_{1}

I propose a mechanism that controls the stock-recruitment relationship (SRR), which should replace traditional SRR models. The difference between the traditional SRR models and the model proposed herein is whether or not a density-dependent effect is assumed. The new mechanism elucidates the following: 1) why clockwise loops or anticlockwise loops are commonly observed in an actual SRR, and 2) the age at maturity (not a density-dependent effect) determines the slope of the regression line for the SRR. These cannot be explained by the traditional SRR models, the basis of which is a density-dependent effect. The new SRR mechanism can well explain the SRRs for Pacific stocks of Japanese sardine, Pacific bluefin tuna and Arctic bluefin tuna. The new SRR mechanism proposed herein should be used as a valid SRR model when the management of fishery resources is discussed. Without the adaptation of this new mechanism, fishery resource management may continue to be troubled by the lack of validity of the traditional SRR models.

The stock-recruitment relationship (SRR) is one of the most important topics in fishery resource management. The traditional SRR models such as the Ricker model [

These traditional models have been commonly used all over the world by not only management organizations of individual countries but also international fishery management bodies such as regional fishery management organizations (RFMOs). However, the models have faced the serious problem of determining a suitable SRR. For example, the International Scientific Committee (ISC) for Tuna and Tuna-like Species in the North Pacific Ocean, which is the international management body for Pacific bluefin tuna, Thunnus orintalis, has not been able to determine a suitable SRR model, because the SRR for Pacific bluefin tuna has no trend and no relationship has been detected between the recruitment and the S for this species. The ISC concluded that the SRR for Pacific bluefin tuna is not yet known [

The maximum sustainable yield (MSY) theory also has a serious defect. The MSY is derived from the SRR. A surplus yield can be calculated by subtracting the replacement yield from the recruitment. The relationship between the surplus yield and S shows a dome shape, and the maximum value of this relationship is the so-called MSY. The MSY is commonly used as the target of fishery resource management. In RMFOs, a “Kobe Chart” is commonly used to judge the appropriateness of the current level of tuna resources and fish intensity [_{current}/B_{MSY} for the x-axis, and F_{current}/F_{MSY} for the y-axis; B_{current} and B_{MSY} denote the current biomass and the biomass level that provides the MSY, respectively. F_{current} and F_{MSY} denote the current fishing mortality coefficient and the value of the fishing mortality coefficient that gives the MSY, respectively. If the current fisheries are located at a region that satisfies B_{current}/B_{MSY} > 1 and F_{current}/F_{MSY} < 1 (that is, the current biomass is above the B_{MSY} and the current fishing mortality coefficient is below the F_{MSY}), then the current fisheries intensity is judged as not overfishing and the level of resources is considered to be safe.

When the current fisheries are located at a region that satisfies B_{current}/B_{MSY} < 1 and F_{current}/F_{MSY} > 1, the current fisheries intensity is judged as overfishing, and the resource level is dangerous. This indicates that the current F should be reduced. However, the International Commission for the Conservation of Atlantic Tunas (ICCAT), which is the international organization for managing tuna in the Atlantic Ocean, suddenly stopped using the Kobe Chart for bluefin tuna resources from this year [

I have noted that environmental factors play an extremely important role in the SRR, and I proposed a new theory for the SRR that does not assume any density-dependent effect [

This new theory can elucidate both the mechanism in which clockwise loops or anticlockwise loops commonly appear in the SRR, and the mechanism underlying the concept that age at maturity determines the slope of the regression line for the SRR. I applied this theory to the Pacific stock of Japanese sardine, Sardinops melanostictus, and Pacific bluefin tuna, and I detected an increasing trend of the SRR with clockwise loops in the SRR for the Pacific stock of Japanese sardine, whereas no trend with anticlockwise loops was detected in the SRR for Pacific bluefin tuna [

However, the previous papers are quite complicated, and it may thus be difficult to understand the essential point of the theory. I strongly feel that a more easily understood explanation of the theory is necessary. Herein I provide a further explanation of the new SRR theory, using simple simulations.

