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 t0. 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 t0 of a bird present at time t0, let us denote y the actual time of departure and Y the associated random variable. Let i be the integer verifying ti-1 £ y < ti; this is always possible if ti can take the value +¥. Then
Pr[stays to y] = Pr[stays to t1]xPr[stays from t1 to t2] ...xPr[stays from ti-1 to y] = f 1 f 2 ... f i-1 exp(-m i (y-ti-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-ti-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 tn(y) < y < tn(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 t0 can now be calculated. Finally, we have