Abstract
A model for the recovery of an endangered species is developed and applied to the red-cockaded woodpecker (Picoides borealis), a species once abundant in the southeastern United States. There is a finite set of recovery actions that might be implemented in each period, with the goal of reaching a population target at some future date. Dynamic programming is used to solve deterministic and stochastic versions of the model. Least-cost recovery plans are found for the deterministic problem where it is possible to attain a population target with certainty. For the stochastic problem, least-cost, adaptive actions are identified. (JEL Q24, Q28)
I. Introduction
The Endangered Species Act (ESA) of 1973 provided a critical legal and strategic framework for saving threatened species from extinction. The ESA charged the U.S. Fish and Wildlife Service (USFWS) and the National Oceanic and Atmospheric Administration (NOAA) with identifying threatened species and formulating recovery plans that would establish and maintain viable populations (NOAA 2012).
Section 4(f) of the ESA states that, at a minimum, species recovery plans must include the following:
A description of site-specific management actions necessary to achieve recovery of the species
Objective, measurable criteria that, when met, would result in a determination that the species be removed from the list
Estimates of the time and costs required to achieve the plan's goal
Under the ESA, the listing of a species, designation of critical habitat, and the formulation of recovery plans were almost exclusively the domain of ecologists and population biologists. In the 1978 case, Tennessee Valley Authority v. Hill,1 the U.S. Supreme Court ruled that the ESA required federal officials to “halt and reverse the trend toward species extinction—whatever the cost” (see Brown and Shogren 1998). Economists were viewed with suspicion. However, given that there may be numerous ways to meet the objective criteria necessary for delisting, economists would advocate for selecting the least-cost recovery plan (Halsing and Moore 2008). We would maintain that recovery resources are scarce, and finding the best, feasible recovery strategy is inherently an optimization problem.
We develop a general model of cost-effective recovery when management actions are discrete variables and then apply both deterministic and stochastic specifications to the red-cockaded woodpecker (Picoides borealis), a federally listed endangered species that was once abundant in the southeastern United States. Our model is based on a recovery strategy where management actions are chosen from finite sets in each time period. This leads to a dynamic optimization problem seeking the least-cost sequence of recovery strategies to achieve a population target at some future date. We show that deterministic and stochastic versions of this problem can be solved using dynamic programming.
There have been numerous studies aimed at accounting for the economic costs of species conservation plans. Conrad and Salas (1993) used stochastic simulation to analyze the economic trade-offs in different land use management strategies and their effect on biodiversity. The authors simulated population dynamics of the Monarch butterfly population in Michoacán state in central Mexico under different levels of timber harvest in the forest areas surrounding the butterfly preserve. This repeated simulation lead to the construction of a production possibilities frontier exhibiting the trade-offs between the present value of net revenue from timber harvest and the population size of the Monarch butterfly.
Montgomery, Brown, and Adams (1994) developed an analytical framework for evaluating recovery programs when a species is faced with the possibility of extinction and the likelihood of species persistence is correlated with management effort/expenditure. Specifically, the authors determined whether the benefits of preserving a species outweigh the costs of preservation. By analyzing alternative management strategies, they estimated the costs associated with species extinction probabilities, leading to the creation of a marginal cost curve for the probability of the species continued viability. The authors argued that the most important policy choice variable is the probability of species survival because one cannot determine with certainty whether a species will persist or go extinct.
Haight (1995) presented a management decision framework wherein managers seek to maximize the present value of current land use practice while satisfying a population viability constraint. This constraint required the probability of achieving an objective population size at the end of the management horizon be greater than or equal to an acceptable lower bound. The framework assigned a cost for each combination of target population size and probabilistic lower bound. Managers could then analyze the trade-offs intrinsic in selecting an acceptable amount of “risk.” Marshall, Homans, and Haight (2000) used this framework to examine the cost-effectiveness of management strategies for Kirtland's warbler. They used stochastic simulation to assign probabilities to possible population responses to management. These probabilities were then used in a chance-constrained optimization framework wherein they sought to minimize management cost (in this case, diminished revenue from current logging practice) while satisfying a population viability constraint.
Another study highlights the importance of accounting for direct wildlife conservation actions in order to find truly cost-effective management plans. Rashford and Adams (2007) developed a bioeconomic model for cost-effective conservation planning that accounted for both land use practice and direct conservation activities, including predator control and habitat restoration. They used a deterministic model to find a total cost function for the production of waterfowl across a variety of landscapes.
This paper contributes to the existing literature by proposing an alternative framework for determining cost-effective species recovery plans in a stochastic, dynamic setting. This framework is particularly suited to the case where management activities are integer variables. We argue that the majority of direct conservation actions are inherently integer valued. Our framework can accommodate primary land use activities, land acquisition, and direct management actions. The model can be used to develop least-cost management plans for individual wildlife populations, which can assist wildlife managers when developing comprehensive species recovery plans to achieve a population target at least cost.
