The Rosa Lee phenomenon and its consequences for fisheries advice on changes in fishing mortality or gear selectivity

Abstract

When size-selective fishing removes faster-growing individuals at higher rates than slower-growing fish, the surviving populations will become dominated by slower-growing individuals. When this “Rosa Lee phenomenon” is ignored, bias may occur in catch and stock projections. In a length-and-age-based model we quantified the effects through simulations of a simplified fishery on a stock that resembles Western Baltic cod. We compared outcomes of runs with and without taking account of the Rosa Lee phenomenon in scenarios of changes in fishing mortality. We found that, when only fishing rate was changed, the biases in predictions of spawning-stock biomass (SSB), yield and catches of undersized fish were relatively small (<10% in absolute values). When the selectivity parameters of the gear were increased, the bias in the prediction of the catches of undersized fish was very substantial (+120 to 160%). When the selectivity parameters were decreased, the biases in the predictions of SSB, yield and catches of undersized fish, were substantial (25–50% in absolute values). With slower mean growth the biases became more pronounced. We conclude that in short-term forecasts, medium-term projections, and MSE simulations featuring selectivity changes, the Rosa Lee phenomenon should be accounted for, ideally by using length-based models.

age–length-structured population model, demographic consequences, Rosa Lee phenomenon, selectivity changes, stock effects

Introduction

In fisheries science, stock projections and simulations are often undertaken to evaluate the consequences to yield and stock status of various fishing scenarios. One example is the short-term forecast in the ICES advice catch-option tables (EU, 2018; ICES, 2018a) where the stock is projected forward 2 years and the predicted catch and stock status are listed against different scenarios of fishing mortality rate. Another example is to predict the consequences of changes in gear selectivity, which is particularly important in the light of the EU landing obligation (discard ban; EU, 2013), which calls for changes in selectivity. In management strategy evaluation (MSE) (Smith, 1994; Punt et al., 2016), projections are longer term, typically a few decades, and the scenarios may involve complex harvest control rules for fishing as well as alternative hypotheses on the non-fishing drivers of the population dynamics, such as climate change. The calculations are usually done in an age x year matrix, using the exponential decay equation determining the decreasing numbers-at-age by cohort and the Baranov catch equation (Baranov, 1918), both based on assumed values of natural mortality at age (M-at-age) and fishing mortality at age (F-at-age). Relative F-at-age is usually referred to as the exploitation or selection pattern. Underlying the selection-at-age pattern is the selection-at-length pattern. This selection pattern reflects the availability of the fish to the fishing gear as well as the size selection of the gear. In trawl fisheries, size selection is described by a sigmoid curve. This means that with a particular configuration of the gear, e.g. mesh size or mesh shape, individual fish of a small length can escape through the net, while the probability of being retained in the net increases with increasing length until, at a sufficiently large length, all individuals are retained in the net (we note that other fishing gears, which are not considered here, can deliver different size-selection patterns, e.g. dome-shaped). Nevertheless, most fish stock projections are done only by age, assuming a constant mean length-at-age.

Mean length-at-age may, however, not be constant over time. First of all, length-at-age is a function of growth, and growth may be influenced by abiotic and biotic factors, such as temperature, prey availability, and density-dependent effects. But even if these factors are assumed to be constant and thus growth is assumed to be constant, length distribution-at-age is likely to vary over time. The “Rosa Lee phenomenon” (Lee, 1912) is the effect that apparent growth rate in fish populations is truncated as a result of size-selective (fishing) mortality. An explanation of this effect is that fish populations exhibit variation in intrinsic individual growth rate and that size-selective fishing (or other sources of similar size-selective mortality) removes faster-growing individuals at a higher rate than slower-growing fish, leaving behind a population that becomes cumulatively dominated by slower-growing individuals in the cohorts. The effect is well documented (Kristiansen and Svåsand, 1998; Catalano and Allen, 2010) and often cited in the context of the estimation of growth-rate parameters of fish populations (Ricker, 1969; Francis, 1988; Hanson and Chouinard, 1992; Beyer and Lassen, 1994; Gudmundsson, 2005). Some studies explored (i) the effect on the estimation of quantities relevant to fisheries advice, such as yield per recruit (Ricker, 1969; Parma and Deriso, 1990; Kvamme and Bogstad, 2007) or (ii) its incorporation into an assessment model (Taylor and Methot, 2013). A number of length-based assessment models exist (Punt et al., 2013). The commonly used assessment model Stock Synthesis has a rarely used feature that can account for the Rosa Lee phenomenon: the normal distribution of length-at-age can be partitioned into three or five overlapping platoons with slow, medium, or fast growth trajectories, which are then tracked separately in the model (Taylor and Methot, 2013, see also Punt et al., 2001). Surprisingly, the effect that the Rosa Lee phenomenon may have in the type of stock projections and simulations mentioned at the start of this article has not received much attention.

