Component and demographic Allee effects

ecology tutorial

population ecology

behaviour

conservation

Component versus demographic Allee effects in R: how a mate-finding shortfall at low density scales up to a population threshold, and when it stays harmless.

Author

Tidy Ecology

Published

2026-07-30

The previous post treated the Allee effect as a property of a whole population: per-capita growth turning negative below a critical density. But populations do not have mating problems; individuals do. An Allee effect at the population level is assembled from Allee effects on the pieces of individual fitness, and the translation from one to the other is not automatic. Keeping the two levels apart is the point of this post (Stephens, Sutherland and Freckleton 1999 Oikos 87(1):185-190).

Two levels

A component Allee effect is positive density dependence in a single component of fitness: the chance of finding a mate, the success of fertilisation, the effectiveness of group defence or cooperative hunting. A demographic Allee effect is positive density dependence in the bottom line, the per-capita growth rate of the whole population. The first is about one part of an individual’s life; the second is about whether the population grows.

Build a demographic rate from a component. Let the component be mate-finding: the probability that a female finds a mate rises with density as N/(N + theta), where theta sets how hard mates are to find. Births are this probability times a maximum rate, and against them stand a density-independent death rate and ordinary crowding.

b <- 1.2; d <- 0.3; a <- 0.006
mate <- function(N, theta) N/(N + theta)                 # the component Allee effect
rN   <- function(N, theta) b*mate(N, theta) - d - a*N    # the demographic growth rate

The same component, two outcomes

The mate-finding curve is a component Allee effect for any positive theta: it always rises with density. What it does to the population depends on how strong it is. Compare mates that are easy to find (theta = 2, the curve saturates almost at once) with mates that are hard (theta = 30, it climbs slowly).

rbind(easy_theta2  = round(rN(c(2, 10, 40), 2), 3),
      hard_theta30 = round(rN(c(2, 10, 40), 30), 3))

               [,1]  [,2]  [,3]
easy_theta2   0.288  0.64 0.603
hard_theta30 -0.237 -0.06 0.146

When mates are easy to find, per-capita growth is already positive at a density of 2 and there is no meaningful threshold: the demographic Allee effect exists in principle but its critical density sits near 0.7, below one individual, of no consequence to a real population. When mates are hard to find, per-capita growth is negative up to a critical density of 14.2, a strong demographic Allee effect at a density a population can easily fall to. The component was the same kind of thing in both cases. Only its strength decided whether it became a population-level threshold.

Ng <- seq(0, 80, 0.4)
comp <- rbind(data.frame(N = Ng, y = mate(Ng, 2),  theta = "easy (theta 2)",  panel = "mate-finding (component)"),
              data.frame(N = Ng, y = mate(Ng, 30), theta = "hard (theta 30)", panel = "mate-finding (component)"))
demo <- rbind(data.frame(N = Ng, y = rN(Ng, 2),  theta = "easy (theta 2)",  panel = "per-capita growth (demographic)"),
              data.frame(N = Ng, y = rN(Ng, 30), theta = "hard (theta 30)", panel = "per-capita growth (demographic)"))
both <- rbind(comp, demo)
ggplot(both, aes(N, y, colour = theta)) +
  geom_hline(yintercept = 0, colour = col_ref, linewidth = 0.3) +
  geom_line(linewidth = 1) +
  facet_wrap(~panel, scales = "free_y") +
  scale_colour_manual(values = c("easy (theta 2)" = col_easy, "hard (theta 30)" = col_hard)) +
  labs(x = "Population density", y = NULL, colour = NULL,
       title = "A component Allee effect becomes demographic only if it is strong enough") +
  theme(strip.text = element_text(size = 9), legend.position = "bottom")

Two panels: on the left, two rising mate-finding curves; on the right, the per-capita growth curves they produce, one staying positive and one dipping well below zero before recovering. — Figure 1: The mate-finding component (left) rises with density for both parameter values; the demographic growth it produces (right) crosses zero at a density worth worrying about only when mates are hard to find.

There is a continuum between these two cases. As mates get harder to find, the demographic threshold climbs from a harmless fraction of an individual to a density that dominates the population’s fate.

th <- seq(1, 60, 1)
dt <- data.frame(theta = th, A = sapply(th, thresh))
ggplot(dt, aes(theta, A)) +
  geom_line(colour = col_hard, linewidth = 1) +
  labs(x = "Mate-finding difficulty (theta)", y = "Demographic critical density",
       title = "Harder mate-finding lifts the threshold into the danger zone")

A rising curve of critical density against the mate-finding difficulty parameter, near zero at low difficulty and climbing steeply as mates become harder to find. — Figure 2: The demographic critical density against how hard mates are to find. Below a mild difficulty the threshold is negligible; past it, the threshold rises steeply into the range a real population occupies.