For Pacific bluefin tuna, the data of recruitment, the spawning stock biomass and fishing mortality coefficient from 1952 to 2012, and the mean weight by age and the growth function were used [

The model used in the simulation is as follows:

R t = cos ( ( π / T ) ∗ t ) + 6 (1)

S t = cos ( ( π / T ) ∗ ( t − m ) ) + 6 (2)

Here, R_{t} and S_{t} denote the recruitment and the S in year t, respectively. T denotes the cycle of environmental conditions and is fixed at 8 years throughout this simulation. The notation m denotes the age at maturity, and cases in which m changes from 0, 1, 2, ・・・ were tested in the simulations. The value 6 in Equations (1) and (2) was added to make R_{t} and S_{t} have positive values.

The weighted mean age at maturities is calculated using the following two methods. First, the biomass by age a, B_{a}, is calculated by Equation (3).

B a = N 3 e − 0.25 ( a − 3 ) w a u a (3)

where N_{3} denotes the number of fish at age 3-year-old, which is the first age at maturity. The notations u_{a} and w_{a} denote the maturity rate at age a and mean weight at age a, respectively. The number 0.25 denotes the natural mortality coefficient. The weighted mean age at maturity is then calculated by Equation (4).

m ¯ = ∑ 3 25 a B a / ∑ 3 25 B a (4)

The length by age a, L_{a}, and the weight by age a, w_{a}, are calculated using the following equations [

L a = 2.54 ( 1 − e − 0.517 ( a − 0.561 ) ) (5)

w a = 1.71 L 3.03 / 100000 (6)

Second, the weighted mean age at maturity is calculated using Equations (7) and (8). The mean weight of 5-year-old fish is 68 kg, at which point each fish spawns approx. one million eggs [

E a = 35.1 ⋅ e 0.209 a (7)

m ¯ = ∑ 3 25 a u a E a N a / ∑ 3 25 u a E a N a (8)

where N_{a}, u_{a} and w_{a} denote the number of fish at age a, the maturity rate at age a, and the mean weight at age a, respectively.

and S can be expressed with the same curve. In this simulation, the years from 4 to 12 are used as one cycle, because this makes the simulation and theory easy to explain. The 8-year trajectories from year 4 - 5, 5 - 6, ・・・, 11 - 12 in one cycle T are divided into two periods. One is the period comprised of 4 years in which the recruitment and the S are both increasing. This period is denoted as P1. The second period is comprised of 4 years in which the recruitment and the S are both decreasing; this is denoted as P2. One cycle is composed of P1 and P2.

_{4}, R_{4}), (S_{5}, R_{5}), (S_{6}, R_{6}), ・・・, (S_{11}, R_{11}) plotted on the S-R plane in the case of m = 0. The point (S_{12}, R_{12}) is not shown because it coincides with the point (S_{4}, R_{4}). The numbers from 4 to 11 attached near the points of the SRR trajectory denote the years that construct one cycle. When m = 0, the slope of the regression line is unity. As shown in the right-hand side of _{4}, R_{4}) moves northeast on the S-R plane along the line of which the slope is unity, and then it moves southwest along the same line.

In P3, the recruitment begins to decrease but the S still increases, and thus the point (S, R) on the S-R plane moves southeast on the S-R plane shown above the trajectories in panel (b). The length of P3 is 1 year. In P4, the recruitment and S both decrease, and therefore, (S, R) on the S-R plane moves southwest as shown on the S-R plane shown above the trajectories in

order, creating a clockwise loop. The lengths of P1 and P3 are both 1 year, and the periods of P2 and P4 are 3 years; therefore, the loop has a long axis for the northeast or southwest direction. The slope of the regression line thus has a positive value.

_{4}, R_{4}), (S_{5}, R_{5}), (S_{6}, R_{6}), ・・・, (S_{11}, R_{11}) on the S-R plane when m = 1. When m = 1, the slope of the regression line is 0.652, the 95% confidence interval (95% CI) of which is 0.245 - 1.058. A significant positive slope is detected and the slope is significantly different from unity.

Therefore, as one cycle passes, a point (S, R) on the S-R plane moves northwest in P1, northeast in P2, southeast in P3, and southwest in P4, in that order. A clockwise loop is thus constructed. The periods of P1 and P3 are 3 years, and the periods of P2 and P4 are 1 year; therefore, the loop has a long axis for the southeast or northwest direction. The slope of the regression line thus has a negative value.

In _{4}, R_{4}), (S_{5}, R_{5}), (S_{6}, R_{6}), ・・・, (S_{11}, R_{11}) are plotted on the S-R plane when m = 3. When m = 3, the slope of the regression line is −0.641, the 95% CI of which is −1.124 to −0.159. A significant negative slope is detected and the slope is not significantly different from minus unity.

