  A population is said to have an ideal free distribution in a spatially
heterogeneous but temporally constant environment if each of its members have
chosen a fixed spatial location in a way that optimizes its individual fitness,
allowing for the effects of crowding. In this paper, we extend the idea of
individual fitness associated with a specific location in space to account for
the full path that an individual organism takes in space and time over a
periodic cycle, and extend the mathematical formulation of an ideal free
distribution to general time periodic environments. We find that, as in many
other cases, populations using dispersal strategies that can produce a
generalized ideal free distribution have a competitive advantage relative to
populations using strategies that do not produce an ideal free distribution. A
sharp criterion on the environmental functions is found to be necessary and
sufficient for such ideal free distribution to be feasible. In the case the
criterion is met, we showed that there exist dispersal strategies that can be
identified as producing a time-periodic version of an ideal free distribution,
and such strategies are evolutionarily steady and are neighborhood invaders
from the viewpoint of adaptive dynamics. Our results extend previous works in
which the environments are either temporally constant, or temporally periodic
but the total carrying capacity is temporally constant.