Why the distinction earns its keep

Two practical consequences follow. First, a component Allee effect can be masked. If low density relaxes some other limit, less competition for food, fewer shared parasites, the gain can offset the mating shortfall, and a population showing a clear mate-finding problem can still have no demographic Allee effect at all (Stephens et al. 1999; Courchamp et al. 1999 Trends in Ecology and Evolution 14(10):405-410). Measuring the component alone can therefore mislead about the population. Second, the reverse: a demographic Allee effect can be built from several weak components, none alarming on its own, that stack. Mate limitation plus weaker group defence plus reduced foraging efficiency can together push per-capita growth negative where no single one would.

So the component is where the mechanism lives and the demographic rate is where the consequence lives, and conservation needs both: the mechanism to know what to fix, the demographic rate to know whether fixing it matters. The next post takes the demographic threshold into a fluctuating world, where it does its real damage.

References

Courchamp F et al. 1999. Trends in Ecology and Evolution 14(10):405-410 (10.1016/S0169-5347(99)01683-3).
Kramer AM et al. 2009. Population Ecology 51(3):341-354 (10.1007/s10144-009-0152-6).
Stephens PA, Sutherland WJ, Freckleton RP 1999. Oikos 87(1):185-190 (10.2307/3547011).

--- title: "Component and demographic Allee effects" description: "Component versus demographic Allee effects in R: how a mate-finding shortfall at low density scales up to a population threshold, and when it stays harmless." date: "2026-07-30 12:00" categories: [ecology tutorial, R, population ecology, behaviour, conservation] image: thumbnail.png image-alt: "A mate-finding curve rising with density, and the per-capita growth it produces, which crosses zero at a meaningful density only when mates are hard to find." --- ```{r setup} #| include: false knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE, dev = "png", dpi = 120, fig.path = "figures/", fig.width = 7, fig.height = 4.4) library(ggplot2) theme_te <- theme_minimal(base_size = 12) + theme(panel.background = element_rect(fill = "white", colour = NA), plot.background = element_rect(fill = "white", colour = NA), panel.grid.minor = element_blank(), plot.title = element_text(face = "bold", size = 12)) theme_set(theme_te) col_easy <- "#3a6b52"; col_hard <- "#b5651d"; col_ref <- "#555555" b <- 1.2; d <- 0.3; a <- 0.006 mate <- function(N, theta) N/(N + theta) # component Allee: mate-finding rN <- function(N, theta) b*mate(N, theta) - d - a*N # demographic per-capita growth thresh <- function(theta) { Ng <- seq(0.05, 120, 0.05); up <- Ng[which(diff(sign(rN(Ng, theta))) > 0)]; if (length(up)) up[1] else NA } A_easy <- thresh(2); A_hard <- thresh(30) ``` The [previous post](../allee-effects-and-thresholds/) treated the Allee effect as a property of a whole population: per-capita growth turning negative below a critical density. But populations do not have mating problems; individuals do. An Allee effect at the population level is assembled from Allee effects on the pieces of individual fitness, and the translation from one to the other is not automatic. Keeping the two levels apart is the point of this post (Stephens, Sutherland and Freckleton 1999 Oikos 87(1):185-190). ## Two levels A **component** Allee effect is positive density dependence in a single component of fitness: the chance of finding a mate, the success of fertilisation, the effectiveness of group defence or cooperative hunting. A **demographic** Allee effect is positive density dependence in the bottom line, the per-capita growth rate of the whole population. The first is about one part of an individual's life; the second is about whether the population grows. Build a demographic rate from a component. Let the component be mate-finding: the probability that a female finds a mate rises with density as `N/(N + theta)`, where `theta` sets how hard mates are to find. Births are this probability times a maximum rate, and against them stand a density-independent death rate and ordinary crowding. ```{r model} b <- 1.2; d <- 0.3; a <- 0.006 mate <- function(N, theta) N/(N + theta) # the component Allee effect rN <- function(N, theta) b*mate(N, theta) - d - a*N # the demographic growth rate ``` ## The same component, two outcomes The mate-finding curve is a component Allee effect for any positive `theta`: it always rises with density. What it does to the population depends on how strong it is. Compare mates that are easy to find (`theta = 2`, the curve saturates almost at once) with mates that are hard (`theta = 30`, it climbs slowly). ```{r outcomes} rbind(easy_theta2 = round(rN(c(2, 