A density projection approach for non-trivial information dynamics: Adaptive management of stochastic natural resources

We demonstrate a density projection approximation method for solving resource management problems with imperfect state information. The method expands the set of partially-observed Markov decision process (POMDP) problems that can be solved with standard dynamic programming tools by addressing dimensionality problems in the decision maker's belief state. Density projection is suitable for uncertainty over both physical states (e.g. resource stock) and process structure (e.g. biophysical parameters). We apply the method to an adaptive management problem under structural uncertainty in which a fishery manager's harvest policy affects both the stock of fish and the belief state about the process governing reproduction. We solve for the optimal endogenous learning policy—the active adaptive management approach—and compare it to passive learning and non-learning strategies. We demonstrate how learning improves efficiency but typically follows a period of costly short-run investment.     

Resolving structural uncertainty in natural resources management using POMDP approaches

In recent years there has been a growing focus on the uncertainties of natural resources management, and the importance of accounting for uncertainty in assessing management effectiveness. This paper focuses on uncertainty in resource management in terms of discrete-state Markov decision processes (MDP) under structural uncertainty and partial observability. It describes the treatment of structural uncertainty with approaches developed for partially observable resource systems. In particular, I show how value iteration for partially observable MDPs (POMDP) can be extended to structurally uncertain MDPs. A key difference between these process classes is that structurally uncertain MDPs require the tracking of system state as well as a probability structure for the structure uncertainty, whereas with POMDPs require only a probability structure for the observation uncertainty. The added complexity of the optimization problem under structural uncertainty is compensated by reduced dimensionality in the search for optimal strategy. A solution algorithm for structurally uncertain processes is outlined for a simple example in conservation biology. By building on the conceptual framework developed for POMDPs, natural resource analysts and decision makers who confront structural uncertainties in natural resources can take advantage of the rapid growth in POMDP methods and approaches, and thereby produce better conservation strategies over a larger class of resource problems.     

Minimise long-term loss or maximise short-term gain?: Optimal translocation strategies for threatened species

Translocation is a useful management option for conservation of threatened animal species. It can be used to increase the range of a species, augment the numbers in a critical population, or establish new populations and hence spread the risk of extinction through local catastrophes. As it is an important and expensive conservation tool, translocation management decisions must be carefully considered, with the objective of the translocation project in mind. By analysing the translocation problem within a decision-theory framework, we find optimal management decisions that are rational and transparent. We illustrate our approach using a case study of the bridled nailtail wallaby (Onychogalea fraenata). Our particular translocation question is: if we have a set number of wallabies to translocate in each time period and two translocation sites, how many wallabies should we put at each site given the state of each population to maximise the benefit to the species? We model the translocated populations with first-order Markov chain stochastic population models, and use stochastic dynamic programming to determine the optimal management decisions. We look at two sites with different growth rates – one increasing and one decreasing – and compare the optimal strategies for two different objective functions. The first is a long-term persistence objective function, which maximises the persistence of translocated populations a large number of time steps after the end of the translocation program. The second maximises total population size at the end of the translocation program. Although these objective functions are similar, they generate surprisingly different optimal translocation strategies. When maximising the long-term persistence of the translocated populations, translocation decisions are not important as long as an increasing population is established. This indicates that site quality – rather than the number and timing of translocations – primarily determines the long-term persistence of populations. When maximising total population size, the optimal strategy is to add to the increasing population unless it is above a size where it is likely to reach its carrying capacity over the planning timeframe. As translocation decisions are important in fulfilling the objective, this objective function is more useful in creating practical advice for translocation managers. The discrepancy between the optimal strategies given by the two objectives demonstrates the importance of careful consideration when specifying the goals of a project. This observation applies not only to translocation programs, but any project where clear decision-making is needed.     

Subpopulation Triage: How to Allocate Conservation Effort among Populations

Abstract:  Threatened species often exist in a small number of isolated subpopulations. Given limitations on conservation spending, managers must choose from strategies that range from managing just one subpopulation and risking all other subpopulations to managing all subpopulations equally and poorly, thereby risking the loss of all subpopulations. We took an economic approach to this problem in an effort to discover a simple rule of thumb for optimally allocating conservation effort among subpopulations. This rule was derived by maximizing the expected number of extant subpopulations remaining given n subpopulations are actually managed. We also derived a spatiotemporally optimized strategy through stochastic dynamic programming. The rule of thumb suggested that more subpopulations should be managed if the budget increases or if the cost of reducing local extinction probabilities decreases. The rule performed well against the exact optimal strategy that was the result of the stochastic dynamic program and much better than other simple strategies (e.g., always manage one extant subpopulation or half of the remaining subpopulation). We applied our approach to the allocation of funds in 2 contrasting case studies: reduction of poaching of Sumatran tigers ( Panthera tigris sumatrae ) and habitat acquisition for San Joaquin kit foxes ( Vulpes macrotis mutica ). For our estimated annual budget for Sumatran tiger management, the mean time to extinction was about 32 years. For our estimated annual management budget for kit foxes in the San Joaquin Valley, the mean time to extinction was approximately 24 years. Our framework allows managers to deal with the important question of how to allocate scarce conservation resources among subpopulations of any threatened species.     

