This is a weird monograph to say the least. Rather than presenting a body of finished work with theorems and proofs and examples and applications, we have presented the problem of how to treat the evolution of densities under that action of delayed dynamics and given no solutions! We have simply illustrated the problem in Chaps. 1–3 and then given all of the reasons why it is difficult mathematically in Chap. 4. Following this we have presented a series of chapters in which we detail various attempts that have been made to solve the problem, all of which have led to naught. This is definitely NOT your standard mathematical monograph! So why have we done this? Simply to lay out a map of what we think are blind alleys for the unwary neophyte starting out in search of a solution.

In our collective opinion, based on years of fruitless attempts, the most promising avenue seems to be that of the Hopf functionals in Chap. 5, based on the work of Capiński (1991), but we have not taken it further than what is detailed here. The second most promising may be that of Sect. 7.2 in which we present analytic results based on the work of Tulcea and Marinescu (1950), but it is unclear how to extend these results. If our suspicions about either of these two approaches turn out to bear fruit then we will be gratified and applaud the appearance of concrete progress.

One aspect that we have not written about is that of stochastic differential delay equations. There is an extensive literature dealing with this, and in our opinion none of it deals effectively with the substantiative mathematical issues raised in Chap. 4. The main paper of interest in this literature is that of Guillouzic et al. (1999) and the review Longtin (2009).