Predictability–standard fisheries models (SFMs)

We evaluated the performance of three SFMs to predict the spawner-recruit relationship in the 134 populations from the Ransom Myers database: density independence, Ricker, and Beverton-Holt⁴. These models assume that the number of recruits is a function of the current stock size. All models can be written in the general form Rt=αStg(St), where R is recruitment, S is stock size, α is the maximum rate of reproduction, and g(St) is a function that accounts for density-dependent processes¹⁴. In the case of the density-independent model, the function g(St)=1 and the model is a straight line that intercepts the origin (0,0) with slope α. The Ricker and Beverton-Holt models introduce the term β, which is proportional to the product of fecundity and density-dependent mortality (see e.g. Quinn & Deriso³³). The three models are presented below.

The Ricker and Beverton-Holt models were fitted on a log scale, re-written so that yt=ln[Rt/St]³³. All models were fit using the function ‘fminsearch’ in Matlab R2015b.

To calculate the predictability achieved by each model, we performed leave-one-out cross validation.The minimum length of any time series in our dataset was 25 years; thus, to ensure sample sizes were consistent across models, for time series with more than 25 points, we randomly selected 25 points as targets to be predicted. For each prediction on a target, the point one timestep before the target and 23 other, randomly selected points were used to fit model parameters, and used to make a prediction on the target. For time series with exactly 25 points, all points were used as targets to be predicted, and the other 24 points were used to fit model parameters on each iteration. Even though the Ricker and Beverton-Holt models were fitted on a log scale, all predictions were made in the original recruitment scale. We then calculated the predictability (ρ) as the Pearson correlation coefficient between the 25 observations and their respective predicted values.