Strength and consistency of density dependence in marine fish productivity

The correct prediction of the shape and strength of density dependence in productivity is key to predicting future stock development and providing the best possible long- term fisheries management advice. Here, we identify unbiased estimators of the relationship between somatic growth, recruitment and density, and apply these to 80 stocks in the Northeast Atlantic. The analyses revealed density- dependent recruitment in 68% of the stocks. Excluding pelagic stocks exhibiting significant trends in spawning stock biomass, the probability of significant density dependence was even higher at 78%. The relationships demonstrated that at the commonly used biomass limit of 0.2 times maximum spawning stock size, only 32% of the stocks attained three quarters of their maximum recruitment. This leaves 68% of the stocks with less than three quarters of their maximum recruitment at this biomass limit. Significantly lower recruitment at high stock size than at intermediate stock size was seen in 38% of the stocks. Density dependence in late growth occurred in 54% of the stocks, whereas early growth was generally density- independent. Pelagic stocks were less likely to exhibit density dependence in recruitment than demersal and benthic stocks. We recommend that both the degree to which productivity is related to density and the


| INTRODUC TI ON
Density-dependent processes are key to avoiding population extinction and explosion (Henle et al., 2004). Understanding the strength, direction and consistency of density dependence is particularly important for exploited populations, for which correctly predicting the effects is essential to avoiding over-or under-exploitation with associated loss of ecological sustainability or social benefits. The theoretical explanations of density dependence are mainly derived from resource limitation of either the species investigated or its predators. Under resource limitation, the abundance of individuals affects individual growth, reproduction and survival as competition for prey increases with abundance (Chesson, 1998). Additionally, survival and growth can be density-dependent as a result of the response of predators to changes in prey abundance in the form of satiation, diet switching or aggregation in areas of high prey abundance (Bax, 1998).
Unfortunately, the processes through which density dependence act are notoriously difficult to identify, even in populations that are clearly regulated (Murdoch, 1994). The difficulty in detection can at least partially be alleviated by using information over a long time series and analysing a large number of populations concurrently. Long time series are widely available for commercially exploited fish populations. These populations are generally managed based on shortterm projections to meet long-term management aims. Information on the level of fishing that would achieve Maximum Sustainable Yield (MSY, UN, 2002) is derived from population models that include density dependence implicitly or explicitly (Quinn & Deriso, 1999). The assumptions made about the characteristics and functional form of density dependence are highly influential in determining the exploitation rate and selectivity pattern that produces MSY. Generally, long-term predictions of yield and sustainability assume density dependence in the abundance of incoming recruits by incorporating a decelerating relationship between spawning stock biomass and recruitment (Cadigan, 2013). In contrast, density dependence in individual growth is rarely incorporated and hence the effect on potential yield of such changes is largely ignored, though some analyses exist (Andersen et al., 2017;Gislason, 1999;Horbowy & Luzeńczyk, 2017). These assumptions raise questions as to how recruitment and growth depend on density within the range of stock biomass modelled and whether these relationships are consistent over time.
The objective of the present study was to determine the strength, direction and consistency of density dependence in recruitment and body growth of fished stocks in the North Atlantic. Recruitment is defined here as the combined effect of spawning output and early life survival. The analyses aim to determine whether density dependence is present, whether the strength and direction differs between species feeding habitat types (pelagic: feeding in the water column, demersal: feeding on the bottom and in the waters above it, benthic: feeding on the bottom), and whether the relationship is consistent over time. The results of these analyses are also investigated in relation to characteristics such as the contrast in the densities observed and the variability around predicted relationships. We note that density dependence is not the only relevant driver, and that the signals from density dependence will be confounded with environmental drivers such as variation in temperature and habitat quality and food abundance, which again may be impacted by density.
An increase in food abundance over time may spark increased recruitment success and increased growth and hence a positive correlation between abundance and individual growth without a causal relationship with density. In contrast, a decrease in food degree to which the relationship changes over time should be investigated. Both of these aspects should be considered in evaluations of whether sustainability and yield can be improved by including density dependence in forecasts of the effects of different management actions.

