Allee Effect: Population Extinction Threshold (2D)

Population Parameters

Growth rate r 0.35

Allee threshold A 20

Carrying capacity K 150

Perturb Population

Live Readouts

Population N

Per-capita growth

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Trend

Growing

How it works

Unlike simple logistic growth, many real populations show positive density dependence at low numbers — mates are harder to find, cooperative defense and foraging break down, and small groups become genetically fragile. This is the strong Allee effect, modelled as:

dN/dt = r · N · (N/A − 1) · (1 − N/K)

  N — current population size
  r — intrinsic growth rate
  A — Allee threshold (critical population size)
  K — carrying capacity

The sign of (N/A − 1) is the key: when N < A the term is negative, so dN/dt < 0 — the population declines toward extinction even with no external pressure. When A < N < K the population grows toward carrying capacity, exactly as in logistic growth. N = 0 and N = K are stable equilibria; N = A is an unstable tipping point between them.

Habitat map — a live top-down view of the colony; drag to pan, scroll/pinch to zoom.
Phase portrait — plots dN/dt against N. The curve crosses zero at the three equilibria (0, A, K); grab the white dot and drag it left/right to set the population directly and watch which side of A it falls on.
r, A, K sliders — reshape the dynamics live; try raising A above the current population to watch the same group tip from recovering to collapsing.
Cull 30% — simulates a poaching event, disease outbreak, or habitat loss; if it pushes N below A, extinction becomes inevitable even without further disturbance.
+10 individuals — simulates reintroduction/translocation, the standard conservation tool for pushing a population back above its Allee threshold.

Real-world relevance: this threshold is why conservation biology defines a minimum viable population — reintroduction programs for species like the whooping crane or Florida panther deliberately release enough founders at once to clear A, because a slow trickle of individuals can go locally extinct before ever reaching self-sustaining numbers.

Above threshold — recovering Below threshold — declining Extinct

drag to pan · scroll to zoom

Phase portrait — drag the dot

N(t) — dashed A / K