Synthesis

INTRODUCTION

Decision science for the management of coastal and marine systems

Environmental disturbances from storms, sea-level rise (SLR), and land use changes are reshaping many coastal habitats and having an impact on important ecosystem services (Hsiang et al. 2017). The loss of ecosystem services has significant impacts on local economies (Hsiang et al. 2017), which has led to increased interest in coastal management activities (Dalyander et al. 2016, Eaton et al. 2019, 2021). Examples of coastal restoration activities include: restoration of oyster reefs, restoration of submerged aquatic vegetation, and barrier island restoration (Dalyander et al. 2016, Yurek et al. 2021). These restoration projects have implications for ecosystem services, such as fisheries, biodiversity, and storm buffering.

Structured decision making (SDM) or decision analysis, provides a framework for addressing these complex problems (Kirkwood 1996, Martin et al. 2009, Gregory et al. 2012). Decision analysis is a formal method for analyzing a decision by breaking down the decision problem into key components: management objectives, potential management actions, models to project the consequences of the actions, an optimization approach to identify decisions that are optimal with respect to the objectives, and finally a monitoring program to evaluate changes in the system (Martin et al. 2009, Gregory et al. 2013). SDM techniques can be used to assist natural resource managers with identifying decisions that are optimal with respect to their management objectives (Kirkwood 1996, Williams et al. 2002, Goodwin and Wright 2009, Martin et al. 2009). Coastal and marine restoration efforts tend to be complex because they involve considerations of oceanography, geology, ecology, economics, and environmental laws, and therefore using a modular approach can be helpful to facilitate collaborations (Fig. 1). Each module can be subdivided into interconnected submodules. For example, the system model can have submodules that deal with different aspects of the system, such as models of physical systems (geomorphological and hydrodynamics models) and ecological models (habitat and population models). This modular structure can facilitate collaboration among scientists with diverse expertise and for team members to focus on modules that fit their expertise. Figure 1 provides an illustration of this modular approach with models (such as coastal morphological models) that predict the effect of restoration on outcomes, such as barrier island evolution, and can have cascading effects on ecological outcomes such as the quality of bird foraging habitats. These outcomes that are affected by the actions can then be converted into utility values, which represent a quantification of the management objectives. An optimization algorithm can be used to identify decisions that are optimal with respect to the objectives and models. A monitoring program can be implemented to keep track of system changes affected by the actions and learn by confronting model predictions with observations in the context of adaptive management (Williams et al. 2002, Martin et al. 2009).

This paper is a synthesis of concepts and techniques to help scientists and managers better integrate their work for efficiency in restoring coastal systems. We focus on a few restoration studies that we have been involved with to demonstrate the process. Our capstone example is based on a recently completed Alabama Barrier Island Restoration Assessment at Dauphin Island, Alabama (Irwin et al. 2020, USACE 2020). We also discuss the example of spatial conservation planning at Cape Romain National Wildlife Refuge, South Carolina (hereafter the Refuge). Our goal is not to provide details on how to implement specific techniques; instead, we seek to highlight key concepts and references. The first four sections can be viewed as an overview of the SDM process for coastal restoration; whereas, the remaining sections are more technical and concentrate on models, optimization techniques, adaptive management, and monitoring.

Capstone example: Dauphin Island

Dauphin Island is a barrier island along the northern Gulf of Mexico coast that provides habitat for numerous imperiled species and protection for coastal resources. In addition, it is a tourist destination and includes 2400 property owners and 1400 permanent residents (Five E’s Unlimited 2007). The island is of low elevation (dune heights are under 3 m), making it vulnerable to the impacts of storms and SLR. The Dauphin study was a collaboration between federal and state agencies to characterize the physical processes affecting the island and their impact on important resources (Irwin et al. 2020, USACE 2020). The goal of the project was to evaluate the ability of potential actions to increase island sustainability while protecting and restoring habitats under scenarios of storminess (variability in the frequency and intensity of storms) and SLR (Enwright et al. 2020, Mickey et al. 2020a, b). This restoration assessment included geomorphological and ecological modeling components to forecast changes in the system over decadal timescales, including island elevations and resulting habitat types. We used this assessment project to show how geomorphological models can be combined with ecological models to help inform management decisions while considering uncertainty about system changes and risk tolerance. Our capstone example highlights a multidisciplinary effort by experts in decision science, ecology, geomorphology, geography, and oceanography. For the sake of illustration, we considered a subset of scenarios of environmental changes and restoration activities (beach nourishment and dune elevation by a maximum of ~3.7 m). We considered three environmental scenarios: low SLR/storminess [S1] and high SLR/storminess [S2] scenarios (Mickey et al. 2019, 2020a, b); a third scenario [SM] was obtained through model averaging of S1 and S2, (see section about Model uncertainty for how model averaging was implemented to account for model uncertainty) and binary management actions (action and no action).

FRAMING THE PROBLEM

Decision statement

A formula for writing a decision statement to help scope the decision problem is: “Decision-maker (D) is trying to decide on (action A) to achieve (objective O) over time (T) and in location (L) considering (constraints C).” (Romito et al. 2015). During this scoping phase, the decision analysts identify the key sources of uncertainty and constraints of the problem. The relevant regulatory frameworks, governance, overlapping jurisdictions could be identified and considered during this phase (Appendix 1; Gregory et al. 2012, Partelow et al. 2020). At this stage, the broad class of decision problem is identified, such as construction project (beach nourishment or repair of structures), land acquisition, or resource protection. After the decision problem has been framed, the next step is to elicit the objectives from the decision makers. Rapid prototyping is an effective approach when analyzing a decision. This prototyping can occur over a few days, as a workshop, and the participants (technical experts, decision makers, and stakeholders) go over each component and develop a prototype, which serves as a blueprint for the remainder of the project (Knapp et al. 2016, Garrard et al. 2017, Runge et al. 2020). This prototype can be subsequently revised, and more details can be added to each component. Prior to the prototyping workshop the organizers select the participants while considering trade-offs about group size. The group should be large enough to include the decision makers and stakeholders, but groups of 10 or more can be difficult to manage. The selected group can then develop the decision statement during pre-workshop meetings by identifying the decision makers (by asking the question: “who are the decision makers?”), scoping the problem, and asking participants “what are impediments to the decision problem?” (Gregory et al. 2012, Runge et al. 2020).

Decision statement for Dauphin Island

An example of a concise decision statement for Dauphin is: “the decision maker is trying to decide on which restoration options to choose (restoration action vs no action) in order to achieve a sustainable barrier island that would provide suitable foraging habitats for shorebird and would promote barrier island sustainability over a 10-year period while considering cost and permitting constraints.” In real applications, the managers, agencies, and stakeholders should be clearly identified and involved in the development of the decision statement, objectives, and alternative options. This was done in the actual decision statement for Dauphin, and in this case the primary decision maker was the commissioner of the Alabama Department of Conservation and Natural Resources (Irwin et al. 2020). Here we used a modified and simplified version of the problem to better illustrate the application of decision analytical tools. As we will see with our numerical examples, the selection of the action is generally driven by both models of system behavior and the value system of managers (Williams et al. 2002). Therefore, to identify the optimal choices, formal elicitation of values is necessary.

