A length-based hierarchical model of brown trout (Salmo trutta fario) growth and production

TL;DR: A hierarchical Bayesian model is presented to estimate the growth parameters, production, and production over biomass ratio (P/B) of resident brown trout populations by investigating the growth and production of a brown trout population by using data collected in the field from 2005 to 2010.

Abstract: We present a hierarchical Bayesian model (HBM) to estimate the growth parameters, production, and production over biomass ratio (P/B) of resident brown trout (Salmo trutta fario) populations. The data which are required to run the model are removal sampling and air temperature data which are conveniently gathered by freshwater biologists. The model is the combination of eight submodels: abundance, weight, biomass, growth, growth rate, time of emergence, water temperature, and production. Abundance is modeled as a mixture of Gaussian cohorts; cohorts centers and standard deviations are related by a von Bertalanffy growth function; time of emergence and growth rate are functions of water temperature; water temperature is predicted from air temperature; biomass, production, and P/B are subsequently computed. We illustrate the capabilities of the model by investigating the growth and production of a brown trout population (Neste d'Oueil, Pyrenees, France) by using data collected in the field from 2005 to 2010.

Summary

Introduction

• Abundance is modeled as a mixture of Gaussian cohorts; cohorts centers and standard deviations are related by a von Bertalanffy growth function; time of emergence and growth rate are functions of water temperature; water temperature is predicted from air temperature; biomass, production, and P/B are subsequently computed.
• The trend is to construct such statistical models within a Bayesian framework (Congdon, 2006).
• The authors primary objective is to provide a layout to compute growth parameters of brown trout populations by using accessible data (namely removal sampling and air temperature data).
• Time scale is daily, spans from January 1st of the year of the first campaign to December 31st of the year of the last campaign, with a total of T days.

Emergence

• Cardinal temperatures are minimum (ymin), optimum (yopt), and maximum (ymax) temperatures required for growth as well as minimum (y0) and optimum (y1) temperatures required for hatching.
• CE50 is the critical value leading to the emergence of 50% of the fry.
• See text for values of multidimensionnal parameters (Li; to; tovio;k).
• Are illustrated in Fig. 1: abundance and growth submodels depend on common parameters (mo;k and so;k, defined later), growth depends on the time of emergence and growth rate, these quantities further depend on the water temperature, fish biomass is the cross-product of fish weight and abundance, and the combination of growth and biomass parameters lead to production.

Weight

• Parameters are shape and rate for gamma distributions, expectation and variance for normal and lognormal distributions, and boundaries for uniform distributions.
• Units which are provided with precisions (e.g. 1=s2l) are units of respective standard deviations (e.g. sl).
• Standard deviations are related to random errors across campaigns (sl, st, sa, sb, sZ, sz), among individuals (sN, s0), and residual (sm, sT, sW).

Abundance

• The abundance submodel is briefly presented and is more thoroughly investigated by Ruiz and Laplanche (2010).
• The motivation to include this additional error term in the model is illustrated and discussed later.
• Model alternatives are defined whether cohort standard deviations so;k (M1 4) and centers mo;k (M1 3) are constrained with a VBGF and whether growth rate Gt (M1 2) and date of emergence temo;k (M1) are temperature-dependent.
• Nevertheless, in the aim of illustrating the modeling of temperature-dependent time of emergence and growth rate, following results are computed by using baseline (M1).
• The model at the current state applies to riverine brown trout (S. trutta fario).

Conflict of interest

• The authors have declared no conflict of interest.
• Growth with seasonally varying temperatures – an expansion of the von Bertalanffy growth model.

Figures

Table 3 Definition of deterministic and stochastic nodes.