The Rosa Lee phenomenon will occur under the following conditions. First, even when growth at the population level is constant, there is random variation at the individual level. Second, these differences in growth, to some extent, persist through life; slow-growing individuals tend to remain slow-growing individuals, and fast-growing individuals tend to remain fast-growing individuals. Random variations that do not persist may also occur [an individual that grows slow this year may grow fast next year and vice versa, “transient variation”, sensuWebber and Thorson (2016)], but these variations may cancel out each other’s effects. Persistent individual growth differences are indeed likely to exist (Vincenzi et al., 2014; Webber and Thorson, 2016) since fish growth is thought to have heritabilities between 0.2 and 0.3 (Gjedrem, 1983; Carlson and Seamons, 2008), where heritability is the fraction of total phenotypic variance that is due to additive genetic variance, leading to individuals having different genetic predispositions for growth rate, all else being equal. Given these conditions, smaller, slower-growing individuals at each age will have a greater chance to escape the size-selective trawled gear and survive another year to the next age, at which time the relative frequency of slower-growing individuals will have increased. As a consequence, the individuals at each successive age in the population that have survived size-selective mortality will consist increasingly of slower-growing individuals, which are thus smaller than this age group would have been without size-selective mortality. In actual fact, the population consists of a select subsample of the fish at birth, namely those that survived cumulative size-selective mortality, i.e. the slow growers. This also explains why mean length-at-age is often seen to decrease across older ages (growth has slowed down while length-dependent removal is still taking place). Note that this is a purely demographic effect within cohorts and should not be confused with any fishery-induced genetic shifts in length-at-age that may take place in addition on an evolutionary (intergenerational) time scale (Michod, 1979; Law and Grey, 1989; Conover and Munch, 2002; Enberg et al., 2012) and which are not considered in this article.

When a population is in equilibrium and size-selective (fishing and natural) mortality is constant over time, the length distribution-at-age will remain constant over time. However, when size-selective mortality changes, size distribution-at-age will change accordingly. For example, when fishing mortality increases substantially, more of the faster-growing individuals will be removed from the population, leading to a greater length truncation at each age, i.e. a smaller average length-at-age in the population and a resulting smaller vulnerability-at-age to the gear. However, fisheries scientists usually assume length-at-age to be a fixed biological property of the population, even under scenarios of changing fishing mortality (such as in the short-term forecast of ICES advice). Another example is a change in mesh size. This will influence the survival of relatively fast- and slow-growing fish differentially, subsequently affecting the length distribution-at-age. This will, in turn, result in a different mesh-related selection-at-age than if the length distribution would be assumed to have remained constant. Because of these changes in the length-at-age distribution that are usually not accounted for, there will also be unaccounted for changes in the population’s weight-at-age as well as maturity-at-age, the latter usually being age- as well as length-dependent (Heino et al., 2002a, b), affecting yield and spawning-stock biomass (SSB) in ways that are normally not accounted for.

These considerations imply that when such demographic effects on length are ignored, projections of catch and SSB can be biased. Here, we explored and demonstrated the magnitude of the bias that may result when the Rosa Lee phenomenon is ignored. We did this under various scenarios of changes in fishing rate or fishing pattern in simulations of a simplified fishery on a stock that resembles western Baltic cod (Gadus morhua). We simulated persistent, but not transient, growth variation [sensuWebber and Thorson (2016)]. For each scenario, we pairwise compared a simulation setting where the Rosa Lee phenomenon is accounted for and one where it is ignored; i.e. in one setting, we mimicked the condition that growth differences persist through life and length-at-age distributions respond to (changes in) fishing as described above, whereas length-at-age was kept constant in the other setting.

Methods

The model

This section describes the basis of the model. Details are in the Supplementary material, and the R scripts are available at https://github.com/sarahbmkraak/Rosa-Lee-paper. The model tracks numbers of individuals in length bins (1 cm wide, range = 1–200 cm) within age bins (range = 1–20 years, no “plus-group”) across time (in monthly time-steps). Natural mortality is age-based. Fishing mortality is length-based according to a logistic selection ogive. Catch and population numbers were calculated according to the Baranov catch equation and exponential decay equation, respectively, using numbers-at-length-at-age at each time-step (N_i,s,a in the Supplementary material). Growth was modelled with six assumed parameters: averages of the two von Bertalanffy parameters $L_{\infty }$ and K, the third von Bertalanffy parameter t₀, the standard deviations (⁠$\sigma$⁠) of $L_{\infty }$ and K, and the Pearson correlation (⁠${\rho }_{L_{\infty },K}$⁠) between $L_{\infty }$ and K, with default values: $L_{\infty }$ = 154.56 cm, K = 0.11 year⁻¹, t₀ = −0.13 year, ${\sigma }_{L_{\infty }}$ = 16.75 cm, ${\sigma }_K$ = 0.025 year⁻¹, and ${\rho }_{L_{\infty },K}$ = 0.7. The first four values are from McQueen et al. (2018), and the latter two we chose to account for variability in growth rate and positive correlation between $L_{\infty }$and K. This correlation was assumed to be positive because of the condition mentioned in the “Introduction” that slower- and faster-growing individuals tend to remain so throughout their life (Vincenzi et al., 2014). Nevertheless, we found that the results of our simulations are largely insensitive to this assumption. The covariation between $L_{\infty }$ and K was calculated by multiplying ${\sigma }_{L_{\infty }}$⁠, ${\sigma }_K$⁠, and ${\rho }_{L_{\infty },K}$⁠. Given these parameters, 10 000 random growth trajectories were simulated (Figure 1). From these trajectories, a growth transition matrix (G in the Supplementary material) was constructed with the age- and length-bin-specific proportions of individuals that transition to particular length bins in the next time-step. Note that, as a consequence, a fish of a certain length that has reached that length at, for example, a relatively late age is a relatively slow grower and will subsequently have a relatively small growth increment compared to a fish that reached the same length at an earlier age (Supplementary Figure S1). These growth increments were applied in each time-step after mortality has taken place. The model was parametrized to resemble the western Baltic cod stock, following the procedures of Haase (2018) and Haase et al. (in prep.). We first projected the stock 20 years forward under constant fishing mortality rate and constant recruitment, so that in year 21 we arrived at an equilibrium population structure that is independent of the starting numbers. Below, we describe the simulation runs that were carried out (see also Table 1).

