**Michael Schaub, Roger Pradel, Lukas Jenni,
and Jean-Dominique Lebreton. 2001. Migrating birds stop over longer
than usually thought: an improved capture-recapture analysis.
Ecology 82: 852-859.**

Appendix. Derivation of formula 2.

When analyzing capture-recapture data, one gets estimates of probabilities of departure over discrete periods of time as represented on the diagram below.

Here we derive a formula for the stopover duration after time
*t*_{0}. It depends not only on the discrete probabilities
of departure f _{i},
but also on the pattern of the instantaneous rate of departure
within each interval. Without any additional information, a reasonable
assumption is that the instantaneous rate of departure m
_{i} is constant within each interval. Then, departures
during some interval follows a Poisson process and m
_{i} = -ln(f _{i})
(see for instance, Seber, 1982 p.3). To calculate the stopover
duration after time *t*_{0 }of a bird present at
time *t*_{0}, let us denote *y* the actual time
of departure and *Y* the associated random variable. Let
*i* be the integer verifying *t*_{i-1}
£ *y* < *t*_{i};
this is always possible if *t*_{i} can take
the value +¥. Then

Pr[stays to *y*] = Pr[stays to *t*_{1}]xPr[stays
from *t*_{1} to *t*_{2}] ...xPr[stays
from *t*_{i-1} to *y*] = f
_{1} f _{2} ... f _{i-1} exp(-m
_{i} (*y*-*t*_{i-1})).

The cumulative distribution function of *Y* being F(*y*)
= Pr[*Y* < *y*] = 1-Pr[stays to *y*], its probability
density function is

f(*y*) = F'(*y*) = m _{i}
f _{1} f
_{2} ... f _{i-1}
exp(-m _{i} (*y*-*t*_{i-1}))

and the mean time of residency is

The following two lemmas, which are easily established, will be useful to calculate the integral above:

If y is any real and *n(y)* the integer verifying *t*_{n(y)}
< *y* < *t*_{n(y)+1}, then

To go further, an assumption is required about the pattern
of departure beyond the study period. We opted for a constant
instantaneous rate of departure m _{n+1}
and derived it from a weighted gliding average of the last three
estimable f 's according to the formula

With m _{n+1} = -ln(f _{n+1}), the stopover duration
after time *t*_{0} can now be calculated. Finally,
we have