# 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

### Data Analysis

**
Density-dependent and Density-independent Life-cycle
Segments
**

Many studies have demonstrated that the major density-dependent
processes in the life-cycle of Atlantic salmon occur during the
freshwater stages (eg Jonsson
*et al*.
*1998;* Gurney
*et al*. 2010) and that the demographic processes between
emigration to sea as a smolt and return to the spawning grounds as
an adult are essentially density-independent (eg Gurney
*et al*. 2010).

In this analysis the density-dependent part of North Esk's salmon population life-cycle was regarded as starting with the return of adult spawners to their natal habitat and ending with emigration of their progeny to sea as smolts. For each distinct sub-stock, this part of the life-cycle was characterised by a separate stock recruitment relationship. The available information about the density-independent parts of the life-cycle was heavily concentrated in the segment between return to the estuary and spawning, so this segment of the life-cycle was treated separately from the segment between smolting and return to the estuary (hereafter referred to as `at sea').

The 'at-sea' portion of the Atlantic salmon life-cycle, while
complex, shows no evidence of density-dependence. It is composed of
fisheries mortality in both distant water (oceanic) and coastal
commercial fisheries and predation mortality in both oceanic and
coastal locations. For all of these there is little or no
information that can be reliably attribute fish to the North Esk.
At-sea survival is also affected by survival and growth processes
that are complex, poorly understood and likely to vary in both
space and time (eg Eriksson 1994; Friedland 2013; McGurk 1996;
Salminen
*et al*. 1995). This study required an objective approach to
characterise the at-sea mortality of
*specific cohorts* (hereafter smolt-year-classes) of
sub-stocks (spent differing periods at-sea and having appreciably
differing body sizes on return). As we could find no suitable
quantitative data for Atlantic salmon that could be generalised to
Scottish fish, we here adapted an allometric approach based on
McGurk's (1996) results for Pacific salmonids.

The sizes, sea-ages and seasonal return times of Scottish
Atlantic salmon vary within seasons and have varied over time
(Bacon
*et al*. 2009): by the 1970s the number of three sea-winter
(3
SW) salmon
at the North Esk (and on the eastern coast of Scotland more
generally) had declined to near zero. The proportion of
previous-spawners is also very low (1.7% on the Scotland east coast
generally, Bacon
*et al*. 2009) and only 0.9% for the North Esk.
Consequently, the sea-age structure was simplified by treating the
repeat-spawners and 3+
SW salmon
as '
MSW' fish, but
characterising their periods at sea as if they were 2
SW
(hereafter 2
SW' ).
Average periods at sea (assuming a median smolt emigration time of
May for all smolts and a median monthly return time for each of the
proposed sub-stocks based on their scale-read ages and counts) were
used for each putative sub-stock, of 16, 22 and 28 months for
grilse, early- and late2
SW' fish
respectively, with associated mean sizes at return of 60, 72.5 and
80 cm fork-length respectively.

**
Estimating Sub-population Smolt Production and
Sea-survival
**

To partition smolts among phenotypic stocks we denoted the
number of individuals of sub-stock
*i* returning to the river in year
*y* by
*R
_{i,y},* the total smolt production that year as

*P*and defined an (unknown) quantity φ

_{y}*to represent the proportion of the year*

_{i,y}*y*smolt output who will, if they survive, return to the river as sub-stock

*i*individuals. Finally we denoted the proportion of sub-stock

*i*individuals going to sea as smolts in year

*y*who will survive to form part of the pre-estuary population in year y + τ

*as*

_{i}*S*so that

_{i,y}

As The φ
*
_{i.y}
* (proportions i of the year

*y*smolt production), must sum to one over all sub-stocks, for each year. Hence we know that for each and every year

*y*

Equation (3) is little use, because it contains as many unknowns
as there are sub-stocks. However, if, for each sub-stock
*i*, we can relate each
*S
_{i,y}* to a single year-class-dependent constant,
ε

*, so that*

_{y}

then equation (3) becomes

which, for each year in which total smolt output
*P
_{y}* and subsequent pre-estuary numbers {R

_{i,y+τ i }} are known, can be solved for the single unknown ε

*. Although few variants of equation (5) have closed-form solutions, its general case is readily treated numerically (we used*

_{y}*optimise*from the R package, R Dev. Team, 2011).

Once we know ε
_{y}, we can calculate the proportional survival from smolt
to returner (
*S
_{i,y}*) of sub-stock

*i*individuals emigrating to sea in year

*y*from equation (4) and the number (

*P*) of such individuals from,

_{i,y}

The time-series of smolt to
PEA survivals
calculated from this process for any given sub-stock are related by
a non-linear transformation defined by our assumption about
**
S
**

*, to those for other sub-stocks whose smolts form part of the un-differentiated smolt-output observations.*

_{i}
**
Modelling Smolt to Pre-estuary Survival
**

One needs an explicit definition of the function
**
S
**

*(ε*

_{i}*) to model smolt to pre-estuary survival. However, a number of plausible alternatives exist, given the paucity of data to test such models.*

_{y}A simple formulation for smolt to pre-estuary survival, which we
called the
*uniform risk-rate (
UR) model*,
assumes that all individuals (of whatever sub-stock) who emigrate
in year
*y* are subject to the same
*per-capita, per-unit-time* mortality rate while at sea. In
this formulation, if sub-stock
*i* individuals spend
*m
_{i}* months at sea, then ε

_{y}is the at-sea mortality rate for individuals emigrating in year

*y*and set

A potentially much more realistic formulation can be found in
the work of McGurk (1996) who examined the marine-mortality of a
number of species of Pacific salmon, and concluded that the marine
survival (
*s*) of a species whose individuals go to sea at weight
*W
_{o}* and return to spawn at weight

*W*can be well described by

_{s}

Although the values that McGurk obtained for his parameters
α
*
_{m}
* and β

*varied considerably between species, the form seemed highly robust, so we shall postulate that the allometry parameter (β*

_{m}*) remains constant over time for a given sub-stock, and identify the ratio (α*

_{m}*/ β*

_{m}*) with our sub-stock independent smolt-year parameter ε*

_{m}_{y}. Hence we write

In this formulation,
*W
_{o}* represents the sub-stock independent smolt weight
and

*W*represents the spawning weight of sub-stock

_{si}*i*individuals.

We examined two variants of this allometric model, which were
differentiated by the value chosen for the allometry parameter
β
*
_{m}
*. Our

*default allometric mortality ( DAM)*model used the value of β

*found by McGurk as most suitable for a composite dataset encompassing all five species of Pacific salmon covered by his study, namely β*

_{m}*=-0.37. As an alternative quite extreme case, but which McGurk (1996) argued was also well supported, we examined a*

_{m}*uniform survival*( US) model, in which all sub-stocks have the same smolt to pre-estuary survival ( N.B. this implies considerably different marine-mortality risk-rates), which we achieved by setting β

*= 0*

_{m}### Contact

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