Figure 1.

100 simulated random growth trajectories with parameters mean $L_{\mathrm{\infty }}$ = 154.56 cm, mean K = 0.11 year⁻¹, t₀ = −0.13 year, ${\sigma }_{L_{\mathrm{\infty }}}$ = 16.75 cm, ${\sigma }_K$ = 0.025 year⁻¹, and ${\rho }_{L_{\infty },K}$ = 0.7. For the experiments of this study, 10 000 random growth trajectories were simulated.

Table 1.

Simulation scenarios.

Scenario	Description	Change in Ffull = 1 year−1 to:	Change in L50 = 30 cm to:	Change in SR = 6 cm to:	Growth other than the default parameters: mean${\mathit{L}}_{\infty }$ = 154.56 cm and mean K = 0.11 year−1 (McQueen et al., 2018)
1.1: (fishing rate)	Default run, decrease in Ffull	Ffull = 0.75 year−1	–	–	–
1.2: (fishing rate)	Smaller decrease in Ffull	Ffull = 0.9 year−1	–	–	–
1.3: fishing rate)	Larger decrease in Ffull	Ffull = 0.5 year−1	–	–	–
1.4: (fishing rate)	Increase in Ffull	Ffull = 1.25 year−1	–	–	–
2.1: (fishing pattern)	Default run, increase in L50	–	L50 = 40 cm	–	–
2.2: (fishing pattern)	Decrease in L50	–	L50 = 20 cm	–	–
2.3: (fishing pattern)	Increase in SR	–	–	SR = 9 cm	–
2.4: (fishing pattern)	Decrease in SR	–	–	SR = 3 cm	–
1.5: Sensitivity to growth assumptions	Slower growth, decrease in Ffull	Ffull = 0.75 year−1	–	–	Slower growth: mean $L_{\mathrm{\infty }}$ = 178.5 cm, mean K = 0.073 year−1
1.6: Sensitivity to growth assumptions	Faster growth, decrease in Ffull	Ffull = 0.75 year−1	–	–	Faster growth: mean $L_{\mathrm{\infty }}$ = 135 cm, mean K = 0.165 year−1
2.5: Sensitivity to growth assumptions	Slower growth, increase in L50	–	L50 = 40 cm	–	Slower growth: mean $L_{\mathrm{\infty }}$ = 178.5 cm, mean K = 0.073 year−1
2.6: Sensitivity to growth assumptions	Faster growth, increase in L50	–	L50 = 40 cm	–	Faster growth: mean $L_{\mathrm{\infty }}$ = 135 cm, mean K = 0.165 year−1
2.7: Sensitivity to timing of length-at-age determination	Mean length-at-age from season 1, increase in L50	–	L50 = 40 cm	–	–
2.8: Sensitivity to timing of length-at-age determination	Mean length-at-age from season 12, increase in L50	–	L50 = 40 cm	–	–
2.9: Sensitivity to time-step resolution	Four quarters, increase in L50	–	L50 = 40 cm	–	–

Experiment 1—change in F at the fully selected age

This experiment started from an equilibrium situation after having applied an F at the fully selected age of F_full = 1 year⁻¹, with the L₅₀ (length at which the retention probability is 50%) at L₅₀ = 30 cm and the selection range (SR, range of lengths between L₇₅ and L₂₅, which are the lengths at which the retention probabilities are 75 and 25%, respectively, denoting the sharpness of the selection process) at SR = 6 cm for 21 years. These F_full, L₅₀, and SR are realistic for the western Baltic cod stock; between 2008 and 2017, the estimated F_bar came down from 1.0 to 0.6 year⁻¹ (ICES, 2018b), but note that this exceeded the proxy for the F that leads to maximum sustainable yield, F_MSY = 0.26 (EU, 2016; ICES, 2018b).

We then projected 10 years forward with a changed F. The default run had F at the fully selected age changed from F_full = 1 to 0.75 year⁻¹. Additional analyses were run with changes from F_full = 1 to 1.25, 0.9, and 0.5 year⁻¹ (Table 1). All years have constant recruitment. For comparison, the resulting time-series of the SSB, the human consumption yield (= catch above the legal minimum conservation reference size, MCRS, of 35 cm; hereafter called “yield”) and the catch below MCRS were also calculated under the assumptions of usual projections where length changes are not accounted for and F is only age-based and relative monthly F-at-age (F_s,a in the Supplementary material) is kept constant.