Figure 1 The overall model comprises eight submodels: Abundance, growth, growth rate, emergence, temperature, weight, biomass, and production. Large frames with rounded corners: submodels with at least one stochastic node. Large frames with right corners: submodels with deterministic nodes only. Within frames: levels, constant parameters (bold), free parameters. Light filled rectangles: variables common to pairs of submodels. Dark filled rectangles: observed variables. Full arrows: stochastic links. Dashed and dotted arrows: deterministic links. Parameters of some submodels (growth rate, emergence, temperature, weight) are precomputed and provided to parent submodels as fixed input values (dotted arrows). Remaining submodels (abundance, growth, biomass, production) form a full HBM.

Figure 2 DAG of the abundance submodel. Frames indicate levels: campaign (o 2 f1; . . . ;Og), cohort (k 2 f1; . . . ;Kg), length class (i 2 f1; . . . ; Ig), and removal (j 2 f1; . . . ; Jog). Rectangles: deterministic nodes; Ellipses: stochastic nodes; Dark filled nodes: observed variables (A, Co,i,j); Light filled nodes: variables common to the growth (mo,k, so,k) and biomass (lo,i,k) submodels. Full arrows: stochastic links. Dashed arrows: deterministic links.

Figure 4 Cohort growth curves. Expected cohort centers (thick solid lines) plus and minus (thin solid lines) cohort standard deviations are computed by using the cohort growth model with the point estimates of the growth parameters which are reported in Table 5. Cohort centers at campaigns ð mo;k and mo,k) are represented (circles and triangles, respectively). Squares: emergence (at time temo;k of length Lo). Observation times to are highlighted with vertical dashed lines.

Table 4 Growth model alternatives, fit (D), and growth point parameter estimates.

Table 1 Values of constant parameters.

Figure 3 Number of caught fish Co,i,j per Dl 5 10mm length class at Saint-Paul from October 2005 (o 5 1) to October 2010 (o 5 8). The number Co,i,j of fish of length class i caught during removal j during campaign o is illustrated as the j-th stacked sub-bar making up the bar of i-th length class (x-axis) of the subplot related to campaign o. Cohort centers (mo,k, triangles) are represented. Expected population sizes per cohort (lo,i,kADl, thin solid lines) and total (lo,iADl, thick solid lines) are computed by using Eq. (1) with point parameter estimates of mo,k, so,k, to,k, and lo.

Table 2 Distributions of free parameters (priors).

Full text

To link to this article: DOI:10.1002/bimj.201100083
http://dx.doi.org/10.1002/bimj.201100083
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Eprints ID: 5610
To cite this version: Lecompte, Jean-Baptiste and Laplanche, Christophe A
length-based hierarchical model of brown trout (Salmo trutta fario) growth and
production. (2012) BiometricalJournal, vol. 54 (n°1). pp. 108-126. ISSN 0323-
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A length-based hierarchical model of brown trout (Salmo trutta
fario) growth and production
Jean-Baptiste Lecomte
1,2
and Christophe Laplanche