MANAGEMENT OBJECTIVES

Objectives and performance measures

The management objectives for coastal restoration are diverse, such as to promote sustainable fisheries and recreational activities, protect native species, foster resistance and resilience of habitats in the face of storms and SLR, maximize carbon sequestration, breakdown of pollutants, mitigate flooding and erosion, support local economies, protect cultural resources, and minimize costs (initial and maintenance cost). Performance measures are criteria used to quantify or assess progress toward each objective. Examples of performance measures for coastal management include: area of piping plover nesting habitat suitability (in km²), probability of nesting success, area of specific habitats (in km²). Constructing an objective hierarchy when identifying objectives and actions can be useful. Fig. 2a shows an objective hierarchy for a single fundamental objective, which consists of “endangered species protection.” The questions “What do the decision-makers want to achieve ultimately?” and “Why is this goal important?” can help identify the fundamental objectives (Gregory et al. 2012). Means objectives, performance measures and actions can then be derived from the fundamental objectives, by asking the question “How do we achieve it?” (Gregory et al. 2012).

Dauphin Island example

Fig. 2b-c shows an objective hierarchy with two fundamental objectives to promote: (1) Shorebird foraging habitats, and (2) Barrier Island sustainability. We used the performance measure “intertidal beach/flat” habitat (as a percentage of total habitat; Enwright et al. 2021) as a proxy for shorebird foraging habitats (Brush et al. 2019). A means objective for “Barrier Island sustainability” may be to increase the resistance of the barrier island to the impacts of storms while also incorporating resilience to SLR (Zinnert et al. 2017). Our performance measures included the volume of sand on the barrier island, such as “Dune height” (in m), barrier “Island width” (m), and whether the island was breached or not, “Breaching.” We also considered the following performance measures: subaerial “Land mass” of the island, storm “Inundation” (amount of time in hours, that the island was flooded during storm events), and “Cost” of the restoration (arbitrary for this example). Most of these measures implicitly reflected preferences related to the sustainability of the island, and its ability to provide ecosystem services such as mainland protection through the attenuation of wave energy. Some objectives may be in conflict, for example, inundation of the island is obviously detrimental to the infrastructures on the island, yet some level of overwash through inundation processes may be beneficial to the maintenance of shorebird foraging habitats. Because the objectives drive everything else in the SDM process (Fig. 1), it is important for the managers to be actively involved in the development of the objective hierarchy (Fig. 2). This can be achieved during the initial phase of a rapid prototyping workshop (Gregory et al. 2012). Once the means objectives and performance measures are identified the next step is to develop models that will project consequences of the actions on the performance measures. Then the analyst can establish the shape of the utility function by eliciting it from the managers and then weight these objectives also elicited from the managers (Kirkwood 1996). The reason it is preferable for the models to be implemented prior to eliciting the value weights is because the weights can be affected by the magnitude of the effects of the actions on performance measures. For example, the weights that managers assign to “Inundation” and “Cost” would be quite different if restoration reduces “Inundation” by 1% for a “Cost” of $1,000,000 compared to a situation in which restoration reduces “Inundation” by 5% for a “Cost” of $100,000. The decision analysts can elicit the weights by asking managers to assign the weights to each performance measure after consideration of the magnitude of the effects.

Linear utility

Objectives can be quantified by using utility functions U(x) to convert performance measures (x) into utility values. Utilities can be viewed as a common currency among performance measures to compare the desirability of alternative actions with respect to specified management objectives (Kirkwood 1996, Goodwin and Wright 2009, Gregory et al. 2012). A utility for x is by convention on a scale from 0 to 1 (with 1 being the most desirable). The simplest type of utility function is linear (Fig. 3). For Dauphin, model projections that lead to a greater proportion of “Intertidal” habitat will also have greater utility values. A simple formula to compute linear utilities when the preference is increasing with x is:

(1)

when the preference is declining with x (Kirkwood 1996):

(2)

Nonlinear utility

Nonlinear utility functions provide more flexibility than linear utility to capture the manager’s preferences. Several functions can be used and could be fit to specific values that were elicited from the managers. For Dauphin, we considered three shapes of nonlinear utility functions (linear, concave up, and concave down) that are based on equations described in Kirkwood (1996; Fig. 3, Appendix 2):

(3)

(4)

Where ρ is a parameter that governs the inflection of the curves and corresponds to the risk tolerance for the utility function; x corresponds to performance measure x, for example “Intertidal habitat.” Eq. 3 is for the case for which the preference increases with x and Eq. 4 for which it is decreasing with x. Alternative utility functions could be used such as piecewise, thresholds, and logistic functions. Utility functions can be used to compute expected utilities for each actions as:

(5)

where EV_x(a) is the expected utility for action a, v_a(x) is the utility value of outcome x of the performance measure, and p_a(x) is the probability of occurrence of outcome x. Actions with greater expected utilities are preferred (Kirkwood 1996, Canessa et al. 2015). When multiple performance measures are considered we can compute the sum of weighted expected utilities for each action. The expected utilities are weighted by “value weights,” which reflect how much a manager values the outcome of a performance over another. The expected utility depends on both how much the managers value the outcomes (“Breaching” of the island or not) associated with a given action and the probability of that outcome occurring. The “value weights” help managers place more emphasis on preferred performance measures (“Cost” vs “Intertidal” habitats). Both the “value weights” and the shape of the utility functions contribute to determining the sum of weighted expected utilities and can be elicited with formal elicitation techniques (Goodwin and Wright 2009). For example, the swing weight method can be used to identify weight values that reflect the managers’ values (Goodwin and Wright 2009). The action that has the greatest sum of weighted expected utilities is preferred (Kirkwood 1996).

POTENTIAL ACTIONS

The actions are preferably derived from the fundamental and means objectives, and an objective hierarchy (Fig. 2) can be useful with this process (Gregory et al. 2012). This section outlines a list of potential actions for coastal restoration; where, when, and how (restoration techniques, amount and type of material to move [sand, oyster shells, reef balls, sediment removal] or organisms [submerged aquatic vegetation, SAV, oysters]) to restore (Appendix 3). In the context of living shorelines, the decision could involve the depth, height, and distance from shoreline of the construction material for reef restoration that would maximize wave attenuation benefits (reduce coastal erosion) or productivity for fisheries. The decisions can also be related to land acquisition strategies (fee-simple purchases, conservation easements, with the option of divestments), or construction projects (alteration of roads that may alter the hydrology of coastal systems; Eaton et al. 2019). Potential actions may also be focused on preserving populations of imperiled species (nest protection). An important consideration is to ensure that the actions are relevant to the fundamental objectives (Fig. 2). For Dauphin, we will explore a decision related to beach nourishment and raising dune elevations. To better understand the impact of the restoration action on a landscape-scale, we extended the study area to include the back-barrier area behind the restoration site (USACE 2020).