Note that the metric of MCRS is similar to the metric of minimum landing size (MLS) prior to the EU landing obligation, when fish below that size were not allowed to be landed and were often discarded. Under the EU landing obligation (EU, 2013) the concept of MLS was replaced with the concept of MCRS; only fish above MCRS can be sold for human consumption and the former discards below MCRS must be landed, but cannot be sold for human consumption.

Experiment 2—change in selection parameters of the gear

This experiment started from the same equilibrium as Experiment 1. We then projected 10 years forward with a changed fishing pattern (selectivity), but the same F at the fully selected age of F_full = 1 year⁻¹. The default run had the L₅₀ changed from L₅₀ = 30 cm to L₅₀ = 40 cm with a constant SR = 6 cm. Additional analyses were run with changes to L₅₀ = 20 cm with a constant SR = 6 cm, a constant L₅₀ = 30 cm with a changed SR = 9 cm, and a constant L₅₀ = 30 cm with a changed SR = 3 cm, respectively (Table 1). Both the range of values of the selectivity parameters used in this experiment and the degree of change applied considered the historical development of codend selectivity in the Baltic Sea trawl fishery during the last three decades (Madsen, 2007; Wienbeck et al., 2011; Herrmann et al., 2013). The assumption of constant F at the fully selected age (a proxy for constant effort) under changed fishing pattern implies that F_bar = F_3–5 was then changed. All years had constant recruitment.

For comparison, the resulting time-series were also calculated under the assumptions of usual projections where F is only age-based. In this case, F-at-age was modified for the new gear according to the assumption of fixed length-at-age. The monthly F-at-age was modified by an age-specific conversion factor (⁠${\theta }_{s,a}^{}$ in the Supplementary material) that is calculated as the ratio between the new selectivity-at-age (⁠$r_{s,a }^{\mathrm{new}}$ in the Supplementary material) and the old selectivity-at-age (⁠$r_{s,a }^{\mathrm{old}}$ in the Supplementary material). The respective selectivities-at-age were calculated from the average length-at-age at the starting equilibrium (year 21; separately for each month; l_s,a in the Supplementary material) and the selection function (Supplementary Table S1).

Sensitivity to the choice of growth parameters

A sensitivity analysis was conducted for default Experiments 1 and 2 to address the influence of the growth parameters on simulation results. For all runs above, we used fixed von Bertalanffy parameters extracted from McQueen et al. (2018). In the sensitivity runs, we replicated the same default runs, but considering faster- and slower-growing populations (mean $L_{\infty }$ = 135 cm and mean K = 0.165 year⁻¹ for the faster-growing population, and mean $L_{\infty }$ = 178.5 cm and mean K = 0.073 year⁻¹ for the slower-growing population), while t₀ stayed constant at t₀ = −0.13 year (Supplementary Figure S2). The parameters for faster and slower growth were chosen such that mean K decreased by one-third each time from 0.165 to 0.11 to 0.073 year⁻¹ and mean $L_{\infty }$ decreased by 13% each time from 178.5 to 154.56 to 135 cm; these factors were chosen by eyeballing to result in plausible curves that are more or less symmetrical around the default curve (Supplementary Figure S2).

Further sensitivity tests

Additional biases may perhaps occur when the traditional method derives the assumed constant length-at-age from fish sampled at only one time of year. In our simulations so far, when implementing the traditional method assuming constant mean length-at-age, we used mean length-at-age in the equilibrium year (year 21) (l_s,a) separately for each of the 12 months to derive old and new retention probabilities and the ratios between these for each of the 12 months separately (Supplementary Table S1). As can be seen in that table, since fish grow over the course of the year, retention probabilities get higher and the differences between the two gears grow smaller over the course of the year. This affected the outcomes of the simulations. To demonstrate this, we ran two additional scenarios resembling the default gear-change scenario (i.e. scenario 2.1 in Table 1), but now with the traditional method based on mean length-at-age in month 1 only (l_1,a) and month 12 only (l_12,a), respectively.

Even with our age–length model, the time-step resolution is important. In reality, growth is a continuous process, but growth in models such as ours takes place in discrete time-steps. This means that size-selective mortality is applied either before or after growth, which affects the numbers of fish that are retained by the gear and their weight. The effect is illustrated through another simulation resembling the default gear-change scenario (i.e. scenario 2.1 in Table 1), with 4 quarterly (S = 4) instead of 12 monthly (S = 12) time-steps per year.

Results

For reference, note that with the settings of this simulation, the unfished SSB at equilibrium amounts to 427 kt, whereas the equilibrium SSB at F_full = 1 year⁻¹ amounts to 2.6% of that, namely 11 kt (Figure 2a).

Figure 2.

