Among-site variability in the stochastic dynamics of East African coral reefs

Data collection

Surveys of 20 spatially distinct reefs in Kenya and Tanzania (Table 1 and Fig. A1) were conducted annually during the period 1991–2013 (generally in November or December prior to 1998, but January or February from 1998 onwards). Sampling dates are shown in Fig. A6–A35. Reefs in the north were typically fringing reefs, 100 m–2,000 m from the shore, while those in the south were typically smaller and more isolated patch reefs, further from the shore (McClanahan & Arthur, 2001). We categorized reefs as either fished or unfished, although there was substantial heterogeneity within these categories, because some fished reefs were community management areas with reduced harvesting intensity (Cinner & McClanahan, 2015), and some unfished reefs had only recently been designated as reserves. Of the 20 reefs, 10 were divided into two sites separated by 20 m–100 m, while the remaining 10 reefs comprised only one site. The selection of sites represents available data rather than a random sample from all the locations at which coral reefs are present in the geographical area (and all of the longest time series are from Kenyan fringing reefs). Below, we use a random term to model the variation in dynamics among these sites. Thus, when we refer to ‘a randomly-chosen site’ we mean ‘a site drawn at random from the distribution describing variability in dynamics among our sites’, which is not the same as a site drawn at random from all coral reef locations in the region.

Each of the 30 sites was visited at least twice (data from sites visited once were omitted), with a maximum of 20 visits. In total, there were 289 site visits. In each, a version of line-intercept sampling (Kaiser, 1983; McClanahan, Muthiga & Mangi, 2001) was used to estimate reef composition. In total, 2,665 linear transects were sampled across all sites and years, with between five and 18 transects (median 9) at each site in a single year. Transects were randomly placed between two points 10 m apart, but as the transect line was draped over the contours of the substrate, the measured lengths varied between 10 m and 15 m. Cover of benthic taxa was recorded as the sum of draped lengths of intersections of patches of each taxon with the line, divided by the total draped length of the line. Intersections with length less than 3 cm were not recorded. Taxa were identified to species or genus level, but for this study cover was grouped into three broad categories: hard coral (scleractinians and Millepora), macroalgae and other (algal turf, calcareous and coralline algae, soft corals and sponges). Millepora were included in the coral category because they are important calcareous framework builders and may have calcification rates similar to those of branching corals (Lewis, 1989). Sand and seagrass were recorded, but excluded from our analysis, which focused on hard substrate. The dynamics of a subset of these data were analyzed using different methods in Żychaluk et al. (2012).

Data processing

The three cover values form a three-part composition, a set of three positive numbers whose sum is 1 (Aitchison, 1986, Definition 2.1, p. 26). Standard multivariate statistical techniques are not appropriate for untransformed compositional data, due to the absence of an interpretable covariance structure and the difficulties with parametric modelling (Aitchison, 1986, chapter 3). To avoid these difficulties, the proportional cover data were transformed to orthogonal, unconstrained, isometric log-ratio (ilr) coordinates (Egozcue et al., 2003). It is of course true that the model presented below for transformed data has an analogous model for untransformed data (Mateu-Figueras, Pawlowsky-Glahn & Egozcue, 2011). However, working with transformed data allows us to use familiar methods.

The transformed data at site i, transect j, time t were represented by the vector y_i,j,t = [y_1,i,j,t, y_2,i,j,t]^T, in which the first coordinate y_1,i,j,t was proportional to the natural log of the ratio of algae to coral, and the second coordinate y_2,i,j,t was proportional to the natural log of the ratio of other to the geometric mean of algae and coral (Section A1). The T denotes transpose: throughout, we work with column vectors. Note that both raw and transformed data are dimensionless.