MODELS

The primary role of models in decision analysis is to project the effect of potential actions on the system under study (Kirkwood 1996, Williams et al. 2002, Martin et al. 2009). Both static and dynamic models can be used to link potential actions (planting of vegetation, beach nourishment, shelling for oyster reef restoration) to the physical systems such as dune geomorphology, which in turn may impact the ecology of organisms, for example, nesting success of shorebirds or sea turtles. Below we discuss both advances in geomorphological and ecological modeling that are relevant to coastal restoration projects.

Geomorphological models

Morphodynamic models can be used to understand how the coastal landscape may evolve in the future under drivers such as storms and SLR. This requires integrated modeling approaches to capture the nonlinear feedbacks and dynamic responses of the bio-geophysical coastal system, and assess resulting social, economic, and ecological impacts (Passeri et al. 2015, Hagen et al. 2017). In the Dauphin example, models of alongshore sediment transport during quiescent conditions, storm-driven morphology, and post-storm dune recovery were integrated to simulate barrier island evolution over decadal timescales, accounting for variations in storminess (intensity and frequency of storms) and SLR (Fig. 4a-c; Mickey et al. 2020a, b, Passeri et al. 2020; Appendix 4). Output from the models showed how island elevations (dune heights, island width) and inundation during storm events may change in the future with “No action” versus scenarios of beach and dune nourishment.

Morphodynamic models can be used to project the consequences (elevation, dune height) of alternative potential actions, which can then be incorporated in habitat models to quantify changes in habitat types (Enwright et al. 2021). This link between potential actions and model outcomes is a key ingredient of system models that can be used for decision analysis. Figure 4 shows model projections for elevation under the alternative actions and the environmental scenario S2.

Ecological modeling

Habitat models

Habitat types can be important performance measures for restoration. They can be modeled with statistical models that incorporate multiple data sources (imagery from satellite and unmanned aerial systems [UAS]). Habitats are dynamic and statistical models, such as multistate models, can be used to model transition probabilities among community states as a function of potential actions or environmental changes such as water levels (Hotaling et al. 2009). For example, Markov chain models and state-and-transition simulation models can be used to project changes in habitat composition over time (Hotaling et al. 2009, Zweig and Kitchens 2014, Daniel et al. 2016).

Modeling habitat dynamics can be challenging, and for many applications, static habitat models can be useful approximations. For Dauphin, we used model results from Enwright et al. (2020) who developed a machine learning-based model to predict barrier island habitats using morphodynamic model outputs (Appendix 5). The results of the habitat model are spatially explicit maps of several barrier island habitats (intertidal beach, beach, dune, barrier flat, woody vegetation, intertidal marsh, intertidal flat). For this decision analysis, we quantified the non-vegetated intertidal habitats (i.e., intertidal beach and intertidal flat) and the total land area (i.e., intertidal habitats and supratidal/upland habitats; Fig. 4d-f).

Population models

Abundance and occupancy states of animal populations or vegetation community states (i.e., state variables) are common model outcomes that can be linked to actions. As discussed for habitat models, many decision problems can be approximated with a static approach. For instance, the probability of occupancy of species at a site or its abundance can be linked to management actions. The equation below considers how an occupancy state for nesting sea turtles can be linked to environmental variables but also some actions, such as nest protection.

(6)

Where ψ_s is the probability of occupancy at site s, β₀ is the parameter for the intercept, and β_i are the parameters for environmental covariates (V_i, temporal [temperature], or spatial [elevation, substrate type]). β_j can then be included in the model to consider an action variable, which in the simplest case is binary, as could be the case for a nest protection action (i.e., with or without protection). Other relevant examples may include restoration of reef systems (coral or oysters) or SAV, in which case ψ could represent the occupancy of these organisms, and the binary action could be to restore or not. A similar approach could be used for multiple occupancy states or abundance (Martin et al. 2011, Fackler et al. 2014).

These measures would be well suited for quantifying the abundance of many coastal species (shorebirds, sea turtles, and other protected species and their habitats). Occupancy and abundance models can account for important sources of errors (imperfect detection, misclassification error, and geographic variation) through proper sampling designs (robust design, stratified random sampling; Williams et al. 2002, MacKenzie et al. 2017). Occupancy models can also be extended to consider multiple ecological states, such as habitat suitability using multistate occupancy models or multistate models; the states may consist of abundance states, or ecological states such as vegetation communities (marshes; Hotaling et al. 2009).

Considering the dynamics of ecological systems can help improve our understanding how the system will respond to potential actions. Here again occupancy states and abundance can be used, but now vital rates that govern the dynamics of populations can be incorporated. In fact, these dynamics can be represented with a matrix formulation. Age or stage structured model (Leslie and Lefkovitch matrix) have been used extensively to model population growth rates whereas Markov chain discrete time models are more commonly used for metapopulations, communities, or ecosystems (Caswell 2001, Hotaling et al. 2009, O’Donnell et al. 2020).

As for the static case, the actions can be linked to state variables, but they can also be linked to vital rates (survival and reproduction in the case of matrix population models, and extinction and colonization rates in the case of occupancy models). For example, we could consider a simple metapopulation/occupancy model:

(7)

where is ψ_t is the occupancy at time t, ε is the local extinction probability and γ the local colonization rate (MacKenzie et al. 2017).

(8)

where β₀ is the intercept, β_i are the model parameters for the covariates Vi (elevation, precipitation, overwash frequency), and β_j the parameters for the action (nest protection). Several examples in the ecological literature have linked actions to occupancy models (Martin et al. 2009, 2010, Tyre et al. 2011, O’Donnell et al. 2020).

Uncertainty and model complexity

Models to predict system response to actions would account for important sources of uncertainty, such as parametric uncertainty (sampling variance associated with estimates of abundance, occupancy, or vital rates), scientific or model uncertainty (competing hypotheses and models about how the system will respond to actions, such as how a barrier island will respond to storms after restoration activities), environmental stochasticity, (vital rates are greater during “good” years than during “bad” years). (See Appendix 6 for other sources of uncertainty).

Model complexity is another important consideration when developing models for decision analysis. Choice about complexity involves trade-offs such as bias and precision and development and run time (Appendix 6; Burnham and Anderson 2002, Williams et al. 2002). Complex models can also make the optimization process challenging (Bellman 1957).

Accounting for uncertainty in the Dauphin Island example

In the Dauphin example there are several sources of uncertainty that can be considered, for example, each combination of SLR/storminess scenarios (three scenarios were considered here: low SLR/storminess [S1] and high SLR/storminess [S2] scenarios; a third scenario [SM] was obtained through model averaging of S1 and S2, see next section) and actions lead to outcomes in terms of change in elevation, inundation, and habitats. If uncertainty is available from the models, it is then possible to: (1) fit distributions to the outcomes, or (2) use the observed distributions produced from the model.

