Effects of an autocorrelated stochastic environment and fisheries on the age at maturity of Chinook salmon

Abstract

Chinook salmon (Oncorhynchus tshawytscha) reproduce only once in their lifetime, and their age at reproduction varies among individuals (indeterminate semelparous). However, the factors that determine their spawning age still remain uncertain. Evidence from recent studies suggests that individual growth and reproduction of Chinook salmon are affected by the rate of coastal upwelling, which is shown to be positively autocorrelated between years. Therefore, the serially autocorrelated environmental is expected to play an important role in determining their spawning age. In the present study, I demonstrate the advantage of an indeterminate maturation strategy under a stochastic environment. I then present theoretical evidence for the advantage of adjusting the maturation probability based on the environment they experienced and demonstrate that fisheries reduce the fitness of the strategy to delay maturation. The results presented herein emphasize the importance of incorporating detailed life-history strategies of organisms when undertaking population management.

Appendix

Appendix A

Fig. A1

Environmental indices and associated correlograms. a Monthly Coastal Upwelling Index (CUI) in June at 42°N 125°W (http://swfsc.noaa.gov/) and b associated Correlogram. c Winter (December, January, February, and March) Pacific Decadal Oscillation (PDO) index (http://www.jisao.washington.edu/pdo/) and d associated correlogram. e Winter North Atlantic Oscillation (NAO) index (http://www.cgd.ucar.edu/cas/jhurrell/nao.stat.winter.html) and f associated correlogram. The lag_one autocorrelations of all three indices are significantly positive (p ≈ 0.0002 for CUI; p ≈ 0.0076 for PDO index; p ≈ 0.0412 for NAO index)

Appendix B

This appendix describes the derivation of Eqs. 1–(2). First, let n _1,t + 1 the number of individuals in Stage 1 at time t + 1. Then, the expected number of individuals in Stage 1 at time t + 1, conditional on the fact that their parents experienced a favorable environment immediately before their reproduction (X _t  = 1), is given by

$$ {\text{E}}{\left( {\left. {n_{{1,t + 1}} } \right|X_{t} = 1} \right)} = F^{{{\left( + \right)}}}_{3} {\text{E}}{\left( {n_{{3,t}} } \right)} + pF^{{{\left( { + + } \right)}}}_{4} {\text{E}}{\left( {n_{{4,t}} } \right)} + {\left( {1 - p} \right)}F^{{{\left( { - + } \right)}}}_{4} {\text{E}}{\left( {n_{{4,t}} } \right)}, $$

(B1)

where E(·) denotes the expectation of the argument and p is the probability that the environment at time t-1 was also favorable, as the environmental model is reversible (see Gallager 1995). Refer to Table 2 for the specifications of $F^{{{\left( \cdot \right)}}}_{k}$. Expressing n _i,t in terms of the number of individuals in Stage 2, Eq. B1 becomes

$$ {\text{E}}{\left( {\left. {n_{{1,t + 1}} } \right|X_{t} = 1} \right)} = F^{{{\left( + \right)}}}_{3} S_{r} S^{{{48} \mathord{\left/ {\vphantom {{48} {52}}} \right. \kern-\nulldelimiterspace} {52}}}_{o} m{\text{E}}{\left( {n_{{2,t - 1}} } \right)} + {\left( {pF^{{{\left( { + + } \right)}}}_{3} + {\left( {1 - p} \right)}F^{{{\left( { - + } \right)}}}_{4} } \right)}S_{r} S^{{{48} \mathord{\left/ {\vphantom {{48} {52}}} \right. \kern-\nulldelimiterspace} {52}}}_{o} S_{o} {\left( {1 - m} \right)}{\text{E}}{\left( {n_{{2,t - 2}} } \right)}. $$

(B2)

Refer to Tables 1 and 2 for descriptions of the parameters in Eq. B2. Assuming a stable stage distribution, $\alpha E{\left( {n_{{1,t + 1}} } \right)} = E{\left( {n_{{2,t - 1}} } \right)} = E{\left( {n_{{2,t - 2}} } \right)}$, where α is a constant relating to the number of individuals in stages 1 and 2. By dividing both sides by $S_{r} S^{{{48} \mathord{\left/ {\vphantom {{48} {52}}} \right. \kern-\nulldelimiterspace} {52}}}_{0} {\text{E}}{\left( {n_{{2,t - 1}} } \right)}$ and equating it to E(${\Re }_{1}$), we obtain,

$$ {\text{E}}{\left( {{\Re }_{1} } \right)} = F^{{{\left( + \right)}}}_{3} m + 0.8{\left( {1 - m} \right)}{\left( {pF^{{{\left( { + + } \right)}}}_{4} + {\left( {1 - p} \right)}F^{{{\left( { - + } \right)}}}_{4} } \right)} = \frac{{E{\left( {n_{{1,t + 1}} \left| {X_{t} = 1} \right.} \right)}}} {{S_{r} S^{{{48} \mathord{\left/ {\vphantom {{48} {52}}} \right. \kern-\nulldelimiterspace} {52}}}_{o} {\text{E}}{\left( {n_{{2,t - 1}} } \right)}}}, $$

(B3)

where S ₀ = 0.8 (Table 1). Similarly, we obtain Eq. 2 as

$$ {\text{E}}{\left( {{\Re }_{0} } \right)} = F^{{{\left( - \right)}}}_{3} m + 0.8{\left( {1 - m} \right)}{\left( {pF^{{{\left( { - - } \right)}}}_{4} + {\left( {1 - p} \right)}F^{{{\left( { + - } \right)}}}_{4} } \right)} = \frac{{E{\left( {\left. {n_{{1,t + 1}} } \right|X_{t} = 0} \right)}}} {{S_{r} S^{{{48} \mathord{\left/ {\vphantom {{48} {52}}} \right. \kern-\nulldelimiterspace} {52}}}_{0} {\text{E}}{\left( {n_{{2,t - 1}} } \right)}}}. $$

(B4)

These quantities are the total number of offspring produced by Stage 3 and 4 divided by a quantity that is proportional to the number of individuals in Stage 2 (which is in turn proportional to the number of individuals in other stages under a stable stage distribution). Thus, E(${\Re }_{1}$) and E(${\Re }_{0}$) are indices of the fertilities assuming a stable stage distribution.