Dynamic reserve site selection under contagion risk of deforestation

Recently, dynamic reserve site selection models based on stochastic dynamic programming (SDP) have been proposed. The models consider a random development pattern in which the probability that a site will be developed is independent of the development status of other sites. However, development often takes the form of a contagion process in which the sites most likely to be developed are near sites that already have been developed. To consider site selections in such cases, we propose improved algorithms that make use of a graph representation of the sites network. The first formulation is an exact, dynamic programming algorithm, with which theoretical and experimental complexities are evaluated. The exact method can be applied only to small problems (less than 10 sites), but real-world problems may have hundreds or thousands of sites, implying that heuristic selection methods must be used. We provide a general framework for describing such heuristic solution methods, and propose a new heuristic method based on a parameterised reinforcement learning algorithm. The method allows us to compute a heuristic function by performing and exploiting many simulations of the deforestation process. We show that the method can be applied to problems with hundreds of sites, and demonstrate experimentally that it outperforms previously proposed heuristic methods in terms of the average number of species conserved.     

Metapopulation persistence in a dynamic landscape: more habitat or better stewardship?

Habitat loss and fragmentation has created metapopulations where there were once continuous populations. Ecologists and conservation biologists have become interested in the optimal way to manage and conserve such metapopulations. Several authors have considered the effect of patch disturbance and recovery on metapopulation persistence, but almost all such studies assume that every patch is equally susceptible to disturbance. We investigated the influence of protecting patches from disturbance on metapopulation persistence, and used a stochastic metapopulation model to answer the question: How can we optimally trade off returns from protection of patches vs. creation of patches? We considered the problem of finding, under budgetary constraints, the optimal combination of increasing the number of patches in the metapopulation network vs. increasing the number of protected patches in the network. We discovered that the optimal trade-off is dependent upon all of the properties of the system: the species dynamics, the dynamics of the landscape, and the relative costs of each action. A stochastic model and accompanying methodology are provided allowing a manager to determine the optimal policy for small metapopulations. We also provide two approximations, including a rule of thumb, for determining the optimal policy for larger metapopulations. The method is illustrated with an example inspired by information for the greater bilby, Macrotis lagotis, inhabiting southwestern Queensland, Australia. We found that given realistic costs for each action, protection of patches should be prioritized over patch creation for improving the persistence of the greater bilby during the next 20 years.     

A state-dependent model for the optimal management of an invasive metapopulation.

Management of invasive species involves choosing between different management strategy options, but often the best strategy for a particular scenario is not obvious. We illustrate the use of optimization methods to determine the most efficient management strategy using one of the most devastating invasive forest pests in North America, the gypsy moth (Lymantria dispar), as a case study. The optimization approach involves the application of stochastic dynamic programming (SDP) to a metapopulation framework with different infestation patch sizes, with the goal of minimizing infestation spread. We use a novel "moving window" approach as a way to address a spatially explicit problem without being explicitly spatial. We examine results for two cases in order to develop general rules of thumb for management. We explore a model with limited parameter information and then assess how strategies change with specific parameterization for the gypsy moth. The model results in a complex but stable, state-dependent management strategy for a multiyear management program that is robust even under situations of uncertainty. The general rule of thumb for the basic model consists of three strategies: eradicating medium-density infestations, reducing large-density infestations, and reducing the colonization rate from the main infestation, depending on the state of the system. With specific gypsy moth parameterization, reducing colonization decreases in importance relative to the other two strategies. The application of this model to gypsy moth management emphasizes the importance of managing based on the state of the system, and if applied to a specific geographic area, has the potential to substantially improve the efficiency and cost-effectiveness of current gypsy moth eradication programs, helping to slow the spread of this pest. Additionally, the approach used for this particular invasive species can be extended to the optimization of management programs for the spread of other invasive and problem species exhibiting metapopulation dynamics.     

