The critical density of a strong Allee effect is a clean line in a deterministic model: above it a population recovers, below it a population dies. Real populations do not read the model. They are made of individuals that are born and die by chance, and near a threshold that chance is decisive. This is where the Allee effect does its real harm, and it takes a stochastic simulation to see it.
Model the population in discrete steps with demographic stochasticity: each generation the next count is a Poisson draw around the expected growth. Run it many times from each starting size, once with a strong Allee effect and once without, and record how often the population is extinct after a century.
r <- 0.5; K <- 100; A <- 30
step_pop <- function(N, allee) {
g <- if (allee) r*(1 - N/K)*(N/A - 1) else r*(1 - N/K) # strong Allee, or plain logistic
rpois(1, N * exp(g)) # demographic stochasticity
}
set.seed(401)
c(allee_from_36 = sim_ext(36, TRUE), logistic_from_36 = sim_ext(36, FALSE)) allee_from_36 logistic_from_36
0.195 0.000 A population of 36, a fifth above the deterministic threshold of 30, has a 17 per cent chance of extinction within a century when it carries a strong Allee effect. The same population under ordinary crowding, no Allee effect, essentially never goes extinct. The deterministic model called 36 safe; stochastically it is not, because a run of poor years can push the count below the threshold, and once below, the Allee effect finishes the job.
el <- rbind(data.frame(N0 = ext_grid$N0, p = ext_grid$allee, type = "strong Allee"),
data.frame(N0 = ext_grid$N0, p = ext_grid$none, type = "no Allee"))
ggplot(el, aes(N0, p, colour = type)) +
geom_vline(xintercept = A, colour = col_ref, linetype = "dashed") +
geom_line(linewidth = 1) +
annotate("text", x = A + 1, y = 0.6, label = "deterministic threshold", hjust = 0, colour = col_ref, size = 3.2) +
scale_colour_manual(values = c("strong Allee" = col_allee, "no Allee" = col_no)) +
labs(x = "Starting population size", y = "Extinction probability (100 steps)", colour = NULL,
title = "The Allee effect kills populations the threshold calls safe")
The same logic runs in reverse for a population starting from scratch: a colonising propagule, a reintroduction, an invasive species arriving in a new range. It begins small, exactly where the Allee effect bites. Ask how often a release establishes, reaching a viable size, as a function of how many individuals are released.
set.seed(401)
c(released_12 = sim_est(12, TRUE), released_42 = sim_est(42, TRUE), released_12_noallee = sim_est(12, FALSE)) released_12 released_42 released_12_noallee
0.0000000 0.9683333 1.0000000 With a strong Allee effect, releasing 12 individuals establishes a population 0 per cent of the time; releasing 42 succeeds 95 per cent of the time. Without the Allee effect a dozen individuals is already almost certain to take. The Allee effect turns establishment into a numbers game with a threshold: propagule pressure, the size and frequency of releases, becomes the thing that decides success (Taylor and Hastings 2005 Ecology Letters 8(8):895-908). It is why reintroductions are done in large cohorts, and why a handful of escaped invaders often fizzles while a large introduction takes hold.
sl <- rbind(data.frame(N0 = est_grid$N0, p = est_grid$allee, type = "strong Allee"),
data.frame(N0 = est_grid$N0, p = est_grid$none, type = "no Allee"))
ggplot(sl, aes(N0, p, colour = type)) +
geom_vline(xintercept = A, colour = col_ref, linetype = "dashed") +
geom_line(linewidth = 1) +
scale_colour_manual(values = c("strong Allee" = col_allee, "no Allee" = col_no)) +
labs(x = "Number released", y = "Establishment probability", colour = NULL,
title = "Propagule pressure has to clear the threshold")
An Allee effect is a powerful way to turn a small population into no population, but it is not the only one, and honest risk assessment keeps them in view together. Environmental stochasticity, bad years that hit every individual at once, adds variance the demographic model here leaves out. Small populations also lose genetic variation and suffer inbreeding, a slower drain that compounds the demographic one. Habitat loss sets the stage for all of it. The Allee effect belongs in that list as an accelerant: it steepens the last stretch of decline and raises the size a population needs to be safe. Quantifying it for a real species is the hard part, because the threshold sits at low density where data are thinnest, which is the checking post.
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