The data used in this study are shown below: 1) catch in weight for pink salmon landed in the Maritime Province of Siberia from 1950 to 2010 [13] ; 2) indices of the Arctic Oscillation (AO) by month from 1948 to 2010 [14] ; and 3) indices of the Pacific Decadal Oscillation (PDO) by month from 1948 to 2010 [15] .

We separated the pink salmon catch data into two groups. One is the catch harvested in the odd-numbered year t, which is denoted with C t O , and another is that harvested in the even-numbered year t, which is denoted with C t E . We calculated the correlation coefficients between C t O and AO and between C t O and PDO in month m (m = 1, 2, …, and 12) of year t − k, (k = 0, 1 and 2), and that between C t E and AO and between C t E and PDO in month m (m = 1, 2, …, and 12) of year t − k, (k = 0, 1 and 2).

The eggs of pink salmon are spawned from September to November in year t, and they hatch from February to March in year t + 1. The fries swim downstream to the ocean from April to May in year t + 1 (Figure 1). They stay in the ocean for about one year, and then they go back to the coastal waters and go upstream in their native rivers from July to August in year t + 2. Then, they lay their eggs on the bottom of the rivers and die. To summarize the life circle, the spawning stock biomass (SSB) in year t reproduces the SSB of their next generation in year t + 2. That is, there is a two-year difference between the two generations.

The abundance of pink salmon in the maritime Province of Siberia has not been estimated, and so we cannot use abundance data directly. However, when the abundance is high, the catches in the coastal waters and in rivers would also be high. Further, when the catches in the coastal waters and in rivers are high, the SSB that has escaped the harvest in the coastal waters and rivers would also be high. Therefore, we can assume that the SSB in year t (SSB_t) is proportional to the catch in year t (C_t). In this study, we assume that C_t is proportional to SSB_t and the catch in year t + 2 (C_t+2), is proportional to the recruitment in year t + 2, (R_t+2), which is reproduced by SSB_t. Therefore, in this study, we assume SSB_t ∝ C_t and R_t+2 ∝ C_t+2 and we analyze the relationship between C_t+2 and C_t as the SRR, which is the relationship between R_t+2 and SSB_t.

This study used three regression methods in plotting ln(C_t+2) against ln(C_t), i.e., simple regression analysis, Deming regression analysis [16] , and Passing and Bablok regression analysis [17] . A simple regression analysis is problematic because it assumes that the independent variable contains no observation errors. Therefore, parameters estimated using a simple regression analysis usually have serious biases [18] [19] [20] [21] [22] . When both independent and dependent variables contain observational errors, the Deming and Passing and Bablok regression analyses can effectively remove the bias inherent in the results of a simple regression analysis. The programs developed by Aoki [23] [24] were used when the Deming and Passing and Bablok regression analyses were applied to the data.

Pink salmon born in odd- and even-numbered years are completely separated genetically, because the years when they born are completely separated. Therefore, there may be some competitive relationship between these two stocks in the same way that different species that inhabit the same area sometimes engage in competition. In order to confirm this possibility, we investigated the relationship in the case when ln ( C t + 1 E ) was plotted against ln ( C t O ) , because there might be a possibility that the fries born in the even-numbered year t + 1 are eaten by one-year-old fish born in the odd-numbered year t. The opposite relationship was also checked. That is, ln ( C t O ) was plotted against ln ( C t − 1 E ) .

According to Sakuramoto [9] [11] [12] , we tested the following model, which reproduces the trajectories of the catch ratio,

C t + 2 / C t = g ( x 1 , x 2 , … , x n ) (1)

where x₁, x₂, …, x_n denote the environmental factors that control C_t+2/C_t, which corresponds to the R per SSB (RPS). In this model, n denotes the number of environmental factors and g(•) denotes the function that determines how environmental factors affect the ratio C_t+2/C_t. We applied stepwise regression analysis using R software, “stepwlm”, to select the optimal model shown in Equation (1). Using the estimated values of C_t+2/C_t, we calculated C_t+2 using the following equation;

C t + 2 = exp [ { ln ( C t + 2 / C t ) } estimated ] { C t } observed (2)

We calculated the correlation coefficients between C t O and AO and between C t O and PDO in month m of year t − k (k = 0, 1 and 2), and those between C t E and AO and C t E and PDO in month m of year t − k (k = 0, 1 and 2). The AOs and PDOs that showed high correlation coefficients with p-values less than 0.10 are shown in Table 1. We used these AOs and PDOs as the candidates for the environmental factors in Equation (1).

Figure 2 shows the relationship between ln ( C t + 2 O ) and ln ( C t O ) , which corresponds to the relationship between ln(R_t+2) against ln(SSB_t) for odd-numbered years. The parameters of regression lines estimated by the simple, Deming and

Passing-Bablok regression methods are shown in Table 2. The 95% confidence intervals of the slope determined by the simple, Deming and Passing-Bablok regression methods were (0.204, 0.852), (0.689, 1.208) and (0.733, 1.470), respectively. That is, the slope estimated by the simple regression analysis was statistically less than unity, and it was judged that a density-dependent effect was detected. However, in the results obtained using the Deming and Passing-Bablok regression methods, neither slope was statistically different from unity, and it was judged that a density-dependent effect was not detected. In Figure 2, a clear clockwise loop appeared, a phenomenon that can be explained by the logic proposed by Sakuramoto [8] [10] [11] . That is, the age at maturity is 2 years old, and it is considered that the cycle of environmental condition is much longer than this age at maturity.

Figure 3 shows the relationship between ln ( C t + 2 E ) and ln ( C t E ) , i.e., the relationship between ln(R_t+2) against ln(SSB_t) for even-numbered years. The parameters of the regression line estimated by the simple, Deming and Passing- Bablok regression methods are shown in Table 2. The 95% confidence intervals of the slope determined by the simple, Deming and Passing-Bablok regression methods were (0.164, 0.876), (0.578, 1.592) and (0.430, 1.571), respectively. That is, the slope estimated by simple regression analysis was statistically less than unity, and it was judged that a density-dependent effect was detected. However, in the results obtained by the Deming and Passing-Bablok regression methods,

the slopes were not statistically different from unity, and it was judged that no density-dependent effect was detected. In Figure 3, a clear anticlockwise loop can be recognized, which is opposite what would be expected according to the logic proposed by Sakuramoto [8] [10] [11] . That is, the age at maturity is the same for pink salmon born in odd-numbered years (2 years old), so that a clockwise loop emerges. We will address in the Discussion section why two opposite phenomena occur in the odd- and even-numbered years.

Figure 4 shows the trajectories of the catches, ln ( C t + 1 E ) and ln ( C t O ) . The trajectories seem to have a negative relationship. Figure 5 shows the plots of ln ( C t + 1 E ) against ln ( C t O ) . The slope of the regression lines estimated using the simple and Deming regression analyses are shown in Table 3. When the dependent variable has a negative relationship to the independent variable, the Passing-Bablok regression method cannot be used. Therefore, only the simple and Deming regression methods were applied in this analysis.

The slope was significantly negative with a 10% significance level (p = 0.0663) when the simple regression analysis was applied, and the slope was significantly negative with a 5% significance level when the Deming regression analysis was applied. That is, we can conclude that ln ( C t + 1 E ) has a negative relationship with ln ( C t O ) . The result of the oppositional relationship, that is, ln ( C t O ) plotted against ln ( C t − 1 O ) is also shown in Table 3. The slope was slightly larger than the 10% significance level (p = 10.6) when the simple regression analysis was applied; however, the slope was significantly negative with a 5% significant level when the Deming regression analysis was applied.

In Equation (1), the candidates for the environmental factors are shown in Table 1. Table 1 shows the AOs and PDOs in month m of year t − k, (k = 0, 1, and 2), for which the p-values of the correlation coefficients for C_t were less than 0.10. Using these monthly AOs and PDOs, we estimated the optimal model using the stepwise regression analysis in R software. Here, we assume that ln (C_t+2/C_t) corresponds to ln(RPS_t).

The results are also shown in Table 1. That is, AOs in April and November in year t and in February and May in year t − 2 were chosen as the environmental factors for the model forecasting the catch ratio in odd-numbered years. The model estimated was as follows,

ln ( R P S t ) = − 0.128 − 0.173 a 2 , t − 2 + 0.504 a 5 , t − 2 + 0.404 a 4 , t − 0.356 a 11 , t (3)

Here a_m,t−k denotes the AO in month m of year t − k. The result is shown in Figure 6, and the Akaike information criteria (AIC) of which was-39.26. The catches forecast by Equation (2) are also shown at the bottom of Figure 6.

The AOs in February and May in year t − 2 and the PDO in January in year t and the PDO in November in year t − 2 were chosen as the environmental factors for the model forecasting the catch ratio in even-numbered years. That is,

ln ( R P S t ) = 0.0810 + 0.116 a 2 , t − 1 + 0.475 a 7 , t − 2 − 0.132 a 1 , t − 0.218 a 12 , t − 2 (4)

Here p_m,t-k denotes the PDO in month m of year t − k. The result is shown in Figure 7, and the AIC of which was-37.98. The catches forecast by Equation (2)

are also shown at the bottom of Figure 7. The ln(RPS_t), which is defined by the ratio of the catch, ln(C_t+2/C_t), in both odd- and even-numbered years, was well reproduced only by these environmental factors, and no density-dependent effect seems to exist in the RPS in both sets of years.