Optimal monitoring and control under state uncertainty: Application to lionfish management

State variables in many renewable resource management problems, such as the abundance of a fish stock, are imperfectly observed over time. In systems characterized by state uncertainty, decision makers often invest in monitoring to learn about the level of a stock. We develop a stochastic bioeconomic model of marine invasive species management under state uncertainty. The decision maker in our model simultaneously evaluates optimal investment in monitoring and population control. Using a recently-devised method for solving continuous-state Partially Observable Markov Decision Processes (POMDPs), we find that the ability to learn through monitoring can alter the role of population control in the optimal policy function, for example by reducing control intensity in favor of monitoring. Optimal monitoring depends on the management context, including in our application lionfish population structure. The rich transient dynamics of our model depend critically on the relationship between the initial conditions for information and invader abundance.     

Optimization model for a wolf-ungulate system

Public outcries against predator control create a need to devise management policies that optimally balance the cost (managerial and environmental) of predator control against the benefit of ungulate harvesting. To address this problem, an optimization procedure utilizing stochastic dynamic programming is described. Through this approach, optimal feedback strategies for a wolf-ungulate system in Alaska are estimated. The dynamic predator-prey model used in the analysis is based on parameter estimates from data collected over an eight-year period in Denali (Mt. McKinley) National Park. Stability analysis of the system revealed that stability properties depend on predator search efficiency. The effects of random fluctuations in winter severity and alternative objective functions are considered in the estimation of optimal feedback strategies. Optimal moose harvesting strategies appear to be dependent on wolf control costs. If no wolf control cost is assessed, optimal moose harvest is independent of wolf density. Optimal wolf control strategies are completely insensitive to moose density. The strategies are compared to current and simplified management policies.     

River water quality management model using genetic algorithm

Conventional mathematical programming methods, such as linear programming, non linear programming, dynamic programming and integer programming have been used to solve the cost optimization problem for regional wastewater treatment systems. In this study, a river water quality management model was developed through the integration of a genetic algorithm (GA). This model was applied to a river system contaminated by three determined discharge sources to achieve the water quality goals and wastewater treatment cost optimization in the river basin. The genetic algorithm solution, described the treatment plant efficiency, such that the cost of wastewater treatment for the entire river basin is minimized while the water quality constraints in each reach are satisfied. This study showed that genetic algorithm can be applied for river water quality modeling studies as an alternative to the present methods.     

Finding all optimal solutions to the reserve site selection problem: formulation and computational analysis

The problem of selecting nature reserves has received increased attention in the literature during the past decade, and a variety of approaches have been promoted for selecting those sites to include in a reserve network. One set of techniques employs heuristic algorithms and thus provides possibly sub-optimal solutions. Another set of models and accompanying algorithms uses an integer programming formulation of the problem, resulting in an optimization problem known as the Maximal Covering Problem, or MCP. Solution of the MCP provides an optimal solution to the reserve site selection problem, and while various algorithms can be employed for solving the MCP they all suffer from the disadvantage of providing a single optimal solution dictating the selection of areas for conservation. In order to provide complete information to decision makers, the determination of all alternate optimal solutions is necessary. This paper explores two procedures for finding all such solutions. We describe the formulation and motivation of each method. A computational analysis on a data set describing native terrestrial vertebrates in the state of Oregon illustrates the effectiveness of each approach.     

A Real Options Approach to the Valuation of a Forestry Investment

The theory of real options is used to model the optimal tree harvesting decision. The value of the option to harvest is estimated using a dynamic programming approach and a general numerical solution technique suitable for any type of stochastic process for prices. The conditions under which the optimal harvest time is independent of price are examined. In addition, the impact of assuming future lumber prices are mean reverting is considered. It is found that option value and optimal cutting time are significantly different under the mean reversion assumption compared to geometric Brownian motion.     

The Quantitative Incidence Function Model and Persistence of an Endangered Butterfly Metapopulation

The incidence function model is derived from a linear first-order Markov chain of the presence or absence of a species in a habitat patch. The model can be parameterized with "snapshot" presence/absence data from a patch network. Using the estimated parameter values the Markov chain can be iterated in the same or in some other patch network to generate quantitative predictions about transient metapopulation dynamics and the stochastic steady state. We tested the ability of the incidence function model to predict patch occupancy using extensive data on an endangered butterfly, the Glanville fritillary ( Melitaea cinxia ) Parameter values were estimated with data collected from a 50-patch network in 1991. In 1993 we surveyed the entire geographic range of the species in Finland, within an area of 50 × 70 km², with 1502 habitat patches (dry meadows) of which 536 were occupied. Model predictions were generated for the 1502 patches and were compared with the observed pattern of occupancy in 1993. The model predicted patch occupancy well in more than half of the study area, but prediction was poor for one quarter of the area, probably because of regional variation in habitat quality and because metapopulations may have been perturbed away from the steady state. The incidence function model provides a practical tool for making quantitative predictions about metapopulation dynamics of species living in fragmented landscapes.     