The areas for each habitat type were obtained from a model developed for the Dauphin study (Enwright et al. 2020, 2021). Because uncertainty estimates were not available for some of the performance measures, for the sake of illustrating how to account for uncertainty we used a CV of 10%. In most applications, we would want to use statistical estimates or expert elicited values (Appendix A8.2).

In order, to compute expected utility it is convenient to discretize distributions. Projected “Intertidal” habitat and “Cost” followed lognormal probability distributions, and we derived the probability for each bin k out of K bins for each performance measures. The discretized distributions of the performance measures were computed with a Monte Carlo approach implemented in R (with a sample size of 10,000; R Core Team 2021; Fig. 5).

Model uncertainty

In the previous section we explained how to obtain probability distributions for each scenario and restoration combination for all the performance measures at Dauphin. These distributions are shown in Fig. 5 and reflect the parametric uncertainty associated with each performance measure. Each distribution associated with a scenario could be viewed as predictions linked to a single model (Mi).

We now illustrate how model uncertainty can be incorporated from the outcomes of model projections, in the case of Dauphin. This allows us to make predictions that consider multiple models (i.e., competing hypotheses about how the system will change over time), based on their support from the data or based on the probability of a given scenario (S1 or S2) occurring. Accounting for model uncertainty can have substantial implications for management, but it is also an essential ingredient of adaptive management (i.e., learning through management for the purpose of better management in the future). Several model averaging techniques are available (see Dormann et al. 2018 for a review), for the sake of simplicity we used a model averaging technique described by Buckland et al. (1997, see also Burnham and Anderson 2002; see Appendix 7, for variance calculation):

(9)

where “theta-bar-hat” is the model averaged estimate, “theta-hat-m” is the estimate for model m (out of M models), w_m is the model weights for model M_m. Model weights represent our belief in a particular hypothesis or model.

We used equal “model weights”, but weights can be calculated by fitting statistical models to empirical data. Indeed, information theoretic model weights such as AIC weights or Bayesian model weights can be derived from statistical models (Dormann et al. 2018). Alternatively, expert elicitation can be used to elicit the initial “model weights” (Appendix A8.2). Then model selection techniques or the implementation of Bayes theorem can be used to update our relative belief in each model, which are reflected by the “model weights” (Williams et al. 2002). “Model weights” can also be used for model selection to identify models that represent a reasonable trade-off between bias (Burnham and Anderson 2002; Appendix 6 and 8).

We used Eq. 9 to compute model averaged estimates (SM) for all performance measures (see Appendix 7 for variance estimation). We assumed equal weights for scenarios S1 and S2 to compute the model averaged estimates for SM. Fig. 5 shows the distributions for each performance measure under S1, S2, and the model averaged estimates (SM) for performance measures (“Intertidal,” “Dune Heights,” “Inundation”). For example, when compared to the no-action case, the action reduced the mean “Inundation” in hours by 98.7% for S1, 56.4% for S2, and 59% for SM. Instead of equal “model weights” we could have used the expected probability of each scenario occurring. As more information becomes available about the likelihood of S1 and S2, the weights and the resulting SM can be updated to better reflect model uncertainty.

FINDING SOLUTIONS: TRADE-OFF AND OPTIMIZATION

Single time step (static) problems with few decisions

Optimization techniques can be separated into two broad classes: static and dynamic optimization problems. The most common techniques to identify trade-off in decision analysis treat the problems as static. Multi-criteria decision analysis (MCDA, e.g., simple multi attribute rating technique [SMART] tables) and decisions trees are among the most used techniques (Kirkwood 1996, Goodwin and Wright 2009, Gregory et al. 2012). Although SMART tables can provide useful insights, most SMART tables do not consider nonlinear utilities and uncertainty. For Dauphin a static MCDA was applied but one that accounts for both uncertainty and nonlinear utilities.

Accounting for uncertainty with MCDA: Dauphin example

We used Monte Carlo simulations to account for uncertainty. Specifically, we discretized probability distributions associated with our performance measures (Fig. 5). Our analyses showed an impact of the scenarios and the actions (Fig. 5) in terms of outcomes on their natural scales. Figure 6 illustrates how utility functions can be used to convert outcomes (proportion of intertidal habitats), and the shape of the utility may reflect the risk attitude of the managers (Fig. 3). Options with greater utilities are more desirable (Kirkwood 1996). Figure 6 shows the weighted expected utilities for seven performance measures for the high sea level rise high storminess/sea level rise [S2] for two options: “action” and “no action” under the concave down utility functions for all performance measures. For the concave down case the sum of weighted utilities was greater for the restoration “action” than for the “no action” (Fig. 6). Figure 7 illustrates how the choice of options were primarily driven by the models (i.e., storminess/SLR scenarios) and the objectives (both in terms of utility functions but also the “value weights” assigned to the performance measures). The “action” option had greater utilities as the level of SLR/storminess increased. It also tended to have greater utilities for the “Concave up” utility function. Regarding the inclusion of uncertainty, adding model uncertainty influenced utilities, we note however, that adding parametric uncertainty to each performance (as opposed to using the mean of the distribution) did not appear to alter the results when utility functions were linear and the probability distributions were symmetrical. For this example, we started by assigning equal weights to the performance measures (Fig. 6). We then varied the weights from 0.25 to 0.75 for one performance measure at a time, the weights for the remaining performance measures were apportioned equally (Fig. 7). Figure 7 shows that the sum of weighted utilities is greater for the action than no action when the weight for “Intertidal” is 0.25 under SM (model averaged scenario) whereas for a weight of 0.5 the “no action” case is preferred based on the sum of weighted utilities. This illustrative example emphasizes the importance of eliciting values from the managers to incorporate how much they value different resources. This valuation can be achieved by eliciting (1) the “value weights” (with swing weights or other weight elicitation techniques) and the shape of the utility functions from managers (Kirkwood 1996). The other component that drives the decision is the model used to predict the effect of the action on the performance measures. In the case where value weights are set at 0.25 for “Inundation,” the decision with the greater sum of weighted utilities is the restore “action” under S1 whereas it switches to “no action” under S2. Thus, because the model predictions about “Inundation” are essential to the decision, managers may want to consider basing their decisions on rigorous predictive model for this performance measure that are both accurate and precise. In this case, a geomorphological model contributed to the modeling of inundation and would be helpful to determine optimal decision to meet the managers’ objectives.

Our example reinforced the importance of these components to the decision-making process and showed how both science (model outcomes) and value considerations (value weights assigned to performance measures and utility functions) can drive the optimal solutions. Implementing MCDA with Monte Carlo simulations (or Bayesian Belief Networks) provides a useful extension to more commonly used deterministic methods such as SMART tables. Key sources of uncertainty can then be incorporated. In our application we showed how to incorporate parametric uncertainty associated with performance measures (e.g., sampling variance). In the section about “Model uncertainty” we explained how to account for model uncertainty associated with alternative SLR and storminess scenarios.