The eight points are plotted on the S-R plane when m = 4 (_{4}, R_{4}) move northwest on the S-R plane according to the line of which the slope is minus unity, and then moves southeast along the same line.

In P3, both the recruitment and the S decreases, and point (S, R) on the S-R plane moves southwest on the S-R plane shown above the trajectories. The period of P3 is 1 year. In P4, the recruitment decreases but the S increases, and the point (S, R) on the S-R plane moves southeast on the plane shown above the trajectories. P4 is 3 years. As one cycle passes, the point on the S-R plane moves northeast in P1, northwest in P2, southwest in P3, and southeast in P4, constructing an anticlockwise loop. The periods of P1 and P3 are 1 year and those of P2 and P4 are 3 years, giving the loop a long axis for the northwest direction. The slope of the regression line thus has a negative value.

The recruitment and S trajectories when the age at maturity is set at 6 are shown in

The trajectories of the recruitment and S when the age at maturity is set at 7 are shown in

In P4, the recruitment decreases but the S increases, and point (S, R) on the S-R plane moves southeast on the S-R plane. P4 is 1 year. Therefore, as one cycle passes, (S, R) on the S-R plane moves northeast in P1, northwest in P2, southwest in P3, and southeast in P4, creating an anticlockwise loop. Since the periods of P1 and P3 are 3 years and those of P2 and P4 are 1 year, the loop has a long axis for the northeast direction and the slope of the regression line has a positive value.

In _{4}, R_{4}), (S_{5}, R_{5}), (S_{6}, R_{6}), ・・・, (S_{11}, R_{11}) are plotted on the S-R plane for the case of m = 7. When m = 7, the slope of the regression line is 0.641, the 95% CI of which is 0.159 - 1.124. A significant positive slope is thus detected and the slope is not significantly different from unity. When m exceeds the cycle, the same pattern repeats. That is, when m = 8, the pattern is same of the case of m = 0, when m = 9, the pattern is same of the case of m = 1, and so on.

The results obtained from the above simulations are summarized in

When the age at maturity is greater than the half of the environmental cycle, an anticlockwise loop appears, and the slope changes from minus to plus as the age at maturity increases. That is, the clockwise or anticlockwise nature of the loop and the slope of the regression line for the SRR are determined by the combination of age at maturity and the cycle of environmental fluctuation. When m exceeds the cycle, the same pattern repeats. That is, when m = 8, the pattern is the same as that in the case of m = 0 (i.e., 8 − T) (where T denotes the environmental cycle and is set at 8 years in the simulation), and when m = 9, the pattern is the same as that in the case of m = 1 (i.e., 9 − T), and so on.

_{a} and E_{a} and the total number of eggs by age.

Age at maturity (m) | Slope of the regression line | Ratio (m/T) | Clockwise or anticlockwise | Corresponding stock | Slope on the SRR | 95% confidence Intervals |
---|---|---|---|---|---|---|

0 | b = 1 | 0/8 = 0 | Clockwise | |||

1 | 0 < b ≤ 1 | 1/8 = 0.125 | Clockwise | Pacific stock of Japanese sardine | 0.764 | (0.634, 0.879) |

2 | b = 0 | 1/4 = 0.25 | Clockwise | |||

3 | −1 ≤ b < 0 | 3/8 = 0.375 | Clockwise | |||

4 | b = −1 | 1/2 = 0.5 | Anticlockwise | |||

5 | −1 ≤ b < 0 | 5/8 = 0.675 | Anticlockwise | |||

6 | b = 0 | 3/4 = 0.75 | Anticlockwise | Pacific bluefin tuna | 0.211 | (−0.048, 0.109) |

7 | 0 < b ≤ 1 | 7/8 = 0.875 | Anticlockwise | |||

8 | b = 1 | (8 − T)/8 = 0 | Clockwise | |||

9 | 0 < b ≤ 1 | (9 − T)/8 = 0.125 | Clockwise | Western Arctic bluefin tuna | 0.719 | (0.478, 0.959) |

The mean age at maturity estimated from the distribution of B_{a} values was 9.79 years old, and that estimated from the distribution of the total number of eggs by age was 13.8 years old.

and medium (green).