K E Y W O R D S
Benthic fish, demersal fish, fisheries, North Atlantic, pelagic fish, stability  Beverton-Holt (1957) and Ricker (1954) functions, both of which through their shape have the disadvantage that the slope of the curve at low densities is highly correlated with the slope at high densities.

| Data
The data used for the analyses were derived for stocks in the North Atlantic from the ICES stock database, published literature and personal contacts (Table S1). The data covered 80 stocks and 25 species in the North Atlantic. Both recruitment and the first reliable age with data on weight at age in the stock data were derived from stock assessments. The term recruitment in this manuscript refers to the number of fish in a cohort that survive to the first age included in the stock assessment model. Recruitment in stock assessment models is determined from survey indices and the age composition of catches. The ages at which cohorts are consistently observed are most influential in determining cohort strength. The inherent assumption in most assessment models is that natural mortality is constant or at least density independent.
Hence, when recruitment in the model is given at age 0, recruitment at this age is estimated under the assumption that all density dependence occurs before this age. The age at recruitment in the assessment was strongly correlated to the age at which Lorenzen and Camp (2019) suggests density dependence has fully acted (20% of asymptotic length, Supplementary Material).
Based on this, we infer that the age at which recruitment occurs in our data is unlikely to bias our conclusions. The first reliable age with weight at age data is often based on samples from commercial catches. However, weight at age was based on survey data for younger ages in some of the stocks. The first reliable weight at age was within one year of the age of recruitment for all but 7 stocks but where the reliable weight at age was 2 years from age at recruitment. We divided the stocks into pelagic, demersal, and  (Takasuka et al., 2019), we chose this approach to ensure consistency across stocks with and without information of total egg production per kg of spawning biomass. We further related recruitment to the abundance of the preceding cohort, assuming that larger fish are at a competitive advantage over smaller fish (Hoare et al., 2000). If food abundance is limited or predation mortality increases with abundance, we expect to see a decrease in recruitment success at high initial cohort size. We note that other environmental variation may also occur.
If food production is constant, the relevant measure of density for impacts on growth is the food consumption by the competing fish. Most fish show pronounced ontogenetic shifts from early life to later life, and hence, the recruiting cohort is likely to compete mainly with individuals in the same cohort (Hoare et al., 2000). While there may be some variation in the weight of individuals within the cohort, this is likely to be much less than the variation in number of recruits  Figure S1).
In addition to the contrast and long-term stock development, a measure of inter-annual variability was derived as the r 2 of a loess fit to the log-transformed density measure (indicating the proportion of variation explained by long-term changes, Spencer & Collie, 1997). We used the loess.as function in the fANCOVA package in R to automate the selection of the 'span' smoothing parameter. We expect that density dependence is most easily determined where contrast and long-term change is large and inter-annual variability is small.

| Recruitment
The dominant models for the relationship between stock size and recruitment, Beverton-Holt and Ricker, are very similar in their prediction of the relationship between recruitment and stock size at low abundance. However, they differ in the predicted relationship at high stock size, the Ricker model predicting a decrease in recruitment, whereas the Beverton-Holt relationship predicts no change or a slight increase in recruitment as stock size continues to increase and carrying capacity is approached. We do not wish to prejudge the shape of this relationship in this study, and therefore we employ a more general model suggested by Cadigan The model was formulated with smoothing parameters k = 20 and sp = 0.01 to provide enough flexibility to capture the trends in the data. In some cases, this resulted in unrealistic model behaviour, with recruitment increasing, then decreasing, and then increasing again as SSB increased. In these cases, sp was set to 0.1 to increase the amount of smoothing. For comparison, we included a model with a proportional relationship between recruitment and spawning stock biomass (R = aSSB) as this is the relationship that should appear in the absence of any density dependence. Subsequently, we evaluated which of the two models resulted in the lowest AIC.
One stock, for which recruitment occurred at age 5 was not included in the analysis of recruitment as by this time, the number of fish in the cohort may have been affected by large differences in mortality due to reasons unrelated to density.
The Cadigan model assumes that the variance of SSB estimates are negligible compared to those of recruitment estimates. If this assumption is not valid this could potentially affect the estimated stock recruitment relationship (Walters and Ludwig, 1981;Kehler et al., 2002;Kope, 2006;Cadigan, 2009). To address this issue, the stock recruitment relationship can be estimated within the assessment model, thereby accounting for variance and covariance of recruitment and SSB in the estimation of the stock recruitment function.
This was possible for a selection of stocks assessed using the statespace assessment model (SAM, Nielsen & Berg, 2014) at stockassessment.org or through personal correspondence. For these cases, a stock recruitment relationship similar to the Cadigan model (compensatory mortality property or CMP spline, Albertsen & Trijoulet, 2020) was estimated internally in SAM and the resulting steepness and overcompensation compared to those derived when estimating the relationship externally from the stock assessment using both the CMP spline and the Cadigan model.
We analysed the probability that a stock is best fitted by a nonproportional (density-dependent) relationship using general linear models assuming binomial distributed observations. We included model effects for species, ecotype and SSB (decline/no trend/increase across all observations) and continuous effects of interannual variation and contrast in SSB.
As additional measures of the strength of density dependence, we estimated the steepness h of the estimated Cadigan relationship: where max(R (SSB) )is the maximum recruitment predicted from the estimated relationship and R (0.2 * max(SSB))is the predicted recruitment at a spawning stock biomass of 0.2 times the maximum observed spawning stock biomass, max(SSB) (Punt & Dorn, 2014).
Steepness can only take values from 0.2 to 1, and h was transformed to produce a measure that is continuous on (−∞, ∞) for further analyses:

