Scottish Marine and Freshwater Science Volume 6 Number 10: At-Sea Turnover of Breeding Seabirds - Final Report to Marine Scotland Science

Appendix A Algorithm for Simulating Foraging Locations

Without Site Fidelity

In the absence of site fidelity the foraging locations on each of the d days are simulated by splitting the study area into a fine resolution regular grid (0.5 x 0.5 degrees) that contains n grid cells. The expected foraging density f _i within each grid square i is estimated from GPS data, and for each bird j the foraging locations g _jk on days k = 1,..., d are then simulated independently from a multinomial distribution with size one and probabilities ( f _1,..., f _n), so that

g _jk~ Multinomial(1, ( f ₁,..., f _n))

This is exactly the same approach as that used in Searle et al., 2014.

With Site Fidelity

We now assume that fidelity operates at the spatial scale of a grid which is coarser than the fine grid (and which is based on aggregating cells of the fine grid, so that the fine grid is nested within the coarse), and that the level of fidelity is given by a parameter that lies between zero (fidelity is the same as that obtained under independence) and one (perfect fidelity). For each bird j we modify the original independently-generated locations g _jk so that they have site fidelity, in the following way:

(1) Find the coarse grid cells c _{j 1,...,} c _jd that are associated with g _{j 1},..., g _jd;

(2) Find the number of these coarse grid cells that are unique - let this be u( j);

(3) Calculate the number of coarse grid cells that would be visited if fidelity were equal to to be;

v( j) = round{1 + (1 - ) ( u( j) - 1)};

(4) Randomly select v( j) from the set of u( j) unique coarse grid cells;

(5) Re-assign each day to lie within one of this set of coarse grid cells. Each coarse grid cell is allocated at least one day; the remaining days are allocated by simulating from a multinomial distribution in which the probability of being allocated to coarse grid cell 'x' on a particular day is proportional to the frequency with which days were allocated to 'x' in the original simulations.

(6) Then use the bird density p _i to simulate the grid cell for each bird on each day, conditional on the coarse grid cell that they are contained in on that day.

One potential problem at step (5) is that the random selection may lead to less than v( j) unique coarse grid cells. We, therefore, begin by assigning each of the unique coarse grid cells randomly to one day each, guaranteeing that this does not happen; the coarse grid cells for the remaining d - v( j) days are then allocated in proportion to their frequencies in the original selections made by each bird (before site fidelity is imposed).