Delayed logistic equation ($u^{\prime }(t)=u(t)[1-u(t-\tau )]$)

This delay differential equation replaces instantaneous logistic inhibition by a population value one delay earlier. A positive history function of a delay differential equation gives positive solutions while they exist. The equilibria are zero and one. Linearization at one gives $x^{\prime }(t)=-x(t-\tau )$, with characteristic equation $\lambda =-e^{-\lambda \tau }$. Delayed sign reversal permits overshoots around one and, at the delayed logistic Hopf threshold, a periodic branch.

A purely imaginary root of $\lambda =-e^{-\lambda \tau }$ has frequency one and first appears at $\tau =\pi /2$. Roots cross with positive real derivative $1/(1+{\tau }_c^2)$; all roots lie in the left half-plane for smaller positive delay, since a hypothetical root $a+ib$ with $a\ge 0$ would require both $|b|\tau \ge \pi /2$ and $|b|\le 1$. A small-amplitude expansion $u=1+a\cos (\omega t)+a^2[\cos (2\omega t)/5+\sin (2\omega t)/10]+\cdots$ yields $\omega =1-3a^2/20+\cdots$ and $\tau =\pi /2+(3\pi -2)a^2/40+\cdots$, verifying a nondegenerate Hopf bifurcation. The limiting period is $2\pi =4{\tau }_c$.