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Critical slowing down is the divergence of a relaxation time as a stable equilibrium point approaches loss of hyperbolicity. For a scalar ordinary differential equation it occurs when $f^{\prime }(N^{\ast })\uparrow 0$ and the recovery time of a stable equilibrium $-1/f^{\prime }(N^{\ast })$ diverges. At the threshold, nonlinear terms may replace exponential decay by algebraic decay; for example $N^{\prime }=-k{N}^3$ has $N(t)\sim (2kt{)}^{-1/2}$ for positive initial data.

Linearizing at the positive steady state gives $n^{\prime }=f^{\prime }(N^{\ast })n$, so a small displacement decays as $n(t)=n(0)e^{-t/\tau }$. The recovery time of a stable equilibrium is thereforeAs $\delta \uparrow r$ with $r,b$ fixed, $N^{\ast }\to 0$ and $\tau \to \infty$. Thus the equilibrium population approaches extinction and recovers increasingly slowly after small perturbations. This is critical slowing down. Exactly at the threshold the leading equation is $N^{\prime }\sim -rb{N}^3$, which gives algebraic decay $N(t)\sim (2rbt{)}^{-1/2}$ instead of exponential recovery.