An optimization model to select redwood stands for the conservation of the marbled murrelet in the Headwaters Forest HCP.

Abstract

This work describes an explicit method by which alternative stand selection scenarios concerning the threatened marbled murrelet (Brachyramphus marmoratus) in the Headwaters Forest Habitat Conservation Plan were generated and evaluated. Preservation goals and timber harvesting goals are simultaneously considered. An integer programming model (classical 1,0 knapsack problem) is applied to identify sets of redwood groves that optimized marbled murrelet nesting habitat value for given values of foregone timber production. Alternatively, the results can be interpreted as minimizing applicant cost for given levels of murrelet habitat value. The outcome of the analysis does not show that a small number of very high quality selections would reserve most of the available murrelet habitat. Instead, the analysis supports the contention that the more old-growth that can be reserved, the better the measures of nesting habitat become. Ultimately, the results of this analysis were not used for negotiating the final Headwaters Forest Habitat Conservation Plan. However, the grove selection optimization model is valuable in that it provides a general method for balancing habitat value and economic value.

Introduction

The Headwaters Forest in northern coastal California is infamous for one of the premiere conservation battles of the 1990s. The contention is over the preservation of old-growth forest, especially redwood habitat. Listed species that are affected include the northern spotted owl (Strix occidentalis caurina), coho salmon (Oncorhynchus kisutch), and marbled murrelet (Brachyramphus marmoratus). This paper focuses on a decision aid that was applied during the negotiations over the amount of private timberland that would be set aside as nesting habitat for the marbled murrelet.

The best-known conservation activities in the Headwaters were associated with the government purchase of a 7,500-acre tract of old-growth redwoods. Lesser known was the development of a Habitat Conservation Plan (HCP) that is a 50-year agreement between the U.S. Government and Pacific Lumber Company (PL) governing activities on approximately 200,000 acres that are owned privately by PL. The decision model described herein was applied in the context of the HCP.

The marbled murrelet is a seabird that forages in marine coastal areas and is dependent for nesting on the large branches found on old-growth trees such as redwood and Douglas-fir (Naslund 1993; McCarthy 1993). In California, redwood-dominated stands relatively near the coast are considered to be of prime importance (Hunter et al. 1998). The severe loss of old-growth forest habitat has reduced populations in California, Oregon, and Washington to such an extent that the marbled murrelet (for those states) was listed as federally threatened in 1992 (USFWS 1992). A difficult conflict inevitably arises from the marbled murrelet's dependence on old-growth redwood trees for successful nesting and reproduction. Because of the high economic value of such forests for timber harvest, habitat reserves are potentially very costly for a company such as PL. The result is a direct conflict between species protection and economic hardship imposed on a private rights holder. The HCP has to balance both interests. This study uses data developed during the HCP negotiations in the late 1990s to identify "optimal" alternatives that achieve the greatest habitat protection for a given amount of timber volume set aside for conservation. Not just a single solution is created. By varying the limit on foregone harvest, a set of points representing an optimal tradeoff curve between marbled murrelet habitat value and applicant cost can be generated.

Data

Of the 200,000 acres of PL property covered by the HCP, fewer than 10,000 acres remained that were considered to be prime habitat for murrelet nesting. At the time (late 1997), these potential reserved groves were delineated as 23 different stands (Table 1). (It must be noted that these stand boundaries were later changed during the course of the negotiations, and thus no correspondence can be drawn between Table 1 and the final HCP negotiated in March 1999). Many stands were composed of a mix of forest types, including second-growth, partially cut old-growth, and pristine old-growth. Thus, murrelet habitat value may be quite variable within stands. The best measure of murrelet reproductive habitat quality available at the time was volume of old-growth timber (both partially cut and pristine) in a stand. An alternative measure was the ratio of a stand's "core" old-growth area to the "edge" old-growth area (again, including all types of old-growth).

Edge is defined as the outer 50-meter strip of old-growth area along the stand periphery surrounding the core. A greater core/edge ratio translates to better habitat value since edge effects are considered to diminish successful reproductive output (due to nest predation by crows, etc.). The "cost" to the company of preserving any stand is the volume of harvestable timber within the stand. While habitat value is measured in terms of the old-growth trees necessary for marbled murrelet nesting, cost to the HCP applicant (PL) is measured in terms of any type of merchantable timber existing within a stand.

Since this model uses sensitive corporate data belonging to PL, it is necessary for purposes of this manuscript to conceal the true volume of timber (old-growth and otherwise) in the respective stands. Accordingly, I have modified timber volume per stand by creating a relative index, which I term board foot index (BFI). BFI is created simply by multiplying board feet by a constant. In this way the relative value of each stand is maintained while maintaining company confidentiality. Only relative, not absolute, habitat and economic values are required by the optimization model.

Defining the model

As a verbal definition, determining which redwood stands on PL property should be set aside for marbled murrelet nest habitat can be stated as:

(1) Maximize reproductive output of the local sub-population,

(2) Minimize cost to the applicant.