Recruitment
No density dependence Mortality in the pre-recruit stage is not density-dependent Recruitment increases proportionally with initial cohort abundance, indicated by spawning stock biomass AIC of a proportional relationship exceeds that of non-proportional relationships Compensation Density dependence leads to increased mortality at higher abundance.
Recruitment increases with initial cohort abundance to an asymptote, after which there is no significant change AIC of a non-proportional relationship exceeds that of proportional relationships. Further, low steepness indicates small distance of the stock from the proportional relationship and vice versa Over-compensation Density dependence leads to decreased survival as abundance increases.
Recruitment increases with initial cohort abundance up to a maximum, after which there is a significant decrease with cohort abundance Predicted recruitment at maximum observed biomass is smaller than lower confidence interval of maximum predicted recruitment Depensation Predator satiation leads to increased survival beyond the satiation level Recruitment at low stock size is lower than expected in the stock recruitment relationship The mean of residuals from the predicted stock recruitment relationship at low SSB significantly less than zero Correlation between density and growth significantly less than zero Apparent positive density dependence Increased food availability leads to increased recruitment, increased growth and increased biomass High recruitment success initially co-occurs with high growth as increased food abundance enhances both. The correlation between the two disappears if the new state persists long enough for food abundance and stock size to once again be in balance around a new level.
Correlation between density and growth significantly greater than zero

TA B L E 1 (Continued)
The effects were analysed in a general linear model assuming the logit was normally distributed, analysing fixed effects of species, ecotype and stock decline/no trend/increase and continuous effects of interannual variation and contrast in SSB.
Overcompensation (OC) of the estimated relationship was defined as: Overcompensation was analysed in two models, one for the probability of a stock exhibiting significant overcompensation (assuming binomial distributed observations) and one for the logit transformed OC: Stocks with OC = 0 were omitted from the analyses using logit transformed data. Both the probability of exhibiting significant overcompensation and the degree of overcompensation were analysed in a general linear model with effects described above. Estimated values were transformed back to OC before presenting the results. In the proportional model, h is 0.2 and OC is 0 by definition.
Overcompensation was recorded as significant if the predicted recruitment at the maximum observed stock size max(SSB) was below the confidence interval of the maximum recruitment predicted within the range of observed spawning stock biomass. After deriving estimates of steepness and the probability of significant overcompensation, these were tested for significant differences between habitat types, asymptotic length of the species (derived from Rindorf et al., 2020), contrast and trend in spawning stock observations using general linear models with a normal and binomial error distribution respectively.
We investigated evidence of depensation (lower recruitment than proportional at low stock size) by testing if the average log residual at SSB less than 0.2 times the maximum observed biomass was significantly less than zero, analysing the 47 stocks that had at least five SSB values below 0.2 max (SSB) . We also tested if recruitment decreased following a large cohort, which can be caused by e.g. competition for low mortality habitat, by estimating the autocorrelation in log recruitment. The length of the time series affects estimates of steepness and overcompensation by changing the maximum observed spawning stock, and hence estimates of temporal changes in steepness and overcompensation were not conducted.

