Allee effects
An Allee effect occurs when population performance is reduced at very low population density. Unlike ordinary logistic growth, being rare can itself be disadvantageous.
A strong Allee-effect model
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\left(\frac{N}{A}-1\right),\]where \(A\) is an Allee threshold with \(0<A<K\).
Threshold behaviour
The equilibria are
\[N^*=0,\qquad N^*=A,\qquad N^*=K.\]If \(0<N<A\), the growth rate is negative and the population moves toward extinction. If \(A<N<K\), the population grows toward \(K\).
Biological causes. Allee effects may arise from difficulty finding mates, cooperative defence, group hunting, pollination failure or loss of social structure at low density.
Key idea. A strong Allee effect creates a critical population threshold below which decline is expected even when resources are abundant.