The model

The true value x_i,t = [x_1,i,t, x_2,i,t]^T of the isometric log-ratio transformation of cover of hard corals, macroalgae and other at site i at time t was modelled by a vector autoregressive process of order 1 (i.e., a process in which the cover in a given year depends only on cover in the previous year), an approach used in other recent models of coral reef dynamics (Cooper, Spencer & Bruno, 2015; Gross & Edmunds, 2015). Unlike previous models, we include a random term representing among-site variation, and explicit treatment of measurement error (making this a state-space model). The full model is (1)${\displaystyle \begin{array}{c}{\mathbf{x}}_{i,t+1}=\mathbf{a}+{\alpha }_i+\mathbf{B}{\mathbf{x}}_{i,t}+{\epsilon }_{i,t},\hfill \\ {\alpha }_i\sim \mathcal{N}\left(\mathbf{0},\mathbf{Z}\right),\hfill \\ {\epsilon }_{i,t}\sim \mathcal{N}\left(\mathbf{0},\Sigma \right),\hfill \\ {\mathbf{y}}_{i,j,t}\sim t_2\left({\mathbf{x}}_{i,t},\mathbf{H},\nu \right),\hfill \end{array}}$ where all variables and parameters are dimensionless.

The column vector a represents the among-site mean proportional changes in x_i,t evaluated at x_i,t = 0. The column vector α_i represents the amount by which these proportional changes for the ith site differ from the among-site mean, and is assumed to be drawn from a multivariate normal distribution with mean vector 0 and 2 × 2 covariance matrix Z. The 2 × 2 matrix B represents the effects of x_i,t on the proportional changes, and can be thought of as summarizing intra- and inter-component interactions such as competition. The column vector ε_i,t represents random temporal variation, and is assumed to be drawn from a multivariate normal distribution with mean vector 0 and covariance matrix Σ. We assume that there is no temporal or spatial autocorrelation in ε, and that ε is independent of the among-site variation α.

The observed transformed compositions y_i,j,t vary around the corresponding true compositions x_i,t due to both small-scale spatial variation in true composition among transects within a site, and measurement error in estimating composition from a transect. We cannot easily separate these sources of variation because transects were located at different positions in each year, and there were no repeat measurements within transects. Observed log-ratio transformed cover y_i,j,t in the jth transect of site i at time t was assumed to be drawn from a bivariate t distribution (denoted by t₂) with location vector equal to the corresponding x_i,t, and unknown scale matrix H and degrees of freedom ν, so that y_i,j,t has mean vector x_i,t if ν > 1, and covariance matrix νH∕(ν − 2) if ν > 2 (Lange, Little & Taylor, 1989). The bivariate t distribution can be interpreted as a mixture of bivariate normal distributions whose covariance matrices are the same up to a scalar multiple (Lange, Little & Taylor, 1989), and therefore allows a simple form of among-site or temporal variation in the distribution of measurement error or small-scale spatial variation, whose importance increases as the degrees of freedom decrease. Preliminary analyses suggested that it was important to allow this variation, because the model in Eq. (1) fitted the data much better than a model with a bivariate normal distribution for y_i,j,t (Section A3).

We make the important simplifying assumptions that B is the same for all sites, and that the causes of among-site and temporal variation are not of interest. A separate B for each site, or even a hierarchical model for B, would be difficult to estimate from the amount of data we have. It might be possible to explain some of the random temporal variation using temporally-varying environmental covariates such as sea surface temperature, and some of the among-site variation using temporally constant covariates such as management strategies (Cooper, Spencer & Bruno, 2015). However, it is not necessary to do so in order to answer the questions listed at the end of the introduction, and keeping the model as simple as possible is important because parameter estimation is quite difficult. Furthermore, some of the relevant environmental variables may be associated with management strategies, making it difficult to separate the effects of environmental variation and management. For example, although some water quality variables were not strongly associated with protection status (Carreiro-Silva & McClanahan, 2012), unfished reefs were designated as protected areas due to their relatively good condition and are generally found in deeper lagoons with lower and more stable water temperatures than fished reefs (TR McClanahan, pers. comm., 2015).

