 Solution

For a purely imaginary root $\lambda =i\omega$ with $\omega >0$, separating real and imaginary parts givesThus $\omega =1$ and $\tau =\pi /2+2\pi m$, $m\ge 0$. The first crossing isFor $0<\tau <\pi /2$ there is no root with nonnegative real part. Indeed, write $\lambda =a+ib$ with $a\ge 0$. The real equation forces $\cos (b\tau )\le 0$, hence $|b|\tau \ge \pi /2$. But the imaginary equation gives $|b|\le e^{-a\tau }\le 1$, contradicting $\tau <\pi /2$.

Differentiate the characteristic equation along the root:The imaginary roots are simple because $1+i{\tau }_c\ne 0$, and they cross into the unstable half-plane as delay increases.

One can also check the nonlinear Hopf bifurcation directly. With $\theta =\omega t$ and small amplitude $a$, substituteThe constant and second harmonics at order $a^2$ give the displayed coefficients. Put $d=\mu +{\tau }_c\nu$. At order $a^3$, absence of resonant first harmonics requires $d+1/20=0$ and $\nu +3/20=0$. HenceThe nonzero coefficient verifies a branch of small periodic solutions for $\tau >{\tau }_c$, consistent with the transverse Hopf bifurcation in the delayed logistic equation. This is the delayed logistic Hopf threshold.