Some High-Profile Endangered Species in the United States
For many endangered species inthe United States, translocation of individuals or breeding pairs from wild populations or breeding facilities (often zoos or government facilities) has played an important role in the recovery or reestablishment of an extirpated population. Translocation, by its very nature, is an integer-valued activity. Because other recovery activities are often oriented toward individuals in an endangered population, they too may be integer variables. Table 1 provides a brief description of translocation and other recovery activities for six, high-profile, endangered species in the United States.
As seen in Table 1, translocation has been used to reintroduce a species to areas where it was extirpated and to increase genetic diversity where an isolated population is showing signs of inbreeding, as with the Florida panther. Other recovery actions are often related to the number of individuals in a population. For example, in the case of the Florida panther and red wolf, vaccination against disease and treatment for parasites has been used to increase the survival of juveniles. For the California condor, “nest watching” and the removal of “microtrash” from the crops of chicks was critical to increasing wild populations in California. When reestablishing the eastern population of whooping crane, ultralight aircraft were used to teach “puppetreared” juveniles the migratory route from the Nacedah National Wildlife Refuge in Wisconsin to an overwintering site in central Florida. For the red-cockaded woodpecker (RCW), translocation has been used to increase both population size and genetic diversity in isolated communities. The construction of artificial cavities has been used to maintain and increase the carrying capacity of RCW habitat.
Six High-Profile Endangered Species in the United States
Not reported in Table 1 are estimates of the total expenditures made to reestablish wild populations of these high-profile species. Accurate estimates are difficult to make because recovery efforts often use money and resources from both private (nonprofit) conservation groups and federal, state, and local governments. The problem is made more difficult because staff and overhead in both nonprofit organizations and government agencies have to be allocated across several activities, only some of which might be dealing with a federally listed species. That said, species recovery costs are generally quite high. It was estimated that $48 million is needed to fund recovery efforts that might lead to the delisting of the whooping crane (USFWS 1994). The Florida panther recovery plan was estimated to cost $17.75 million for five years, 2008-2012 (USFWS 2008). Tobin and Dusheck (2005) estimated that the red wolf recovery plan has cost $1 million per year since 1974. Estimates of the per-wolf costs for the reintroduction of the gray wolf range from $200,000 to $1,000,000 (Daley and Trevis 2005). The cost per year to save the California condor has been estimated at $5 million (Barlow 2008).
II. The General Model
Suppose there is a finite list of qualitative actions that might be employed to hasten the recovery of an endangered species on a preserve owned by a government or conservation organization, i ∈ I, each metered out in a quantitative amount in each period, Xi,t∈ Xi∙, for I∈ {1,2,...,Ī} and Xi∈ {0,1,...,Xi,MAX}. For example, we will designate the number of RCW breeding pairs translocated in period t by X1,t. Other actions, i ≠ 1, might increase carrying capacity on the property or reduce mortality from predators or disease.
The dynamics of an endangered species might be modeled as a stochastic map. Let Nt ≥ 0 denote the number of individuals or breeding pairs in the preserve in period t. The population in t +1 is a realization of the stochastic map, Nt+1 = F(Nt,Kt,Xt,εt+1), where Kt is the preserve's carrying capacity in period t (discussed in greater detail below), Xt = [X1,t,X2,t,...,X,Ī,t] is the recovery strategy in period t, and εt+1 is an independently and identically distributed (iid) random variable from the distribution, φ(εt+1).2 The recovery strategy is Markovian—management actions in period t are chosen based solely on the current state, (Nt, Kt), and are independent of all previous states and actions.
The Penalty Function in the Terminal Time Period, t=T
Let 
be a recovery target in the terminal period t = T. In the stochastic problem we seek the sequence of recovery strategies, 
, or the recovery plan, that will minimize the expected discounted cost of recovery actions over the interval t = 0,1,2,...,T-1 plus the discounted penalty (reward) for failing to reach (exceeding) the target. The penalty function attempts to capture the decision-maker's subjective risk preferences for failing to meet the population target. The penalty function may be written as 
and a possible shape for the penalty function is shown in Figure 1. The function is strictly decreasing in NT and takes on the value of zero at 
. Realized population levels in t = T that exceed the recovery target serve to reduce the discounted cost of recovery plans.
Carrying capacity, Kt, might be enhanced by some actions. This leads to a second deterministic map Kt+1 = min[G(Kt,Xt),KMAX], implying that carrying capacity in period t+1 is the minimum of Kt+1 = G(Kt,Xt) or KMAX, interpreted as the maximum carrying capacity when the preserve has been “fully enhanced.” In each period t + 1, the population level, Nt+1, and the carrying capacity, Kt+1, are updated based on the level of management actions, Xi,t, executed in period t, and a realization for εt+1.