Scenario 1.1. Development of SSB (a), catch above (b), and catch below MCRS (c) compared to status quo and mean length-at-age (d) over time when, from simulation year 22 onwards, F_full is reduced from 1 to 0.75 year⁻¹. In (a–c), the solid line is when no change occurs, the stippled line represents results from the dynamic length-based model, and the dashed line represents results under the assumption of fixed length-at-age. In (d), the solid lines represent mean lengths in month 12 at ages 1–15 years from bottom to top. The biases amount to −3, −4, and +3% of the actual SSB, yield, and catch below MCRS, respectively, in year 31 (year 10 after the change).

Experiment 1

As expected, after a reduction in F at the fully selected age (F_full) from 1 to 0.75 year⁻¹, SSB increases, yield experiences a dip in the first year, but then increases because of the increased stock size, and the catches below MCRS are reduced (Figure 2a–c). All three changes are slightly underestimated with the traditional method compared to the method that accounts for changes in mean length-at-age (Figure 2a–c); the biases amount to −3, −4, and +3% of the actual SSB, yield and catch below MCRS, respectively, in year 31 (year 10 after the change). At all ages, the mean length slightly increases (Figure 2d) because size truncation is slightly released. This increase in size (and thus weight) is not accounted for by the traditional method and, therefore, it slightly underestimates the SSB and yield. Similarly, slightly fewer of the 1- and 2-year old fish are below MCRS than assumed according to the traditional method.

These changes are greater and the biases stronger after stronger reductions in F (from F_full = 1 to 0.9, 0.75, and 0.5 year⁻¹, respectively) and the changes switch sign after an increase in F from F_full = 1 to 1.25 year⁻¹ (compare Figure 2 and Supplementary Figures S3–S5; for all comparisons, note that the scale of the y-axis may differ).

Experiment 2

As expected, after an increase in L₅₀ from 30 to 40 cm, SSB increases, yield experiences a dip in the beginning, but then increases because of the increased stock size, and the catches below MCRS are strongly reduced (Figure 3a–c). The change in SSB is overestimated (Figure 3a), whereas the yield is underestimated (Figure 3b) with the traditional method compared to the method that accounts for changes in mean length-at-age. The catches below MCRS are substantially overestimated with the traditional method (Figure 3c). The biases relative to the length-and-age-based SSB, yield, and catch below MCRS amount to +6, −3, and +150%, respectively, in year 31 (year 10 after the change); the latter bias is already substantial in the first year of the change. At the first age, the mean length slightly increases. At the second age, the mean length dips slightly and then slightly increases to above the initial level. At all older ages, the mean length dips and then increases, but stays below the initial level, and increasingly so for older ages (Figure 3d). Mean length goes down because the small individuals are less fished, so more of them remain in the population. These changes are not accounted for by the traditional method. Because older (larger) fish become smaller than the traditional method assumes, it overestimates SSB. The younger (small) fish are slightly larger than the traditional method assumes and, therefore, more of them belong to the fraction above MCRS and fewer of them to the fraction below MCRS than the traditional method estimates.

Figure 3.

Scenario 2.1. Development of SSB (a), catch above (b), and catch below MCRS (c) compared to status quo and mean length-at-age (d) over time when, from simulation year 22 onwards, L₅₀ is increased from 30 to 40 cm. In (a–c), the solid line is when no change occurs, the stippled line represents results from the dynamic length-based model, and the dashed line represents results under the assumption of fixed length-at-age. In (d), the solid lines represent mean lengths in month 12 at ages 1–15 years from bottom to top. The biases amount to +6, −3, and +150% of the actual SSB, yield, and catch below MCRS, respectively, in year 31 (year 10 after the change).

The effect of an increase in SR from 6 to 9 cm is qualitatively very similar to the effect of an increase in L₅₀ from 30 to 40 cm (compare Figure 3 and Supplementary Figure S6). The effects of a decrease in SR from 6 to 3 cm and a decrease in L₅₀ from 30 to 20 cm resemble each other qualitatively (compare Figure 4 and Supplementary Figure S7). In these latter cases, the mean length-at-age increases, SSB and yield go down substantially, whereas catches below MCRS go up substantially, and these changes are substantially overestimated with the traditional method. The traditional method assumes the fish to be smaller than they actually are and, therefore, estimates SSB and yield too low and the fraction of fish below MCRS too high. The biases in catch below MCRS and SSB are already substantial in the first and second year, respectively, after the gear change.

Figure 4.

Scenario 2.4. Development of SSB (a), catch above (b), and catch below MCRS (c) compared to status quo and mean length-at-age (d) over time when, from simulation year 22 onwards, SR is reduced from 6 to 3 cm. In (a–c), the solid line is when no change occurs, the stippled line represents results from the dynamic length-based model, and the dashed line represents results under the assumption of fixed length-at-age. In (d), the solid lines represent mean lengths in month 12 at ages 1–15 years from bottom to top. The biases amount to −43, −42, and +44% of the actual SSB, yield, and catch below MCRS, respectively, in year 31 (year 10 after the change).

