_{1}

^{*}

This study shows that the stock-recruitment relationship (SRR) for Pacific bluefin tuna and the Pacific stock of Japanese sardine can be expressed by the same SRR model. That is,
(environmental factors), where
*R*
_{t} and
*S*
_{t-1} denote the recruitment in year t and spawning stock biomass in year
*t* - 1, and f(.) is a function that evaluates the effect of environmental factors in year
*t*. The simulations showed that when the fluctuation in environmental factors cyclically changed, 1) the shape of the apparent SRR assumed clockwise loops for the shorter maturity age of fish, and 2) the apparent SRR comprised scattered anticlockwise loops for the longer maturity age of fish. These features coincided well with those observed. This finding gives us a new paradigm in SRR, which is far different from the concept that has predominated in the field for more than 60 years.

Recently, the abundance of Pacific bluefin tuna, Thunnus thynnus, has decreased and it is necessary to rehabilitate the stock [

That is, a shape or a model of SRR is one of the key issues to elucidate when we discuss a management procedure. The Ricker [

Recently, however, Sakuramoto [

where R_{t} and S_{t−}_{1} denote the recruitment in year t and spawning stock biomass in year t − 1, and f(.) denotes a function that evaluates the effects of environmental factors in year t. The variable

representing the environmental factors, 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, R_{t} is proportionally determined by S_{t−}_{1}, and simultaneously, R_{t} is affected by environmental factors in year t.

The purpose of this study is to determine whether the SRR model proposed for the Pacific stock of Japanese sardine [

For Pacific bluefin tuna, data of recruitment and spawning stock biomass from 1952 to 2012 were used [

The following four models were assumed, based on Equation (1):

Model 1 is the basic SRR model, which is the case when environmental effects can be neglected. That is, f(x_{t}) in Equation (1) can be assumed to be unity. That is,

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

For simplicity, m denotes the age at maturity and longevity of the fish. That is, fish reach maturity at age m; then, they spawn their eggs and die. In Equation (3), _{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. That is,

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. On the contrary, when environmental factors are bad for the stock, r takes negative values (−1 < r < 0) and R decreases. In this model, the survival process is the same of that shown in Equation (3).

Model 3 is the case when r in Equation (4) changes cyclically. It can be expressed by a sine curve as defined below:

Thus,

Here,

Model 4 is the case when Model 2 and Model 3 are combined: i.e., f(x_{t}) in Equation (1) changes cyclically with an increasing or decreasing trend. That is,

Here,

In Model 3, the values of α,

_{t}, ln(R_{t}), against that of spawning stock biomass, ln(S_{t−}_{1}), for Pacific bluefin tuna. The plots widely scatter and it seems difficult to find any relationship between ln(R_{t})

and ln(S_{t−}_{1}). However, this apparent lack of relationship is caused by the wrong approach, as Sakuramoto pointed out [_{t}) and ln(S_{t−}_{1}) should be expressed by a 3- or more than 3-dimensional model as expressed in Equation (1).

The result of a principle component analysis showed that AO in June was located at the nearest point to that of ln(R). Hereafter, I will show the results when AO in June is used as a representative of the environmental factors. ln(S) was nearly located to ln(R) next to AO in June, July and November.

high values in recruitment from 1953 to 1956 correspond to the high values in SSB from 1958 to 1961. However, a great reduction in R in 1958 and 1959 caused the reduction in SSB that began from around 1963. That is, the trajectory in ln(S) follows that in ln(R) with about 5-year lag, because the maturity age of Pacific bluefin tuna is 5 years and older.

ln(R) decreased greatly from 1957 to 1959 according to the drastic reduction of AO in June from 1957 to 1958. However, ln(R) increased greatly from 1960 to 1963 according to the great increases of the values of AO in June from 1959 to 1962. That is, the recruitment was strongly affected by environmental factors.

ln(S) and AO in June.

