*Background*

ICES (ICES 2001) routinely produces for management authorities 10-year projections of population abundance for a large number of fish populations, including sprat in the Baltic Sea. These projections are produced during annual assessments of stock status and use existing knowledge of fish biology (e.g., growth, mortality, maturation rates, recruitment) and fishing mortality rates as inputs to the calculations. Because variations in fish biology in the future are unknown and cannot presently be predicted, the biological inputs assume mean values with random variation based on historical data. As a result the biological inputs represent the overall biological response to average environmental conditions observed in the recent past. The projections therefore represent stock development under environmental conditions observed during the period represented by the biological variables. In the case of sprat, the projections assume no functional relationships between any of the key biological variables and abiotic factors; hence the effects of climate variability on stock development cannot be simulated. Nevertheless major changes in ecosystem structure and functioning occur and are known as “regime shifts” (Beamish et al. 2000) . In such instances it is useful to know how a fish population will react (e.g., increase vs. decrease, or whether the population might fall below predefined critical levels) during a different regime and under different levels of exploitation.

Notably one of the major uncertainties in projections of fish population development (especially for sprat in the Baltic Sea) is a functional relationship between recruitment and any other variable, including spawner biomass and environmental factors. In the absence of a functional relationship, assessment working groups often must assume a mean recruitment with random variability, or a highly uncertain relationship (i.e., one that explains little variation) between spawner biomass and recruitment. As a result, projections based on these inputs are also highly uncertain.

We have observed that recruitment
for sprat in the Baltic Sea has co-varied with temperature for at least 45 years.
We incorporated our temperature–recruitment relationship into a population-projection
model used by the assessment working group (ICES 2001) and
calculated the expected spawner biomass trend for the next 10 years. In
addition, we estimated the probability for different temperature and exploitation
scenarios that the spawning biomass would fall below the level that would require
ICES to recommend major reductions in fishing quotas. This level of biomass
is known as the “precautionary-approach biomass” (*B*_{PA};
ICES 1998 ).

*Methods*

The projections used the same assumptions
of sprat biology (e. g., growth, maturity, natural mortality rates) and exploitation
as used by ICES when making stock projections as part of its routine stock assessment
procedures. The only modifications we made involved the functions used
to generate recruitment estimates (see below). Natural mortality rates
were age-specific and derived from a multi-species virtual population analysis
(MSVPA) that calculates sprat mortality rates due to predation by cod.
These rates were assumed constant throughout the projection period to be consistent
with ICES assessment practice (ICES 2001) . The applied exploitation pattern
used age-specific fishing mortality rates for the years 1998-2000 (ICES
2001) . The projections assumed random variation associated with initial
estimates of age-specific sprat abundance as derived from virtual population
analyses (VPA; ICES 2001) . Calculations also assumed
that growth rates were variable by allowing weights-at-age to vary randomly
within ranges observed in historical data (ICES 2001) . The
ICES projection used a Beverton-Holt stock-recruitment model to derive recruitment
estimates; the model does not explain significant variation and no environmental
forcing was included in this projection (ICES 2001) . We
conducted six series of projections to explore how stock development might react
to changes in temperature (three levels; cold, average and warm) and exploitation
(i.e., status quo and 1.2 × status quo). The higher exploitation
rate corresponds to the “precautionary fishing mortality” (*F*_{PA};
ICES 2001) and enables us to evaluate whether a sustainable
stock development is possible under different environmental scenarios. The simulations
of stock development for all scenarios were repeated 200 times to generate distributions
of spawner biomass, recruitment and fishing yield. These distributions
used the random variations associated with the temperature–recruitment–spawner
biomass functions (see below) and other inputs (e.g., growth) to generate realistic
levels of recruitment variability. Based on the computed spawner biomass
distributions we estimated several percentiles to examine the probability that
spawner biomass would fall below *B*_{PA}
for different temperature and exploitation scenarios. These probabilities
were visualized by producing time-dependent contour plots of spawner biomass.

*Computational
details regarding recruitment–environment–spawner biomass relationship*

We chose to include in our stock
projections a functional relationship among recruitment, spawner biomass, and
environmental variability, even though our statistical analyses indicated only
a small and usually insignificant effect of spawner biomass on recruitment for
our data. The type of relationship we used (see below) can represent stock
dynamics at low spawner biomass levels, should certain combinations of exploitation
and environmental variability cause the simulated population to fall to low
levels. We emphasize that our environmentally based models of recruitment
(Fig. 2 of the article; Table B1) may not apply
in situations where spawner biomass (*S*) is lower than that observed in
our time series.

We assumed a piecewise spawner biomass–recruitment
relationship in which recruitment for *S* > *S*_{crit.}
and *S* < *S*_{crit.} had two different functions (see
also ICES 2003*a*, *b*). Simulation studies
with real spawner biomass–recruitment data for several fish populations
show that implementation of a piecewise model (“hockey stick”) is acceptable
or even preferable to traditional models in cases where stock dynamics at low
*S* are unknown or highly uncertain (Barrowman and Myers
2000) .

When *S* > *S*_{crit.},
we assumed that recruitment was more strongly influenced by environmental variables
than by variations in spawner biomass. This assumption is supported by
our statistical analyses (e.g., Fig. 2 of the article; Table
B1).

When *S* < *S*_{crit.},
the relationship between recruitment and spawner biomass is poorly known because
there are few data points in this range. We assumed for *S* <
*S*_{crit.} that recruitment was more strongly influenced by spawner
biomass than by environmental variables and that the relationship between the
variables is linear. We assumed that recruitment increased from 0 to the
long-term geometric mean (here labelled *R*_{max}) for the range
*S* < *S*_{crit}.

Since no recruits can be produced
if *S* = 0, the slope *A* of a line from the origin to R_{max}
over the range of *S* < *S*_{crit.} is assumed to
be *R*_{max}/*S*_{crit.} and

*R*_{i} = *R*_{max}/*S*_{crit.} × *S*_{i}
for *S*_{i} < *S*_{crit}.

We chose *S*_{crit.}
= *B*_{PA} = 275 × 10^{8} kg
(ICES 2001) . Other values could be determined using least-squares (Barrowman
and Myers 2000) ; however *B*_{PA} is
a reasonable choice for our purpose as it assumes impaired recruitment due to
low spawning stock size.

Recruitment predictions derived from these functions were configured to include realistic levels of variability. This variability was defined on the basis of historical recruitment variability observed under three different levels of temperature (i.e., cold, average, and warm). Observations indicate that cold and warm periods do occur in the Baltic Sea (Fig. 1 of the article): for example, mean temperatures during 1976–1987 and 1989–1999 were, respectively, 2.6 ± 1.0 °C and 4.4 ± 0.9 °C (mean ± 1 SD).

We ranked our temperature data from
coldest to warmest and then grouped the data into three categories whose means
approximated the (1) overall mean temperature (3.7 °C), (2) overall mean temperature
– 1 SD (2.4 °C), and (3) overall mean temperature + 1
SD (5.0 °C). We assigned temperatures to only one
category to ensure that each category had a unique set of recruitment-temperature
observations. For each of the three subgroups, we estimated the mean and
standard deviation of log recruitment. The subgroup standard deviations
were then used to estimate the variation in recruitment (*R*) expected
at different temperatures (Table C1).

For example, for *S* > *S*_{crit.}
and for the mean temperature situation (category 2):

R(10^{9})
= exp(4.02 ± 0.761) |

For S < S_{crit.}
and the mean temperature situation (category 2): |

R = exp(log R_{max})/275 × S |

= exp(4.02)/275 × S
(mean log R_{max} = 4.02) |

= 0.202 × S
(10^{9}). |

The amount of *variation* in
*R* associated with this prediction cannot be directly estimated from observed
data because there are too few observations of *S* < *S*_{crit.}
(*N* = 4 of 25). In this case we assumed that the amount of variation
in *R* scales with spawner biomass and can be estimated from the observed
variation in log *R* at each temperature category:

For example, consider temperature
category 2, *S* = 100 and *S*_{crit.} = *B*_{PA}
= 275 × 10^{8} kg:

1 SD of log
*R*_{s} = (SD of log *R* for temperature
category_{i} ) × *S*/*S*_{crit.}

= 0.761 * S/S_{crit} |

= 0.277. |

Therefore

R(10^{9})
= [(exp(4.02 ± 0.761×S/S_{crit.}))
/ S_{crit.} ] × S |

= [(exp(4.02 ± 0.761×100/275)) / 275] × 100 |

= [(exp(4.02 ± 0.277)) / 275] × 100 |

= 15.4; 20.2; 26.7 (respectively for mean - 1 SD, mean, mean + 1 SD) |

These approximations allow us to
estimate the combined effects of temperature and spawning stock biomass on recruitment
in sprat for a wide range of temperatures and spawner biomasses, including the
special case *S* < *S*_{crit.} where observations are lacking.

The output from the simulations
is summarized in Fig. 4 (article), and Figs. C1 and C2
(below). Biomass is highest in the warmest scenario after 10 years, even if
exploitation rates increase 20%. The stock is expected to remain at current
levels only in the warm environment scenario. Note that the population
has a 15% chance of falling below *B*_{PA}
(275 × 10^{8} kg) in the low-temperature,
high-exploitation scenario. If temperatures remain near the long-term
mean, the population has a high probability (> 95%) of remaining above *B*_{PA},
even if exploitation rates increase by 20%. As might be expected the projection
for the mean temperature and status quo exploitation is nearly identical to
that produced by the assessment working group (ICES 2001),
which excluded environmental variability.

Table C1. Temperature categories and recruitment means ± SD for input to medium term projections of sprat stock development in the Baltic Sea.

Temperature Category

Mean Temperature (°C)

Mean log

R(10

^{9})1 SD log

RExp(mean log

R)(10

^{9})1

2.4

3.71

0.946

41.0

2

4.0

4.02

0.761

55.7

3

5.1

4.34

0.792

76.7

Notes: The temperature categories were derived from the overall mean (3.7° C) and standard deviation (1.3° C) for the entire time series. Category means are derived from subsets of temperature data containing observations whose means equate to the overall mean or overall mean ± 1 SD. (N= 10, 7, and 8 years per category.)

FIG.
C2. Percentiles of simulated distributions of projected sprat spawner
biomass (SSB; 10^{6} kg) under different
temperature scenarios with exploitation rates 20% higher than those observed
in 1998–2000 (i.e., 1.2 × status quo). Calculations
assume a hockey-stick (Barrowman and Myers 2000) relationship
between recruitment and spawner biomass with a break at 275 × 10^{8}
kg (B_{PA}). |

Barrowman, N. J., and R. A. Myers.
2000. Still more spawner-recruitment curves: the hockey stick and its generalizations.
Canadian Journal of Fisheries and Aquatic Sciences **57**:665–676.

Beamish, R. J., D. J. Noakes, G.
A. McFarlane, W. Pinnix, R. Sweeting, and J. King. 2000. Trends in coho marine
survival in relation to the regime concept. Fisheries Oceanography **9**:114–119.

ICES [International Council for Exploration of the Sea]. 1998. Report of the Study Group on Management Strategies for Baltic Fish Stocks. ICES CM 1998/ACFM:11.

ICES [International Council for Exploration of the Sea]. 2001. Report of the Baltic Fisheries Assessment Working Group. ICES CM 2001/ACFM:18.

ICES [International Council for
Exploration of the Sea]. 2003*a*. Report of the Study Group on the
Further Development of the Precautionary approach to fishery management.
ICES CM 2003/ACFM: 9.

ICES [International Council for
Exploration of the Sea]. 2003*b*. Report of the Study Group on Precautionary
Reference Points for Advice in Fishery Management. ICES CM 2003/ACFM:
15