= Solution
For real $\lambda\ge0$, draw $y=\lambda+\beta$ and $y=-\gamma e^{-\lambda}$. The first is nonnegative and increasing, while the second is negative and approaches zero from below. They cannot intersect on the nonnegative real axis. At zero the <characteristic> <function> equals $\beta+\gamma>0$, so a stationary zero-root crossing is excluded as well.
Consequently a loss of stability from a stable regime cannot occur by a real <eigenvalue> crossing through zero. It must involve a <complex conjugate>[conjugate pair] crossing the <imaginary axis> at nonzero <frequency>, yielding \b[oscillatory instability]. This does not mean that every <characteristic> root is nonreal: negative real roots may exist and are harmless for stability. The graph and imaginary-axis conditions are shown together below.
\Image[/past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2009-respiration-stability.png]
{title=Respiratory delay feedback: no growing real root and the exact first oscillatory threshold}
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