Let the cost of strategy Xt be given by a function C(Xt). Let ρ = 1/(1+ δ) be a discount factor, where δ > 0 is the rate of discount. Then, our dynamic optimization problem seeks to
where E0[ · ] is the expectation operator in t = 0. In the deterministic problem, the stochastic map for the endangered species is replaced by a deterministic map, Nt+1= F(N t,K t,X t). The penalty function is dropped in the deterministic model because it is possible to determine all achievable values of NT = NT* In solving for the optimal feedback policy, we will determine the optimal recovery strategy, XT*, for a given population target, initial condition, time horizon, and probability distribution for εt+1.3
The optimal recovery problem might have been formulated with a chance constraint, a common approach in population viability analysis (Haight 1995; Marshall, Homans, and Haight 2000). This type of constrained optimization problem is often solved by generating a deterministic approximation to the stochastic problem and then using a coordinate-search method along with a penalty function to account for the constraint (Bazaraa and Shetty 1979). The main issue with this solution method is the likelihood of convergence to a local optimum. We did not adopt this approach because dynamic programming with a penalty function provides optimal feedback policies for all instances when the terminal population target can and cannot be met. Furthermore, by simulating the model with the optimal decision rules, one can construct a distribution of the terminal population in period t = T.
III. The Red-Cockaded Woodpecker
RCWs are the primary excavators of tree cavities used by at least 27 vertebrate species (USFWS 2003). Alteration of the fire regime and degradation of longleaf pine habitat in the southeastern United States has led to severe declines in RCW populations and resulted in the RCW being listed as an endangered species in 1973. In addition, existing populations are highly fragmented and often isolated from other populations (Conner, Rudolph, and Walters 2001). As a result, habitat conservation and management are crucial to the continued viability of the RCW.
RCWs are cooperative breeders. They live in breeding groups consisting of a breeding pair and up to four helpers. These helpers forego reproduction and assist in raising the group's fledglings until they are able to fill a breeding vacancy in their current (or an adjacent) breeding group. RCW breeding groups occupy a territory consisting of nesting and foraging habitat and will typically cover 40 to 160 ha (USFWS 2003). Each RCW territory contains a collection of cavity trees, called a cavity cluster, and each group member occupies its own cavity (Walters, Doerr, and Carter 1988). The construction of a RCW cavity takes many years, and the time to completion can vary significantly (Conner and Rudolph 1995). The RCW prefer stands of pine 80 years or older, as these stands are often infected with red heart fungus, which softens the wood pulp and makes cavity excavation easier. Due to the significant time required to construct a cavity, new territory creation is rare, and it is advantageous for an individual woodpecker to compete to fill a breeding vacancy in an existing territory. The number of suitable territories (occupied and unoccupied) comprising a population is referred to as the population's carrying capacity. A decrease in carrying capacity occurs when a cluster becomes unsuitable for occupation and is often the result of cavity tree mortality, cavity enlargement (most commonly by pileated woodpeckers), and/or hardwood midstory encroachment (Conner and Rudolph 1989).
Some interesting population dynamics arise from the cooperative breeding behavior of the RCW. The existence of a large nonbreeding class (helpers) serves to buffer variations in breeder mortality or fecundity (Connor, Rudolph, and Walters 2001; Walters, Crowder, and Priddy 2002). The size of the breeding population is not severely affected by a decline in the number of fledglings or an increase in breeder mortality. As a result, the number of breeding pairs is commonly used as the measure of RCW population size (USFWS 2003). In addition, because new territory creation in most instances is rare, the number of breeding pairs is limited by the number of suitable territories in the population. These dynamics have important implications for RCW management.
Red-cockaded Woodpecker Recovery
Research has resulted in a suite of management activities that are currently employed to maintain and enhance RCW habitat. These include but are not limited to translocation, artificial cavity construction, and prescribed burning. The use of translocation serves to augment the size of the destination population and helps to increase genetic diversity within that population. Artificial cavity construction involves the drilling (or installation) of artificial cavities in desired locations. Artificial cavities allow managers to replace cavities lost due to tree mortality or cavity enlargement,4 and/or create new cavity clusters/territories in previously unoccupied habitat. Territories constructed in previously unoccupied habitat should be located within 3 km of an occupied territory (in order to facilitate colonization) and include at least four artificial cavities (to create a cavity cluster) (USFWS 2003). Prescribed burning is a management activity that can effectively control hardwood encroachment and improve foraging habitat (USFWS 2003).
Rudolph, Connor, and Walters (2004) describe management techniques available to RCW land managers based on years of research into RCW population dynamics. Rudolph, Connor, and Walters argue that the decline in RCW habitat carrying capacity is the primary reason for the decline in RCW breeding groups and that most populations, even those in decline, contain a level of potential breeding groups at or near carrying capacity. The authors advocate for the use of management techniques aimed at maintaining or increasing carrying capacity: (1) prescribed burning should be employed in order to keep nesting and foraging habitat suitable, (2) artificial cavities should be constructed to replace cavities lost due to cavity tree mortality in order to maintain the number of suitable cavity clusters and, (3) artificial cavities should be constructed in previously unoccupied habitat to increase overall carrying capacity.