Simple rules for ranking and optimally managing metapopulations

We present two ‘rules of thumb’ for metapopulation management. The first identifies an explicit formula for the persistence time of the population, and thus enables the population manager to form a priority species ranking by identifying those species most at risk of extinction. The second identifies an optimal management strategy that gives direction on how to alter the colonisation rate (creation or improvement of habitat corridors) and local extinction rate (restoring habitat quality or expanding habitat) in order to maximise the persistence time under a budgetary constraint. We employ a simple stochastic version of Levins (1969) metapopulation model, which is first calibrated to a more realistic spatial model. Our rules are tested on computer-generated patch networks and a model for malleefowl (Leipoa ocellata) in the Bakara region of South Australia.     

Why are metapopulations so rare?

Roughly 40 years after its introduction, the metapopulation concept is central to population ecology. The notion that local populations and their dynamics may be coupled by dispersal is without any doubt of great importance for our understanding of population-level processes. A metapopulation describes a set of subpopulations linked by (rare) dispersal events in a dynamic equilibrium of extinctions and recolonizations. In the large body of literature that has accumulated, the term "metapopulation" is often used in a very broad sense; most of the time it simply implies spatial heterogeneity. A number of reviews have recently addressed this problem and have pointed out that, despite the large and still growing popularity of the metapopulation concept, there are only very few empirical examples that conform with the strict classical metapopulation (CM) definition. In order to understand this discrepancy between theory and observation, we use an individual-based modeling approach that allows us to pinpoint the environmental conditions and the life-history attributes required for the emergence of a CM structure. We find that CM dynamics are restricted to a specific parameter range at the border between spatially structured but completely occupied and globally extinct populations. Considering general life-history attributes, our simulations suggest that CMs are more likely to occur in arthropod species than in (large) vertebrates. Since the specific type of spatial population structure determines conservation concepts, our findings have important implications for conservation biology. Our model suggests that most spatially structured populations are panmictic, patchy, or of mainland-island type, which makes efforts spent on increasing connectivity (e.g., corridors) questionable. If one does observe a true CM structure, this means that the focal metapopulation is on the brink of extinction and that drastic conservation measures are needed.     

Conservation planning for successional landscapes

The systematic conservation planning literature invariably assumes that the biodiversity features being preserved in sites do not change through time. We develop a conservation planning framework for ecosystems where disturbance events and succession drive vegetation dynamics. The framework incorporates three key attributes of disturbance theory: heterogeneity in disturbance rates, spatial correlation between disturbance events and different impacts of disturbance. In our conservation problem we wish to maximise the chance that we represent a certain number of successional types given a cap on the number of sites we can conserve. Correlation between disturbance events dramatically complicates the problem of choosing the optimal suite of sites. However, in our problem we discover that spatial correlation in disturbances affects the optimal reserve network very little. The reason is twofold: (i) through our probabilistic framework we focus on the long-term effectiveness of reserve networks and (ii) in the dynamics considered in our model the state of a site is not only affected by the most recent (correlated) disturbance event but also by the site's long-term stochastic history which blurs the impact of spatial correlation. If successional states are the conservation target rather than individual species then, conserving a site can only contribute to meeting one target. However, given that correlation of disturbance events may be ignored, we show that if the number of candidate reserves is sufficiently large the statistical dependence of different conservation targets may be ignored, too. We conclude that the computational complexity of reserve selection methods for dynamic ecosystems can be much simpler than they first appear.     

Conservation Corridors and Contagious Disease: A Cautionary Note

Recent conservation proposals frequently include the establishment of corridors to connect isolated patches of wildlife habitat. Much attention has been focused on the potential benefits of corridors with little note given to potentially adverse consequences. A simulation model is developed here to study the effect of corridors on the survival of a metapopulation in the presence of a fatal disease that is spread by direct contact between susceptible and infected individuals. For the disease modeled here, a landscape of patches connected by corridors generally suffers fewer metapopulation extinctions than a landscape of isolated patches. However, under a narrow range of conditions, results suggest that corridors may dramatically increase the probability of metapopulation extinction. This occurs when disease-induced mortality is low enough to allow infected individuals to spread the disease, but high enough to reduce population levels to the point that random demographic and environmental events cause frequent metapopulation extinctions. This has important implications for the design and management of conservation reserve networks. Although discussion focuses primarily on conservation corridors, the model results apply to any management techniques that increase the movement of individuals among populations.     

