Quantifying parameter uncertainty in a coral reef model using Metropolis-Coupled Markov Chain Monte Carlo

Coral reefs are threatened ecosystems, so it is important to have predictive models of their dynamics. Most current models of coral reefs fall into two categories. The first is simple heuristic models which provide an abstract understanding of the possible behaviour of reefs in general, but do not describe real reefs. The second is complex simulations whose parameters are obtained from a range of sources such as literature estimates. We cannot estimate the parameters of these models from a single data set, and we have little idea of the uncertainty in their predictions.We have developed a compromise between these two extremes, which is complex enough to describe real reef data, but simple enough that we can estimate parameters for a specific reef from a time series. In previous work, we fitted this model to a long-term data set from Heron Island, Australia, using maximum likelihood methods. To evaluate predictions from this model, we need estimates of the uncertainty in our parameters. Here, we obtain such estimates using Bayesian Metropolis-Coupled Markov Chain Monte Carlo. We do this for versions of the model in which corals are aggregated into a single state variable (the three-state model), and in which corals are separated into four state variables (the six-state model), in order to determine the appropriate level of aggregation. We also estimate the posterior distribution of predicted trajectories in each case.In both cases, the fitted trajectories were close to the observed data, but we had doubts about the biological plausibility of some parameter estimates. We suggest that informative prior distributions incorporating expert knowledge may resolve this problem. In the six-state model, the posterior distribution of state frequencies after 40 years contained two divergent community types, one dominated by free space and soft corals, and one dominated by acroporid, pocilloporid, and massive corals. The three-state model predicts only a single community type. We conclude that the three-state model hides too much biological heterogeneity, but we need more data if we are to obtain reliable predictions from the six-state model. It is likely that there will be similarly large, but currently unevaluated, uncertainty in the predictions of other coral reef models, many of which are much more complex and harder to fit to real data.     

A numerical model of wave- and current-driven nutrient uptake by coral reef communities

We developed a numerical model capable of simulating the spatial zonation of nutrient uptake in coral reef systems driven by hydrodynamic forcing (both from waves and currents). Relationships between nutrient uptake and bed stress derived from flume and field studies were added to a four-component biogeochemical model embedded within a three-dimensional (3-D) hydrodynamic ocean model coupled to a numerical wave model. The performance of the resulting coupled physical-biogeochemical model was first evaluated in an idealized one-dimensional (1-D) channel for both a pure current and a combined wave-current flow. Waves in the channel were represented by an oscillatory flow with constant amplitude and frequency. The simulated nutrient concentrations were in good agreement with the analytical solution for nutrient depletion along a uniform channel, as well as with existing observations of phosphate uptake across a real reef flat. We then applied this integrated model to investigate more complex two-dimensional (2-D) nutrient dynamics, firstly to an idealized coral reef-lagoon morphology, and secondly to a realistic section of Ningaloo Reef in Western Australia, where nutrients were advected into the domain via alongshore coastal currents. Both the idealized reef and Ningaloo Reef simulations showed similar patterns of maximum uptake rates on the shallow forereef and reef crest, and with nutrient concentration decreasing as water flowed over the reef flat. As a result of the cumulative outflow of nutrient-depleted water exiting the reef channels and then being advected down the coast by alongshore currents, both reef simulations exhibited substantial alongshore variation in nutrient concentrations. The coupled models successfully reproduced the observed spatial-variability in nitrate concentration across the Ningaloo Reef system.     

Using an individual-based model to quantify scale transition in demographic rate functions: Deaths in a coral reef fish

Scientifically informed population management requires quantitatively accurate demographic rate functions that apply at the spatial scale at which populations are actually managed, but practical constraints confine most field measurements of such functions to small study plots. This paper employs an individual-based population growth model to extrapolate the death rate function in a well-studied coral reef fish, the bridled goby Coryphopterus glaucofraenum, from the scale of coral reef “cells” at which it was measured to the larger scale of an entire coral reef. Density dependence in the whole-reef function actually proves stronger than in the local function because high goby density occasionally arises in local patches with few refuges from predators, producing very high mortality there. This IBM-based approach extends the reach of scale transition theory by examining considerably more realistic models than standard analytical methods can presently handle.     