In our MCDA for Dauphin, there was no need for a complex optimization algorithm because there were only two restoration options. Many decision problems in coastal systems involve many potential decisions. This is the case in the context of spatial conservation planning in which the decision may be to acquire land or not. In that case the number of options can be calculated as 2ⁿ, where n is the number of sites (Kirkwood 1996). In most applications the number of possible options is greater than two. Examples of spatial conservation planning that involve nonbinary actions include: (a) land acquisition or protection decisions (fee-simple, fixed, or rolling conservation easements as well as divestment options), (b) levels of boat speed regulations to protect protected species, (c) allocation of law enforcement activities to enforce regulations in protected areas (Eaton et al. 2019, Udell et al. 2019, Moore et al. 2021). These higher dimensional problems can lead to millions of options to choose from. In Appendix 9, we present a detailed example of spatial conservation planning for coastal restoration at the Refuge (Eaton et al. 2021). In this example the managers identified several threats faced by the Refuge, notably, SLR, storms, and urban expansion. Spatial optimization algorithms can be applied to help managers decide which parcels to potentially acquire to help mitigate these threats.

Dynamic optimization

Many decision problems involve sequential decisions, that is, a decision made in the present has consequences for future decisions. Dynamic optimization algorithms can be applied to identify optimal decisions for sequential decision problems. In theory, both the Dauphin and Refuge problems could have been addressed with dynamic optimization approaches. The main impediment to implement such techniques is the “curse of dimensionality,” which emerges because of the exponential increase in problem size as the number of variables increase (Bellman 1957, Fackler 2018). This explains why many decisions that would more naturally be treated as sequential are simplified and static optimization methods are used instead (Appendix 10).

Time horizon, discounting, and non-stationarity

Considerations about the time horizon is unavoidable in the case of problems that involve sequential decisions and dynamic programming. The managers have to decide whether to view the problem in the context of sustainability (e.g., long term or even infinite time horizons) or consider finite time horizons (Miranda and Fackler 2002). In some instances, using a finite horizon approach can also help address the issue of non-stationarity, which may be induced by climate change processes (e.g., Martin et al. 2011). Nevertheless, most coastal restoration projects, even when they are treated as static, deserve some careful considerations of the time horizon and discounting of the future (Appendix 11).

Solutions that consider risk

Risk can be described as involving two components: (i) the probability of an event (e.g., hazard such as a hurricane) occurring and (ii) the consequences if this event occurs (e.g., in this case the consequences could be the breaching of a barrier island; Wideman 1992, Dale et al. 2013, Game et al. 2013, Ludwig et al. 2018). Game et al. (2013) proposed metrics to quantify risk for the purpose of decision analysis, which can be applied to coastal and marine management issues. But there are numerous ways to consider risk attitude of managers. For example, Eaton et al. (2019) applied the concept of modern portfolio theory (MPT is a mathematical approach for determining a portfolio of assets in which the expected return is maximized for a level of risk) to address risk for natural resource applications (Low et al. 2016). MPT in their case was treated as a “mean-variance portfolio problem” where the benefits and risk of investing resources under market uncertainty was quantified in terms of a portfolio’s expected return and variance (Eaton et al. 2019; Appendix 9). In the Refuge study the portfolio was a blend of potential management actions. The MPT approach can help managers identify solutions that maximize returns for the lowest level of variance (a proxy for risk within MPT). This type of approach is applicable to resource allocation problems, for example, which parcel of lands to acquire with various degrees of risk associated with potential threats such as SLR or urbanization.

A Pareto efficient frontier is the set of Pareto optimal solutions in which no preference criterion (objective) can be made better off without making at least one preference criterion worse off (Debreu 1954). It can be constructed to help visualize the options and plot returns (return can be expressed as the amount of desirable habitats and ecosystem services) and risk (expressed as variance on the returns; Fig. 8). Figure 8 shows the solution obtained by Ghasemi-Saghand et al. (2021) with software SiteOpt, which identifies the Nash bargaining solution (Nash bargaining solution represents a compromise between low risk-low return and high-risk-high return solutions; Appendix 12).

We note that the concept of Pareto efficient frontier is useful to address many management problems and does not require MPT. For example, one could build a Pareto optimal frontier by plotting the return of optimal reserves (y-axis) for alternative budgets (x-axis).

As was the case with the MPT example, managers can then use the Pareto frontier as a tool to identify trade-offs that consider the risk attitude of managers and stakeholder groups. The concept of Pareto frontier has been applied to a variety of problems such as the optimal design of speed zones for marine mammals and the optimal allocation of law enforcement activities to reduce the risk of poaching (Udell et al. 2019, Moore et al. 2021).

In some applications, managers may prioritize solutions that avoid the worst outcomes over solutions that are considered optimal. This is the realm of robust optimization and other methods that provide techniques that guard against bad outcomes (Regan et al. 2005, Ben-Haim 2006, van der Burg and Tyre 2011). In some cases, optimal solutions may help managers identify solutions that maximize returns on average, but may not guard as effectively from avoiding rare but potentially catastrophic events (Regan et al. 2005, Ben-Haim 2006, van der Burg and Tyre 2011, Bonneau et al. 2017).

ADAPTIVE MANAGEMENT AND MONITORING

When an action is taken to restore a coastal system there is a learning opportunity that can be leveraged to inform decisions in the future. Adaptive management (Williams et al. 2002) provides a formal framework for implementing this learning process while managing coastal systems. Adaptive management is a special case of SDM, which involves linked decisions (sequential decisions) and competing hypotheses (approximated mathematically with alternative models) about how the system would respond to alternative potential actions (Williams et al. 2009). In the context of adaptive management, the primary role of monitoring involves keeping track of the system states but also to confront predictions from each model to observations in order to update the belief in each model (“model weights”). This update can be implemented with Bayes theorem, although (Appendix 13) other approaches for considering alternative models have been proposed (Williams et al. 2002).

MONITORING, RESEARCH AND EXPECTED VALUE OF INFORMATION

There are two broad classes of monitoring that are relevant to the management of coastal and marine systems, surveillance and targeted monitoring (Nichols and Williams 2006). Surveillance monitoring consists of broad data collection that can serve many purposes, and the monitoring goals and the data may be analyzed retrospectively to address a variety of questions.

Targeted monitoring is more directly relevant to decision science and the goals are more clearly defined (Nichols and Williams 2006). There are three primary roles to monitoring in the context of decision making. The first role is to parameterize the system models that project the consequences of the actions, for example, monitoring of vital rates of species of interest that are impacted by a coastal restoration project. Second, to keep track of the system state, as for example, the case of harvested species, the optimal removal is a function of the abundance of the stock (determined through monitoring; e.g., Nichols et al. 2015). The third role was discussed in the context of adaptive management and consists of using monitoring to update our beliefs in alternative hypotheses (“model weights”) by confronting predictions and observations.

