= Delayed negative feedback model
{title2=$d$}
A <feedback delay> is a time interval between an event and its returning influence. Multiple biochemical steps can introduce an effective lag; a fixed delay is an idealization, not a universal description of <signal transduction>. To isolate timing, let $x$ be a signed displacement from an <equilibrium point> and use normalized <dimensionless variables>:
$$
\frac{dx}{dt}=-x(t)-2x(t-d),\qquad d\geq0.
$$
The local removal is $-x$ and the returning influence is negative at every delay. At $d=0$ the <ordinary differential equation> reduces to the decay already illustrated under <signaling feedback>. For $d>0$ a <delay differential equation> requires a history on $[-d,0]$, not just an <initial condition> at one instant. Negative values of $x$ are displacements, not negative chemical <concentrations>.
An <exponential function> $x=e^{\lambda t}$ gives the <characteristic equation of a delay differential equation>
$$
\lambda+1+2e^{-\lambda d}=0.
$$
Putting $\lambda=i\omega$ and equating real and imaginary parts gives $\cos(\omega d)=-1/2$ and $\omega=2\sin(\omega d)$. The first crossing with $\omega>0$ therefore occurs at
$$
\omega=\sqrt3,\qquad d_c=\frac{2\pi}{3\sqrt3}\simeq1.20920.
$$
For this particular <mathematical model>, the <equilibrium point> is <asymptotically stable> for $0\leq d<d_c$. No nonnegative real <delayed feedback characteristic root> can satisfy the equation; the first loss of <asymptotic stability> is the indicated pair of imaginary <delayed feedback characteristic roots>. Differentiating the <characteristic equation of a delay differential equation> along a root yields
$$
\frac{d\lambda}{dd}=-\frac{\lambda(\lambda+1)}{1+d(\lambda+1)},\qquad \operatorname{Re}\left(\frac{d\lambda}{dd}\right)_{d=d_c}=\frac{3}{(1+d_c)^2+3d_c^2}>0.
$$
Thus the pair moves into growth as the delay increases through $d_c$. The neutral mode at $d_c$ is a <periodic function>, but its amplitude depends on history: this linear <mathematical model> does not select an attracting <limit cycle>.
The figure plots $x(t)=e^{at}\cos(bt)$ using a <delayed feedback characteristic root> $\lambda=a+ib$ and that same formula on the history interval. At $d=0.7$ the displayed mode decays; at $d=1.5$ it grows. These are exact modes up to numerical root accuracy, not step responses with identical histories. Molecular bounds and a changed return law would be needed to interpret what happens after an unstable biological response becomes large.
Back to article page