To understand the features of dynamics common to all sites, we plotted the back-transformations from ilr coordinates to the simplex of the overall intercept parameter a and the columns a₁ and a₂ of a matrix A, which is related to B and describes the effects of current reef composition on the change in reef composition from year to year (Cooper, Spencer & Bruno, 2015). We plotted A rather than B because it leads to a simpler visualization of effects (Section A4). For example, a point lying to the left of the line representing equal proportions of coral and algae (the 1:1 coral-algae isoproportion line) corresponds to a parameter tending to increase coral relative to algae.

Parameter estimation

We estimated all model parameters and checked model performance using Bayesian methods implemented in the Stan programming language (Stan Development Team, 2015), as described in Section A5. Stan uses the No-U-Turn Sampler, a version of Hamiltonian Monte Carlo, which can converge much faster than random-walk Metropolis sampling when parameters are correlated (Hoffman & Gelman, 2014). For most results, we report posterior means and 95% highest posterior density (HPD) intervals (Hyndman, 1996), calculated in R (R Core Team, 2015). We showed using simulations that we were able to estimate parameters with reasonable accuracy, and that our estimated credible intervals had close to the specified coverage (Section A5.2 and Fig. A3). In order to investigate the effects of number of time points per site on uncertainty in site-specific parameters, we plotted the sample generalized variance of α_i (the determinant of the sample covariance matrix over Monte Carlo iterations) against number of time points. It is worth emphasizing that all parameters other than α_i and x_i,t are common to all reefs, and thus estimated from all 289 site visits.

Long-term behaviour

In the long term (as t → ∞), the true transformed composition x^∗ of a randomly-chosen site will converge to a stationary distribution, provided that all the eigenvalues of B lie inside the unit circle in the complex plane (e.g. Lütkepohl, 1993, p. 10). If the eigenvalues of B are complex, the system will oscillate as it approaches the stationary distribution. Details of long-term behaviour are in Section A6.

This stationary distribution is the multivariate normal vector (2)${\displaystyle {\mathbf{x}}^{\ast }\sim \mathcal{N}\left({\mu }^{\ast },{\Sigma }^{\ast }+{\mathbf{Z}}^{\ast }\right),}$ whose stationary mean μ^∗ depends on B and a, and whose stationary covariance is the sum of the stationary within-site covariance Σ^∗ (which depends on B and Σ) and the stationary among-site covariance Z^∗ (which depends on B and Z).

For a fixed site i, the value of α_i is fixed and the stationary distribution is given by (3)${\displaystyle {\mathbf{x}}_i^{\ast }\sim \mathcal{N}\left({\mu }_i^{\ast },{\Sigma }^{\ast }\right),}$ whose stationary mean ${\mu }_i^{\ast }$ depends on B, a and α_i, and whose stationary covariance matrix is Σ^∗. Note that B, which describes intra- and inter-component interactions on an annual time scale, affects all the parameters of both stationary distributions, and therefore affects both within- and among-site variability in the long term. Also, the back-transformation of the stationary mean μ^∗ of the transformed composition, rather than the arithmetic mean vector of the untransformed composition, is the appropriate measure of the centre of the stationary distribution (Aitchison, 1989).

How important is among-site variability?

The covariance matrix of the stationary distribution for a randomly-chosen site (Eq. (2)) contains contributions from both among- and within-site variability. To quantify the contributions from these two sources, we calculated (4)${\displaystyle \rho ={\left(\frac{\left|{\Sigma }^{\ast }\right|}{|{\Sigma }^{\ast }+{\mathbf{Z}}^{\ast }|}\right)}^{1∕2},}$ (Section A7), which is the ratio of volumes of two unit ellipsoids of concentration (Kenward, 1979), the numerator corresponding to the stationary distribution in the absence of among-site variation (or for a fixed site, as in Eq. (3)), and the denominator to the full stationary distribution of transformed reef composition in the region. The volume of each ellipsoid of concentration is a measure of the dispersion of the corresponding distribution. Thus ρ provides an indication of how much of the total variability would remain if all among-site variability was removed. A similar statistic was used by Ives et al. (2003) to measure the contribution of species interactions to stationary variability.