Accounting for important sources of errors when designing sampling designs for monitoring programs can improve accuracy (Appendix 8). The value of investing in specific monitoring programs can be quantified. For example, the expected value of perfect information (EVPI and its extensions such as expected value of sampling information) provide a formal methodology to consider the value of reducing important sources of uncertainty such as partial observability (our inability to estimate state variables such as abundance with certainty) or model uncertainty (Canessa et al. 2015). Reduction of these sources of uncertainty can be achieved through monitoring or research (e.g., experimentation). In fact, EVPI approach can in some cases provide a specific dollar value to invest in a research or monitoring program in order to meet management objectives. EVPI corresponds to the increase in expected management performance if model uncertainty could be eliminated (Williams 2001, Runge et al. 2011, O’Donnell et al. 2020).

CONCLUSION

In this paper we discussed the integration of decision science, ecology, and geomorphology to address restoration decision making in coastal environments. There are some exciting avenues in combining these disciplines that can inform management of coastal habitats. Using holistic and interdisciplinary approaches to management that integrate all components of the decision-making process can be beneficial to coastal management over the long term. The modular nature of the framework that we presented is well suited to facilitate complex interdisciplinary projects. Many coastal restoration projects are logistically challenging, expensive, involve permitting requirements and conflicting objectives by stakeholder groups, and therefore may take multiple years of planning (e.g., coral reef restoration in the context of disease management). This was also the case in the Dauphin study, which we used as an example. As seen in the Dauphin example the model outputs about island width, inundation breaching, and habitat change can be valuable to project the effects of the restoration action and to weight the cost and benefits considered by the managers (quantified with expected utilities). The process that we described is applicable to numerous coastal restoration projects around the world. We note however, that complex models are not always necessary, and we want to avoid the misconception that decision analysis is always time consuming or necessarily involves complex models. Indeed, simple techniques such as SMART tables combined with an expert elicitation can be done in a matter of days (Runge et al. 2020; Appendix A8.2). Nevertheless, the rapid advances in technology, geomorphological, and ecological modeling and optimization algorithms offer opportunities to integrate disciplines and monitoring programs to inform coastal management.

LITERATURE CITED

Bellman, R. 1957. Dynamic programming. Princeton University Press, Princeton, New Jersey, USA.

Ben-Haim, Y. 2006. Information-gap decision theory: decisions under severe uncertainty. Second edition. Academic, London, UK. https://doi.org/10.1016/B978-0-12-373552-2.X5000-0

Bonneau, M., F. A. Johnson, B. J. Smith, C. M. Romagosa, J. Martin, and F. J. Mazzotti. 2017. Optimal control of an invasive species using a reaction‐diffusion model and linear programming. Ecosphere 8:10:01979. https://doi.org/10.1002/ecs2.1979

Brush, J. M., R. A. Pruner, and M. J. L. Driscoll. 2019. GoMAMN strategic bird monitoring guidelines: shorebirds. Pages 171-202 in R. R. Wilson, A. M. V. Fournier, J. S. Gleason, J. E. Lyons, and M. S. Woodrey, editors. Strategic bird monitoring guidelines for the northern Gulf of Mexico. Mississippi Agricultural and Forestry Experiment Station Research Bulletin 1228, Mississippi State University, Starkville, Mississippi, USA.

Buckland, S. T., K. P. Burnham, and N. H. Augustin. 1997. Model selection: an integral part of inference. Biometrics 53:603-618. https://doi.org/10.2307/2533961

Burnham, K. P., and D. R. Anderson. 2002. Model selection and multimodel inference: a practical information-theoretic approach. Springer, New York, New York, USA.

Canessa, S., G. Guillera-Arroita, J. J. Lahoz-Monfort, D. M. Southwell, D. P. Armstrong, I. Chadès, R. C. Lacy, and S. J. Converse. 2015. When do we need more data? A primer on calculating the value of information for applied ecologists. Methods in Ecology and Evolution 6:1219-1228. https://doi.org/10.1111/2041-210X.12423

Caswell, H. 2001. Matrix population models: construction, analysis, and interpretation. Sinauer Associates. Sunderland, Massachusetts, USA.

Dale, A., K. Vella, R. L. Pressey, J. Brodie, H. Yorkston, and R. Potts. 2013. A method for risk analysis across governance systems: a Great Barrier Reef case study. Environmental Research Letters 8(1):015037. https://doi.org/10.1088/1748-9326/8/1/015037

Dalyander, P. S., M. Meyers, B. Mattsson, G. Steyer, E. Godsey, J. McDonald, M. Byrnes, and M. Ford. 2016. Use of structured decision-making to explicitly incorporate environmental process understanding in management of coastal restoration projects: case study on barrier islands of the northern Gulf of Mexico. Journal of Environmental Management 183:497-509. https://doi.org/10.1016/j.jenvman.2016.08.078

Daniel, C. J., L. Frid, B. M. Sleeter, and M. J. Fortin. 2016. State-and-transition simulation models: a framework for forecasting landscape change. Methods in Ecology and Evolution 7:1413-1423. https://doi.org/10.1111/2041-210X.12597

Debreu, G. 1954. Valuation equilibrium and Pareto optimum. Proceedings of the National Academy of Sciences of the United States of America 40:588-592. https://doi.org/10.1073/pnas.40.7.588

Dormann, C. F., J. M. Calabrese, G. Guillera‐Arroita, E. Matechou, V. Bahn, K. Bartoń, C. M. Beale, S. Ciuti, J. Elith, K. Gerstner, J. Guelat, P. Keil, J. J. Lahoz‐Monfort, L. J. Pollock, B. Reineking, D. R. Roberts, B. Schröder, W. Thuiller, D. I. Warton, B. A. Wintle, S. N. Wood, R. O. Wüest, and F. Hartig. 2018. Model averaging in ecology: a review of Bayesian, information‐theoretic, and tactical approaches for predictive inference. Ecological Monographs 88:485-504. https://doi.org/10.1002/ecm.1309

Eaton, M. J., F. A. Johnson, J. Mikels-Carrasco, D. J. Case, J. Martin, B. Stith, S. Yurek, B. Udell, L. Villegas, L. Taylor, Z. Haider, H. Charkhgard, and C. Kwon. 2021. Cape Romain partnership for coastal protection. Open-File Report 2021-1021. U.S. Geological Survey, Reston, Virginia, USA. https://doi.org/10.3133/ofr20211021

Eaton, M. J., S. Yurek, Z. Haider, J. Martin, F. A. Johnson, B. J. Udell, H. Charkhgard, and C. Kwon. 2019. Spatial conservation planning under uncertainty: adapting to climate change risks using modern portfolio theory. Ecological Applications 29(7):e01962. https://doi.org/10.1002/eap.1962

Enwright, N. M., H. Wang, P. S. Dalyander, and E. Godsey. 2020. Predicting barrier island habitats and oyster and seagrass habitat suitability for various restoration measures and future conditions for Dauphin Island, Alabama. Open-File Report 2020-1003. U.S. Geological Survey, Reston, Virginia, USA. https://doi.org/10.3133/ofr20201003

Enwright, N. M., L. Wang, P. S. Dalyander, H. Wang, M. J. Osland, R. C. Mickey, R. L. Jenkins, and E. S. Godsey. 2021. Assessing habitat change and migration of barrier islands. Estuaries and Coasts 44:2073-2086. https://doi.org/10.1007/s12237-021-00971-w

Fackler, P. L. 2018. Stochastic dynamic programming without transition matrices. CEnREP Working Paper No. 18-018. Center for Environmental and Resource Economic Policy, Raleigh, North Carolina, USA.

