  The bifurcation diagram of a model stochastic differential equation with
delayed feedback is presented. We are motivated by recent research on
stochastic effects in models of transcriptional gene regulation. We start from
the normal form for a pitchfork bifurcation, and add multiplicative or
parametric noise and linear delayed feedback. The latter is sufficient to
originate a Hopf bifurcation in that region of parameters in which there is a
sufficiently strong negative feedback. We find a sharp bifurcation in parameter
space, and define the threshold as the point in which the stationary
distribution function p(x) changes from a delta function at the trivial state
x=0 to p(x) ~ x^alpha at small x (with alpha = -1 exactly at threshold). We
find that the bifurcation threshold is shifted by fluctuations relative to the
deterministic limit by an amount that scales linearly with the noise intensity.
Analytic calculations of the bifurcation threshold are also presented in the
limit of small delay tau -> 0 that compare quite favorably with the numerical
solutions even for tau = 1.
