Appendix E. Details of procedures used in uncertainty and elasticity analyses.
Uncertainty analysis
We conducted an uncertainty analysis to assess the amount of variation in the sea urchin (Strongylocentrotus droebachiensis) model predictions (i.e., urchin density and rate of advance of the front), given the natural variation observed in the parameters, by running 200 Monte Carlo simulations of the model. At the beginning of each simulation, we randomly selected a value for each parameter of the model (r, c, g, σ) from their corresponding distribution. For r and c, we used a normal distribution based on the mean and standard error of each parameter estimate (see Appendix G). For g, we used a uniform distribution between the range of observed values because we did not have information on the variance associated with grazing rates. For σ, we generated a distribution of values using a bootstrap procedure by redrawing individual urchin displacements (Manly 1997). We did not include the feeding threshold ratio (T) and the kelp carrying capacity (K) as we did not have information on their distribution. We used the percentile method (Manly 1997) to evaluate the 95 % confidence interval of the model output.
Elasticity analysis
Elasticity analysis is a type of sensitivity analysis that uses small proportional changes in parameter values to measure the response of the model (Caswell 2001). For Model 1 (Oreaster reticulatus), we calculated the elasticity of the mean rate of advance of the sea star front between t = 500 and t = 1000 to changes in parameter k of the von Mises distribution, and the ratio between the time needed to feed on a cell and the travelling speed (feeding time:travel time). We ran each simulation with 1000 sea stars, and ran 200 replicate simulations to account for variation inherent to the stochastic nature of the displacement process.
For Model 2 (Strongylocentrotus droebachiensis), we evaluated the elasticity of mean rate of advance of the urchin front between day 100 and 200 to changes in σ, g, exp(r), K and T (Appendix G ). We used exp(r), rather than r, so that the parameter had similar units (exp(r) is in d-1).
Elasticity is defined as:
(E.1) |
where Ep (%) is the elasticity of the output to a 1 % increase in parameter p, X0 is the output of the original model and Xp is the output of the model with parameter p modified (Barbeau and Caswell 1999).
LITERATURE CITED
Barbeau, M. A., and H. Caswell. 1999. A matrix model for short-term dynamics of seeded populations of sea scallops. Ecological Applications 9:266–287
Caswell, H. 2001. Matrix Population Models. Sinauer Associates, Sunderland, Massachusetts, USA.
Manly, B. F. J. 1997. Randomization, bootstrap and Monte Carlo methods in biology, Second edition. Chapman and Hall/CRC, New York, New York, USA.