Automatic sensitivity analysis of a finite volume model for two-dimensional shallow water flows

Given a numerical model for solving two-dimensional shallow water equations, we are interested in the robustness of the simulation by identifying the rate of change of the water depths and discharges with respect to a change in the bottom friction coefficients. Such a sensitivity analysis can be carried out by computing the corresponding derivatives. Automatic differentiation (AD) is an efficient numerical method, free of approximation errors, to evaluate derivatives of the objective function specified by the computer program, Rubar20 for example. In this paper AD software tool Tapenade is used to compute forward derivatives. Numerical tests were done to show the robustness of the model and to demonstrate the efficiency of these AD-derivatives.     

Estimating the critical phosphorus loading of shallow lakes with the ecosystem model PCLake: Sensitivity, calibration and uncertainty

There is a vast body of knowledge that eutrophication of lakes may cause algal blooms. Among lakes, shallow lakes are peculiar systems in that they typically can be in one of two contrasting (equilibrium) states that are self-stabilizing: a ‘clear’ state with submerged macrophytes or a ‘turbid’ state dominated by phytoplankton. Eutrophication may cause a switch from the clear to the turbid state, if the P loading exceeds a critical value. The ecological processes governing this switch are covered by the ecosystem model PCLake, a dynamic model of nutrient cycling and the biota in shallow lakes. Here we present an extensive analysis of the model, using a three-step procedure. (1) A sensitivity analysis revealed the key parameters for the model output. (2) These parameters were calibrated on the combined data on total phosphorus, chlorophyll-a, macrophytes cover and Secchi depth in over 40 lakes. This was done by a Bayesian procedure, giving a weight to each parameter setting based on its likelihood. (3) These weights were used for an uncertainty analysis, applied to the switchpoints (critical phosphorus loading levels) calculated by the model. The model was most sensitive to changes in water depth, P and N loading, retention time and lake size as external input factors, and to zooplankton growth rate, settling rates and maximum growth rates of phytoplankton and macrophytes as process parameters. The results for the ‘best run’ showed an acceptable agreement between model and data and classified nearly all lakes to which the model was applied correctly as either ‘clear’ (macrophyte-dominated) or ‘turbid’ (phytoplankton-dominated). The critical loading levels for a standard lake showed about a factor two uncertainty due to the variation in the posterior parameter distribution. This study calculates in one coherent analysis uncertainties in critical phosphorus loading, a parameter that is of great importance to water quality managers.     

Ecosystem network analysis indicators are generally robust to parameter uncertainty in a phosphorus model of Lake Sidney Lanier, USA

Understanding how data uncertainty influences ecosystem analysis is critical as we move toward ecosystem-based management. Here, we investigate how 18 Ecological Network Analysis (ENA) indicators that characterize ecosystem growth, development, and condition are affected by uncertainty in an ecosystem model of Lake Sidney Lanier (USA). We applied ENA to 122 plausible parameterizations of the ecosystem developed by Borrett and Osidele (2007, Ecological Modelling 200, 371-387), and then used the coefficient of variation (CV) to compare system indicator variability. We considered Total System Throughput (TST) as a measure of the underlying model uncertainty and tested three hypotheses. First, we hypothesized that non-ratio indicators whose calculation includes the TST would be at least as variable as TST if not more variable. Second, we postulated that indicators calculated as ratios, with TST in the numerator and denominator would tend to be less variable than TST because its influence will cancel. Last, we expected the Average Mutual Information (AMI) to be less variable than TST because it is a bounded function. Our work shows that the 18 indicators grouped into four categories. The first group has significantly larger CVs than the CV for TST. In this group, model uncertainty is amplified rendering these three indicators less useful. The second group of four indicators shows no significant difference in variability with respect to TST. Finally, there are two groups whose CV values are significantly lower than that for TST. The least variable group includes the ratio-based indicators and Average Mutual Information. Due to their low variability, we conclude that these indicators are the most robust to the parameter uncertainty and most useful for ecosystem assessment and comparative ecosystem analysis. In summary, this work suggests that we can be as certain, or more certain, in most of the selected ENA indicators as we are in the parameters of the model analyzed.     

