  A proper discretization of the logistic differential equation, which is
preserving these two distinct equilibrium solutions and their unstability and
stability, suggest that we need to examine the time delay of the logistic map.
According to Murray, the effect of delay in models is "usually" to increase the
potential for instability. However the word "usually" is really ambiguous. In
this paper, we mathematically formulate and prove the two conjectures about
stability and instability.

Dynamical Systems