Rudolph, Connor, and Walters (2004) also contend that management techniques intended to increase fecundity rates or decrease mortality rates are ineffective at increasing the number of RCW potential breeding groups. They argue that these techniques will result in an increased number of individual birds (nonbreeding helpers in particular), but are not effective in increasing the number of potential breeding groups within a population. As a result, control of predators and kleptoparasitism (invasion of cavities by other species such as flying squirrels and rat snakes) is not recommended.
The framework advocated by Rudolph, Connor, and Walter presents land managers with management actions that are the most effective at achieving an increase in the number of potential breeding groups. Their analysis, however, does not attempt to quantify the effect of management activities on a RCW population nor does it discuss the design of optimal recovery strategies.
IV. Application of the General Model to the RCW
The Conservation Fund established the Palmetto Peartree Preserve in northeastern North Carolina in 1999. The Palmetto Peartree Preserve population is listed as an essential RCW support population (USFWS 2003). The majority of the preserve consists of pine forest (primarily loblolly pine) and mixed forest stands. While 80-year-old pine trees are scarce, there is an abundance of pine large enough to accommodate artificial cavity construction.
We solved the following specification of the general, cost-effective recovery problem, where I = {1,2}↔{translocation, artificial cavity construction}:
Our analysis focused solely on translocation and artificial cavity construction because the effects of these management actions are straightforward to quantify. The management activities are integer variables with constant marginal costs given by c1> c2>0, respectively. We assume that the penalty function is 
and 
. if 
If then 
, and the penalty function serves to decrease discounted total cost by 
dollars.5
Our model explicitly accounts for management activities on the population level and carrying capacity in time t. We adopt the patch-based modeling approach proposed by King and Ashwood (1998), by employing a stochastic, logistic growth function to model the natural growth of the RCW population (colonization of suitable unoccupied cavity clusters) with an intrinsic growth rate, r >0. The use of the logistic growth function allows us to impose density dependent growth and account for possible changes in carrying capacity, Kt, measured by the number of cavity clusters suitable for occupancy.6 We assumed that translocation of a breeding pair serves to directly augment the number of breeding pairs in the population subject to a first-period survival rate of 1>s >0. To account for variation in the natural growth rate we multiply our logistic equation by the iid random variable, εt+1>0.
The number of RCW cavity clusters (or territories) is subject to an upper bound, KMAX, based on the nesting and foraging habitat required for each breeding pair and the amount of suitable habitat available. We made the assumption that artificial cavity construction, X2,t, serves to directly increase the number of suitable available cavity clusters, or carrying capacity. We also assumed the number ofcavity clusters can decrease over time if they are occupied by another species, are encroached by hardcover growth, or the tree containing the cavity is felled by insect infestation, wind, or disease. In our model, cavity clusters become unsuitable at rate 1 >α >0 and require the construction of four artificial cavities (one cavity cluster) to become suitable for colonization. This results in the map Kt+1 = min[(1 - a)Kt+ X2,t,KMAX]. In each period the number of cavity clusters must equal or exceed the number of nesting pairs, Kt≥ Nt.
We used dynamic programming to solve the deterministic and stochastic instances of our optimization problem. The state variables in the model are integer values for carrying capacity, Kt, and the number of breeding pairs, Nt, where 0 ≤ Kt ≤ KMAX and 0 ≤ Nt≤Kt for all t= 0,1,...,T−1. In order to solve the problem with dynamic programming, it is necessary to discretize the state space. However, the state transition functions frequently do not generate integer values for Kt+1 and Nt+1, therefore it was necessary to define probability weights for potential future states. In the stochastic instance, we used bilinear interpolation to determine probability weights for the possible future states (Nt+1,Kt+1) defined by the transition functions (see the Appendix). In the deterministic instance, we rounded noninteger values of Kt+1 and Nt+1 to the nearest integer.7
The Matlab code used to solve the stochastic problem8 with the above penalty function where the random variable, εt+1, is drawn from a discrete distribution where 
9
The Conservation Fund provided yearly data on carrying capacity and RCW breeding pairs from 1999 to 2008. We used nonlinear least squares to estimate the value of the intrinsic growth rate resulting in the estimate, r = 0.13. The upper bound on carrying capacity, KMAX, was calculated by dividing the amount of available suitable habitat by the average size of the home range of an RCW breeding pair (approximately 50 ha). Connor et al. (1991) estimated the annual cavity tree mortality rate of loblolly pine (the most abundant species in the Palmetto Peartree Preserve) to be 0.06. We set the annual rate of decline in carrying capacity to α = 0.10, a value greater than the annual cavity tree mortality rate, in order to account for the potential of additional cavity losses due to cavity enlargement and hardcover encroachment. Our model required a specific criterion be met for the translocation to be considered a success; both translocated individuals must have remained at the target cluster, followed by pairing and nesting. Previous estimates for the translocation success rate of RCW breeding pairs ranged from 33% (Costa and Kennedy 1994) to 13% (Edwards and Costa 2004). In this analysis, we set s = 0.25, as it lies within the previously reported range. The U.S. Fish and Wildlife Service (USFWS 2003) determined the cost of translocating one breeding pair to be c1 = $3,000, and the cost of constructing one artificial cavity cluster (four artificial cavities) to be c2 = $800. Initially, we assigned the penalty parameter in the final function to be Q = $40,000, and the bonus parameter to be R = $5,000. The penalty parameter needed to be sufficiently large relative to the cost ofX1,MAX and X2,MAX for the penalty to have “bite.” Numerically, for the penalty function to be an appropriate disincentive one would want Q>c1X1,MAX+c2X2,MAX. Finally, we arbitrarily assigned a positive discount rate of δ = 0.05, a time horizon of T = 10, and a feasible population target in the stochastic problem of NTT = 42. We have provided a summary of parameter estimates in Table 2.