Simulating the fate of Florida Snowy Plovers with sea-level rise: Exploring research and management priorities with a global uncertainty and sensitivity analysis perspective

Changes in coastal habitats due to sea-level rise provide an uncertain, yet significant threat to shoreline dependent birds. Rising sea levels can cause habitat fragmentation and loss which can result in considerable reduction in their foraging and nesting areas. Computational models and their algorithmic assumptions play an integral role in exploring potential mitigation responses to uncertain and potentially adverse ecological outcomes. The presence of uncertainty in metapopulation models is widely acknowledged but seldom considered in their development and evaluation, specifically the effects of uncertain model inputs on the model outputs. This paper was aimed to (1) quantify the contribution of each uncertain input factor to the uncertainty in the output of a metapopulation model which evaluated the effects of long-term sea-level rise on the population of Snowy Plovers (Charadrius alexandrinus) found in the Gulf Coast of Florida, and (2) determine the ranges of model inputs that produced a specific output for the purpose of formulating environmental management decisions. This was carried out by employing global sensitivity and uncertainty analysis (GSA) using two generic (model independent) methods, the qualitative screening Morris method and a quantitative variance-based Sobol’ method coupled with Monte Carlo filtering. The analyses were applied to three density dependence scenarios: assuming a ceiling-type density dependence, assuming a contest-type density dependence, and assuming that density dependence is uncertain as to being ceiling- or contest-dependent. The sources of uncertainty in the outputs depended strongly on the type of density dependence considered in the model. In general, uncertainty in the outputs highly depended on the uncertainty in stage matrix elements (fecundity, adult survival, and juvenile survival), dispersal rate from central areas with low current populations (the “Big Bend” area of Florida) to the northern, panhandle populations, the maximum growth rate, and density dependence type. Our results showed that increasing the maximum growth rate to a value of 1.2 or larger will increase the final average population of Snowy Plovers assuming a contest-type density dependence. Results suggest that studies that further quantify which density dependence relationship best describes Snowy Plover population dynamics should be conducted since this is the main driver of uncertainty in model outcomes. Furthermore, investigating the presence of Snowy Plovers in the Big Bend region may be important for providing connection between the panhandle and peninsula populations.     

Speciation of nitrogen in the coral reef sedimentary environment of Lakshadweep archipelago,Indian ocean

This paper focuses on the spatial distributional profiles of different species of the important micronutrient element, nitrogen (nitrite, nitrate and total nitrogen) in various coral-reef sedimentary environment of Lakshadweep Archipelago. The relative abundance of the three forms of nitrogen was in the order, total-N???nitrate-N???nitrite-N. Relatively very low levels of nitrite in the different microenvironments of the islands are an indication of a higher rate of nitrification, so as to produce thermodynamically most stable form of nitrogen, namely nitrate under the condition of well-oxygenated shallow coastal/lagoon waters. A lagoon-ward enrichment pattern of total nitrogen in the lagoon transects of Agathy, Minicoy, Kadamath and Kiltan Islands also reflected the fact that the rate and space available for nitrogen fixation in shallow zones of the lagoon are high. Further, the nitrogenous waste materials produced from the reefs and surrounding environments have limited exchange with the sea, and all these factors, together with the organic nitrogen retention capacity of sediment types, contribute to total nitrogen.     

Spatial variation of phosphorus fractionation in the coral reef sediments of Lakshadweep Archipelago,Indian Ocean

Surface sediments were collected from the shore and lagoons of Kavaratti, Kadamat and Agatti islands of Lakshadweep Archipelago during May 2015 and analysed for the spatial distribution of the micronutrient element, phosphorus. Phosphorus was separated by sequential extraction procedure into five fractions – exchangeable (Ex-P), iron bound, (Fe-P), calcium bound (Ca-P), organic and residual fractions (OP) and total phosphorus (TP). The average relative contribution of each P species to TP was: OP?>?Ca –P?>?Ex – P?>?Fe – P. The high concentration of organic and residual phosphorus (87–96%) compared to inorganic phosphorus is particularly evident at stations characterised by higher total phosphorus concentrations. Among the three forms of IP in the sediments, Ca-P was dominant at all stations. The OC/OP ratio ranged from 3 to 163 in the sediments, suggesting that the organic matter in sediments had been subjected to degradation. Hence, the major contribution towards organic and residual phosphorus form is from the residual fraction comprising biologically resistant or non-available phosphorus form composed of refractory materials. The concentration of phosphorus reported in the present study is higher than that of the earlier studies in Lakshadweep, indicating a terrestrial and anthropogenic in?uence on the sediment.     

