eat a contiguous interval", which is what lets it produce omnivores, cannibals and the occasional loop. ```{r} #| label: three-models cascade_web <- function(S, C) { p <- 2 * C * S / (S - 1) # link probability that matches connectance C A <- matrix(0L, S, S) for (i in 2:S) for (j in 1:(i - 1)) if (runif(1) < p) A[i, j] <- 1L A } random_web <- function(S, C) matrix(rbinom(S * S, 1, C), S, S) troph <- function(A) { # prey-averaged trophic level, or NA if undefined S <- nrow(A); W <- A / pmax(rowSums(A), 1) tryCatch(solve(diag(S) - W, rep(1, S)), error = function(e) rep(NA_real_, S)) } props <- function(f, S = 30, C = 0.15, reps = 1000) { z <- t(replicate(reps, { A <- f(S, C); g <- rowSums(A); v <- colSums(A); tl <- troph(A) c(connectance = sum(A) / S^2, no_basal = sum(g == 0) == 0, basal_frac = mean(g == 0), tl_undef = any(!is.finite(tl)) | any(tl > 100)) })) colMeans(z) } set.seed(2024) summ <- rbind(niche = props(niche_web), cascade = props(cascade_web), random = props(random_web)) round(summ, 3) ``` Every model hits the target connectance: the realised values are all near 0.15. What separates them is whether the web they build is a web at all. Look at the `no_basal` column, the fraction of generated webs with no basal species. For the niche and cascade models it is `r round(summ["niche","no_basal"], 3)`: every web has a producer. For the random model it is `r round(summ["random","no_basal"], 3)`. At a connectance typical of real food webs, a random graph of 30 species almost always leaves every single species eating something, so there is no bottom to the web and no energy source. It is not a worse food web; most of the time it is not a food web. ```{r} #| label: fig-no-basal #| fig-cap: "Fraction of 1000 generated webs that lack any basal species, by model. The random graph fails to root the web most of the time; the niche and cascade models always leave a producer." #| fig-alt: "Bar chart with three bars: niche and cascade at zero, random near 0.79, showing that only the random model regularly produces webs with no basal species." #| fig-width: 6 #| fig-height: 4 nb <- data.frame(model = rownames(summ), no_basal = summ[, "no_basal"]) nb$model <- factor(nb$model, levels = c("niche", "cascade", "random")) ggplot(nb, aes(model, no_basal, fill = model)) + geom_col(width = 0.6) + scale_fill_manual(values = c(niche = "#4c9f70", cascade = "#3d6cb0", random = "#c1523f"), guide = "none") + labs(x = NULL, y = "Fraction of webs with no basal species") + ylim(0, 1) + theme_minimal(base_size = 12) + theme(panel.grid.minor = element_blank(), panel.grid.major.x = element_blank(), plot.background = element_rect(fill = "white", colour = NA), panel.background = element_rect(fill = "white", colour = NA)) ``` The mean basal fraction tells the same story from the other side. The niche model puts about `r round(100 * summ["niche","basal_frac"])`% of species at the base, in the range real webs show. The cascade model roots the web too but keeps fewer species basal, around `r round(100 * summ["cascade","basal_frac"])`%. The random model averages under 1%: not just often unrooted, but structurally top-heavy in a way no real community is. ## What the model gets wrong None of this makes the niche model true. It is a null model for structure, not a mechanism: it says nothing about who actually eats whom, only that if you assume one feeding axis and interval diets, the coarse counts come out right. Two failures are worth stating plainly. First, cannibalism: the example web had `r sum(diag(A))` self-eaters, and across the ensemble the niche model produces far more cannibalism than real webs do, because nothing stops a feeding range from covering its owner's niche value. Second, loops. The interval rule permits feeding cycles, and at this connectance about `r round(100 * summ["niche","tl_undef"])`% of niche webs contain a cycle that leaves the prey-averaged trophic level undefined. The cascade model, being strictly hierarchical, never does: its `tl_undef` is `r round(summ["cascade","tl_undef"], 3)`. That is the trade. The cascade model is cleaner but rigid; the niche model is messier but reproduces the omnivory and looping that real webs actually contain. So the niche model earns its place not by being right but by being the simplest rule that fails in interesting ways. A random graph fails boringly, by not making a web. The niche model makes a web and then makes the same kinds of mistakes a real web does. That is why it became the baseline against which structural claims are tested, and why the next post can ask a sharper question: given a web with this structure, is it stable? ## Where to go next The [connectance and trophic level](../food-web-structure-connectance/) post defines the roles and measures used here. The next post, [stability and complexity](../stability-and-complexity-in-food-webs/), takes a web of a given size and connectance and asks whether it can persist. For the general machinery of testing observed structure against a generated baseline, see [null models for interaction networks](../null-models-for-interaction-networks/). ## Related tutorials - [Food web structure: connectance and trophic level](../food-web-structure-connectance/) - [Stability and complexity in food webs](../stability-and-complexity-in-food-webs/) - [Checking a food web analysis](../checking-a-food-web-analysis/) - [Null models for interaction networks](../null-models-for-interaction-networks/) ## References Williams RJ, Martinez ND 2000 Nature 404(6774):180-183 (10.1038/35004572) Cohen JE, Newman CM 1985 Proceedings of the Royal Society of London B 224(1237):421-448 (10.1098/rspb.1985.0042) Williams RJ, Martinez ND 2004 American Naturalist 163(3):458-468 (10.1086/381964) Dunne JA, Williams RJ, Martinez ND 2002 Proceedings of the National Academy of Sciences 99(20):12917-12922 (10.1073/pnas.192407699)