How much variability is there among sites in the probability of low coral cover?

For a given coral cover threshold κ, we define q_κ,i as the long-term probability that site i has coral cover less than or equal to κ. This can be interpreted either as the proportion of time for which the site will have coral cover less than or equal to κ in the long term, or as the probability that the site will have coral cover less than or equal to κ at a random time, in the long term. We set κ = 0.1, which has been suggested as a threshold for a positive net carbonate budget, based on simulation models and data from Caribbean reefs (Kennedy et al., 2013; Perry et al., 2013; Roff, Zhao & Mumby, 2015). We calculated q_0.1,i for each site numerically (Section A8). In order to determine whether differences in q_0.1,i were related to current coral cover, we plotted q_0.1,i against the corresponding sample mean coral cover for each site, over all transects and years. In order to determine whether differences in q_0.1,i had obvious explanations, we distinguished between fished and unfished reefs, and patch and fringing reefs.

In order to determine how the amount of among-site variability affects the strength of the relationship between q_0.1,i and sample mean coral cover, we plotted this relationship for simulated data sets with different amounts of among-site variability (Section A9). In order to determine whether there was strong spatial pattern in the probability of low coral cover, we calculated spline correlograms (Bjørnstad & Falck, 2001) for a sample from the posterior distribution of q_0.1,i (Section A10).

Which model parameters have the largest effects on the probability of low coral cover?

For a given coral cover threshold κ, we define q_κ as the long-term probability that a randomly-chosen site has coral cover less than or equal to κ. This is equal to the expected long-term probability that coral cover is less than or equal to κ over the region, and can be calculated numerically (Section A8). To find the parameters with the largest effects on q_κ, we calculated its derivatives with respect to each model parameter. As above, we concentrated on κ = 0.1. However, we also compared results from κ = 0.05 and κ = 0.20. The probability q_κ is a function of 12 parameters: all four elements of B; both elements of a; elements σ₁₁, σ₂₁ and σ₂₂ of Σ; and elements ζ₁₁, ζ₂₁ and ζ₂₂ of Z. The negative of the gradient vector of derivatives of q_κ with respect to these parameters describes the direction of movement through parameter space in which the probability of low coral cover will be reduced most rapidly, and the elements of this vector with the largest magnitudes correspond to the parameters to which q_κ is most sensitive. To understand why q_κ responds to each model parameter, note that q_κ depends on the parameters μ^∗, Σ^∗ and Z^∗ of the stationary distribution (Eq. (2)), which are in turn affected by the model parameters. We therefore used the chain rule for matrix derivatives (Magnus & Neudecker, 2007, p.108) to break down the derivatives into effects of μ^∗, Σ^∗ and Z^∗ on q_κ, and effects of model parameters on μ^∗, Σ^∗ and Z^∗ (Section A11). We also calculated elasticities of q_κ with respect to each parameter, which measure the rate of relative change in q_κ with respect to relative change in the parameter (Section A12).

How informative is a snapshot about long-term site properties?

In a stochastic system, how much can a “snapshot” survey at a single point in time tell us about the long-term behaviour of the system? For example, are differences among sites that appear to be in good and bad condition likely to be maintained in the long term? To make this question more precise, suppose that we draw a site at random from the region, and at one point in time, draw the true state of the site at random from the stationary distribution for the site. This scenario matches Diamond’s definition of “natural snapshot experiments” as “comparisons of communities assumed to have reached a quasi-steady state” (Diamond, 1986). For simplicity, we assume that we can estimate the true state accurately (for example, by taking a large number of transects). To quantify how informative this is about the long term properties of the site, we computed the correlation coefficients between corresponding components of the true state at a given site at a given time and of stationary mean for that site (Section A13). If these correlations are high, then a snapshot will be informative about long term properties.