Description and Estimate of Parameters for the Deterministic and Stochastic Problem Specifications of Red-cockaded Woodpecker Recovery in the Palmetto Peartree Preserve
The Deterministic Problem
We solved the deterministic problem where Pr(εt+1 = 1.00) = 1. In the deterministic problem, the penalty function was unnecessary because we used dynamic programming to solve for the least-cost sequence of recovery plans that will precisely reach a feasible target, NT*. Given the initial conditions, N0 = 20 and K0 = 30, the set of feasible population targets greater than our initial population is NT* ∈ {21,22, …, 47}. We determined the upper bound by running the population model with the values of all recovery actions set equal to their maximum in each year, X1,t = X1,MAX and X2,f = X2,MAX, where t= 0,1,2, …, 9. We generated optimal (least-cost, present-value) recovery plans,
, for all feasible population targets in the deterministic problem. We present the least-cost recovery plans for the population targets NT* = 42,43, …, 47 in Table 3. It is interesting to note that when analyzing the optimal recovery actions for consecutive population targets, the optimal sequence of translocations is nearly identical, differing only in one time period. For example, for the population targets NT* = 42 and NT* = 43, the optimal sequence of translocations is identical in every time period with the exception of period six, when the larger target requires four translocations and the smaller target requires none. For the targets NT* = 44 and NT* = 45, the sequences are identical except in period four, when, once more, the larger target requires four translocations and smaller target none. With a translocation success rate of 0.25, the additional delayed translocations are the least-cost way to achieve the higher target.
We also calculated the smallest, discounted total cost recovery plans for each feasible target, NT*. For the optimal (least discounted cost) recovery plan, we plot 
as a function of NT*. C* is increasing in NT* (Figure 2).
Discounting provides incentive to delay the adoption of relatively costly recovery actions. The least-cost recovery plan for a population target of 
= 42 and a zero discount rate (δ = 0) is
= {(3,8),(6,10),(1,10), (0,7),(0,5),(0,5),(1,5),(1,5),(1,4),(6,0)}. The least-cost recovery plan with an identical population target and a positive discount rate (δ = 0.05) is
= {(0,7),(2,10),(1,10), (0,6),(0,6),(0,6),(0,2),(5,8),(5,4),(6,0)}. With a positive discount rate, the majority of translocations are delayed until the final three years of the 10-year time horizon.
The Least-Cost Recovery Actions, 
and 
for All t = 0, 1,..., 0 and 
= 42, 43, …, 47
Minimum Discounted Total Cost Required to Precisely Achieve the Population Target in the Deterministic Problem Specification
The Stochastic Problem
The stochastic problem lead to a series of state and time dependent thresholds, where translocation was set at the maximum level if Nt fell below the threshold, and set to zero if Nt was above the threshold.10 The optimal policy for the construction of artificial cavities was more complex. Cavity clusters were constructed so as to maintain a breeding pair to cavity cluster ratio of 1:2. This serves to maximize the growth rate of the RCW population. Because this ratio holds in all time periods, we will explore optimal recovery actions in t = 0 and t = 9 for NTT = 42.
In Figure 3a and 3b, X*1,t = 0 (zero translocations) is indicated by a black grid cell and X*1,t = X1,MAX = 6 is denoted by a light gray cell. The population threshold in the initial time period, t = 0, is 14 breeding pairs (Figure 3a), and in final time period, t = 9, the population threshold is 42 breeding pairs (Figure 3b). The population threshold is increasing in t, and population thresholds for all periods t = 0,1,2,...,9 are listed in Table 4.
Because the optimal policy for the construction of cavity clusters attempts to maintain a ratio of Nt/Kt = ½, the optimal policy for the construction of cavity clusters will take on all possible values of X*2,,t = 0,1,2,...,10 and is indicated by a grid cell shaded from black (zero cavity clusters) to light gray (10 cavity clusters) in Figure 4a and 4b. In t =0 it was optimal to construct the maximum number of artificial cavity clusters, X*2,0 = X2,MAX = 10, because the initial ratio was N0/K0 = 2/3 > ½. However, if the number of breeding pairs is small relative to the number of available cavity clusters, it is optimal to construct fewer artificial cavity clusters. As one approaches the terminal time period, say, t= 9, the optimal number of cavity clusters needs to equal or slightly exceed the number of breeding pairs because building excessive cavity clusters near the terminal time only adds to cost and does not appreciably increase growth given the limited time remaining (Figure 4b).
