# 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 (1 min read)

### 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.

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