Overall dynamics

At all sites, the back-transformed posterior mean true states from the model (e.g., Fig. 1, grey lines) closely tracked the centres of the distributions of cover estimates from individual transects, although there was substantial among-transect variability at a given site in a given year (e.g., Fig. 1, circles). Figure 1 shows two examples, and time series for all sites are plotted in Figs. A6–A35. There were also substantial differences in patterns of temporal change among sites. For example, Kanamai1 (Figs. 1A–1C), a fished site, had consistently low algal cover and no dramatic changes in cover of any component. In contrast, Mombasa1 (Figs. 1D–1F), an unfished site, had a sudden decrease in coral cover in 1998, and algal cover was high from 2007 onwards. As a result, Mombasa1 was unusual in that the current estimate of true algal cover was well above the stationary mean estimate (Fig. 1E: black circle at end of time series). For most other sites, current estimated true cover was close to the stationary mean (Figs. A6–A35, black circles at ends of time series). The uncertainty in true states (Fig. 1, grey polygons represent 95% highest posterior density (HPD) credible bands) was higher during intervals with missing observations (e.g., 2008 in Fig. 1). Uncertainty in true states (grey polygons) and stationary means (black bars at end of time series) was high for sites with few observations (e.g., Bongoyo1, Fig. A6). This is expected because these quantities are estimated at the site level. Similarly, the generalized variance of α_i was high for sites with only two or three observations, but declined quickly as the number of observations increased (Fig. A36).

The overall intercept parameter a (Fig. 2, green), which describes the dynamics of reef composition at the origin (where each component is equally abundant) was consistent with the observed low macroalgal cover in the region (e.g., Figs. 1B and 1E). The back-transformation of a lay close to the coral-other edge of the ternary plot, and slightly above the 1:1 coral-other isoproportion line. It therefore represented a strong year-to-year decrease in algae, and a slight increase in other relative to coral, at the origin.

Current reef composition acts on year-to-year change in composition (through matrix A) so as to maintain fairly stable reef composition, with an increase in each transformed component tending to be counteracted in the following year-to-year change. Each column of A represents the effect of a unit increase in a component of transformed composition on the year-to-year change in transformed composition (Section A4). These effects can be back-transformed to compositions and plotted on a ternary diagram (Fig. 2). The effects can be interpreted by looking at the position of the resulting point relative to three 1:1 isoproportion lines (Fig. 2, grey lines), along each of which the relative proportions of two components do not change: the 1:1 coral-algae isoproportion line, representing no change in coral relative to algae (from the the middle of the coral-algae edge to the other vertex); the 1:1 algae-other isoproportion line, representing no change in algae relative to other (from the middle of the algae-other edge to the coral vertex); and the 1:1 coral-other isoproportion line, representing no change in coral relative to other (from the middle of the coral-other edge to the algae vertex).

The back-transformation of the first column a₁ of A, which represents the effects of the transformed ratio of algae to coral on year-to-year change in composition, lay to the left of the 1:1 coral-algae isoproportion line, above the 1:1 other-algae isoproportion line, and below the 1:1 coral-other isoproportion line (Fig. 2, orange). Thus, increases in algae relative to coral resulted in decreases in algae relative to coral and other, and increases in coral relative to other, in the following year. The back-transformation of the second column a₂ of A, which represents the effects of the transformed ratio of other to algae and coral on year-to-year change in composition, lay on the 1:1 coral-algae isoproportion line, below the 1:1 other-algae isoproportion line, and below the 1:1 coral-other isoproportion line (Fig. 2, blue). Thus, increases in other relative to algae and coral resulted in little change in the ratio of coral to algae, but decreases in other relative to both coral and algae. Consistent with this evidence of a tendency towards stability in year-to-year dynamics, every set of parameters in the Monte Carlo sample led to a stationary distribution, since both eigenvalues of B lay inside the unit circle in the complex plane (Section A14). The magnitudes of these eigenvalues were smaller than those for a similar model for the Great Barrier Reef (Cooper, Spencer & Bruno, 2015), indicating more rapid approach to the stationary distribution. There was some evidence for complex eigenvalues of B, leading to rapidly-decaying oscillations in both components of transformed reef composition on approach to this distribution. This contrasts with the Great Barrier Reef, where there was no evidence for oscillations (Cooper, Spencer & Bruno, 2015).

