Appendix A. Methodological details.
Power-law tail probability density function
After setting the maximum step length for a particular simulation (e.g., 100 km), the power-law tail probability density function for a particular step length li was:
where = 1 (normalizing constant), µ = 2 (idealized Lévy flight power exponent), li = a step length value in the 1…i…k range. Minimum daily step length was set to approximately 1 % of the maximum to avoid problems associated with zero values. R code (R Development Core Team 2004) for this procedure is provided in the Supplement.
To simulate an animal’s movement track based on the power-law tail probability density function described above, we first set the number of days of tracking (e.g., 365) and assumed that each step length sampled from Pr(li) represented a daily movement. For each day, we sampled with replacement a li from the Pr(li) function and a random turn angle between 0 and 360º. Starting at a grid co-ordinate of x,y = 0,0, we calculated the new co-ordinate as:
where = turn angle in radians. R code (R Development Core Team 2004) for this procedure is provided in the Supplement.
Because the estimation of µ is highly sensitive to the histogram binning procedure used in comparing the log10 of the step length bin frequency and the log10 of the step length (Sims et al. In press), we modified the binning procedure such that the bin widths were set to increase exponentially relative to the number of k bins; here, the vector of bin widths = 2k (Viswanathan et al. 1996, Newman 2005):
where [w1 … wk] = the width-adjusted histogram bin vector and b1 = the start bin size. The number of bins (e.g., interval between bins) and b1 were determined iteratively to avoid too many bins with zero frequencies. For example, in the simulation with 100 km maximum li, the start bin was set to 0 with an interval of 0.15 km. After determining the optimal [w1 … wk], the bin frequencies must still be divided by the bin widths of [w1 … wk] to normalize the probability density. See the R code in Supplement 1 for more detail.
Incorporating location error
Errors associated with each tracking technology (Table 1, main text) were incorporated into the track simulations using a random normal deviate based on a mean = 1 and the standard deviation estimated for each error level. The true x,y co-ordinate for each time step was simulated as outlined above, but in this case the appropriate random normal deviate (north-south or east-west) was added to each x and y co-ordinate to generate a new (incorrect) location. The new error-blurred step length was then calculated using the Pythagorean Theorem:
where eli = error-blurred step length, and x" and y" = co-ordinates for the error-blurred (incorrect) location.
Newman, M. E. J. 2005. Power laws, Pareto distributions and Zipf’s law. Contemporary Physics 46:323351.
R Development Core Team. 2004. R: A language and environment for statistical computing. in. R Foundation for Statistical Computing, Vienna, Austria.
Sims, D. W., D. Righton, and J. W. Pitchford. In press. Minimising errors in identifying Lévy flight behavior of organisms. Journal of Animal Ecology.
Viswanathan, G. M., V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley. 1996. Levy flight search patterns of wandering albatrosses. Nature 381:413415.