10, 40), 2), 3), hard_theta30 = round(rN(c(2, 10, 40), 30), 3)) ``` When mates are easy to find, per-capita growth is already positive at a density of `r sprintf("%.0f", 2)` and there is no meaningful threshold: the demographic Allee effect exists in principle but its critical density sits near `r sprintf("%.1f", A_easy)`, below one individual, of no consequence to a real population. When mates are hard to find, per-capita growth is negative up to a critical density of `r sprintf("%.1f", A_hard)`, a strong demographic Allee effect at a density a population can easily fall to. The component was the same kind of thing in both cases. Only its strength decided whether it became a population-level threshold. ```{r fig-mate} #| fig-cap: "The mate-finding component (left) rises with density for both parameter values; the demographic growth it produces (right) crosses zero at a density worth worrying about only when mates are hard to find." #| fig-alt: "Two panels: on the left, two rising mate-finding curves; on the right, the per-capita growth curves they produce, one staying positive and one dipping well below zero before recovering." Ng <- seq(0, 80, 0.4) comp <- rbind(data.frame(N = Ng, y = mate(Ng, 2), theta = "easy (theta 2)", panel = "mate-finding (component)"), data.frame(N = Ng, y = mate(Ng, 30), theta = "hard (theta 30)", panel = "mate-finding (component)")) demo <- rbind(data.frame(N = Ng, y = rN(Ng, 2), theta = "easy (theta 2)", panel = "per-capita growth (demographic)"), data.frame(N = Ng, y = rN(Ng, 30), theta = "hard (theta 30)", panel = "per-capita growth (demographic)")) both <- rbind(comp, demo) ggplot(both, aes(N, y, colour = theta)) + geom_hline(yintercept = 0, colour = col_ref, linewidth = 0.3) + geom_line(linewidth = 1) + facet_wrap(~panel, scales = "free_y") + scale_colour_manual(values = c("easy (theta 2)" = col_easy, "hard (theta 30)" = col_hard)) + labs(x = "Population density", y = NULL, colour = NULL, title = "A component Allee effect becomes demographic only if it is strong enough") + theme(strip.text = element_text(size = 9), legend.position = "bottom") ``` There is a continuum between these two cases. As mates get harder to find, the demographic threshold climbs from a harmless fraction of an individual to a density that dominates the population's fate. ```{r fig-threshold} #| fig-cap: "The demographic critical density against how hard mates are to find. Below a mild difficulty the threshold is negligible; past it, the threshold rises steeply into the range a real population occupies." #| fig-alt: "A rising curve of critical density against the mate-finding difficulty parameter, near zero at low difficulty and climbing steeply as mates become harder to find." th <- seq(1, 60, 1) dt <- data.frame(theta = th, A = sapply(th, thresh)) ggplot(dt, aes(theta, A)) + geom_line(colour = col_hard, linewidth = 1) + labs(x = "Mate-finding difficulty (theta)", y = "Demographic critical density", title = "Harder mate-finding lifts the threshold into the danger zone") ``` ## Why the distinction earns its keep Two practical consequences follow. First, a component Allee effect can be masked. If low density relaxes some other limit, less competition for food, fewer shared parasites, the gain can offset the mating shortfall, and a population showing a clear mate-finding problem can still have no demographic Allee effect at all (Stephens et al. 1999; Courchamp et al. 1999 Trends in Ecology and Evolution 14(10):405-410). Measuring the component alone can therefore mislead about the population. Second, the reverse: a demographic Allee effect can be built from several weak components, none alarming on its own, that stack. Mate limitation plus weaker group defence plus reduced foraging efficiency can together push per-capita growth negative where no single one would. So the component is where the mechanism lives and the demographic rate is where the consequence lives, and conservation needs both: the mechanism to know what to fix, the demographic rate to know whether fixing it matters. The next post takes the demographic threshold into a fluctuating world, where it does its real damage. ## Related tutorials - [Allee effects and thresholds in R](../allee-effects-and-thresholds/) - [Allee effects and extinction risk](../allee-effects-and-extinction-risk/) - [Checking an Allee analysis](../checking-an-allee-analysis/) - [GLM for count data and abundance](../glm-count-data-abundance/) ## References - Courchamp F et al. 1999. Trends in Ecology and Evolution 14(10):405-410 (10.1016/S0169-5347(99)01683-3). - Kramer AM et al. 2009. Population Ecology 51(3):341-354 (10.1007/s10144-009-0152-6). - Stephens PA, Sutherland WJ, Freckleton RP 1999. Oikos 87(1):185-190 (10.2307/3547011).