Good question!

In my experience, a delay could be interpreted as the time needed to observe a measurable change of the object of study. For a better reference, please see the links below:

http://arxiv.org/pdf/0910.1219.pdf

http://www.di.unipi.it/~milazzo/papers/Purely_Delayed.pdf

http://www.math.nthu.edu.tw/~sbhsu/Biologial%20Science.pdf

I hope this will be helpful

In addition to @Erick Arguello's incisive obervation, it is appropriate to consider the role of delay differential equation models of delay in biological phenomena.

For example, see

J.E. Forde, Delay Differential Equation Models in Mathematical Biology, Ph.D. thesis, The University of Michigan, 2005:

http://www.math.utah.edu/~forde/research/JFthesis.pdf

A delay differential equation provides a discrete delay model of delay in biological phenomena.   It is observed that discrete delay models are only an approximation of what actually happens in biological phenomena (p. 89).