| Growth
The growth measure used must be responsive to changes caused by density in any individual year but preferably not the previous or subsequent year. This is particularly important for species where the biomass may fluctuate greatly between years, as is often the case for short lived species. Furthermore, it is important that the measure has a high power to detect density dependence with high precision. We distinguish between measures to detect growth changes early in life (growth to the age of first occurrence in the fishery data) and subsequent growth (growth after the age of first occurrence in the fishery). This eliminates any impact of growth later in life on estimates of early growth, in contrast to using methods based on estimating cohort growth curves (e.g. von Bertalanffy, 1938). Furthermore, it allows us to use all data regardless of the number of times a cohort has been observed whereas estimation of the von Bertalanffy curves rely on the observation of a minimum of three age groups (one for each parameter to be estimated). To determine the most appropriate combination of growth and density measures, we first conducted a simulation study to determine the combination that provides high power to detect density dependence when it occurs and the lowest risk of The effect of density dependence on juvenile growth was analysed by relating the weight at age of each species at the youngest age consistently observed to recruitment numbers of the cohort. To obtain reliable results, weight at age must be consistently sampled and reflect cohort growth rather than differences in catchability due to changes in survey time relative to the growth period. This was ensured by eliminating ages for which weight at age was not significantly correlated to the weight at age of the same cohort in the subsequent year. The analyses were performed on log transformed data as variance of both numbers and weight at age increased with the mean. A drawback to this method is that the initial age will depend on how sampling was conducted as well as the biology of the fish stock. Recruitment and weight at age were matched to be that of the cohort. This measure has the advantage of integrating annual effects in a single measure. However, by doing so, the measure tends to smear out cohort effects caused by rapid or slow growth in the first year of life. To avoid this affecting longer time periods, we used only three age classes for all stocks. The ages were chosen as 1-3 ages above the youngest age with consistent weight at age data.

| Analyses of density dependence of growth
The presence of density dependence in growth (weight at the youngest age and lnR for early growth and G late and lnB* for late growth) was investigated in two analyses: (1) estimation of correlation between the growth and the density measure and (2)  We repeated the correlation analyses for a moving window of 20 years, resulting in measures revealing the long-term variation in the density dependence of growth. The moving window analyses will identify positive relationships between survival and growth without reacting to long-term correlations between the number of recruits and the mean weight at age of the youngest age caused by environmental changes increasing both factors simultaneously.
In addition to the above analysis of early growth and density, we investigated the evidence for positive effects of growth on survival of juveniles by relating mean weight to residuals from the stockrecruitment relationship, as positive residuals indicate greater than average survival. After deriving estimates of correlation between growth and density for each of the two measures as well as the probability of significant negative and positive correlations, these were tested for significant differences between habitat types, range and trend in density using general linear models with a normal and binomial error distribution.

| Recruitment
Among the stocks investigated, the relationship between recruitment and spawning stock biomass was better described by a density-dependent (non-proportional) relationship in 54 out of 79 stocks, corresponding to significant density dependence in 68% of the stocks (Figure 2, Table S2, Figure S2)   The confidence intervals of the estimated curves were slightly larger in internal than external CMP. In conclusion, there may be minor effects of estimating the stock recruitment relationship externally, with overcompensation and the occurrence of significant overcompensation likely to be slightly more affected than steepness.
Only two of the 79 stocks (2.5%) showed significantly negative residuals below 20% of maximum spawning stock, a pattern consistent with depensation, which supports our decision not to include a depensatory recruitment model, as the proportion is less than the 5% expected by type 1 error. The results are summarised in Table 2.