Environ indicator sensitivity to flux uncertainty in a phosphorus model of Lake Sidney Lanier, USA

Effective environmental impact assessment and management requires improved understanding of the organization and transformation of ecosystems in which independent agents are linked through an intricate network of energy, matter, and informational interactions. While advances have been made, we still lack a complete understanding of the processes that create, constrain, and sustain ecosystems. Network environ analysis (NEA) provides one approach for building novel ecosystem insights, but it is model dependent. As ecological modeling is an imprecise art, often complicated by inadequate empirical data, the utility of NEA may be limited by model uncertainty. Here, we investigate the sensitivity of NEA indicators of ecosystem growth and development to flow and storage uncertainty in a phosphorus model of Lake Sidney Lanier, USA. The indicators are total system throughflow (TST), total system storage (TSS), total boundary input (Boundary), Finn cycling index (FCI), ratio of indirect-to-direct flows (Indirect/Direct), indirect flow index (IFI), network aggradation (AGG), network homogenization (HMG), and network amplification (AMP). Our results make two primary contributions. First, they demonstrate that five of the indicators – FCI, Indirect/Direct, IFI, AGG and HMG – are relatively robust to the flow and storage uncertainty in the Lake Lanier model. This stability lets us draw robust conclusions about the Lake Lanier ecosystem organization (e.g., phosphorus flux in the lake is dominated by internal processes) in spite of uncertainties in the model. Second, we show that the majority of the indicators co-vary and that most of their common variation could be mapped onto two latent factors, which we interpret as (1) system integration and (2) boundary influences.     

Model of coral population response to accelerated bleaching and mass mortality in a changing climate

We model coral community response to bleaching and mass mortality events which are predicted to increase in frequency with climate change. The model was parameterized for the Arabian/Persian Gulf, but is generally applicable. We assume three species groups (Acropora, faviids, and Porites) in two life-stages each where the juveniles are in competition but the adults can enter a size-refuge in which they cannot be competitively displaced. An aggressive group (Acropora species) dominates at equilibrium, which is not reached due to mass mortality events that primarily disadvantage this group (compensatory mortality, >90% versus 25% in faviids and Porites) roughly every 15 years. Population parameters (N individuals, carrying capacity) were calculated from satellite imagery and in situ transects, vital rates (fecundity, mortality, and survival) were derived from the model, field observations, and literature. It is shown that populations and unaltered community structure can persist despite repeated 90% mortality, given sufficiently high fecundity of the remaining population or import from connected populations. The frequency of disturbance determines the dominant group—in low frequency Acropora, in high frequency Porites. This is congruent with field observations. The model of an isolated population was more sensitive to parameter changes than that of connected populations. Highest sensitivity was to mortality rate and recruitment rate. Community composition was sensitive to spacing of disturbances and level of catastrophic mortality. Decreased mortality led to Acropora dominance, increased mortality led to Acropora extinction. In nature, closely spaced disturbances have severely disadvantaged Acropora populations over the last decade. Unless a longer (>10 years) disturbance-free interval can be maintained, a permanent shift away from Acropora dominance will be observed. A mortality rate of 99% in Acropora, as observed in 1996, is not sustainable if repetitive and neither is a disturbance frequency <15 years—each leading to population collapse. This shows that the severity and/or the spacing of the 1996–1998–2002 disturbances were unusual in frequency and duration.     