Optimal Number of Translocations for a Population Target of 42 Breeding Pairs; Translocation Thresholds Indicated by Dashed Lines in (a) t = 0; (b) t =9
Translocation Thresholds for All t =0,1,2,...,9
Optimal Number of Artificial Cavity Clusters Constructed for Population Target of 42 Breeding Pairs in (a) t = 0; (b) t = 9
In general, the optimal decision rules are consistent across a diverse set of parameter values. The translocation threshold in a given time period will increase (decrease) if the intrinsic growth rate is increased (decreased), the time horizon is increased (decreased), the probability of translocation success is increased (decreased), or the population target is decreased (increased).
We generated 10,000 realizations (rounding the number of breeding pairs and carrying capacity to the nearest integer) using the optimal feedback policies generated by our model. Discounted total management cost for optimal recovery strategies when NT* = 42 ranged from a minimum of $45,251 to a maximum of $173,190, with an average total management cost of $115,430 (Figure 5).
Sensitivity of the Results to the Penalty Function
The results presented thus far serve to characterize the form of the optimal decision rules resulting from our model. However, the results relied on a subjective assignment of the penalty parameter, Q = $40,000. In addition, we considered only a piecewise linear form for the penalty function. The penalty function is critical in shaping the optimal recovery plan. The proportion of realizations achieving the population target increases as you increase the penalty parameter, Q. In addition, the average discounted total management costofthe resulting recovery plan increases as one increases the penalty parameter. This makes intuitive sense. If the penalty for falling short of the target increases, one will be willing to invest more heavily in management actions to avoid incurring the stiffer penalty. Here we apply an algorithm to determine the cost-effective recovery plan(s) from the set of recovery plans generated from our model using a range of candidate penalty parameters and forms of the penalty function.
Cost-effectiveness analysis (CEA) is a method to determine the efficiency of alternative projects that utilize scarce resources to produce societal benefits. It is most prevalent in the health care literature, where decision makers compare various interventions with different costs and health effects by calculating an incremental cost-effectiveness ratio, ICER (Weinstein and Stason 1977). We used CEA to examine the effects and costs associated with the optimal recovery plans resulting from running our model with a wide range of penalty parameters and two different forms of the penalty function. We considered a piecewise linear functional form, 
if 
and a quadratic form, 
if 
. In all instances we set the reward parameter, R, to zero. We solved the stochastic model with a population target, N*T = 42, for a range of candidate penalty parameters for each candidate penalty functional (Q1 ={0,1500,3000,...,313500, 315000} and Q2= {0,100,200,...,20900, 21000}). We then generated n = 100,000 realizations for each optimal recovery plan. These realizations produced distributions of discounted management costs and the number of breeding pairs in time T. In our CEA, we used the average discounted management cost as the measure of cost and the percentage of realizations that meet or exceed the population target (or success rate) as our effect. We generated all values for the random variable εt+1 by randomly generating a single matrix E of size n× T and then used this matrix to generate the stochastic realizations for each candidate recovery plan. This enabled us to compare the costs and effects across all recovery plans. The average discounted management costs and percentage of realizations exceeding the population target of all recovery plans generated from our model with the linear and quadratic penalty function specifications are shown in Figure 6a and 6b.
Distribution of Discounted Total Management Cost (Excluding Penalties and Rewards) for Population Target of 42 Breeding Pairs
We arranged all recovery plans in ascending order of success rate. We eliminated those recovery plans that were “strongly dominated”—those plans that had increased costs or reduced success rates when compared with at least one other alternative. We then calculated the 
for each adjacent pair of remaining recovery plans, where 
is the average discounted management cost of recovery plan i and Ei is the success rate of recovery plan i. We eliminated those recovery plans that were “weakly dominated”—those plans with a lower success rate but with a higher ICER when compared with the next highest ranked plan. We then recalculated the ICER for the remaining recovery plans and eliminated those plans that were weakly dominated. We repeated this until we were left with a set of recovery plans with strictly increasing ICER. This analysis found 12 nondominated, cost-effective recovery plans (Table 5).
We plot all generated recovery plans based on their average discounted management cost and success rates in Figure 7. The 12 costeffective recovery plans were all generated with the linear penalty function11 and lie on the convex hull of the plotted data in Figure 7. It should be noted that weakly dominated recovery plans that lie near the convex hull are also feasible candidate recovery plans.
Average Discounted Total Management Cost and Percentage of Realizations That Achieve the Population Target (Success Rate) for Recovery Plans Generated from All Potential Values of the Penalty Parameter of the (a) Linear Penalty Function and (b) Quadratic Penalty Function
The Cost-effective Candidate Recovery Plans
V. Conclusion
The general model we developed can be a useful tool for determining the optimal (leastcost, present-value) recovery plan for a population of an endangered species. When applied across multiple populations of the same species, managers can choose which populations to manage in order to optimally achieve delisting criteria. In addition, budget estimates and probabilities of achieving the delisting criteria can also be generated. We applied the general model to the recovery of the RCW in the Palmetto Peartree Preserve in North Carolina.