| Early growth
Only 17 of the 69 stocks (25%) exhibited a significant correlation between recruitment and weight of the cohort at the youngest age observed (Figure 3, Table S4, Figure S4). Among these, negative relationships were more frequent than positive relationships (11 negative and 6 positive significant relationships). The effect on early growth was closely related to the correlation coefficient. There was no significant effect of trend in recruitment, species, ecotype or stock development on the correlation (p > 0.05). The probability of achieving a negative correlation between early growth and recruitment was significantly higher than the 0.05 expected from type-1 error (probability 0.17, 95% confidence interval (0.10, 0.28)) with no difference between species, ecotypes or stock development (p > 0.05). The probability of achieving a significant positive correlation did not exceed that expected by type-1 error significantly (probability 0.09, 95% confidence interval (0.04, 0.18)). There was no significant over-occurrence of above-average growth at below average abundance ( Figure S5).
The correlation between growth and abundance of benthic and demersal stocks was centered around 0 for the stocks having sufficient observations to perform the moving window analyses (Figures   4 and 5). In contrast, the correlation of pelagics showed considerable changes over time (range of loess 0.7 on average), varying from a correlation close to 0 to a substantial negative correlation ( Figure 5).
Investigating the evidence for individual growth enhancing cohort survival, there was a significant positive relationship between growth and residuals from the stock recruitment curve in four out of 69 stocks, corresponding to 5.8% and hence the level expected by type 1 error alone. In contrast, 14 of the 69 stocks showed a significant negative relationship between growth and residuals from the stock recruitment curve, indicating that a higher than expected year class generally experienced less than average growth. There were no clear tendencies when comparing fish groups (Supplementary Material, Fig. S6).

| Late growth
Thirty-seven of the 61 stocks (61%) investigated exhibited a significant correlation between total stock biomass and late growth ( Figure 6, Table S5). Negative relationships were much more frequent than positive relationships (33 negative and 4 positive significant relationships) ( Figure 6). The probability of obtaining a significant negative correlation was 0.54 (95% confidence interval (0.42, 0.66), significantly larger than the 5% expected from type-1 error, and it did not differ significantly between any of the factors tested. The probability of significant positive correlations was 0.07 (95% confidence interval (0.03, 0.14)), which is not significantly different from the 5% expected from type-1 error.
The quadrant analysis supported the occurrence of strong density dependence in late growth in many stocks, as 45 of the 61 showed lower than average growth at higher than average biomass (benthic: 13 out of 14 stocks, demersal: 19 out of 27 stocks and pelagic: 13 out of 20 stocks). At below average abundance, an overoccurrence of above average growth was seen in 40 out of 61 stocks (benthic: 12 out of 14 stocks, demersal: 15 out of 27 stocks and pelagic: 13 out of 20 stocks) ( Figure S7).
The correlations of the demersal stocks having sufficient numbers of observations to perform the moving window analyses showed a very large variation in the correlation over time, with both large negative and moderate positive correlations (Figure 7). In contrast, the correlation of benthic and pelagic stocks changed little over time and were almost exclusively negative (Figure 7).
There was a tendency for the correlations between density and growth to be positively correlated for early and late growth of a stock, but the correlation was not significant (p > 0.08, Figures   S8-S12). There were no significant correlations between steepness, overcompensation and the change in relative growth of a stock at observed densities (p > 0.39, Figures 8, S12). Of 60 stocks, The results of all analyses are summarised in Table 2.

