A stochastic approach to marine reserve design: Incorporating data uncertainty

Abstract

Marine reserves, or protected areas, are used to meet an array of biodiversity and conservation objectives. The design of regional networks of marine reserves is concerned with the problem of where to place the marine protected areas and how to spatially configure them. Quantitative methods for doing this provide important decision support tools for marine managers. The central problem is to balance the costs and benefits of the reserve network, whilst satisfying conservation objectives (hence solving a constrained optimization problem). Current optimization algorithms for reserve design are widely used, but none allow for the systematic incorporation of data uncertainty and its effect on the reserve design solutions. The central purpose of this study is to provide a framework for incorporating uncertain ecological input data into algorithms for designing networks of marine reserves.

In order to do this, a simplified version of the marine reserve design optimization problem is considered. A Metropolis–Hastings random search procedure is introduced to systematically sample the model solution space and converge on an optimal reserve design. Incorporation of the uncertain input data builds on this process and relies on a parametric bootstrapping procedure. This allows for the solution (i.e. the marine reserve design) to be expressed as the probability of any planning unit being included in the marine reserve network. Spatial plots of this acceptance probability are easily interpretable for decision making under uncertainty. The bootstrapping methodology is also readily adapted to existing comprehensive reserve design algorithms. Here, a preliminary application of the algorithm is made to the Mesoamerican Barrier Reef System (in the Caribbean Sea) based on satellite-derived and mapped conservation features (from Landsat).

Introduction

Marine Protected Areas (MPAs) have been established worldwide to conserve and protect highly valued ecosystems such as coral reefs (Agardy et al., 2000). The design and selection of a network of MPAs in physically and biologically diverse areas, taking account a variety of ecological factors such as biodiversity and fisheries (Lubchenco et al., 2003), as well as social and economic considerations (Richardson et al., 2006). Conservation and protection must be balanced against potential economic losses such as natural resources (food and pharmaceuticals), coastline and property protection, as well as tourism (Costanza et al., 1997). For instance, the MPA designer is confronted with the question of Single Large Or Several Small (SLOSS) MPAs, and how far apart the MPAs should be and where the MPAs should be situated (Palumbi et al., 2002). Such considerations provide the reasoning behind using a systematic, quantitative approach to reserve design. Optimization models provide a formalized mechanism for balancing the many competing criteria for MPA network design, and application tools have been developed and are enjoying increasing use (McDonnell et al., 2002, Leslie et al., 2003). In this paper we develop a novel stochastic approach to this problem, and apply it to coral reef ecosystems of the Mesoamerican Barrier Reef System.

One popular optimization algorithm for reserve design is implemented in the MARXAN software (Possingham et al., 2000). This provides decision-makers with a rigorous method for locating a set of conservation areas that meet quantitative conservation criteria, taking account environmental and economic costs, thereby providing an objective basis for further discussion with stakeholders. Examples of reserve designs provided by MARXAN exist for the Gulf of California (Sala et al., 2002), and the central coast of British Columbia (Ardron et al., 2002). Leslie et al., (2003) also developed a siting algorithm that used simulated annealing and “irreplaceability analysis” to site marine reserves in the Florida Keys. Roberts et al. (2003) developed a method of examining ecological criteria to evaluate candidate models for reserve design.

Quantitative reserve design is generally treated as a constrained optimization algorithm. It involves the minimization of some measure of the cost of implementing and managing the reserve system, while at the same time meeting an array of biodiversity goals or constraints. Optimization algorithms currently used in reserve design are complex and often based on simulated spatial annealing algorithms (Kirkpatrick et al., 1983, Halpern et al., 2006). These proceed by creating random spatial distributions of reserves within the domain of interest, and then iteratively examine possible reserve placement solutions by making progressive changes to the reserve system. In this way, the minimization of a user-defined cost function is carried out, constraints are satisfied, and an optimal solution reserve design can be obtained, often with the aid of criteria such as irreplaceability (Carwardine et al., 2007).

Such reserve design algorithms are based on the assumption that data inputs, which are used to define the cost function and constraints, are known, accurate and measured without error. As a consequence, the problem is treated from the perspective of deterministic optimization. Clearly, the input ecological or economic data is most often uncertain, and there has been much recent interest in the effect of this uncertainty (Moilanen et al., 2006, Pressey et al., 2007, Grand et al., 2007). Hence, the central motivation for our study is to quantify the effect of data uncertainty on the design of networks of MPAs by developing algorithms for stochastic simulation that can properly incorporate uncertainty into model predictions. These provide an effective tool for quantifying the consequences of uncertainty in the reserve which are useful for management decision and policy makers.

Towards this end, we carry out stochastic simulation for the general problem of marine reserve design. This relies on a (stochastic) optimization algorithm, the Metropolis–Hastings algorithm (Hastings, 1970), to solve the deterministic constrained optimization algorithm. To account for data uncertainty this optimization is embedded within a bootstrap procedure.

In this way, uncertainty in the input data can be used to assess the uncertainty in the resultant reserve design. In fact, the goal is to provide the user with a solution that yields the probability of including a particular area in the final reserve design, in other words the probability distribution for the reserve design solution. In this way, the robustness of the reserve design solution is quantified. As well, it provides a convenient means to make rational choices for alternative marine reserve designs. Most notably, the ideas presented here are fully generalizable and can be directly used with current optimization based reserve design algorithms such as MARXAN.

Here we discussed the problem of quantitative marine reserve design and the effect of uncertainty in the input data. Approaches to marine reserve design are outlined from both deterministic and stochastic perspectives. We use an idealized example to illustrate the issues and procedure before presenting a more realistic application of the model to the Mesoamerican Barrier Reef System (Fig. 1), using geospatial data derived from the analysis of Landsat-7 satellite imagery. Uncertainty associated with these estimates of the areas of various conservation features is quantified and used to illustrate how it can be dealt with probabilistically in the optimization process. Practical implementation issues are highlighted and results interpreted in the context of current applications of statistical models to the design of networks of marine reserves.