,1,2
1
Universite
´
de Toulouse; INP, UPS, EcoLab (Laboratoire Ecologie Fonctionnelle
et Environnement); ENSAT, Avenue de l’Agrobiopole, 31326 Castanet Tolosan, France
2
CNRS, EcoLab, 31326 Castanet Tolosan, France
We present a hierarchical Bayesian model (HBM) to estimate the growth parameters, production,
and production over biomass ratio (P/B) of resident brown trout (Salmo trutta fario ) populations.
The data which are required to run the model are removal sampling and air temperature data which
are conveniently gathered by freshwater biologists. The model is the combination of eight submodels:
abundance, weight, biomass, growth, growth rate, time of emergence, water temperature, and pro-
duction. Abundance is modeled as a mixture of Gaussian cohorts; cohorts centers and standard
deviations are related by a von Bertalanffy growth function; time of emergence and growth rate are
functions of water temperature; water temperature is predicted from air temperature; biomass,
production, and P/B are subsequently computed. We illustrate the capabilities of the model by
investigating the growth and production of a brown trout population (Neste d’Oueil, Pyre
´
ne
´
es,
France) by using data collected in the field from 2005 to 2010.
Keywords: Growth; Hierarchical Bayesian model; Production; Removal sampling; Salmo
trutta.
1 Introduction
In order to evaluate, compare, and predict the status of fish populations, freshwater biologists have
considered numerous descriptors of fish populations. Such variables can characterize fish stocks
(Pauly and Moreau 1997; Kwak and Waters 1997; Ruiz and Laplanche 2010) durati on and rates of
success through life-history stages (Hutchings 2002, Klemetsen et al., 2003), phenotypic or geno-
typic traits (Ward, 2002; Shinn, 2010). Descriptors of fish populations can be examined separately,
depending on the population aspect under focus. For instance, the impact of surface water con-
tamination by pesticides on fish can be investigated by assessing DNA da mage to blood cells
(Polard et al., 2011). Aquatic resource management would rather focus on fish stock variables, such
as abundance (number of fish per unit area of stream), biomass (mass of fish per unit area),
production (mass of fish produced per unit area per unit time), or production over biomass ratio
P/B (Pauly and Moreau, 1997). The drawback of examining a single population variable is its
limited, descriptive prospect. A countermeasure would be to relate a population variable to cov-
ariates, e.g. environm ental variables or descri ptors of coexisting populations. As an illustration,
growth of salmonids has been related to environmental factors such as temperature (Mallet et al.,
*
Corresponding author: e-mail: laplanche@gmail.com, Tel: 133-534-323-973, Fax: 133-534-323-901

1999) and stream flow (Jensen and Johnsen 1999; Daufresne and Renault, 2006). Descriptors of a
population can also be examined jointly. The examination of multiple variables has a more potent,
explicative prospect and could improve our understanding of population dynamics (Nordwall et al.
2001).
Brown trout ( Salmo trutta) is indigenous to Eurasia. Brown trout has been introduced to non-
Eurasian freshwaters for fishing purposes. It can either grow in oceans and migrate to freshwaters
for reproduction (S. trutta trutta), or live in lakes (S. trutta lacustris), or be stream-resident (S. trutta
fario). As a result of such adaptation capabilities, brown trout has successfully colonized fresh-
waters to a world-wide distribution (Elliott, 1994; Klemetsen et al., 2003). Although ecologically
variable, brown trout is demanding in terms of habitat and water quality. As a result, brown trout is
a relevant bioindicator of the quality of freshwaters at a global scale (Lagadic et al 1998; Wood
2007). Moreover, a fundamental environmental variable driving brown trout life history is tem-
perature (Jonsson and Jonsson, 2009), hence using brown trout as a bioindicator of climate change.
Temperature affects grow th (as aforementioned) as well as life-stage timing (Webb and McLay,
1996; Armstrong et al., 2003). Key life-stages of brown trout are egg laying (oviposition), hatching
of larvae, emergence of fry, reproduction of adults, and death. In our case, we will focus on the
effect of temperature on growth and on time between oviposition and emergence of riverine brown
trout (S. trutta fario). We will also consider several variables characterizing stocks (abundance,
biomass, production).
Abundance of riverine fish species is conveniently assessed through removal sampling: (i) a reach is
spatially delimited (later referred to as a stream section), (ii) fractions of fish are successively removed
of the section by electrofishing and counted, (iii) captured fish are released altogether (Lobo
´
n-Cervia
´
,
1991). Abundance is typically computed by using the method suggested by Carle and Strub (1978): the
main statistical assumption was that the probability of capturing fish (referred to in the following as
catchability) would be equal for all fish. Some contributions have shown, however, that the use of
more advanced statistical models is recommendable in the aim of lowering estimation bias (Peterson
et al., 2004; Riley and Fausch, 1992). The trend is to construct such statistical models within a
Bayesian framework (Congdon, 2006). Recent hierarchical Bayesian models (HBMs) relate abun-
dance to environmental covariates (Rivot et al., 2008; Ebersole et al., 2009), include heterogeneity of
the catchability (Ma
¨
ntyniemi et al., 2005; Do-razio et al., 2005; Ruiz and Laplanche, 2010), and can
handle multiple sampling stream sections (Wyatt, 2002; Webster et al., 2008; Laplanche, 2010). The
reason of popularity of HBMs over the last decade is their ability to handle complex relationships
(multi-level, non-linear, mixed-effect) between variables with heterogeneous sources (relationships,
data, priors) of knowledge. Freshwater biologists can statistically relate multiple descriptors of fish
populations together with covariates within a single HBM framework.
We present an HBM of riverine brown trout growth and production. Our primary objective is to
provide a layout to compute growth parameters of brown trout populations by using accessible data
(namely removal sampling and air temperature data). Our second objective is to use such a layout to
compute interval estimates of brown trout production. To fulfill the first objective, we extend the
abundance model of Ruiz and Laplanche (2010) with a growth module. The abundance model
performs a multimodal decomposition of length-abundance plots. Our growth module constrains
parameters of the multimodal decomposition with a growth function. The interest is twofold: to
guide the decomposition of length-abundance plots with a growth function and to use length-
abundance plots to estimate growth parameters. Ruiz and Laplanche (2010) also created a module
which computes fish biomass. To fulfill the second objective, we extend their biomass module with a
production model. The overall model is the combination of eight HBMs, three (abundance, weight,
biomass) related to the contribution of Ruiz and Laplanche (2010) and five to growth and pro-
duction (growth, growth rate, emergence, temperature, production), which are presented succes-
sively. We show the capability of the overall model to estimate the growth parameters and the
production of a brown trout population with a data set collected in the field. We discuss possible
extensions of the current model.