Sensitivity to the choice of growth parameters

When average fish growth is slower, the changes in SSB and yield following a reduction in F at the fully selected age are smaller, but the biases relatively larger (compare Figures 2a–c, 5a–c, and 6a–c). This occurs because, under slower average growth, the increase in mean length-at-age following a reduction in F is larger (compare Figures 2d, 5d, and 6d), reflecting the notion that the fish remain longer in the size window of differential selection between fast and slow growers, whereby the effect of release from truncation through reduced F is enhanced.

Figure 5.

Scenario 1.5. Development of SSB (a), catch above (b), and catch below MCRS (c) compared to status quo and mean length-at-age (d) over time when, from simulation year 22 onwards, F_full is reduced from 1 to 0.75 year⁻¹ under slow growth (mean $L_{\mathrm{\infty }}$ = 178.5 cm, mean K = 0.073 year⁻¹ as opposed to the default values mean $L_{\mathrm{\infty }}$ = 154.56 cm, mean K = 0.11 year⁻¹). In (a–c), the solid line is when no change occurs, the stippled line represents results from the dynamic length-based model, and the dashed line represents results under the assumption of fixed length-at-age. In (d), the solid lines represent mean lengths in month 12 at ages 1–15 years from bottom to top. The biases amount to −4, −7, and +4% of the actual SSB, yield, and catch below MCRS, respectively, in year 31 (year 10 after the change).

Figure 6.

Scenario 1.6. Development of SSB (a), catch above (b), and catch below MCRS (c) compared to status quo and mean length-at-age (d) over time when, from simulation year 22 onwards, F_full is reduced from 1 to 0.75 year⁻¹ under fast growth (mean $L_{\mathrm{\infty }}$ = 135 cm, mean K = 0.165 year⁻¹ as opposed to the default values mean $L_{\mathrm{\infty }}$ = 154.56 cm, mean K = 0.11 year⁻¹). In (a–c), the solid line is when no change occurs, the stippled line represents results from the dynamic length-based model, and the dashed line represents results under the assumption of fixed length-at-age. In (d), the solid lines represent mean lengths in month 12 at ages 1–15 years from bottom to top. The biases amount to −2, −3, and +3% of the actual SSB, yield, and catch below MCRS, respectively, in year 31 (year 10 after the change).

The changes in mean length-at-age following an increase in L₅₀ are much more pronounced when average fish growth is slower (compare Figures 3d, 7d, and 8d). This again reflects the notion that the fish remain longer in the size window of differential selection between fast and slow growers, whereby the accumulation of slow growers under increased L₅₀ is greater. Note that under slow growth, mean length of the older ages becomes smaller than that of intermediate ages (Figure 7d). As mentioned in the “Introduction” section, this is sometimes observed and can be explained because growth has slowed down at old ages while length-dependent removal is still taking place, removing the larger fish at a higher rate. These more pronounced changes in mean length-at-age under slower average growth result in slightly larger biases in SSB and yield under slower average growth (compare Figures 3a and b, 7a and b, and 8a and b). The increases in SSB and yield following increased L₅₀ are slightly smaller under slow average growth because the slower-growing fish are smaller (compare Figures 3a and b, 7a and b, and 8a and b). The youngest fish remain very small under slow average growth, and the bias in estimation of catch below MCRS by the traditional method is substantial and slightly larger than with medium growth and almost as large as with fast growth (compare Figures 3c, 7c, and 8c).

Figure 7.

Scenario 2.5. Development of SSB (a), catch above (b), and catch below MCRS (c) compared to status quo and mean length-at-age (d) over time when, from simulation year 22 onwards, L₅₀ is increased from 30 to 40 cm under slow growth (mean $L_{\mathrm{\infty }}$ = 178.5 cm, mean K = 0.073 year⁻¹ as opposed to the default values mean $L_{\mathrm{\infty }}$ = 154.56 cm, mean K = 0.11 year⁻¹). In (a–c), the solid line is when no change occurs, the stippled line represents results from the dynamic length-based model, and the dashed line represents results under the assumption of fixed length-at-age. In (d), the solid lines represent mean lengths in month 12 at ages 1–15 years from bottom to top (as in year 21). The biases amount to +10, −7, and +157% of the actual SSB, yield, and catch below MCRS, respectively, in year 31 (year 10 after the change).

Figure 8.

Scenario 2.6. Development of SSB (a), catch above (b), and catch below MCRS (c) compared to status quo and mean length-at-age (d) over time when, from simulation year 22 onwards, L₅₀ is increased from 30 to 40 cm under fast growth (mean $L_{\mathrm{\infty }}$ = 135 cm, mean K = 0.165 year⁻¹ as opposed to the default values mean $L_{\mathrm{\infty }}$ = 154.56 cm, mean K = 0.11 year⁻¹). In (a–c), the solid line is when no change occurs, the stippled line represents results from the dynamic length-based model, and the dashed line represents results under the assumption of fixed length-at-age. In (d), the solid lines represent mean lengths in month 12 at ages 1–15 years from bottom to top. The biases amount to +5, −1, and +159% of the actual SSB, yield, and catch below MCRS, respectively, in year 31 (year 10 after the change).