Fackler, P. L., K. Pacifici, J. Martin, and C. McIntyre. 2014. Efficient use of information in adaptive management with an application to managing recreation near Golden Eagle nesting sites. PLoS ONE 9(8):e102434. https://doi.org/10.1371/journal.pone.0102434

Five E’s Unlimited. 2007. Dauphin Island Strategic Plan—20 Year Vision: final report & implementation recommendations. Mississippi-Alabama, Sea Grant Consortium Number: MASGP07-023. Seattle, Washington, USA. http://www.eeeee.net/dauphin_island/di_final_report10-07.pdf

Game, E. T., J. A. Fitzsimons, G. Lipsett‐Moore, and E. McDonald‐Madden. 2013. Subjective risk assessment for planning conservation projects. Environmental Research Letters 8:45027. https://doi.org/10.1088/1748-9326/8/4/045027

Garrard, G. E., L. Rumpff, M. C. Runge, and S. J. Converse. 2017. Rapid prototyping for decision structuring: an efficient approach to conservation decision analysis. Pages 46-64 in N. Bunnefeld, E. Nicholson, and E. J. Milner-Gulland, editors. Decision-making in conservation and natural resource management: models for interdisciplinary approaches. Cambridge University Press, Cambridge, UK. https://doi.org/10.1017/9781316135938.003

Ghasemi-Saghand, P., Z. Haider, H. Charkhgard, M. Eaton, J. Martin, S. Yurek, B. J. Udell. 2021 SiteOpt: an Open-source R-package for Site Selection and Portfolio Optimization. Ecography 44:1678-1685. https://doi.org/10.1111/ecog.05717

Goodwin, P., and G. Wright. 2009. Decision analysis for management judgment. Wiley, Chichester, UK.

Gregory, R., J. Arvai, and L. R. Gerber. 2013. Structuring decisions for managing threatened and endangered species in a changing climate. Conservation Biology 27:1212-1221. https://doi.org/10.1111/cobi.12165

Gregory, R., L. Failing, M. Harstone, G. Long, T. McDaniels, and D. Ohlson. 2012. Structured decision making: a practical guide to environmental management choices. Wiley-Blackwell, Chichester, UK. https://doi.org/10.1002/9781444398557

Hagen, S. C., D. L. Passeri, M. V. Bilskie, D. E. DeLorme, and D. Yoskowitz. 2017. Systems approaches for coastal hazard assessment and resilience. Oxford Research Encyclopedia of Natural Hazard Science. https://doi.org/10.1093/acrefore/9780199389407.013.28

Hotaling, A. S., J. Martin, and W. M. Kitchens. 2009. Estimating transition probabilities among Everglades wetland communities using multistate models. Wetlands 29:1224-1233. https://doi.org/10.1672/09-014S

Hsiang, S., R. Kopp, A. Jina, J. Rising, M. Delgado, S. Mohan, D. J. Rasmussen, R. Muir-Wood, P. Wilson, M. Oppenheimer, K. Larsen, and T. Houser. 2017. Estimating economic damage from climate change in the United States. Science 356:1362-1369. https://doi.org/10.1126/science.aal4369

Irwin, E. R., K. O. Coffman, E. S. Godsey, N. M. Enwright, C. Lloyd, K. Joyner, and Q. Lai. 2020. Decision analysis of restoration actions for faunal conservation and other stakeholder values: Dauphin Island, Alabama. U.S. Geological Survey, Reston, Virginia, USA. https://doi.org/10.3133/70228873

Kirkwood, C. W. 1996. Strategic decision making. Wadsworth, Belmont, California, USA.

Knapp, J., J. Zeratsky, and B. Kowitz. 2016. Sprint: How to solve big problems and test new ideas in just five days. Simon and Schuster, New York, New York, USA.

Low, R. K. Y., R. Faff, and K. Aas. 2016. Enhancing mean-variance portfolio selection by modeling distributional asymmetries. Journal of Economics and Business 85:49-72. https://doi.org/10.1016/j.jeconbus.2016.01.003

Ludwig, K. A., D. W. Ramsey, N. J. Wood, A. B. Pennaz, J. W. Godt, N. G. Plant, N. Luco, T. A. Koenig, K. W. Hudnut, D. K. Davis, and P. R. Bright. 2018. Science for a risky world—a U.S. Geological Survey plan for risk research and applications. U.S. Geological Survey Circular 1444. https://doi.org/10.3133/cir1444

MacKenzie, D. I., J. D. Nichols, J. A. Royle, K. H. Pollock, L. Bailey, and J. E. Hines. 2017. Occupancy estimation and modeling: inferring patterns and dynamics of species occurrence. Second edition. Academic, San Diego, California, USA. https://doi.org/10.1016/C2012-0-01164-7

Martin, J., S. Chamaillé-Jammes, J. D. Nichols, H. Fritz, J. E. Hines, C. J. Fonnesbeck, D. I. MacKenzie, and L. L. Bailey. 2010. Simultaneous modeling of habitat suitability, occupancy, and relative abundance: African elephants in Zimbabwe. Ecological Applications 20:1173-1182. https://doi.org/10.1890/09-0276.1

Martin, J., P. Fackler, J. D. Nichols, B. Lubow, M. J. Eaton, M. C. Runge, B. M. Smith, and C. A. Langtimm. 2011. Structured decision making as a proactive approach to dealing with sea level rise in Florida. Climatic Change 107:185-202. https://doi.org/10.1007/s10584-011-0085-x

Martin, J., M. C. Runge, J. D. Nichols, B. L. Lubow, and W. L. Kendall. 2009. Structured decision making as a conceptual framework to identify thresholds for conservation and management. Ecological Applications 19:1079-1090. https://doi.org/10.1890/08-0255.1

Mickey, R. C., E. Godsey, P. S. Dalyander, V. Gonzalez, R. L. Jenkins III, J. W. Long, D. M. Thompson, and N. G. Plant. 2020b. Application of decadal modeling approach to forecast barrier island evolution, Dauphin Island, Alabama. Open-File Report 2020-1001. U.S. Geological Survey, Reston Virginia, USA. https://doi.org/10.3133/ofr20201001

Mickey, R. C., R. L. I. Jenkins, P. S. Dalyander, D. M. Thompson, N. G. Plant, and J. W. Long. 2020a. Dauphin Island decadal forecast evolution model inputs and results. U.S. Geological Survey, Reston, Virginia, USA. https://doi.org/10.5066/P9PDM1OJ

Mickey, R. C., J. W. Long, P. S. Dalyander, R. L. Jenkins III, D. M. Thompson, D. L. Passeri, and N. G. Plant. 2019. Development of a modeling framework for predicting decadal barrier island evolution. Open-File Report 2019-1139. U.S. Geological Survey, Reston Virginia, USA. https://doi.org/10.3133/ofr20191139

Miranda, M. J., and P. L. Fackler. 2002. Applied computational economics and finance. MIT Press, Cambridge, Massachusetts, USA.