_{t} was proportionally determined by S_{t−}_{1}; however, the apparent SRR showed three similar-shape clockwise loops increasing in size. The overall slope of the regression line was less than unity in response to the cyclic environmental factors (

Maturity age | b | 95% confidence limit | P-value | Clockwise or anticlockwise | Judgment of slope b | |
---|---|---|---|---|---|---|

Lower | Higher | |||||

1 | 0.946 | 0.864 | 1.027 | 2.20 (10^{−16}) | Clockwise | b = 1 |

2 | 0.800 | 0.646 | 0.953 | 7.10 (10^{−15}) | Clockwise | 0 < b < 1 |

3 | 0.579 | 0.369 | 0.789 | 8.39 (10^{−7}) | Clockwise | 0 < b < 1 |

4 | 0.311 | 0.065 | 0.556 | 1.39 (10^{−2}) | Clockwise | 0 < b< 1 |

5 | 0.050 | −0.208 | 0.308 | 0.699 | Clockwise | b = 0 |

6 | −0.244 | −0.494 | 5.66 (10^{−3}) | 5.53 (10^{−2}) | Clockwise | b = 0 |

7 | −0.266 | −0.522 | −0.010 | 0.042 | Both | b < 0 |

8 | −0.380 | −0.608 | −0.152 | 1.47 (10^{−3}) | Both | b < 0 |

9 | −0.124 | −0.376 | 0.129 | 0.331 | Both | b = 0 |

10 | 1.04 (10^{−4}) | −0.311 | 0.311 | 0.9995 | Both | b = 0 |

11 | −0.143 | −0.408 | 0.122 | 0.284 | Both | b = 0 |

12 | −0.241 | −0.494 | 1.15 (10^{−2}) | 0.061 | Anticlockwise | b = 0 |

13 | −0.098 | −0.331 | 0.331 | 0.404 | Anticlockwise | b = 0 |

14 | 0.247 | 4.12 (10^{−3}) | 0.489 | 0.39 | Anticlockwise | 0 < b < 1 |

15 | 0.390 | 0.149 | 0.631 | 1.96 (10^{−3}) | Anticlockwise | 0 < b < 1 |

16 | 0.598 | 0.419 | 0.776 | 8.79 (10^{−9}) | Anticlockwise | 0 < b < 1 |

17 | 0.901 | 0.755 | 1.048 | 2.20 (10^{−16}) | Anticlockwise | b = 1 |

18 | 1.147 | 1.009 | 1.266 | 2.20 (10^{−16}) | Anticlockwise | 1 < b < 2 |

19 | 1.344 | 1.264 | 1.423 | 2.20 (10^{−16}) | Anticlockwise | 1 < b < 2 |

Sardine | 0.764 | 0.634 | 0.874 | 2.20 (10^{−16}) | Clockwise | 0 < b < 1 |

Bluefin tuna | 0.211 | −0.048 | 0.471 | 0.109 | Anticlockwise | b = 0 |

anticlockwise loops or mainly anticlockwise loops emerged; however, the slopes of the regression lines were not different from zero. When the maturity age was greater than or equal to 14 years, all the SRR trajectories showed increasing anticlockwise loops.

For the case of Model 2, the SRR shows a very simple pattern. When r is positive, SRR shows a line of which the slope is unity and the intercept is

SRRs for the Pacific stock of Japanese sardine and that for Pacific bluefin tuna seem to be quite different. The former has a positive relationship between ln(R) and ln(S), but the latter seems to have no relationship between ln(R) and ln(S). However, simulations conducted in this paper showed that those apparent SRRs could be reproduced by the same model shown in Equation (1).

For species with a short reproductive cycle, such as sardines, the cycle of environmental conditions will be longer than the reproduction cycle for the species. Therefore, the apparent SRRs for the species will show increasing loops. This can also be seen in the SRRs for anchovies [

Generally, in species with a long period of maturation, the behavior of the apparent SRRs will be much more complicated, because the maturity age is biologically determined species by species whereas the cycle of environmental conditions can easily change depending on circumstances. When the maturity age is long, the half- cycle of environmental conditions is usually shorter than the maturity age; however, it can easily occur in a certain period that the half-cycle of environmental conditions happens to be longer than the maturity age. Further, the survival process will also be changed in response to the intensity of harvesting. Therefore, in cases when the maturity age is long, the apparent shape of the SRR is much more complicated than when the maturity age is short.