This definition of the redwood stand selection problem in the area of the Headwaters Forest can be translated to a mathematical form. This mathematical form is an integer programming model that can be solved to yield a number of alternatives. Each alternative represents a different combination of stands (groves) that would be protected. A stand combination corresponds to a total amount of foregone harvest in terms of total BFI, and a total amount of nesting habitat value for the marbled murrelet. The latter is expressed as total volume of old-growth in BFI, or as the total ratio of core area to edge area among the selected stands, depending on which measure of habitat value is being used in the model.

Model notation

The set of 23 stands eligible for selection as reserves in the HCP is indexed with i.

[P.sub.i] = marbled murrelet nesting habitat measure for stand i. (Here, [P.sub.i] is either the volume of old-growth redwood, measured in BFI, or the ratio of core area to edge area.)

[F.sub.i] = BFI of extractable timber in stand i.

T = maximum level of extractable BFI allowed to be set aside in the set of selected stands (an adjustable parameter, the measure of cost to PL of foregone harvest).

[X.sub.i] = 1 if stand i is selected for protection in the HCP; 0 if not.

Model formulation

The following model is designed to select that set of stands that attains the maximum total stand habitat value for the marbled murrelet given a limit on the maximum amount of timber that the applicant must set aside. By varying this limit (T), a set of points representing the optimal tradeoff between marbled murrelet habitat value and timber lost to PL can be generated. A high value of T corresponds to a relatively large loss of harvestable timber to the applicant (high amount of timber is allowed to be set aside in selected reserves). A low value of T represents allowing relatively little timber to be set aside in reserves, meaning a smaller financial loss to the applicant (but also relatively little reproductive habitat set aside for the murrelet). This integer programming model is an application of a classic mathematical form known as the 1,0 knapsack problem (Wagner 1975). The model is formulated in Box 1.

Box 1: Model Formulations

(1) Maximize total marbled murrelet nesting habitat value across the set of selected stands.

Maximize [Summation.sub.i][P.sub.i][X.sub.i] subject to:

(2) Total BFI of harvestable timber set aside in the selected stands cannot exceed T.

[Summation.sub.i][F.sub.i][X.sub.i] [less than or equal to] T

(3) Integer restrictions for each i.

[X.sub.i] [member of] {0,1} for each i

The objective function (1) sums to the total habitat measure associated with a set of selected redwood stands or groves. The 1,0 variable X/will be valued at 1 if grove i is selected and 0 otherwise. Thus, in (1), the value [P.sub.i] is incurred for each selected stand. At optimality, a set of stands will have been selected and had their respective [X.sub.i] variables set to 1 such that the objective will sum to the maximum total murrelet nesting habitat measure possible given the problem constraints. While any redwood stand that is selected contributes an amount ([P.sub.i]) to species habitat value, it also contributes an amount ([F.sub.i]) to applicant cost in the form of lost timber. Constraint (2) limits total applicant cost by limiting the total BFI among the selected reserved stands to a maximum value of T. T is a parameter that can be set at any desired value. By solving the knapsack problem for various values of T, many different alternative combinations of total habitat value protected and total timber set aside can be generated. Constraint (3) restricts the decision variables to being binary integers. Thus, any solution must contain whole stands, not portions of stands.

Note that [P.sub.i] can represent different measures of murrelet habitat value. For our application here, we use two different measures: quantity of old-growth redwood (in BFI) present in a stand and the core-area/ edge-area ratio of the stand. Having two different measures of the value of each stand means that we will solve two sets of knapsack problems, one for each habitat quality measure. Each set of problems is created by using a range of different values for T, which represents the maximum applicant cost for a solution. In each set of problems, the measure used for merchantable BFI in a stand ([F.sub.i]) is unchanged. Note that the volume-based [P.sub.i] differs from [F.sub.i] due to the mix of timber types within a stand. For example, a stand that contains 60 BFI of harvestable redwood timber may only contain 40 BFI of old-growth redwood, both uncut and residual, the remainder consisting of second-growth timber.

Model results

All solutions were obtained by applying commercial off-the-shelf math programming software (CPLEX), and all solution attempts easily reached optimality. The relatively small size of the Headwaters problem made this approach very viable. Figure 1 graphs thirty runs of the 1,0 knapsack problem where [P.sub.i] is defined as the BFI of old-growth redwood in stand i, and the parameter T ranges from 20 BFI to 600 BFI in increments of twenty BFI. Old-growth volume is one of the most basic measures that can be used to estimate marbled murrelet nesting habitat value. Each point in Figure 1 corresponds to a set of stands selected by the model. The horizontal axis represents, from left to right, increasing values of T, the amount of timber in reserved stands. For a given constraint on the volume of BFI on the horizontal axis, the 1,0 knapsack model identifies the set of stands that yields the maximum amount (in BFI) of old-growth redwood. For example, one solution identified in Figure 1 is generated by setting T to 200 BFI. The optimal solution consumes 199 BFI of harvestable timber and reserves 154 BFI of old-growth redwood. This solution consists of eleven selected stands. The reason that it takes 199 BFI on the horizontal axis to protect 154 BFI on the vertical axis is that the values on the horizontal axis include all harvestable timber, not just old-growth. Not surprisingly, