# Scottish Marine and Freshwater Science Vol 6 No 12: The demography of a phenotypically mixed Atlantic salmon (Salmo salar) population as discerned for an eastern Scottish river

This report investigates the potential for assessment of fish populations at a sub-river

scale. A sophisticated mathematical model was used to separate salmon from a

single river (North Esk, eastern Scotland) into three sub-stocks, based on the

number

### Estimating Stock Recruitment Relationships

**
Relating stock and recruits
**

For each of the sub-stock decompositions, the data described above either provides directly, or allows one to infer, yearly time-series of both spawning stocks and their subsequent smolt outputs. To use these data as the basis of a stock-recruitment relation requires one to associate spawning stock in one year with recruitment (smolt output) at some later time - a process which is materially complicated by individual to individual variability in time taken to develop from ova to smolt.

The actual freshwater age distributions of cohorts of North Esk
smolts going to sea is well established from large samples of smolt
scales obtained from Kinnaber lade (for details see Todd
*et al*. 2012). The smolt-age distribution at the North Esk
(which is rather typical of east-coast Scottish fish (Bacon
*et al*. 2011)) has changed slightly over the duration of
our study period (Todd
*et al*. 2012), and the annual age-stratified estimates (
i.e. including the trend, its annual
fluctuations and any recording errors) were hence used here, with
one simplification; to maximise data-points (by reducing data-loss
from 'end-effects'
^{
[1]
}) consequent on collating smolt contributions over four
freshwater-emigrant age-classes, from S1s to S4s), the smolt-ages
considered were amalgamated into S2's and S3's (S2'=S1+S2; S3'=
S3+S4), ignoring the small average contributions of S1s and S4s
(6.9% and 0.5% respectively).

To determine the total smolt output (
*O
_{i,y}*) associated with the year

*y*spawning stock for a particular sub-stock (

*A*), we define

_{i,y}*ψ*

_{d}_{,y}as the (sub-stock independent but calendar year dependent) proportion of the yearly smolt production (

*p*) which smolt in the

_{i,y}*d*year after ova deposition and thus see that

^{th}

**
Bayesian estimation of a stochastic stock-recruitment
relation
**

Standard Bayesian methods were used to determine the parameters
of a stochastic representation of the spawning stock to smolt
production relationship (Gurney
*et al*. 2010 B); see the Discussion section for an
explanation of why an ova-to-smolt relationship was not used
instead. The expected relationship between stock (
*A*) and recruits (
*O*) is here represented by a conventional Beverton-Holt (
BH; Beverton and Holt,
1957) curve:

where
*O
_{max}* represents the maximum possible output
(recruitment, here smolts) and

*H*represents the spawner population needed to produce an expected recruitment of half the maximum. A further assumption made when setting up the model was that actual output values were Negative Binomially distributed around the expected value with a shape factor

*θ*

*,*which is approximately the inverse of the square of the coefficient of variation.

The maximum number of smolts produced per individual spawning fish was defined as:

β can subsequently be decomposed into the product of the
proportion of the spawning population which is female (F), the ova
fecundity of the females (B
_{o}) and the ova to smolt survival (L
_{os}); see Supplementary Material, part I, for an
illustration of how, and why this is useful.

As the life-history stages characterized by this adult-to-smolt
stock recruitment relation are highly density-dependent, we used
the prior-knowledge prescription of Gurney
*et al*. (2010 B) to estimate initial informative `control
priors' to define the prior parameter distribution from which the
Markov-Chain Monte-Carlo process calculated a posterior
distribution, representing the sum of knowledge provided by our new
data and the prior. If the data do not show the saturation implied
by the informative prior, then the estimated curve will be more
saturated than if estimated with an uninformative prior, and the
precision of the parameter estimates will be a little
*impaired* (see Gurney
*et al*. (2010 B) for details).

The resulting posterior distributions provide estimates of both
the most plausible values for the model parameters (
*O
_{max}*

*, H, θ*), the most probable expected stock recruitment relation ( ESR) and their credibility limitsAfter some experimentation, it was found that this process explored the posterior distribution rather efficiently, and that using a burn-in of 10

^{4}iterations, followed by a MCMC sequence of 2 x 10

^{4}iterations produced results which scarcely differed from those produced by an MCMC sequence a factor of 10 longer.

**
Changes in Fisheries Practice During the Study Period
**

A number of changes in fisheries practices occurred during the
study period. Those which might be expected to generate changes in
parameters reported in this paper include: reductions and
subsequent phasing out of the distant-water fisheries in West
Greenland (1971-1993) and the introduction (1978) and subsequent
decline (ca. 1981-1992) of the Faroes fishery; change of the
'Morphie Dyke' net and coble fishery in the lower North Esk to a
rod fishery in 1991; delayed start to the estuarine net-fishery
season from mid-Feb to 1
^{st} April in 2000, and to 1
^{st} May in 2005; the gradual introduction of
catch-and-release practices in the rod-fishery from the late 1990s
(records kept since 1994) and, particularly, from 2001 onwards.

The first of these changes should alter the marine survival from
smolt to pre-estuary adult (
PEA) for
sub-stocks visiting the fishery areas, while the remaining three
will change the proportion
*of pre-estuary adults who survive to spawn.* The listed
management changes should all produce enduring rather than
transient parameter alterations, so our strategy for detecting such
effects in the presence of strong year-to-year variability was to
look for alterations in the decadally smoothed mean values of the
parameter concerned.

**
Evaluation Criteria
**

The adult time-series data, both pre-fishery and spawner
numbers, are contrasted between the single- and sub-stock-scenario
results to assess whether the resulting time-trends differed. Those
direct adult count data were particularly reliable. In contrast,
enumeration of the smolt numbers derived from much less direct and
more variable estimates. Comparison of smolt production was
accordingly made as follows. An index of smolt productivity, per
female per putative sub-stock, i (and thus independent of sub-stock
size), was obtained from the appropriate sub-stock's
SR curve by estimating
β (eqn. 12). We then took ratios, for different pairs of the
β
_{i} sub-stocks, to assess whether one stock was relatively
more productive than another
^{
[2]
}. However, as the sub-stocks differred in average female
body-sizes, and fecundity is size dependent, those
SR derived
productivity ratios were then themselves divided by body-length
predicted fecundity ratios for the same pair of sub-stocks. The
procedure resulted in a sub-stock comparison index whose
expectation is 1.00 if the
SR derived ratio and
size-expectation ratio are identical. The size-expectations were
calculated from North Esk results reported in Bacon
*et al*. (2012). Expectations are here presented for two
ova-productivity criteria (total ova volume and total ova numbers),
with the ova numbers being calculated both from the standard
(historic, upland) equation and also with an egg-number adjustment
for their actual upland/lowland breeding locale, where appropriate;
the full calculations are explained and illustrated in
Supplementary Material, part E.

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