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

## 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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