Spatio-temporal analysis of alpine ecotones: A spatial explicit model targeting altitudinal vegetation shifts

There is general agreement in literature that Alpine vegetation belt ecotones have shown a trend of upward migration in the last few decades. Despite the potential of such shifts as indicators of global change effects in mountain ecosystems, there are relatively few works focused on their assessment in a systematic and spatially explicit way. In this work our aim is to quantify the altitudinal shifts and analyse the spatial pattern dynamics of mountain ecotones. We developed a novel procedure to delineate the current and former state of three characteristic mountain ecotones, which we formalised as forest, tree and tundra lines. Our approach is based on the recognition of altitudinal extreme outposts identified with ecotone locations at a slope scale. The integration of multi-temporal datasets allows the identification and quantification of altitudinal advances and retreats in the outpost locations for a given period. We tested the method in a section of the Italian Alps for the period 1957-2003. Results show a general trend of an increase in altitude for the three ecotones, despite the occurrence of occasional decreases. We estimate decadal altitude increments of 25 m for forest line, 13 m for treeline and 11 m for tundra line. We also identified changes in ecotone spatial morphology between the two dates, with significant implications in connectivity and colonisation dynamics.     

Connectivity or demography: Defining sources and sinks in coral reef fish metapopulations

The identity of an individual patch as a source or a sink within a metapopulation is a function of its ability to produce individuals and to disperse them to other patches. In marine systems patch identity is very often defined by dispersal ability alone—upstream patches are sources—while issues of variable habitat quality (which affects local production) are ignored. This can have important ramifications for the science of marine reserve siting. This study develops a spatially explicit source–sink metapopulation model for reef fish and uses it to evaluate the relative importance of connectivity versus demography and how this depends upon the level of local larval retention and the strength of density-dependent recruitment. Elasticity analyses indicated that patch contribution (source or sink) was more sensitive to demographic parameters (particularly survival) than connectivity and this effect was conserved even under strong levels of density-dependence and was generally strengthened as local retention increased. Variability in the relationship between parameter elasticity and local retention was shown to be dependent upon the magnitude of connectivity for an individual patch relative to a critical connectivity value. The proportion of larvae lost due to transport processes was an important parameter which directly affected the magnitude of this critical connectivity value. Patches with connectivity values less than the critical value contributed to the metapopulation largely via production (i.e., local demographics most important). As local retention increased, so did the importance of demographic parameters in these patches. Patches with connectivity values greater than the critical value contributed largely via dispersal of larvae and thus the importance of local demographics decreased as local retention increased.     

An agent-based model of red colobus resources and disease dynamics implicates key resource sites as hot spots of disease transmission

The effect of anthropogenic landscape change on disease in wildlife populations represents a growing conservation and public health concern. Red colobus monkeys (Procolobus rufomitratus), an endangered primate species, are particularly susceptible to habitat alteration and have been the focus of a great deal of disease and ecological research as a result. To infer how landscape changes can affect host and parasite dynamics, a spatially explicit agent-based model is created to simulate movement and foraging of this primate, based on a resource landscape estimated from extensive plot-derived tree population data from Kibale National Park, Uganda. Changes to this resource landscape are used to simulate effects of anthropogenic forest change. With each change in the landscape, disease outcomes within the simulated red colobus population are monitored using a hypothetical microparasite with a directly transmitted life cycle. The model predicts an optimal distribution of resources which facilitates the spread of an infectious agent through the simulated population. The density of resource rich sites and the overall heterogeneity of the landscape are important factors contributing to this spread. The characteristics of this optimal distribution are similar to those of logged sections of forest adjacent to our study area.     

Dealing with uncertainty in qualitative models with a semi-quantitative approach based on simulations. Application to the Gironde estuarine food web (France)

In the context of ecosystem approach to fisheries, it is a critical issue to build management tools able to predict the possible trajectories of ecosystems under various human pressure or environmental variations, but also capable to point out influent and sensitive components.     

COSMOS,a spatially explicit model to simulate the epidemiology of Cosmopolites sordidus in banana fields

A stochastic individual-based model called COSMOS was developed to simulate the epidemiology of banana weevil Cosmopolites sordidus, a major pest of banana fields. The model is based on simple rules of local movement of adults, egg laying of females, development and mortality, and infestation of larvae inside the banana plants. The biological parameters were estimated from the literature, and the model was validated at the small-plot scale. Simulated and observed distributions of attacks were similar except for five plots out of 18, using a Kolmogorov–Smirnov test. These exceptions may be explained by variation in predation of eggs and measurement error. An exhaustive sensitivity analysis using the Morris method showed that predation rate of eggs, demographic parameters of adults and mortality rate of larvae were the most influential parameters. COSMOS was therefore used to test different spatial arrangements of banana plants on the epidemiology of C. sordidus. Planting bananas in groups increased the time required to colonise plots but also the percentage of banana plants with severe attacks. Spatial heterogeneity of banana stages had no effect on time required to colonise plots but increased the mean level of attacks. Our model helps explain key factors of population dynamics and the epidemiology of this tropical pest.     

