Allee Effects & Critical Population Size

The Allee Effect

The Allee effect describes a phenomenon in population biology where individual fitness increases with population density at low densities. This creates a critical population threshold below which populations decline to extinction, leading to bistability between extinction and persistence.

Mathematical Model

The population dynamics with a strong Allee effect can be modeled as:

• dN/dt = rN(1 - N/K)(N/A - 1)
• r: Intrinsic growth rate
• K: Carrying capacity (upper equilibrium)
• A: Allee threshold (lower critical threshold)
• N: Population size

Strong vs Weak Allee Effects

• Strong Allee effect: Per-capita growth rate is negative below threshold A. Populations below A always go extinct. This creates three equilibria: 0 (stable), A (unstable), and K (stable).
• Weak Allee effect: Per-capita growth rate is positive but reduced at low densities. Growth is slower at low densities but populations can still recover. Only two equilibria: 0 (unstable) and K (stable).
• No Allee effect: Standard logistic growth. Population always grows toward K from any positive density.

Biological Mechanisms

Allee effects arise from various mechanisms:

• Mate limitation: Difficulty finding mates at low density (sexual species)
• Cooperative defense: Reduced predation risk in larger groups
• Cooperative hunting: Better foraging success with more individuals
• Inbreeding depression: Reduced fitness from mating with relatives
• Genetic diversity loss: Reduced adaptive potential in small populations
• Environmental modification: Insufficient individuals to modify habitat favorably

Conservation Implications

The Allee effect has critical implications for conservation biology. It means that endangered species may require a minimum viable population size to avoid an "extinction vortex." Simply preventing hunting or habitat loss may be insufficient if populations have already fallen below the Allee threshold. Active intervention (captive breeding, translocations, population supplementation) may be necessary to push populations above the critical threshold.

Examples in Nature

• Passenger pigeon: Extinction possibly accelerated by Allee effects after population reduction
• African wild dogs: Pack hunting creates cooperative Allee effects
• Gypsy moths: Mate-finding Allee effects affect invasion dynamics
• Plants with specialist pollinators: Pollination success depends on population density
• Marine invertebrates: Broadcast spawners require sufficient density for fertilization

Invasion Biology

Allee effects also affect biological invasions. Invasive species must establish populations above the Allee threshold to successfully invade. This creates a "critical patch size" for invasion: introductions below this size will fail, while larger introductions can succeed. Understanding Allee thresholds can inform biosecurity strategies for preventing invasions.