## Abstract

Worm *et al*. (Research Articles, 3 November 2006, p. 787) reported an increasing proportion of fisheries in a “collapsed” state. We show that this may be an artifact of their definition of collapse as a fixed percentage of the maximum and that an increase in the number of managed fisheries could produce similar patterns as an increase in fisheries with catches below 10% of the maximum.

Based on an analysis of commercial catch data from 64 large marine ecosystems (LMEs) spanning from 1950 to 2003, Worm *et al*. (*1*) reported that an increasing proportion of fisheries were in a “collapsed” state. Extrapolation of these data further suggested the potential for the world's fisheries to collapse by 2048. We contend that their analysis and subsequent projections are inappropriate because the reported pattern of increasing prevalence of collapsed fisheries is largely an artifact of their definition of collapse. Furthermore, the analysis does not account for the increase in the number of stocks managed.

Worm *et al*. (*1*) compared the catch of each species within an LME in each year to the maximum catch of that species over the entire time series. They defined a fishery as collapsed in a single year if the observed catch occurred after the maximum catch and was less than 10% of the recorded maximum catch. They then fit a trend line to the proportion of collapses over time and found a significant increase. Implicit in Worm *et al*.'s analysis is the null hypothesis that the proportion of collapses should not increase over time if catches are not decreasing. However, a notably similar pattern of increasing collapses to that shown in figure 3A in (*1*) can be generated by applying their definition of collapse to stationary (constant mean and variance) time series of random numbers (see Fig. 1A). To produce appropriate expectations for the null model (i.e., evaluate consequences of the Worm *et al*. definition of collapse), we generated 5000 time series of random numbers, 34 to 54 years long (*2*), from an autocorrelated log-normal process with a coefficient of variation (CV) of 80% and an autocorrelation coefficient of 0.75 (*3*). These time series of random numbers represent catch time series for which catches are not declining on average. Our analyses make no assumptions about underlying population dynamics or fishing effort. Rather, we assume only that catch data from a non-trending fishery are log-normally distributed, with a constant mean and variance. We used the same definition of collapse as Worm *et al*. and scored a time series as collapsed in a given year if the value was less than 10% of the observed maximum and the value occurred after the maximum.

This analysis indicates that the null expectation of applying the Worm *et al*. definition is an increasing proportion of collapsed time series (i.e., below 10% of the maximum). This occurs because the expected value of the maximum is an increasing function of the length of a time series (*4*). The rate of increase in the proportion of collapsed time series depends on the CV of the time series. The curvature in the proportion of collapsed time series occurs because the time series begin at different times. Thus, about half of the observed collapses reported by Worm *et al*. in (*1*) are expected simply by chance in randomly generated time series with fixed mean values (without decreasing trends). Increasing the CV to 110% produced a similar magnitude of proportions of collapsed stocks to Worm *et al*. (Fig. 1B). Simply stated, applying Worm *et al*.'s definition of collapse produces an increasing pattern of collapses over time even when time series are not declining on average. Thus, their finding of a significant increase in collapses over time could be due to chance rather than to declining populations.

By comparing all subsequent catches to the maximum, Worm *et al*. seem to suggest that maximum historical catch represents an achievable and sustainable target for fisheries management. However, maximum historical catches are not likely to be sustainable and are therefore not ideal measures of sustainable ecosystem services or targets for fisheries management. Fisheries brought under management to reduce overexploitation would tend to have reductions in catch, which Worm *et al*.'s approach could erroneously categorize as being collapsed but which in fact represent an improvement in fisheries management. Without examining each, or at least many, of the time series for the proportion that have come under management, it is impossible to determine whether decreases in catch are due to management, overfishing, or other causes.

We believe people should be concerned about conserving the world's marine resources, and we are not arguing that some fisheries have not or are not collapsing. However, the analysis of Worm *et al*. may exaggerate the magnitude of the problem; ad hoc measures of overexploitation need to be evaluated to determine whether patterns are actually due to putative causes.