| DISCUSS ION
Density dependence in recruitment and late growth occurred in the majority of stocks, whereas density dependence was uncommon in early growth. Benthic stocks were more likely to exhibit densitydependent growth than demersal and pelagic stocks, and pelagic stocks with trends in stock size had lower probabilities of density dependence in their stock-recruitment relationship than benthic and demersal stocks. If habitat size of pelagic species is larger than that of benthic species, the pattern in density dependence confirms the predictions of Andersen et al. (2017) that density dependence of larger species in medium sized habitats may occur later than that of smaller species in larger habitats.
Density dependence in stock recruitment was by far the most common relationship. Across all stocks, 68% exhibited significant density dependence, and the probability of occurrence was even higher (78%) when pelagic stocks exhibiting significant trends in spawning stock biomass were excluded. This level is the same order of magnitude as that found in a study of 16 marine and freshwater fish populations (Lorenzen, 2008) and a meta-analysis of recruits per spawning biomass as a function of spawning biomass (assuming a Ricker curve, Zimmermann et al., 2018). The relationships estimated here were significantly density-dependent when tested against the null model of proportional recruitment. Considering the degree of curvature of the stock-recruitment relationships, 13% of the stocks of the stocks had steepness values above 0.9 and 32% were above 0.75. For the remaining 55% of the stocks, recruitment would therefore already be seriously impaired before biomass fell below 20%B max and presumably therefore also 20% of the average unfished biomass, a reference point that is frequently used as lower limit (AFMA, 2007). Likewise, limits of 0.40-0.60 times virgin biomass would not ensure full recruitment of all stocks (AFMA, 2007;Pikitch et al., 2012). In fact, in 22%-32% of the stocks, there was no density dependence in recruitment success, indicating that a decrease in stock density would lead to a decrease in average recruitment regardless of the density at which it occurred. We did not find a F I G U R E 3 Relationship between weight at the youngest observed age and cohort abundance. Top left: correlations estimated between growth and recruitment by species group. Remaining plots: Change in growth from minimum to maximum observed recruitment as a function of correlation (top right, red indicates significant correlations (p < 0.05)), by species groups for all species (bottom left) and by species groups where the correlation is significant (bottom right). Ranged from +34% to −33% with neither of these extremes being significant correlations. The effect on early growth ranged from −5% to −29% among the significant negative correlations significant effect of asymptotic length on steepness or overcompensation, in accordance with Thorson (2020) but contrary to Goodwin et al. (2006), whose analysis examined derivatives of fitted Ricker and 'hockey stick' stock recruitment relationships. It is unclear if the difference in results is related to our use of a more flexible stock recruitment relationship and hence in effect a structural effect rather than an underlying biological difference. There was no significant evidence for depensation occurring more frequently than what is expected by chance, confirming previous results of Myers et al. (1995) and Hilborn et al. (2014).
Only half the pelagic stocks exhibited significant density dependence in recruitment and only 8% showed significant overcompensation. This difference from benthic and demersal stocks seemed to be linked to lower density dependence when there were significant trends in stock size. These trends can occur when there are changes in stock productivity, effectively meaning that two stock recruitment relationships are overlayed in the relationship for the full period. If one productivity period has low SSB and low recruitment and the other high SSB and high recruitment, overlaying the two in one relationship gives the appearance of a linear relationship (Szuwalski et al., 2015). Hence, a higher frequency of changes in early survival rates of pelagic fish than occur in the more stable demersal communities may be the cause of the difference between the ecotypes.
Overcompensation occurred most frequently for benthic and demersal stocks. These stocks often settle to bottom habitat that is already inhabited by older conspecifics. Furthermore, most demersal stocks feed on larger items than pelagic fish of the same size, which allows them to be opportunistic cannibals (Bogstad et al., 1994;Link et al., 2009;Uzars & Plikshs, 2000). Pelagic fish often school together with individuals of similar size (Hoare et al., 2000), thereby effectively limiting the interaction between adults and juve- F I G U R E 5 20-year moving window analyses of correlation coefficients between early growth and cohort abundance. Range denotes the range (maximum-minimum) of the loess of correlation as a function of start year of the moving window, residual standard deviation is the variation around the loess and finally, average is the average correlation across all moving window analyses for a stock. Numbers above boxes denote number of stocks included thereby potential overcompensation. In benthic fish, overcompensation must be linked to competition for either food or space or both (Biro et al., 2003) as most benthic fish have insufficient gape width to cannibalise settling juveniles.
Lack of density dependence in early growth is consistent with the food competition hypothesis if food abundance varies greatly between years for reasons unrelated to fish density. Most fish stocks feed on zooplankton at some stage during the recruitment phase, but only pelagic planktivores continue this diet throughout their life span. Zooplankton biomass is often considered highly variable, even if the variation between years seems lower after accounting for the various sampling artefacts (Beaugrand & Reid, 2003). The recent interdecadal changes in planktonic food abundance (Mackas & Beaugrand, 2010) (Lorenzen, 2008) have addressed this issue by predicting the biomass in the absence of fishing, but as there are generally no data from periods with sustained lack of fishing, this requires predictions of population dynamics well outside the observed range of densities, potentially introducing structural errors (Miller & Brooks, 2021).