In each period, the blue and red arrows indicate the vector of R and S, the lengths of which are determined by the number of years in the period. For the upper panel, 1) in period P1, both R and S increase; then the direction of the combined vector takes the northeastward direction. P1 is composed of 9 years; then the length of vectors is described by long arrows. 2) In period P2, R decreases but S increases; then the direction of the combined vector takes the southeastward direction. P2 is composed of only 3 years, and the length of vectors is described by short arrows. 3) In period P3, both R and S decrease; then the direction of the combined vector takes the southwestward direction. 4) In period P4, R increases but S decreases; then the direction of the combined vector takes the northwestward direction. The periods occur in the order P1, P2, P3 and P4; then, the trajectory of SRR shows a clockwise loop.

The anticlockwise loop occurs when the phase between R and S is greater than the half-length of the cycle (m = 13) as shown in

Under the condition that the cycle of the environmental condition was 20 years and the maturity age was 13, the shape of the SRR derived from the simulation coincided well with the SRR for Pacific bluefin tuna, although,

in the simulations, if the length of the environmental cycle is longer than 20 years, the maturity age will also be longer than 13 years. Therefore, it is generally considered that the maturity age of Pacific bluefin tuna is 5 years and older; however, the average maturity age must be much greater than 5-years. If the average maturity age is more than 10 years, another interpretation for

In the simulations conducted in this paper, the S is assumed to be composed of only one age-class (m-year-old fish). However, both the S of sardine and bluefin tuna are composed of several age classes. Therefore, simulations assuming iteroparous species should also be conducted, although the essential results will not be largely different.

In this study, the effects of process and/or observed errors that surely exist in both R and S are not discussed. Generally, these errors would have the effect of hiding the real relationship between R and S [

It should be emphasized that the essential mechanism in SRR is very simple. R and S are inseparable. That is, R in the t-th generation produces S in the (t + m)-th generation, and the S in the (t + m)-th generation produces the R in the (t + m + 1)-th generation as shown in _{t} and R_{t+m+}_{1}, or S_{t−}_{1} and S_{t+m} as shown in _{t} (or S_{t}) fluctuates cyclically in response to environmental factors, the mechanism in SRR is only the relationship between the two points, R_{t} and R_{t+m+}_{1} or S_{t-}_{1} and S_{t+m} on the same curve.

Stock-recruitment relationships for the Pacific stock of Japanese sardine and that for Pacific bluefin tuna can be expressed by the same model shown in Equation (1). That is, _{t} and S_{t−}_{1} denote the recruitment in year t and spawning stock biomass in year t − 1, and f(.) denotes a function that evaluates the effects of environmental factors in year t. The variable

The cyclic environmental conditions strongly affect the stock-recruitment relationships. For species with a short reproductive cycle, such as sardines, the cycle of environmental conditions will be longer than the reproduction cycle for the species. In this case, the apparent SRRs for the species will show increasing loops. On the contrary, for species with a long reproduction cycle, the apparent SRRs will scatter widely and will show anticlockwise loops with no trend, such as that observed in bluefin tuna.

I thank Drs. Masanori Miyahara, Tokio Wada, Tatsu Kishida, Jiro Suzuki, Rikio Sato, Seizo Hasegawa, Yukimasa Ishida, Mitsuo Sakai and Naoki Suzuki for their useful comments that improved this manuscript.

KazumiSakuramoto, (2015) A Stock-Recruitment Relationship Applicable to Pacific Bluefin Tuna and the Pacific Stock of Japanese Sardine. American Journal of Climate Change,04,446-460. doi: 10.4236/ajcc.2015.45036