Further sensitivity tests

With the traditional method based on mean length-at-age in month 1 only and month 12 only, respectively, this leads to different outcomes. Now the traditional method calculates larger or smaller differences, respectively, between the new and the old gear (Supplementary Table S1) causing larger or smaller biases (compare Figure 3 and Supplementary Figures S8 and S9).

The differences in outcomes between simulations with 4 and 12 time-steps per year are negligible (compare Figure 3 and Supplementary Figure S10).

Discussion

This study brings to the table the Rosa Lee phenomenon, which consists of demographic changes in fish populations under varying mortality rates. Here, we specifically focus on changes in mortality caused by variations in fishing rate and pattern. The results presented herein illustrate the relative importance of accounting for the Rosa Lee phenomenon, with a view to improving the accuracy of stock and catch projections for exploited fish populations. In general, we found relatively small bias (always <10% in absolute values) when only fishing rate (F at the fully selected age) is changed. Here, the bias can probably be considered negligible in light of uncertainties in the estimates from various sources; the Rosa Lee effect can be ignored without erring too much. Nevertheless, with large changes in F and slow-growing fish populations, the bias becomes bigger and caution is necessary. In contrast, when scenarios involve reductions in vulnerability to the gear, by increasing L₅₀ or SR, bias in the prediction of catches below MCRS is always very substantial (+120 to 160%) and should not be ignored. Likewise, when scenarios involve increased vulnerability to the gear, by decreasing L₅₀ or SR, biases in the predictions of all three quantities (SSB, yield, and catches below MCRS) are substantial (25–50% in absolute values) and should not be ignored. The biases in scenarios with gear selectivity changes are already apparent in the first or second year; this implies that short-term forecasts are affected as well as more longer-term projections.

An accumulating number of selectivity gears have been developed and tested to reduce bycatch of undersized fish and/or unwanted species (Madsen, 2007; Catchpole and Revill, 2008; Nikolic et al., 2015), many of them having been adopted in commercial fisheries in the EU through technical regulations (EC, 1998; EU, 2005, 2010; STECF, 2012). Avoidance of unwanted catch through improved selectivity has become even more pertinent in Europe since the introduction of the landing obligation (EU, 2013). Many studies, however, that set out to explore through simulation the bio-economic consequences of gear selectivity changes (Kuikka et al., 1996; Macher et al., 2008; Garcia et al., 2011; Prellezo et al., 2017), including simulation studies directly used for fisheries advice (ICES, 1999—section 1.7), assumed constant length distribution-at-age. These studies ignored the Rosa Lee phenomenon and would, therefore, have given biased results.

Already in the early 2000s, it was recognized that taking account of length in age–length-based models might be important (Frøysa et al., 2002; ICES, 2003; Kvamme and Frøysa, 2004). It was argued that many biological and fishery-related processes in fish populations are length-based and thus age–length-based models are a better representation of these processes. Obviously, natural mortality from predation depends more on length than on age. Maturation is often size- as well as age-dependent (Heino et al., 2002a, b; Grift et al., 2003). Fecundity and growth may also be length-dependent (ICES, 2002, 2003; Birkeland and Dayton, 2005). Clearly, fishing through the removal of fish with nets is length-dependent, and gear studies relate catch to fish length (Wileman et al., 1996; Millar and Fryer, 1999). In the usual age-structured models, these length-based processes have to be transformed into age-based processes (through age–length keys and mean length-at-age), leading to loss of accuracy and loss of information about variation (McGarvey et al., 2007). It has been noted that for some fish stocks in boreal systems the use of age–length-structured assessment models is especially relevant because such stocks experience large interannual variation in growth (ICES, 2003; Kvamme and Frøysa, 2004). Explicitly accounting for the Rosa Lee phenomenon in stock assessment or stock projection models is, however, rare (but see Punt et al., 2013; Taylor and Methot, 2013). To our knowledge, only Kvamme and Frøysa (2004) used an age–length model to assess the consequences of gear-selectivity changes.

Our study did not aim at quantifying the consequences of changes in fishing rate or gear selectivity in western Baltic cod (as Kvamme and Frøysa, 2004 did for Northeast Arctic cod); for that, we refer to Haase et al. (in prep.). We also did not try to emulate all possible length-dependent processes in the most realistic way. For example, we did not assume natural mortality to be length-dependent, and we did not consider that growth rate might diminish after maturation (Barot et al., 2004; Charnov, 2008; Alós et al., 2010). We also did not consider that recruitment may depend on the length and age structure of the population (reviewed by Hixon et al., 2014), nor did we consider density-dependent effects on growth or fisheries-induced evolution of maturation (Heino et al., 2002a, b; Grift et al., 2003; Barot et al., 2004) or growth (Enberg et al., 2012). Our model also lacked realism in terms of fishing; we ignored that a proportion of the western Baltic cod catch is taken by gillnets with a dome-shaped selection curve (but see Haase, 2018). All we aimed at in this study was to explore the extent of bias that may arise from ignoring the Rosa Lee phenomenon when not accounting for those very changes in the length-at-age distribution that are triggered by changes in fishing pressure (rate and/or pattern). For this reason, we simulated relatively simple scenarios with just enough realism to tackle the question. We compared simple scenarios of fishing and growth in models that do and do not consider length-at-age, keeping all else constant.