2 Materials and methods
2.1 Notations and measured variables
A single section (of area A,m
2
) of a stream populated with S. trutta far io is sampled by electrofishing.
Electrofishing is spread over several years in a series of O campaigns with J
o
removals per campaign
(index over campaigns is o 2f1; ...; Og). The number of campaign(s) per year as well as the number of
removal(s) per campaign can be variable. Time scale is daily, spans from January 1st of the year of the
first campaign to December 31st of the year of the last campaign, with a total of T days. Times of
campaigns are noted t
o
(day). Let C
o;j
be the number of fish caught during removal j 2f1; ...; J
o
g at
time t
o
and C
o
¼
P
j
C
o;j
be the total number of fish caught at time t
o
.Thelengthandweightofthe
caught fish are measured and are noted L
o;j;f
(mm) and W
o;j;f
(g) (f 2f1; ...; C
o;j
g). As done by Ruiz and
Laplanche (2010), fish are grouped by length class: D
l
(mm) is the length class width, I is the number of
length classes, [(i 1)D
l
, iD
l
[ are the classes (with [a,b[ denoting an interval including the lower limit a
and excluding the upper limit b), and L
i
5 (i1/2) D
l
are the class centers. Given class centers L
i
and fish
lengths L
o;j;f
, the number of fish of length class i caught during removal j at time t
o
can be computed and
is labeled C
o;i;j
(i 2f1; ...; Ig). The air and water temperatures on day t are noted y
a
t
and y
w
t
, respec-
tively (t 2f1; ...; T g).ThemeasuredvariablesoftheHBMarefishlengthandweight(L
o;j;f
and W
o;j;f
),
catch (C
o;i;j
), stream section area (A),airandwatertemperatures(y
a
t
and y
w
t
). Constant (known)
parameters, free (unknown) parameters of interest, and remaining (unknown) nuisance parameters are
provided together in Tables 1–3, respectively.
2.2 Model structure
As briefly introduced, the overall HBM is the combination of eight submodels. Submodels are
interconnected as a consequence of sharing subsets of parameters, connections between submodels
Table 1 Values of constant parameters.
Parameter Notation Value Unit
Abundance
Area A 574.5 m
2
Length class center L
i
–mm
Growth
Date of campaign t
o
– day
Growth rate
Cardinal temperature y
min
3.6 1C
Cardinal temperature y
max
19.5 1C
Cardinal temperature y
opt
13.1 1C
Emergence
Cardinal temperature y
0
2.8 1C
Cardinal temperature y
1
22.4 1C
Date of oviposition t
ovi
o;k
– day
Critical value CE
50
76.2 day
Cardinal temperatures are minimum (y
min
), optimum (y
opt
), and maximum (y
max
) temperatures required for growth as well as
minimum (y
0
) and optimum (y
1
) temperatures required for hatching. CE
50
is the critical value leading to the emergence of
50% of the fry. See text for values of multidimensionnal parameters (L
i
; t
o
; t
ovi
o;k
).