We used dynamic programming to find the optimal sequence of recovery actions that might achieve a population target over a given time horizon. For the deterministic model, it was possible to determine the feasible population targets, and the least-cost recovery plans that would precisely achieve those targets. In the stochastic model, it was not possible to guarantee that a population target could be reached with certainty. We solved the stochastic problem by specifying a penalty function where the population target might serve as a kink separating the penalty line segment from the reward line segment (Figure 1).
Intuitively, the optimal decision rules for the RCW recovery actions make sense. Translocation is relatively expensive. Therefore, with a positive discount rate, it is optimal to delay translocation (if possible) until later in the time horizon. As t approaches T, it is necessary to perform translocation to reach or exceed the population target in order to avoid incurring a penalty. If a population level is achieved that will reach the population target with high probability, there is no incentive to perform translocation if the reward does not exceed the cost. Conversely, artificial cavity construction is relatively inexpensive. Instead of delaying, it is optimal to construct artificial cavity clusters early in the time horizon. By constructing cavity clusters early on, one can inexpensively increase net growth. We can summarize the optimal decision rules as follows: if it is not possible to achieve the population target through an increase in carrying capacity, then it is necessary to resort to translocation.
All Recovery Plans Generated from Both Specifications of the Penalty Function Plotted Based on Success Rate (x-axis) and Average Discounted Total Management Cost (y-axis); Nondominated, Cost-effective Recovery Plans Lie on the Convex Hull of the Plotted Data
Our RCW population model makes several simplifying assumptions. The model accounts only for the spatial characteristics of the population in assigning the limits on carrying capacity and the maximum number of artificial clusters that can be constructed in a given time period. The model does not explicitly account for the spatial attributes that may influence the RCW population at a particular site. Our model assumes that managers will follow the recommended guidelines for locating artificial cavity clusters within three km of an existing RCW breeding group in order to facilitate colonization (USFWS 2003).
The application of the general model to the Palmetto Peartree Preserve included only two RCW management activities. Prescribed burning was not included due to the fact that it is difficult to quantify the effect it has on RCW population growth. We performed an analysis that included prescribed burning, assuming that burns were performed annually in proportion to the number of cavity clusters (carrying capacity). All other model parameters remained identical to those in Table 2. Based on the preliminary analysis, prescribed burning may result in delaying the construction of artificial cavities until later in the time horizon. This is due to the fact that more cavity clusters require more prescribed burning, thus raising the de facto cost of cavity clusters. As a result, translocation becomes relatively less expensive early in the time horizon and the translocation threshold in each time period shifts down, thus making translocation a more viable management action.
In addition, the model did not include land use activities or the option for land acquisition. These types of management options can easily be incorporated in our decision framework, given that relevant data are available regarding the response of the species to the management action.12 This highlights the need for more research on species response to management activities. For example, RCW management planning could be greatly enhanced if it were known how the population responded to different frequencies of prescribed burning. When adequate data are available, econometric analysis can be employed to estimate the effects of recovery actions on species population dynamics (Marshall, Homans, and Haight 2000; Rashford and Adams 2007).
It may not be possible to quantify the benefits and costs of endangered species recovery plans in economic terms. Furthermore, assigning values to the penalty (and reward) parameters in our model is likely to be problematic. However, these parameters play a crucial role in determining the likelihood of achieving the terminal population target. These parameters are subjective; they reflect the manager's (or agency's) risk preferences for failing to meet the delisting criteria (terminal population targets) in the time allotted. By changing the value of the penalty and/or the reward, managers can influence the probability that the optimal recovery actions generated by the model will achieve the population target. To increase the likelihood of achieving a population target, managers will need to invest more heavily in costly management actions. There is a fundamental trade-off here: how much is one willing to pay to increase the likelihood of achieving a population target?
In order to facilitate the selection of a suitable penalty parameter, we presented an algorithm for cost-effectiveness analysis that can determine the most efficient penalty parameter(s) for achieving the given delisting criteria. When given a set of cost-effective recovery plans resulting from cost-effectiveness analysis, managers must choose which plan to implement based on their risk preferences and/or available budget. Even without penalties or rewards, when implementing any species recovery plan, managers must analyze the trade-offs between management costs, delisting criteria, and the likelihood of achieving population targets.
Acknowledgments
We thank participants at the 2010 Canadian Resource and Environmental Economics (CREE) study group, especially G. Gaudet, for their helpful comments. We also thank P. Fackler for checking our dynamic programming results and solving the stochastic problem with bilinear interpolation. In addition, we thank the Conservation Fund, specifically Ole Amundsen, Will Allen, and Buck Vaughan, for generously providing data and support. We also wish to thank two anonymous reviewers for their valuable comments. Finally, we greatly appreciate the support of the NSF Expeditions in Computing grant on Computational Sustainability (Award Number 0832782).