The role of habitat disturbance and recovery in metapopulation persistence

Classical metapopulation theory assumes a static landscape. However, empirical evidence indicates many metapopulations are driven by habitat succession and disturbance. We develop a stochastic metapopulation model, incorporating habitat disturbance and recovery, coupled with patch colonization and extinction, to investigate the effect of habitat dynamics on persistence. We discover that habitat dynamics play a fundamental role in metapopulation dynamics. The mean number of suitable habitat patches is not adequate for characterizing the dynamics of the metapopulation. For a fixed mean number of suitable patches, we discover that the details of how disturbance affects patches and how patches recover influences metapopulation dynamics in a fundamental way. Moreover, metapopulation persistence is dependent not only on the average lifetime of a patch, but also on the variance in patch lifetime and the synchrony in patch dynamics that results from disturbance. Finally, there is an interaction between the habitat and metapopulation dynamics, for instance declining metapopulations react differently to habitat dynamics than expanding metapopulations. We close, emphasizing the importance of using performance measures appropriate to stochastic systems when evaluating their behavior, such as the probability distribution of the state of the metapopulation, conditional on it being extant (i.e., the quasistationary distribution).     

Ungulate population dynamics and optimization models

Optimal harvesting strategies for an ungulate population are estimated using stochastic dynamic programming. Data on the Llano Basin white-tailed deer (Odocoileus virginianus) population were used to construct a 2-variable population dynamics model. The model provided the basis for estimating optimal harvesting strategies as a feedback function of the current values of the state variables (prefawning older deer and juveniles). Optimal harvest strategies were insensitive to assumptions about the probability distributions of the stochastic variable (rainfall). The response of the population components to harvesting and the returns obtained from applying optimal strategies were explored through simulation. Mean annual harvest is about 15% of the population. Simplified harvesting strategies based on age-ratios as well as a simplified version based on optimal strategies—but assuming persisting equilibrium juvenile deer density—were compared to optimal strategies through examining values of information. Simplified harvesting strategies lead to a lower harvest over a 50-year simulation period.     

Human perspectives and conservation of grizzly bears in Banff National Park, Canada

Some conservation initiatives provoke intense conflict among stakeholders. The need for action, the nature of the conservation measures, and the effects of these measures on human interests may be disputed. Tools are needed to depolarize such situations, foster understanding of the perspectives of people involved, and find common ground. We used Q methodology to explore stakeholders' perspectives on conservation and management of grizzly bears (Ursus arctos horribilis) in Banff National Park and the Bow River watershed of Alberta, Canada. Twenty-nine stakeholders participated in the study, including local residents, scientists, agency employees, and representatives of nongovernmental conservation organizations and other interest groups. Participants rank ordered a set of statements to express their opinions on the problems of grizzly bear management (I-IV) and a second set of statements on possible solutions to the problems (A-C). Factor analysis revealed that participants held 4 distinct views of the problems: individuals associated with factor I emphasized deficiencies in goals and plans; those associated with factor II believed that problems had been exaggerated; those associated with factor III blamed institutional flaws such as disjointed management and inadequate resources; and individuals associated with factor IV blamed politicized decision making. There were 3 distinct views about the best solutions to the problems: individuals associated with factor A called for increased conservation efforts; those associated with factor B wanted reforms in decision-making processes; and individuals associated with factor C supported active landscape management. We connected people's definitions of the problem with their preferred solutions to form 5 overall problem narratives espoused by groups in the study: the problem is deficient goals and plans, the solution is to prioritize conservation efforts (planning-oriented conservation advocates); the problem is flawed institutions, the solution is to prioritize conservation efforts (institutionally-oriented conservation advocates); the problems have been exaggerated, but there is a need to improve decision-making processes (optimistic decision-process reformers); the problems have been exaggerated, but managers should more actively manage the landscape (optimistic landscape managers); and the problem is politicized decision making, solutions vary (democratizers). Although these 5 groups differed on many issues, they agreed that the population of grizzly bears is vulnerable to extirpation, human use of the area should be designed around ecological constraints, and more inclusive decision-making processes are needed. We used our results to inform a series of workshops in which stakeholders developed and agreed on new management strategies that were implemented by Parks Canada. Our research demonstrates the usefulness of Q method to illuminate people's perspectives and identify common ground in settings where conservation is contested.