Bayesian models of age and growth in sharks : a thesis submitted in partial fulfilment of the requirements for the degree of Master of Information Sciences, School of Mathematical and Computational Sciences, Massey University, Auckland, New Zealand

Abstract

Marine ecosystems are under increasing pressure from factors such as overfishing, that have lead to reported declines in chondrichthyan populations across the globe. Approximately 80% of New Zealand chondrichthyian species have no specific management or monitoring, and a lack of species-specific information has led to uncertainty concerning population trends over time. Biological parameters such as growth rates are typically incorporated into fisheries growth models, the most common of which is the von Bertalanffy growth model (VBGM). The common VBGM is monophasic, and is therefore a continuous and monotone curve over all stages of growth. The VBGM has received some criticism concerning the assumption of monophasic growth. One critique is that a single curve fails to account for changes in energy reallocation and growth that occur during the transition between the juvenile and mature life stages of the shark. Conversely, a biphasic growth model may better model the patterns of both juvenile and mature sharks. The biphasic VBGM (BVBGM) has offered promising results when applied to chondrichthyans in previous growth studies. Estimating the ages of chondrichthyans is typically performed by trained readers who count the growth bands deposited within the vertebral centra. However, the reading of vertebrae is subjective, error-prone, and time consuming. While vertebral band pairs are the most commonly used structure for age estimation, there are various sources of bias and uncertainty inherent to the reading process. Deep learning models such as convolutional neural networks (CNNs) have demonstrated promise for the automated interpretation of bony fish otolith growth zones, though the application of CNNs for the automated reading of chondrichthyan vertebrae has yet to be explored. The principal goal of this thesis is to advance methodologies for estimating the growth parameters of sharks via the application of Bayesian models, thereby offering more robust management and conservation of chondrichthyes against various factors such as overfishing. We first explored the potential of biphasic growth models on five species of New Zealand chondrichthyes: Centrophorus squamosus; Isurus oxyrinchus; Lamna nasus; Mustelus lenticulatus; and Prionace glauca. We compared two monophasic and two biphasic growth models using the Pareto-smoothed importance sampling approximation of leave-one-out cross-validation (PSIS-LOO) metric. Biphasic models appeared to provide superior fit for both males and females in the majority of cases, and we were able to improve upon prior examples from the literature where parameter estimates were noted to be biased or poor. Our overall results demonstrated that the popular monophasic VBGM should not be chosen a priori as the only candidate model to describe the growth of chondrichthyans. Instead, we should consider multiple alternative growth models, as informed by statistical evidence and domain expertise. We next explored the feasibility of automating the age estimation of chondrichthyans by training CNNs on an image dataset of Isurus oxyrinchus vertebrae. We evaluated three Bayesian deep learning methods for uncertainty quantification: DeepEnsembling, mixture of Laplace approximations (MoLA), and multi-stochastic weight average Gaussian (MultiSWAG) in terms of predictive power and model calibration. We found that MultiSWAG offered marginally superior predictive performance and model calibration relative to DeepEnsembling and MoLA. Moreover, predictions produced by MultiSWAG typically closely matched the estimates provided by human readers, though the performance of these deep learning models tended to degrade for older age classes. We argue that our results demonstrate promise for emulating trained readers, leading to potential efficiency gains and cost savings. However, we note that there is a lack of evidence that our models are directly counting the bands of the vertebrae, and that further refinement of our CNNs may be required. Our findings have demonstrated the ability of Bayesian methods to perform principled uncertainty quantification and parameter estimation within the context of age and growth modelling. Our Bayesian growth models were able to quantify the epistemic uncertainty of our parameter estimates, offering more robust estimation of growth parameters and superior model fit. Additionally, we were able to incorporate prior information into our growth models, as informed by the available literature and domain expertise. Likewise, the application of Bayesian deep learning facilitated the quantification of epistemic and aleotoric uncertainty for our age estimates, while also offering superior predictive performance and well-calibrated prediction intervals. We showed that Bayesian CNNs could be used to efficiently automate the interpretation of vertebral growth bands for the purposes of age estimation. This study has contributed to the research of New Zealand sharks, marine conservation, and fisheries management by improving methods used to measure and interpret growth parameters. We hope it contributes to the management and persistence of these species for future generations.