The dynamics described by (37) are as follows: the rate of change of flow on a path
connecting an O/D pair is equal to the difference between the travel disutility for that O/D
pair and the cost on that path at that instance in time. If the path cost exceeds the travel
disutility, then the flow on the path will decrease; if it is less than the disutility, then the
flow on that path will increase. The projection operator in (37) guarantees that the flow
on the paths will not be negative, since this would violate feasibility. Hence, the path flows
(and incurred travel demands) evolve from an initial path flow pattern at time zero given by
x(0) until a stationary point is reached, that is, when ˙x = 0; at which point we have that
for that particular x∗: