_{1}

^{*}

The aim of this paper is to discuss the validity of 20% B
_{0}
, 20% B
_{unfished}
and B
_{MSY}
as reference points for managing fisheries resources. I reanalyzed eight stock-recruitment relationship (SRR) sets of data that were analyzed by Myers
et al
. in 1994, and showed that the theory proposed by Sakuramoto could be applied to the above SRR data. The results showed that: 1) clockwise loops or anti-clockwise loops emerged in the plot
s of SRR, and the results coincided well with those of 25 stocks that lived around Japan and the stock of Pacific bluefin tuna; 2) the slopes of the regression lines (
b
) drawn on SRR planes for above 34 stocks had a negative relationship with the age-at-maturity (
m
). That is,
b
= 0.995 - 0.211·
m
. Therefore, the results of this paper indicate that the SRR is governed by a mechanism that is quite different from that which has been traditionally accepted, and in which the main factor is believed to be a density-dependent effect. The results also indicated that 20% B
_{0}
, 20% B
_{unfished}
and B
_{MSY}
, which are derived from traditional SRR models, do not have any scientific basis as reference points for managing fisheries resources. Empirical reference points seem to be more reasonable measures as Hilborn and Stokes emphasized in 2010.

Determining reference points is considered one of the most important tasks in fisheries resource management, and they have been broadly used not only by many international organizations but also in many domestic management procedures [_{0}, 20% B_{unfished}, and B_{MSY} as reference points (1) and (2). Where B_{0}, B_{unfished} and B_{MSY} denote the biomass at the initial stage, the biomass was calculated assuming that the resource was not harvested and that the biomass of the maximum sustainable yield (MSY) was achieved.

However, I strongly oppose the use of reference points (1) and (2), including parameters such as 20% B_{0}, 20% B_{unfished} and B_{MSY}. The reasons I oppose the use of 20% B_{0}, 20% B_{unfished} and B_{MSY} are as follows:

1) B_{0} has never been clearly defined in a scientific manner;

2) B_{unfished} can easily be calculated; however, the relationship between B_{0} and B_{unfished} has not been explained, nor has the meaning of B_{unfished} been explained from a biological point of view;

3) There is no clear biological basis for the choice of 20%;

4) B_{MSY} does not exist, because the concept of B_{MSY} itself is not valid.

Here I will explore these problems in detail. Baumgartner et al. [_{0}. Also, it is not possible to conclude that the declines in abundance occurred due to fishing and the increases in abundance occurred in response to fisheries regulations. It is clear that B_{0} cannot be defined and does not have any scientific basis as a reference point. That is, B_{0} is meaningless.

When the recruitment (R) is estimated using, for instance, virtual population analysis, the B_{unfished} can easily be calculated. However, the relationship between B_{0} and B_{unfished} cannot be adequately explained. Scientists who believe that B_{MSY} is valid tend to believe in the existence of a density-dependent effect; however, they do not incorporate a density-dependent effect when they calculate B_{unfished}. This is a fatal contradiction. If, as stated above, B_{0} itself is meaningless, then B_{unfished} and B_{MSY}, which is a fraction of B_{0}, are also meaningless. Further, the stock-recruitment relationship (SRR) itself forms the basis of B_{MSY}, as B_{MSY} is derived from SRR models such as those of Ricker [

Myers uploaded the dataset [

The report by Myers et al. [_{0} as a threshold for overfishing. However, Hilborn and Stokes [_{0} as a threshold for overfishing, this is a serious misinterpretation of the results of that paper. Myers et al. [_{0} is a useful threshold for defining overfishing. Indeed Myers et al. [_{0}. I strongly support the views of Hilborn and Stokes’ [_{0} is meaningless and further, I strongly believe that the traditional interpretation of the mechanism that controls SRR is itself invalid.

Sakuramoto proposed a new mechanism that may underlie SRR [

Sakuramoto [_{0} has no scientific basis.

The aim of this paper is to show evidence that the 20% B_{0} has no scientific basis using the loop theory. That is, this paper reanalyzes the SRR data for eight stocks that Myers et al. [_{0}, 20% B_{unfished} and B_{MSY} are not meaningful reference points for managing the stocks of fisheries.

I reanalyzed eight data sets of R and SSB, which were shown in

Sakuramoto [

R t = α S t − d ⋅ f ( x t ) . (1)

where R_{t}, S_{t−}_{d} and f (.) denote the recruitment in year t, spawning stock biomass in year t − d, and a function that evaluates the effects of environmental factors in year t. The notation d denotes the recruitment age. The vector x t = [ x t , 1 , ⋯ , x t , k ] is a list of environmental factors that affect the strength in R, which comprised not only of physical factors such as water temperature, but also biological interactions such as prey-predator relationships. Parameters α and k denote a proportional constant and the number of environmental factors, respectively. That is, Equation (1) implies that R_{t} is proportionally determined by S_{t−d}, and simultaneously, R_{t} is affected by environmental factors in year t.

Model 1 is the case when environmental effects can be neglected [

R t = α S t − d . (2)

where α denotes the recruitment per spawning stock biomass (RPS). The survival process is expressed by

S t + m = γ R t . (3)

For simplicity, m denotes the age at maturity and longevity of the fish [_{t} and S_{t+m} are constant regardless of year (

Model 2 is the case in which when f(x_{t}) in Equation (1) can be expressed by 1 + r [

R t = α ( 1 + r ) S t − d . (4)

The increasing or decreasing rate, r, is determined by environmental factors. When environmental factors are good for the stock, r takes positive values (r > 0) and R increases (

Model 3 is the case when r in year t in Equation (4), r_{t}, changes cyclically [

r t = β sin ( ω t ) . (5)

Thus,

R t = α ( 1 + β sin ( ω t ) ) S t − d . (6)

Here, β and ω denote the amplitude of the sine curve and angular velocity, respectively (

Generally, the spawning stock biomass in year t − d (S_{t}_{−d}) produces the recruitment in year t (R_{t}), and the R_{t} becomes the pawning stock biomass in year t + m (S_{t}_{+m}). Then the pawning stock biomass in year t + m (S_{t}_{+m}) produces the recruitment in year t + m + d (R_{t+}_{m+d}). This cycle repeats infinitely as shown in

R t + m + d = α ( 1 + r ) S t + m . (7)

As shown in

R t + m + d = ( 1 + r ) R t . (8)

That is, the relationship from S_{t}_{+m} to R_{t+}_{m+d} is replaced by the relationship from R_{t} to R_{t+m+d}. Equations (7) and (8) reveal an important fact that is hidden behind a SRR. That is, the relationship from S_{t}_{+m} to R_{t+m+d}, which is the SRR itself, is the relationship from R_{t} to R_{t+m+d}, which is so to speak “R to R relationship” (

R t + m + d = α ( 1 + β sin ( ω ( t + m + d ) ) ) S t + m = α ( 1 + β sin ( ω ( t + m + d ) ) ) ( 1 + β sin ( ω t ) ) S t − d = ( 1 + β sin ( ω ( t + m + d ) ) ) R t . (9)

Therefore, when the environmental factors cyclically fluctuate, such as a sine curve, the SRR simply means the relationship between R_{t+}_{m+d} and R_{t}. In other words, the SRR shows only a relationship between two different points at t and t + m + d on the same sine curve (

According to Sakuramoto [_{t}) against ln(SSB_{t}_{−d}). Further, I investigated the directions of the loops depending on the age at maturity.

I applied the rule [

the directions were concluded to be anticlockwise, anticlockwise and anticlockwise [

Sakuramoto [

In this study, I estimated the slope of the regression line that plots ln R t * against ln S S B t − d * . The eight slopes estimated in this study were plotted in

The left panels in Figures 5(a)-(h) show the trajectories of the natural logarithms of R and SSB, i.e., ln R and ln SSB, and their 3-year moving averages,

ln R* (shown by a red line) and ln SSB* (shown by a blue line), for each stock, respectively. The bottom left panels in Figures 5(a)-(h) simultaneously show the trajectories of ln R* (shown by red line) and ln SSB* (shown by blue line) in the same panel for each stock. The arrows indicate the vertexes observed in the trajectories. The vertexes observed in ln R* were projected to the vertexes observed in ln SSB* with certain time lags corresponding to the age at maturity. The time lags are shown by horizontal red arrows. The right panels in Figures 5(a)-(h) show the auto-correlations of ln R* and ln SSB* and the cross-correlations between ln R* and ln SSB*, respectively.

The third panel on the right in

Stock | Age at recruitment | Age at maturity | Time lags (5% significant ) | Cycle in R | Clockwise | Anti-clockwise | Slope | Lower limit | Upper limit | p-value | |
---|---|---|---|---|---|---|---|---|---|---|---|

a | Saithe-Iceland | 3 | 5 | 5, 6, 7, 8, 9, 10 | 10 | 16 | 10 | −0.663 | −1.073 | −0.254 | 2.67 (10^{−3}) |

b | SilverHake-NAFO5Ze | 1 | 2 | 0, 1, 2, 3, 4, 5, 6, 7, 8 | 26 | 22 | 7 | 0.755 | 0.561 | 0.948 | 1.05 (10^{−9}) |

c | Sardine-South Africa | 1 | 2 | 0, 1, 2, 3, 4, 5, 6 | 14 | 13 | 14 | 0.301 | −0.012 | 0.614 | 5.86 (10^{−2}) |

d | Saithe-Faroe | 3 | 5 | 4, 5, 6, 7 | 10 | 24 | 4 | −1.756 | −2.253 | −0.958 | 1.11 (10^{−4}) |

e | US Atlantic Menhaden | 1 | 2.5 | 2, 3, 4, 5, 6, 7 | 18 | 27 | 9 | 0.064 | −0.120 | 0.247 | 0.484 |

f | Herring-North Sea | 1 | 3 | 0, 1, 2, 3, 4, 5, 6, 7 | 18 | 29 | 9 | 0.495 | 0.381 | 0.609 | 1.39 (10^{−9}) |

g | Haddock-NAFO4TVW | 1 | 4.5 | 0, 1, 2, 3, 4, 5, 6, 7, 8 | 18 | 20 | 14 | 0.520 | 0.271 | 0.768 | 1.63 (10^{−4}) |

h | Cod-NAFO2J3KL | 3 | 7 | 0, 1, 2 | 10 | 8 | 14 | 0.235 | −0.104 | 0.574 | 0.162 |

and were plotted with black closed circles, and 2 stocks were estimated by Sakuramoto [

b = 0. 995 − 0. 211 m . (10)

Here, b and m denote the slope of the regression line and the age at maturity, respectively. The 95% confidence intervals of the slopes were (−0.343, −0.097), and those of the intercept were (0.541, 1.450). The p-values of the slope and intercept were 1.02 × 10^{−4} and 2.70 × 10^{−3}, respectively. That is, a significant negative relationship was detected between the slopes of the regression lines and the age at maturity. This result coincided well with the results of the simulations proposed by Sakuramoto [

When we discuss the SRR, we usually plot R against SSB or ln R against ln SSB, and try to adapt an SRR model, such as the Ricker model, the Beverton and Holt model, or another model. However, we seldom connect the continuous years from t to t + 1. If we draw the SRR as a line connecting the years from t to t + 1, an extremely different profile emerges. That is, clockwise loops or anticlockwise loops are commonly observed in the SRR [

The loop theory proposed by Sakuramoto [

In this study, I did not specify the typical environmental factors that controlled the fluctuations in the eight stocks analyzed in this paper, as an analysis of those factors would have been too time-consuming. However, some examples have already been investigated. For instance, the environmental factors that would control the fluctuations in the Pacific stock of Japanese sardines were already investigated in detail [

In this study, I applied the loop theory to the data that Myers et al. analyzed in 1994 [

It is difficult to explain in detail why the exception occurred at this stage; however, some of the possible reasons include the following. Hasegawa et al. [

The age at maturity seems to be determined on a species-by-species or stock-by-stock basis. However, even in the same species, the environmental conditions that affect the population fluctuations are different in different habitats. Further, even in the same habitat, the length of the environmental cycle itself changes era by era. Therefore, the number of line segments that with a clockwise or anticlockwise direction differed by stock and by era. In the case of Saithe Faroe, the age at maturity is high, 5 years; however, the number of line segments that show the clockwise loop is much greater than the number of line segments that show the anti-clockwise loop. In order to discuss these phenomena, we should further examine the environmental conditions and species interactions for each stock.

The results obtained for 24 stocks that live around Japan [

Sakuramoto noted that the slopes of the regression lines adapted for SRR were determined by the age at maturity and the cycle of environmental factors [_{t}) is plotted against ln(SSB_{t}_{−d}), shows a positive value, although the slope is statistically less than unity. When the age at maturity is close to half of the length of the cycle of environmental factors, the slope of the regression line is close to zero and is not statistically different from zero. Furthermore, the age at maturity becomes greater than the length of half of the cycle of environmental factors, the slope of the regression line becomes negative. Sakuramoto [

The intercept of the regression line was 0.995 and the 95% confidence intervals were 0.541 and 1.450. This can be anticipated from the loop theory, the principle of which is shown in

I set a rule that determines the direction of the line segment from year t to year t + 1; however, this rule is only a tentative example. A much more reasonable rule must exist; however, the aim of making the rule is to avoid doing the definition arbitrarily. In any case, if someone determines the direction of the line without any rule, the results will not be very different.

Hilborn and Stokes [_{unfished} to determine the reference points as follows. The standard approaches to biomass reference points in the United States and Australia fail to make clear that they are tied to an almost unknowable quantity, the unfished biomass B_{unfished}. The target and limits for fisheries management based on historical stock sizes and stock productivity have the advantage that they are based on experience, are easily understood, and are not subject to the vagaries of model assumptions. The results of this paper strongly support the warning noted by Hilborn and Stokes [

In the scientific committee of the International Whaling Commission, major discussions and simulation studies have been conducted for more than 5 years in order to develop a revised whale management procedure [_{MSY} and B_{0}, etc. That procedure only required a relative value for population abundance. The simulation program used the catch per unit of effort (CPUE) as the relative value of population abundance. The procedure commenced using two indexes: the present CPUE level compared to the target CPUE level, and the increasing or decreasing trend of the CPUE. If the present CPUE is lower than the target CPUE, the catch quota is reduced and vice versa. Further, if the CPUE shows an increasing trend, the catch quota is increased, and vice versa. The target CPUE can be determined, for instance, as follows: if the CPUEs of 10 to 15 years ago were reasonable, the average CPUE for those years can be used as the target level for the management. When this procedure is applied to actual fish stocks, very detailed simulation trials should be done to determine which parameters are reasonable, by species. However, previous studies [_{0}, B_{unfished} and B_{MSY}, the scientific basis of which is unclear.

The results elucidated in this paper can be summarized as follows:

1) Loop theory can be applied to the eight stocks that Myers et al. [

2) When the age at maturity was low enough compared to the length of the cycle of the environmental factors, clockwise loops were dominant, and when the age at maturity was high compared to the length of the cycle of the environmental factors, anticlockwise loops were dominant.

3) The slope of the regression line for SRR had a negative correlation with the age at maturity. When the age at maturity was low, the slope was positive but less than unity. When the age at maturity became high, the slope decreased to zero. In this case, no relationship was observed between R and SSB.

4) Reference points derived from the traditional SRR model do not have any scientific basis. B_{0}, B_{unfished} and B_{MSY} should not be used as the reference points.

5) Empirical reference points should be used, as Hilborn and Stokes [

I greatly appreciated Dr. Myers and his group for their contributions to producing the database from which many kinds of fish stock data can easily obtained. The database has played a very important role in fisheries sciences and it will also play an important role in the future. The study could only be done thanks to this database. I greatly appreciate Dr. Myers and his group for their great contributions. I also thank Drs. M. Melnychuk and S. Suzuki and Ms. K. Tanaka for their useful comments.

The author declares no competing financial interests.

Sakuramoto, K. (2017) Are 20% B_{0}, 20% B_{unfished}, and B_{MSY} Valid as Reference Points for Fisheries Resource Management? Open Access Library Journal, 4: e3897. https://doi.org/10.4236/oalib.1103897