Appendix
In the stochastic problem, we used bilinear interpolation to assign a positive probability to the potential future states, ⎡Kt+ 1 ⎤, ⎣Kt+ 1 ⎦, ⎡Nt+ 1 ⎤, and ⎣Nt+ 1 ⎦, generated by the state transition functions (where ⎣x⎦ is the largest integer less than x and ⎡x⎤ is the smallest integer greater than x). We assign the probability of achieving state ⎡Kt+ 1 ⎤ to be {Kt+ 1}, and the probability of achieving state ⎣Kt+ 1⎦ to be 1 — {Kt+ 1} (where {x} is the fractional part of a real number x). We use the same methodology to assign probabilities to ⎡Nt+1⎤ and ⎣Nt+1⎦. Because (⎣Ki+χ⎦,⎡Nt+1⎤) could result in an infeasible state, where Kt+1<Nt+1, we use Nt +1 = min(Nt +1, ⎣Kt +1⎦) to adjust Nt +1.
Footnotes
The authors are, respectively, graduate student and professor, Charles H. Dyson School of Applied Economics and Management, Cornell University, Ithaca, New York.
Additional material is available as an online appendix at http://le.uwpress.org.
↵1 U.S. Supreme Court, 437 U.S. 187, 184 (1978).
↵2 From the perspective of period t, the random draw εt+1 is not known but will be revealed in period t +1.
↵3 Contingent valuation might be used to estimate a benefit function, B(Nt). Knowing B(Nt) and B(NT), one might attempt to solve a stochastic optimization problem with
as an objective functional. A contingent valuation survey was beyond the scope of this research project. A second impediment to a contingent valuation survey is the ESA itself. It requires recovery plans that achieve terminal population targets, with the ultimate goal of recovery in the wild and delisting. While the accumulation of a benefit stream over the time horizon is a boon, it is rather insignificant when compared to the ultimate goal of delisting of the species. It should also be noted, if we could estimate the benefit function, B(Nt), then specifying a target, NT*, as a terminal condition could lead to lower discounted net benefits if NT* is so large as to incur excessive recovery costs and reduce discounted net benefits.↵4 Artificial cavities are fitted with metal restrictor plates to protect cavities from enlargement from other species, namely, pileated woodpeckers (Saenz et al. 1998).
↵5 In this analysis, the penalty function takes a simple piecewise linear form. Nonlinear penalty functions (where the penalty increases in
) may also be appropriate. A longer horizon, T > 10, may also be considered, but real-world recovery plans are often revisited every 5 to 10 years, so a longer horizon may be unnecessary. In addition, the objective of the ESA is, ultimately, recovery in the wild and delisting, so management actions should be viewed as temporary interventions.↵6 There have been several past attempts to model RCW population dynamics. We have chosen to use a simple logistic growth function to model the growth of the RCW population via colonization of suitable unoccupied territories. This model shares the assumption of density dependent growth in the RCW population model described by Hooten et al. (2009). The age-structured model developed by Heppel, Walters, and Crowder (1994) presented no obvious way to incorporate management actions aimed at specific cohorts. The spatial model presented by Letcher et al. (1998) is an agent-based model and does not lend itself to optimization.
↵7 One might allow all variables to be continuous and ignore the inherent discrete nature of the controls (recruitment clusters, translocated breeding pairs) and RCW population. One might then solve for the optimal feedback policy and then round real valued variables to the nearest integer and adopt those rounded values as the optimal policy. In our case, the discrete nature of cavity clusters and translocated breeding pairs is easily accommodated, but the RCW population requires rounding or bilinear interpolation.
↵8 The Matlab code is available as an online appendix at http://le.uwpress.org.
↵9 The distribution of εt+1used in our model was chosen arbitrarily. Further extensions of this analysis may include a small probability of a population collapse (possibly from a hurricane or pine beetle infestation) where, for example, Pr(εt+1= 0.20) = 0.05.
↵10 This is often referred to as a “bang-bang” solution: the optimal feedback policy is restricted to the upper and lower bound on the number of breeding pairs that can be translocated in a given time period. Bang-bang solutions often occur when the Hamiltonian is linear in the control variable (Sonneborn and Van Vleck 1965).
↵11 It should be noted that the cost-effectiveness analysis presented examined a specific risk preference wherein the manager is concerned only with the rate of achieving the population target. However, one could imagine a set of risk preferences where a manager was interested not only in the rate of achieving the population target, but also, in those instances when the target was not met, the relative size of the terminal population. In this instance, it is quite feasible that the recovery plans produced from the model with the quadratic penalty function would outperform those generated from the model with the linear penalty function.
↵12 The acquisition of an adjacent land parcel can be represented by a binary decision variable, and the type and extent of land use (such as timber management) can easily be modeled as a discretized variable. One must note that as the number of management options in the model increases, solving the problem becomes more computationally expensive.