In biological terms, the above results mean that every site would, if current environmental conditions were maintained in the long term, approach a predictable probability distribution of composition, whatever the initial conditions. However, as described below, these distributions differed substantially among sites.

How important is among-site variability?

There was substantial among-site variability in the locations of stationary means (Fig. 3, dispersion of points). Stationary mean algal cover was always low, but there was a wide range of stationary mean coral cover. Although our primary focus is not on the causes of among-site variability, there was a tendency for most of the reefs with highest stationary mean coral cover to be patch reefs (Figs. 3A and 3C). In the light of these observations, we experimented with a model in which reef type was included as an explanatory variable. Although the estimated effects of patch reef type were consistent with lower long-term probabilities of coral cover ≤0.1, including reef type did not improve the expected predictive accuracy of the model (Chong, 2016), probably because only 482 out of 2,665 transects were from patch reefs, and all but one patch reefs had only very short time series (Table 1). The stationary means did not clearly separate by management (Figs. 3A and 3B versus Figs. 3C and 3D). The long-term temporal variability around the stationary means was also substantial (Fig. 3, green lines). The ρ statistic (Eq. (4)), which quantifies the posterior mean contribution of within-site variability to the total stationary variability in reef composition in the region, was 0.29 (95% HPD interval (0.20, 0.39)), or approximately one-third. Thus, while within-site temporal variability around the stationary mean was not negligible, among-site variability in the stationary mean was more important in the long term. As noted above, uncertainty in the location of the stationary means (Fig. 3, grey dashed lines) was much higher for reefs with few observations than for reefs with many observations. Nevertheless, most parameters of the model are not reef-specific, and data from reefs with few observations contribute to the estimation of these.

For all three components of variability (within-site, among-site, and measurement error/small-scale spatial variability), variation in algal cover was larger than variation in coral or other. This can be seen in the shapes of the back-transformed unit ellipsoids of concentration (Fig. 4: within-site, green; among-site, orange; measurement error and small-scale spatial variability, blue) which were all elongated to some extent along the 1:1 coral-other isoproportion line. This was similar to, but less extreme than, the pattern observed in the Great Barrier Reef (Cooper, Spencer & Bruno, 2015). The among-site ellipsoid almost entirely enclosed the within-site ellipsoid, consistent with the estimate above that within-site variability contributed only around one-third of the total stationary variability in reef composition. The large estimated measurement error/small-scale spatial variability component was consistent with the substantial observed variability in cover among transects at any given site and time (Fig. 1, circles and Figs. A6–A35, circles). The low estimated degrees of freedom ν for the bivariate t distribution of measurement error/small-scale spatial variability (posterior mean 2.99, 95% HPD interval (2.64, 3.35)) suggested that some aspect of the process leading to variation in measured composition among transects at a given site was varying substantially over space or time, although we cannot determine the mechanism.

How much variability is there among sites in the probability of low coral cover?

There was also substantial among-site variability in the probability of low coral cover. For a randomly-chosen site, the posterior mean probability of coral cover less than or equal to 0.1 (q_0.1) in the long term was 0.12 (95% credible interval (0.04, 0.21)). The corresponding site-specific probabilities q_0.1,i varied from 8 × 10⁻⁵ to 0.52 but were low for most sites, with a strong negative relationship between probability of low coral cover and observed mean coral cover (Fig. 5).