In this case, I arbitrarily separated the trajectories of recruitment and S into two periods. The first is 1970 to 1988 (19 years) in which the levels of recruitment and S change from high to low. The second is 1989 to 2010 (22 years) in which the levels of recruitment and S are low.

In this simulation, to simplify the discussion, I discuss only the process from the recruitment to the S; that is, only the survival process from the recruitment to S is considered, not the reproductive process from the S to the recruitment. The SRR is comprised of the reproductive process from spawning stock biomass (S) to the recruitment, but when the SRR is discussed, we should treat simultaneously not only the reproductive process from the S to the recruitment, but also the survival process from the recruitment to the S. In other words, those two processes, survival and reproductive processes, cannot be separated in the discussion. I have discussed the details of considering both processes together [

R t = α S t × f ( x 1 , x 2 , ⋯ , x k ) (9)

S t + m = β R t (10)

Here, x 1 , x 2 , ⋯ , x k denotes the environmental factors and f ( ⋅ ) is the function that determines the effects caused by environmental factors. That is, the recruitment is proportionally determined by the S and simultaneously affected by environmental conditions.

The survival process from the recruitment to the S is shown above in Equation (10). Here, β is the spawning stock biomass per the recruitment (SPR), which is commonly used in the model analysis. The results of the simulations using Equations (9) and (10) are somewhat complicated, but the essential conclusion is the same as that obtained using Equations (1) and (2). That is, when the age at maturity is low compared to the cycle of the environmental fluctuations, the slope of the regression line in the SRR is near unity with clockwise loops, and when the age at maturity becomes higher, the slope of the regression line in the SRR is near zero or minus. As the age at maturity grows higher compared to the cycle of the environmental fluctuations, the slope of the regression line in the SRR becomes near zero or positive values with anticlockwise loops.

To simplify the discussion, we assume the use of semelparous fish such as salmon, but iteroparous fishes are common. Therefore, it is better to consider that the age at maturity assumed in the simulation is the mean age at maturity, and it does not mean the first age at maturity. The mean age at maturity for long-longevity fishes must therefore be much longer than the first age at maturity. For example, considering the age at maturity for Pacific bluefin tuna, the maturity rates are 20%, 50% and 100% for ages 3, 4 and ≥5 years old, respectively. However, the fish live more than 20 years. As shown in

Western Atlantic bluefin tuna and Pacific bluefin tuna are different species, and their habitats differ. However, the ecological similarity of these two species seems to be much higher than that of the Pacific stock of Japanese sardine, even though the shape of the SRR for western Atlantic bluefin tuna is similar to that for the Pacific stock of Japanese sardine, but not similar to the SRR for Pacific bluefin tuna. I believe that this phenomenon can only be explained by the mechanism proposed in this paper.

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, whether or not the SRR has clockwise or anticlockwise loops is determined not only by the stock but also by its era. That is, even for same species, the shape of the SRR can be changeable depending on the era. When a period in which the environmental conditions are relatively stable is long, the slope of the regression line on the SRR is close to unity and the direction of the loop tends to be clockwise. Conversely, when a period with stable environmental conditions is short, the slope of the regression line on the SRR decreases to zero or minus, and the direction of the loop tends to be anticlockwise. Therefore, when we analyze actual data, we should focus on the cycle of the environmental conditions that affect the fluctuation in recruitment.

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

1) When the age at maturity is less than half of the environmental cycle, the SRR curve shows one clockwise loop for one cycle. When the age at maturity is greater than half of the environmental cycle, the SRR curve shows one anticlockwise loop for one cycle.

2) The slope of the regression line for the SRR is determined by the age at maturity. When the age at maturity is low compared to the cycle of the environmental fluctuations, the slope of the regression line has a positive value with clockwise loop, and when the age at maturity is high compared to the cycle of the environmental fluctuations, the slope of the regression line has no trend with anticlockwise loop.

3) The SRR for the Pacific stock of Japanese sardine corresponds to the case when m = 1 in the simulation. The SRR for the Pacific bluefin tuna corresponds to the case when m = 6 in the simulation. The SRR for the Arctic bluefin tuna corresponds to the case when m = 9 in the simulation.

Sakuramoto, K. (2018) The True Mechanism That Controls the Stock-Recruitment Relationship. Open Access Library Journal, 5: e4341. https://doi.org/10.4236/oalib.1104341