Moore, J. L., A. E. Camaclang, A. L. Moore, C. E. Hauser, M. C. Runge, V. Picheny, and L. Rumpff. 2021. A framework for allocating conservation resources among multiple threats and actions. Conservation Biology 35:1639-1649. https://doi.org/10.1111/cobi.13748

Nichols, J. D., F. A. Johnson, B. K. Williams, and G. S. Boomer. 2015. On formally integrating science and policy: walking the walk. Journal of Applied Ecology 52:539-543. https://doi.org/10.1111/1365-2664.12406

Nichols, J. D., and B. K. Williams. 2006. Monitoring for conservation. Trends in Ecology and Evolution 21:668-673. https://doi.org/10.1016/j.tree.2006.08.007

O’Donnell, K. M., P. L. Fackler, F. A. Johnson, M. N. Bonneau, J. Martin, and S. C. Walls. 2020. Category count models for adaptive management of metapopulations: case study of an imperiled salamander. Conservation Science and Practice 2:4:e180. https://doi.org/10.1111/csp2.180

Partelow, S., A. Schlüter, D. Armitage, M. Bavinck, K. Carlisle, R. Gruby, A.-K. Hornidge, M. Le Tissier, J. Pittman, A. M. Song, L. P. Sousa, N. Văidianu, and K. Van Assche. 2020. Environmental governance theories: a review and application to coastal systems. Ecology and Society 25(4):19. https://doi.org/10.5751/ES-12067-250419

Passeri, D. L., M. V. Bilskie, S. C. Hagen, R. C. Mickey, P. S. Dalyander, and V. M. Gonzalez. 2021. Assessing the effectiveness of nourishment in decadal barrier island morphological resilience. Water 13:944. https://doi.org/10.3390/w13070944

Passeri, D. L., S. C. Hagen, S. C. Medeiros, M. V. Bilskie, K. Alizad, and D. Wang. 2015. The dynamic effects of sea level rise on low-gradient coastal landscapes: a review. Earth’s Future 3:159-181. https://doi.org/10.1002/2015EF000298

R Core Team. 2021. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

Regan, H. M., Y. Ben-Haim, B. Langford, W. G. Wilson, P. Lundberg, S. J. Andelman, and M. A. Burgman. 2005. Robust decision-making under severe uncertainty for conservation management. Ecological Applications 15:1471-1477. https://doi.org/10.1890/03-5419

Romito, A. M., J. F. Cochrane, M. J. Eaton, and M. C. Runge. 2015. Problem definition. Module 04 in M. C. Runge, A. M. Romito, G. Breese, J. F. Cochrane, S. J. Converse, M. J. Eaton, M. A. Larson, J. E. Lyons, D. R. Smith, A. F. Isham, editors. Introduction to structured decision making. U.S. Fish and Wildlife Service, National Conservation Training Center, Shepherdstown, West Virginia, USA.

Runge, M. C., S. J. Converse, and J. E. Lyons. 2011. Which uncertainty? Using expert elicitation and expected value of information to design an adaptive program. Biological Conservation 44(4):1214-1223. https://doi.org/10.1016/j.biocon.2010.12.020

Runge, M. C., S. J. Converse, J. E. Lyons. and D. R. Smith, editors. 2020. Structured decision making: case studies in natural resource management. Johns Hopkins University Press, Baltimore, Maryland, USA.

Tyre, A. J., J. T. Peterson, S. J. Converse, T. Bogich, D. Miller, M. Post van der Burg, C. Thomas, R. Thompson, J. Wood, D. C. Brewer, and M. C. Runge. 2011. Adaptive management of bull trout populations in the Lemhi Basin. Journal of Fish and Wildlife Management 2:262-281. https://doi.org/10.3996/022011-JFWM-012

Udell, B. J., J. Martin, R. J. Fletcher, M. Bonneau, H. H. Edwards, T. A. Gowan, S. K. Hardy, E. Gurarie, C. S. Calleson, and C. J. Deutsch. 2019. Integrating encounter theory with decision analysis to evaluate collision risk and determine optimal protection zones for wildlife. Journal of Applied Ecology 56:1050-1062. https://doi.org/10.1111/1365-2664.13290

U.S. Army Corps of Engineers (USACE). 2020. Final Alabama barrier island restoration assessment report. USACE, Washington, D.C., USA. https://gom.usgs.gov/DauphinIsland/data/ALBarrierIslRestoFinalRpt_2020.pdf

van der Burg, M. P., and A. J. Tyre. 2011. Integrating info‐gap decision theory with robust population management: a case study using the Mountain Plover. Ecological Applications 21:303-312. https://doi.org/10.1890/09-0973.1

Wideman, R. M. 1992. Project and program risk management: a guide to managing risks and opportunities. Project Management Institute, Upper Darby, Pennsylvania, USA.

Williams, B. K. 2001. Uncertainty, learning, and the optimal management of wildlife. Environmental and Ecological Statistics 8:269-288. https://doi.org/10.1023/A:1011395725123

Williams, B. K., J. D. Nichols, and M. J. Conroy. 2002. Analysis and management of animal populations. Academic, San Diego, California, USA.

Williams, B. K., R. C. Szaro, and C. D. Shapiro. 2009. Adaptive management: the U.S. Department of the Interior Technical Guide. Adaptive Management Working Group, U.S. Department of the Interior, Washington, D.C., USA.

Yurek, S., M. J. Eaton, R. Lavaud, R. W. Laney, D. L. DeAngelis, W. E. Pine, M. K. La Peyre, J. Martin, P. Frederick, H. Wang, M. R. Lowe, F. A. Johnson, E. V. Camp, and R. Mordecai. 2021. Modeling structural mechanics of oyster reef self-organization including environmental constraints and community interactions. Ecological Modelling 440:109389. https://doi.org/10.1016/j.ecolmodel.2020.109389

Zinnert, J. C., J. A. Stallins, S. T. Brantley, and D. R. Young. 2017. Crossing scales: the complexity of barrier-island processes for predicting future change. BioScience 67:39-52. https://doi.org/10.1093/biosci/biw154

Zweig, C. L., and W. M. Kitchens. 2014. Reconstructing historical habitat data with predictive models. Ecological Applications 24:196-203. https://doi.org/10.1890/13-0327.1