Sensitivity of system stability to model structure

A community is stable, and resilient, if the levels of all community variables can return to the original steady state following a perturbation. The stability properties of a community depend on its structure, which is the network of direct effects (interactions) among the variables within the community. These direct effects form feedback cycles (loops) that determine community stability. Although feedback cycles have an intuitive interpretation, identifying how they form the feedback properties of a particular community can be intractable. Furthermore, determining the role that any specific direct effect plays in the stability of a system is even more daunting. Such information, however, would identify important direct effects for targeted experimental and management manipulation even in complex communities for which quantitative information is lacking. We therefore provide a method that determines the sensitivity of community stability to model structure, and identifies the relative role of particular direct effects, indirect effects, and feedback cycles in determining stability. Structural sensitivities summarize the degree to which each direct effect contributes to stabilizing feedback or destabilizing feedback or both. Structural sensitivities prove useful in identifying ecologically important feedback cycles within the community structure and for detecting direct effects that have strong, or weak, influences on community stability. The approach may guide the development of management intervention and research design. We demonstrate its value with two theoretical models and two empirical examples of different levels of complexity.     

The effects of mating system on male mate choice in a coral reef fish

Summary We show how mate limitation appears to be critical in determining whether or not males exercise mate choice among available females. Thalassoma bifasciatum is a Caribbean reef fish with two distinct mating patterns: group-spawning and pair-spawning. In both mating systems, female fecundity is variable and size dependent, and female availability is high. However, sperm competition among group-spawning males apparently limits the number of effective matings in which a male may engage, whereas territorial pair-spawning males have little or no such limitation. Group-spawning males should be discriminating in their choice of mates and our data confirm this: there is strong evidence for assortative mating in group-spawns, with more large males joining in mating groups around large females. In contrast, pair-spawning males show no indication of mate preferences, and spawn with all females who arrive at their territories.     

Relative influence of environmental factors and fishing on coral reef fish assemblages

Understanding whether assemblages of species respond more strongly to bottom-up (availability of trophic resources or habitats) or top-down (predation pressure) processes is important for effective management of resources and ecosystems. We determined the relative influence of environmental factors and predation by humans in shaping the density, biomass, and species richness of 4 medium-bodied (10–40 cm total length [TL]) coral reef fish groups targeted by fishers (mesopredators, planktivores, grazer and detritivores, and scrapers) and the density of 2 groups not targeted by fishers (invertivores, small fish ≤10 cm TL) in the central Philippines. Boosted regression trees were used to model the response of each fish group to 21 predictor variables: 13 habitat variables, 5 island variables, and 3 fishing variables (no-take marine reserve [NTMR] presence or absence, NTMR size, and NTMR age). Targeted and nontargeted fish groups responded most strongly to habitat variables, then island variables. Fishing (NTMR) variables generally had less influence on fish groups. Of the habitat variables, live hard coral cover, structural complexity or habitat complexity index, and depth had the greatest effects on density, biomass, and species richness of targeted fish groups and on the density of nontargeted fishes. Of the island variables, proximity to the nearest river and island elevation had the most influence on fish groups. The NTMRs affected only fishes targeted by fishers; NTMR size positively correlated with density, biomass, and species richness of targeted fishes, particularly mesopredatory, and grazing and detritivorous fishes. Importantly, NTMRs as small as 15 ha positively affected medium-bodied fishes. This finding provides reassurance for regions that have invested in small-scale community-managed NTMRs. However, management strategies that integrate sound coastal land-use practices to conserve adjacent reef fish habitat, strategic NTMR placement, and establishment of larger NTMRs will be crucial for maintaining biodiversity and fisheries.