We emphasize that the results shown are only valid for the range of chosen settings of fishing pressure. We chose these settings to resemble the western Baltic cod stock and its recent history of F and gear selectivity. The stock has been fished at levels much higher than the proxy for F_MSY, and SSB has been a small fraction of that at unfished levels. Additional simulations with F closer to F_MSY (not shown) indicated that the effects of changes in fishing rate and pattern as well as the biases when using the traditional method compared to the method accounting for length changes were different (not shown). Some of the effects that we report may be caused by growth overfishing. For example, with our relatively low L₅₀ = 30 cm, when we made the selection steeper (SR decreased from 6 to 3 cm, Figure 4), this resulted in lower SSB and yield, and because the L₅₀ was smaller than MCRS (= 35 cm), higher catches below MCRS resulted. In contrast, in simulations with the same model, but with higher L₅₀ (>38 cm), a decrease in SR led to higher SSB and yield and lower catches below MCRS (Haase et al., in prep.). Our results can thus not be extrapolated and are only illustrative of the potential for bias caused by the Rosa Lee phenomenon.

An interesting alternative for the study of these effects would be an individual-based model tracking the recruits as “super-individuals” each with their own von Bertalanffy curves and where each “super-individual” would be subject to probabilistic mortality. The outcomes of such a model could then be compared to our results. It should be remembered that our results are dependent on the assumption that random differences in individual growth persist through life to some extent (slow-growing individuals tend to remain slow-growing individuals and fast-growing individuals tend to remain fast-growing individuals). There will certainly also be random variations in growth that occur independently of previous individual growth history (Webber and Thorson, 2016). These have not been considered in this study, but we expect that these effects will cancel each other.

One of the next steps could be to look for changes in mean length-at-age concurrently with changes in fishing pressure (rate or pattern) in fished populations, which would indicate that the Rosa Lee phenomenon is actually occurring. Nevertheless, it would be difficult to separate noise from real effects and to attribute any observed changes in mean length-at-age to changes in fishing pressure. This is because there is a lag between the change in fishing and the change in mean length, which even differs between age groups (Figure 3d). Moreover, in contrast to our simulations, where we implemented only one change in the entire simulated time-period of 31 years, fishing rate and pattern in the real world are frequently changing (even in direction), and equilibrium is seldom reached. Within the lag for a particular age, the trend in F may have already reversed, and it is hard to predict what pattern in mean length-at-age to expect. Only if a stock has experienced large and sustained changes, may it be possible to find concurrent changes in mean length-at-age.

We want to highlight another consequence of ignoring the Rosa Lee phenomenon. Above, we alluded to the concept of “big old fat fecund female fish” (BOFFFF, Hixon et al., 2014), also referred to as “maternal effects”. Hixon et al. (2014) reviewed the evidence that in numerous fish species, weight-specific annual fecundity increases with female age and/or size; egg quality or larval quality may also increase with female age and/or size. Furthermore, older and/or larger females may have more extensive spawning seasons, providing a bet-hedging life-history strategy helping to ensure that some larvae are spawned at times of favourable environmental conditions (Hixon et al., 2014). Larger and/or older females may thus contribute more to reproductive potential and hence recruitment. To the extent that these positive relations are with length itself and not only with age (i.e. that reproductive potential would increase with length-at-age, which can generally not be deduced from the review by Hixon et al., 2014), it is important to note that our study demonstrates changes in mean length-at-age under changes in fishing pressure. To be precise, reduction in fishing rate results in larger mean length-at-age, especially among older fish and especially in populations with slow mean growth. However, the onset of selectivity at larger fish length—which is thought to be desirable for the avoidance of catching undersized fish—has the negative side effect that it reduces mean length-at-age, especially among older fish and especially in populations with slow mean growth. If the maternal effects relate to length itself, increased gear selectivity would thus result in a negative effect on reproductive potential and hence recruitment. Therefore, on a per-stock basis, it should be considered whether these BOFFFF effects are indeed related to length-at-age, and if so, this should be taken into consideration when deciding on management strategies for the stock in question. In these cases, maternal effects should also be considered in MSE simulations using length- and age-based models.

Conclusions

The Rosa Lee phenomenon, when not accounted for, may cause bias in projections of stock and catch in scenarios of changed fishing mortality, especially when such changes involve a shift in exploitation pattern. Thus, in short-term forecasts, medium-term projections, and MSE simulations featuring selectivity changes, the Rosa Lee phenomenon should be accounted for, ideally by using length-based models such as ours (Haase et al., in prep.) or others (Frøysa et al., 2002; Begley and Howell, 2004; Kvamme and Frøysa, 2004; Howell and Bogstad, 2010).

Acknowledgements

We are grateful for the comments on an earlier draft from André Punt and an anonymous reviewer through which we were able to improve the paper. We thank Michala Ovens for providing information promptly when asked.

Funding

S.K. was partly funded by the European Maritime and Fisheries Fund (EMFF) of the European Union (EU) under the Data Collection Framework (DCF, Regulation 2017/1004 of the European Parliament and of the Council).