**Aim** – Estimate amounts of nutrients released in the enlightened zone of oceans by whole cetacean communities via their waste products (feces, urine)

**How ?** **Bioenergetic model** estimating prey ingestion, nutrient ingestion and nutrient release

$n$ = nutrient

<figure><img src="image_1.png" /></figure>

$c$ = cetacean population

<figure><img src="image_2.png" /></figure>

$$R_{n,c} = \beta_c \times 293,1 \times BM_c \times \frac{1}{AE \times E_c} \times x_{n,c} \times r_n \times A_c \times \text{ndays}_c$$

<table><tr><td>β<sub>c</sub></td><td>metabolic parameter</td><td>estimated by <strong>indirect method</strong> for 11 out of 38 species</td><td>Individual variation Random error</td></tr><tr><td>BM<sub>c</sub></td><td>body mass</td><td>estimated for 34 out of 38 species from <strong>regression equations</strong></td><td>Individual variation Random error</td></tr><tr><td>AE</td><td>Assimilation rate of energy</td><td><em>typically</em> 0,8, <em>rarely measured</em> on cetaceans</td><td>Individual variation Epistemic uncertainty</td></tr><tr><td>E<sub>c</sub></td><td>mean energy in diet</td><td>estimated from mean diet <strong>inferred from literature</strong></td><td>Individual variation Random error</td></tr><tr><td>x<sub>n,c</sub></td><td>mean nutrient</td><td>+ <strong>dataset of nutritional content of prey</strong></td><td>Epistemic uncertainty</td></tr><tr><td>r<sub>n</sub></td><td>nutrient release rate</td><td><strong>never measured</strong> on cetaceans, for all nutrients</td><td>Individual variation Epistemic uncertainty</td></tr><tr><td>A<sub>c</sub></td><td>population abundance</td><td>estimated from <strong>distance sampling surveys</strong></td><td>Random error</td></tr><tr><td>ndays<sub>c</sub></td><td>nb of day spent in area</td><td>ranges <strong>inferred from literature</strong>, from few individuals of few species</td><td>Individual variation Epistemic uncertainty</td></tr></table>

**Problem** – Several sources of uncertainty (random error, natural variation, epidemic uncertainty), few existing data and dataset available

## Objectives

1. Account for uncertainty in model inputs to obtain interval estimates of output

2. Identify influential model inputs

# Dealing with several sources of uncertainty in modelling of community-level processes

Lola Gilbert<fnref n="1" />,<fnref n="2" />, Jérôme Spitz<fnref n="1" />,<fnref n="2" />, Tiphaine Jeanniard-du-Dot<fnref n="1" />, Nicolas Bousquet<fnref n="3" />, Matthieu Authier<fnref n="1" />,<fnref n="2" />

<figure><img src="image_3.png" /></figure>

## Method

### Monte-Carlo simulations

A **distribution** and **variation factor** was assigned to each input, based on literature or common practice:

$$ \beta_c, BM_c, AE \rightarrow \text{Normal, e.g.} \quad \beta_c \sim N(\bar{\beta}_c, \frac{(\bar{\beta}_c+0,5) - (\bar{\beta}_c-0,5)}{4}) $$

$$ r_c, \text{ndays}_c \rightarrow \text{Uniform, e.g.} \quad r_{Fe} \sim U(0.85, 0.95) $$

$$ A_c \rightarrow A_c \sim \log N(\log(\frac{A_c}{\sqrt{1+CV_{A_c}^2}}), \sqrt{\log(1+CV_{A_c}^2)}) $$

### Kernel-based bootstrap

Kernel-based **procedure** to estimate each prey group composition cumulative density function, sample from it and obtain **independant samples**

Composition of prey
Samples simulated from dataset

$$ \frac{\text{\% of prey group in diet (constant)}}{E_c, r_{n,c}} \times \left( \begin{array}{c} \text{prey group 1} \\ \vdots \\ \text{prey group 10} \end{array} \right) $$

### Global sensitivity analysis

* **Global** i.e. effect of an input on the output is estimated with all other inputs varying

* **Variance decomposition** to attribute uncertainty in output to uncertainty in inputs

* **Sobol first-order** and **total** (i.e. including interactions with other inputs) indices with R sensitivity package

<figure><img src="image_4.png" /></figure>

## Results

### Uncertainty in the output

<figure><img src="image_5.png" /></figure>

Ranges of estimates allow comparison between areas

### Sensitivity to the inputs

<figure><img src="image_6.png" /></figure>

2 influential inputs

≠ between macro & micro-nutrients

* Accounting for uncertainty has a **computational cost** (nb of simulations)

* Kernel-based bootstrap → still a dependence on the **empirical basis** (dataset and nb of samples per prey group)

* Abundance is the **most influential input**, yet it is already resulting of a well proofed **complex statistical design and analysis** → hard to be more precise

* Most never measured inputs are only slightly influential

* **Ranges of variation** driven by **researcher's degree of freedom** setting up input variation parameters, but does it accurately reflect the process of interest ?

## Conclusions

1. **Monte-Carlo simulations** combined with **kernel-based bootstrap** allowed to formalize assumptions and obtain interval estimates of a community-level process associated to many sources of uncertainty

2. **Global sensitivity analysis** to get a clear understanding of **how the output is affected** by uncertainties in the inputs

## References

Cariboni, J., Gatelli, D., Liska, R., & Saltelli, A. (2007). The role of sensitivity analysis in ecological modelling. *Ecological Modelling*, 203(1-2), 167-182. https://doi.org/10.1016/j.ecolmodel.2005.10.045

Saltelli, A., Aleksankina, K., Becker, W., Fennell, P., Ferretti, F., Holst, N., Li, S., & Wu, Q. (2019). Why so many published sensitivity analyses are false: A systematic review of sensitivity analysis practices. *Environmental Modelling and Software*, 114(January), 29-39. https://doi.org/10.1016/j.envsoft.2019.01.012

Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., & Tarantola, S. (2010). Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. *Computer Physics Communications*, 181(2), 259-270. https://doi.org/10.1016/j.cpc.2009.09.018

Sobol, I. M. (2001). Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. *Mathematics and Computers in Simulation*, 55(1-3), 271-280. https://doi.org/10.1016/S0378-4754(00)00270-6

<footnote n="1">¹Centre d'Etudes Biologiques de Chizé-La Rochelle, UMR7372 Université de La Rochelle - CNRS, La Rochelle, France</footnote>

<footnote n="2">²Observatoire Pélagis, UAR 3462 Université de La Rochelle - CNRS, La Rochelle, France</footnote>

<footnote n="3">³Sorbonne Université, Laboratoire de Probabilités, Statistiques et Modélisation, Paris, France</footnote>