There was no clear distinction in the probability of low coral cover between fished and unfished reefs (Fig. 5, open symbols fished, filled symbols unfished). However, probability of low coral cover appeared to be systematically lower on patch reefs, which were mainly in Tanzania (Figs. 5 and A1, circles: median of posterior means 2 × 10⁻³, first quartile 4 × 10⁻⁴, third quartile 0.04) than on fringing reefs (Figs. 5 and A1, triangles: median of posterior means 0.08, first quartile 0.04, third quartile 0.11). One site (Ras Iwatine) had a much higher probability of low coral cover than all others, and is one of two relatively eutrophic sites (the other being Kanamai), probably due to pollution (Carreiro-Silva & McClanahan, 2012).

Although a negative relationship between the probability of low coral cover and observed mean coral cover (Fig. 5) was expected, the strength of this relationship depends on the amount of among-site variability. Using simulated data, we showed that when there was much less among-site variability than estimated from the real data, this relationship was very weak, and the probability of low coral cover was small for all sites (Fig. A39). As the amount of among-site variability increased, the probability of low coral cover increased quickly for sites with low mean coral cover, but remained close to zero for sites with high mean coral cover.

There was little evidence for strong spatial autocorrelation in the probability of low coral cover, because the 95% envelope for the spline correlogram included zero for all distances other than 261 km to 322 km (Fig. A40). The general lack of strong spatial autocorrelation reflects the substantial variation in probability of coral cover less than or equal to 0.1 (q_0.1,i) among nearby sites, while the possibility of negative spatial autocorrelation at scales of around 300 km may reflect the generally low values of q_0.1,i for Tanzanian patch reefs, separated from sites in the north of the study area with generally higher q_0.1,i by approximately 300km (Fig. A1).

Which model parameters have the largest effects on the probability of low coral cover?

Both among-site variability and internal dynamics, particularly of other relative to algae and coral (component 2), were important in determining the probability q_0.1 of coral cover ≤0.1 in the region. Figure 6 shows the direction in parameter space along which the probability of low coral cover will reduce most rapidly (the estimated gradient vector of q_0.1 with respect to all the model parameters). The four parameters to which q_0.1 was most sensitive were (in descending order: Fig. 6) ζ₂₁ (among-site covariance between transformed components 1 and 2), b₂₂ (effect of component 2 on next year’s component 2), ζ₂₂ (among-site variance of component 2), and b₁₂ (effect of component 2 on next year’s component 1). Although there was substantial variability among Monte Carlo iterations in the values of these derivatives, the rank order of magnitudes was fairly consistent (Fig. A41). All four most important parameters had positive effects on q_0.1 (Fig. 6), so reducing these parameters will reduce q_0.1. The effects of within-site temporal variability on the probability of low coral cover were relatively unimportant (Fig. 6, derivatives of q_0.1 with respect to σ₁₁, σ₂₁ and σ₂₂ all had posterior means close to zero). The signs of the effects of each parameter on q_0.1, sensitivities for coral cover thresholds 0.05 and 0.2, and elasticities, are discussed further in Sections A15 and A16.

How informative is a snapshot about long-term site properties?

For both components of transformed composition, a snapshot of reef composition at a single time on a randomly-chosen site will be informative about the stationary mean (correlations between true value at a given time and stationary mean: component 1 posterior mean 0.84, 95% HPD interval (0.75, 0.91); component 2 posterior mean 0.82, 95% HPD interval (0.73, 0.90)). This is consistent with the negative relationship between long-term probability of coral cover ≤0.1 and observed mean coral cover (Fig. 5). Thus, while long-term monitoring of East African coral reefs is important for other reasons, it should be possible to identify those with high conservation value (in terms of benthic composition) from a single survey.