Joint estimation of crown of thorns (Acanthaster planci) densities on the Great Barrier Reef

FMP calibration

First, for the FMP data we assumed, due to repeated evidence of comparability between transects and point counts (Samoilys & Carlos, 2000), that detectability would be the most important source of potential bias relative to the known CoTS population size. Therefore, bias B_FMP was estimated by dividing densities observed by the FMP team at each site by the average site-level detectability ($\overline{{\varphi }_j}$), checking for bias relative to the model-calibrated density: (12)${\displaystyle B_{\mathrm{FMP}}=\frac{{\mathrm{OBS}}_{\mathrm{FMP}}}{\overline{{\varphi }_j}}-{\rho }_j}$

with B_FMP being centered on zero taken as substantial evidence that detectability accounts for potential bias present in the FMP observations.

AMPTO calibration

While FMP data is broadly comparable to the mark-recapture data in form, the AMPTO CPUE data is a different measure of abundance, where the total number of CoTS observed is standardized over some unit of effort; in this case, the duration of the dive over which the counts were made. CPUE is known from fisheries science to be a notoriously inconsistent index of abundance (Hartley & Myers, 2001) that can either decline quickly (hyperdepletion) or remain high (hyperstability) as true abundance declines. Hyperstability is expected to occur where populations are highly clustered and handling time to catch or kill individuals dominates the time spent searching for them. A common model for CPUE data is: (13)${\displaystyle {\mathrm{CPUE}}_t=q{N}_t^B}$

where CPUE is proportional to true abundance (N_t) given a catchability coefficient (q) and scaling parameter (B). In general, both q and N_t are difficult to estimate; however, given the mark-recapture estimates from our finite survey area, catchability is given by detectability and true abundance is assumed known. Therefore, we estimated the relationship between log(CPUE) and true abundance as: (14)${\displaystyle \log \left({\mathrm{CPUE}}_{jk}\right)\sim N\left({\theta }_j,{\tau }_{\theta }\right)}$ (15)${\displaystyle {\theta }_{jk}=\log \left(\overline{{\varphi }_j}\right)+B\ast \log \left(N_j\right)}$ (16)${\displaystyle B\sim U\left(0.0,10\right)}$ (17)${\displaystyle {\tau }_{\theta }={\sigma }_{\theta }^{-2}}$ (18)${\displaystyle {\sigma }_{\theta }\sim U\left(0,1000\right).}$

This parameterization essentially defines a log-Normal relationship between true densities and CPUE, estimating parameters on a log–log scale.

Joint zero-inflated survey model

Because the FMP conducts a two-part monitoring program whereby entire reefs are coarsely surveyed to detect the presence of outbreaks using manta-tows and counts of CoTS outbreaks are made by smaller-scale underwater visual count (UVC) surveys, we developed a two-part mixture model that included explicit outbreak (occupancy) and count (abundance) components for each manta-tow section surveyed on each reef (Fig. 1D). We parameterized this mixture using a zero-inflated Poisson model (ZIP): (19)${\displaystyle y_{rs}∕\bar{\varphi }\sim \left\{\begin{array}{c}0\mathrm{with}\mathrm{probability}{\pi }_{rs}\hfill \\ \mathrm{Pois}\left({\lambda }_{rs}\right)\mathrm{with}\mathrm{probability}1-{\pi }_{rs}\hfill \end{array}\right\}}$

with the response being the observed count y_rs on each manta-tow segment (s) within each reef (r), calibrated by the average detectability ($\bar{\varphi }$) estimated from the mark-recapture model.

The first submodel component was a hierarchical zeros model to estimate probability of a CoTS outbreak—defined as three or more CoTS or feeding scars per manta-tow (Doherty et al., 2015)—occurring on any given segment: (20)${\displaystyle \mathrm{logit}\left({\pi }_{rs}\right)\sim N\left({\beta }_r,{\tau }_{{\beta }_r}\right)}$ (21)${\displaystyle {\beta }_r\sim N\left({\beta }_0+{\beta }_{1}DLI,{\tau }_{{\beta }_0}\right)}$ (22)${\displaystyle {\beta }_{0,1}\sim N\left(0.0,0.001\right)}$ (23)${\displaystyle {\tau }_{\beta },{\tau }_{{\beta }_r}={\sigma }_{\beta }^{-2},{\sigma }_{{\beta }_r}^{-2}}$ (24)${\displaystyle {\sigma }_{\beta }^{-2},{\sigma }_{{\beta }_r}^{-2}\sim U\left(0,1000\right).}$

The model included the distance of each reef from Lizard Island (DLI), reflecting the hypothesized source of CoTS outbreaks in the Cairns region (Pratchett et al., 2014).

Similarly, the count submodel was conceived hierarchically, with a covariate (κ₁) to account for potential differences between point counts and timed swim methods used within the FMP surveys: (25)${\displaystyle {\lambda }_{rs}\sim e^{N\left({\delta }_{rs},{\tau }_{\delta }\right)}}$ (26)${\displaystyle {\delta }_{rs}={\delta }_{r0}+{\kappa }_{1}TS}$ (27)${\displaystyle {\delta }_{r0}\sim N\left({\kappa }_0,{\tau }_{\kappa }\right)}$ (28)${\displaystyle {\tau }_{\delta },{\tau }_{\kappa }={\sigma }_{\delta }^{-2},{\sigma }_{\kappa }^{-2}}$ (29)${\displaystyle {\sigma }_{\delta }^{-2},{\sigma }_{\kappa }^{-2}\sim U\left(0,1000\right).}$

Because the AMPTO CPUE data was not collected from the same segments within a reef we elected to model individual CPUE records (l) within a reef as Poisson samples, using their corresponding reef-scale averages (β_r), after first calibrating using the detectability and CPUE scaling parameter (B) estimates from the mark-recapture model: (30)${\displaystyle e^{\frac{\log \left({\mathrm{CPUE}}_{rl}\right)-\log \left(\bar{\varphi }\right)}{B}}\sim \mathrm{Pois}\left({\lambda }_r\right)}$ (31)${\displaystyle {\lambda }_r\sim e^{{\delta }_{r0}}.}$

In this way, the AMPTO CPUE observations were considered informative of reef-scale average CoTS densities within the Cairns sector, adding information to that present in the FMP surveys.

All models were run using the Metroplis-Hastings algorithm for 10⁶ iterations, with a 900,000 burn in period, using the PyMC2 package (Patil, Huard & Fonnesbeck, 2010) for the Python programming language. Model convergence was assessed using Gelman–Rubin statistics from multiple model runs (Gelman & Rubin, 1992) and model fit was evaluated using Bayesian p-values (Brooks, Catchpole & Morgan, 2000), with scores lower than 0.025 and 0.975 providing substantial evidence for lack of model fit.