Negativity of delayed induced oscillations in a simple linear DDE (Q2430062)

Oscillatory behaviour appearing in the equation \[ \frac{dx}{dt} = A-Bx(t)-Cx(t-\tau) \] with \(B < C\) and \(\tau \geq (c^2-B^2)^{-1/2}\cos^{-1}(-B/C)\) is studied. It is shown that, for a solution with \(x(s)=0\) for \(s<0\) and \(x(0)=x^0\geq 0\), and some other initial values, there exists always a \(t\in (0, 4\tau)\) such that \(x(t)<0\). This shows that the proposed Cauchy problem is not a proper description of biochemical reactions or of other biological and physical quantities.