  In this article we use a criterion for the integrability of paths of
one-dimensional diffusion processes from which we derive new insights on
allelic fixation in several situations. This well known criterion involves a
simple necessary and sufficient condition based on scale function and speed
measure. We provide a new simple proof for this result and also obtain explicit
bounds for the moments of such integrals. We also extend this criterion to
non-homogeneous processes by use of Girsanov's transform. We apply our results
to multi-type population dynamics: using the criterion with appropriate time
changes, we characterize the behavior of proportions of each type before
population extinction in different situations.