Furthermore, the estimated biomass in the absence of fishing is highly dependent on the estimated density dependence, thus leading to circularity in conclusions. In addition to the average relationships between SSB and recruitment, some stocks show occasional very high recruitments which rather unexpectedly succeed in bringing the stock size back from very low levels. Such patterns were not specifically accounted for in our analysis. In addition, our analysis does not correct for the effect that changes in size selectivity in the fishery may have had on both survival and observed mean weight at age (Kvamme & Bogstad, 2007). Where fishing pressure has varied significantly, overall biomass of a stock may be confounded with changes in the age structure, which may result in changes in the nature of any density-dependent relationship. Finally, any changes in productivity related to an expansion of the distribution area need to be carefully examined to determine how the expansion affects density dependence.
Density dependence affects the management of exploited stocks through effects on productivity and reference points.
Interestingly, the approach to estimating recruitment in the shortterm forecast for the coming one or two years is often a simple long-term geometric mean, even for stocks that are below levels at which recruitment is expected to be impaired. This assumption results in higher estimates of future recruitment and stock size than accounting for the expected decline in recruitment. The higher estimated biomass leads to higher estimates of future catch opportunities, which if implemented work to maintain the stock at low stock size if the recruitment as expected is below the geometric F I G U R E 8 Change in late weight at age relative to average weight at age from minimum to maximum observed total stock biomass as a function of steepness (left) and overcompensation (middle) and change in relative weight and recruitment as density changes from 0.2 to 1 times maximum observed stock density (right). Red: density dependence in growth (declines only) and recruitment (left) and growth and overcompensation (right) significant. Green: Only density dependence in growth (declines only) significant. Blue: Density dependence in recruitment (left) and overcompensation (right) but no density dependence in growth. Open: No significant density dependence in growth (declines only) or recruitment mean in the coming year. Pelagic stocks showed lower levels of density dependence in recruitment. Hence, pelagic stocks will exhibit greater fluctuations and lower self-regulatory capabilities than benthic and demersal stocks unless mortality is lower at low density and this effect is greater in pelagic than benthic and demersal stocks. Though not considered in this analysis, there is accumulating evidence of time-varying productivity (Tableau et al., 2019), which may affect the estimation of density dependence in recruitment (Claireaux et al., 2022). Time-varying productivity can be implemented in the management system through dynamic reference levels, which may improve performance in terms of both sustainability and yield (Zhang et al., 2020).
While the long-term simulations used to derive reference points such as F MSY and B MSY usually account for density dependence in recruitment, there is typically no accounting for density-dependent growth.
A key reason for this may be the effect that including density-dependent growth has on exploitation levels providing MSY; including density dependence means that the gains of having a large population are less. As a result, the level of exploitation that produces MSY is higher than previously thought (Gislason, 1999;Kovalev & Bogstad, 2005;Sparholt et al., 2021) and the selectivity that attains this may involve catching a higher proportion of smaller and younger individuals than at present, particularly in cases where density dependence occurs late in life (Gemert 2018a, 2018b). While increasing the exploitation rate at high stock size may be desirable, including density-dependent growth at low stock size is risky because it will lead to predictions of higher than average growth in the coming two years and higher estimates of catch opportunities, thereby potentially keeping the stock at low levels of biomass for prolonged periods. Hence, it may be preferable for precautionary reasons to assume that density dependence in growth acts at high densities only (equivalent to the 'weak density dependence' in growth examined here). It should also be noted that the negative bias of assuming constant growth may cancel the positive bias of assuming constant recruitment when projecting catch opportunities at low stock size. An improved prediction of stock response to catch levels relies on an adequate understanding of the historical development in the stock, but also requires acknowledging that density-dependent relationships may change over time (Claireaux et al., 2022;Howell et al., 2013).
Density-dependent growth implies food limitation, which depends, not only on consumer density, but also on prey density, which may change over time. Based on the results of this study, we recommend routine testing for density dependence in recruitment and growth, especially the late growth period. As even significant relationships between density and growth do not necessarily impact the stocks greatly (Lorenzen, 2008;Stawitz & Essington, 2019), we further recommend evaluating the potential importance of the different sources of density dependence on both productivity and variability (Rose et al., 2001). When warranted, both density-dependent recruitment and growth should be included in calculating reference points and providing short-term catch advice in a way that is compatible with policy principles such as the precautionary principle and MSY.

DATA AVA I L A B I L I T Y S TAT E M E N T
The data used area available at https://doi.org/10.11583/ DTU.16887769 (Rindorf & van Deurs, 2021).