are illustrated in Fig. 1: abundance and growth submodels depend on common parameters (m
o;k
and
s
o;k
, defined later), growth depends on the time of emergence and growth rate, these quantities
further depend on the water temperature, fish biomass is the cross-product of fish weight and
abundance, and the combination of growth and biomass parameters lead to production. Submode ls
are also connected to sub sets of measured variables: Fish weight (W
o;j;f
) is predicted from fish length
(L
o;j;f
), water temperature (y
w
t
) is predicted from air tempe rature (y
a
t
), and abundance is related to
removal sampling catch (C
o;i;j
) plus stream section area (A). The model is structured into five levels :
campaign (o 2f1; ...; Og), day (t 2f1; ...; Tg), length class (i 2f1; ...; Ig), removal (j 2f1; ...; J
o
g),
plus an additional level, cohort (k 2f1; ...; Kg), which is defined later. The temperature submodel is
dealt with in Appendix A (Supporting Infomation), remai ning submodels are successively presented
below.
Table 2 Distributions of free parameters (priors).
Parameter Notation Prior Unit
Abundance
Abundance l Gamma(0.001,0.001) per m
2
Cohort proportion t
0
k
Uniform(0,1) –
Slope a Normal(0,1000) per mm
Intercept b Normal(0,1000) –
Precision 1=s
2
l
Gamma(0.001,0.001) per m
2
Precision 1=s
2
t
Gamma(0.001,0.001) –
Precision 1=s
2
a
Gamma(0.001,0.001) per mm
Precision 1=s
2
b
Gamma(0.001,0.001) –
Growth
Optimal growth rate G
opt
Lognormal(7.25, 0.30) per day
Asymptotic length L
N
Lognormal(6.23,0.30 ) mm
Length on t
em
1;1

m
0
k
jL
1
Eq. (9) mm
Precision 1=s
2
1
Gamma(0.001,0.001) mm
Precision 1=s
2
0
Gamma(0.001,0.001) mm
Precision 1=s
2
m
Gamma(0.001,0.001) mm
Temperature
Maximum water temperature a
y
Normal(0,1000) 1C
Inflection b
y
Normal(0,1000) 1C
Inflection g
y
Normal(0,1000) per 1C
Minimum water temperature m
y
Normal(0,1000) 1C
Precision 1=s
2
y
Gamma(0.001,0.001) 1C
Weight
Allometric Z Gamma(0.001,0.001) g/L
Allometric z Normal(0,1000) –
Precision 1=s
2
Z
Gamma(0.001,0.001) g/L
Precision 1=s
2
z
Gamma(0.001,0.001) –
Precision 1=s
2
W
Gamma(0.001,0.001) –
Parameters are shape and rate for gamma distributions, expectation and variance for normal and lognormal distributions,
and boundaries for uniform distributions. Units which are provided with precisions (e.g. 1=s
2
l
) are units of respective
standard deviations (e.g. s
l
). Standard deviations are related to random errors across campaigns (s
l
, s
t
, s
a
, s
b
, s
Z
, s
z
),
among individuals (s
N
, s
0
), and residual (s
m
, s
T
, s
W
).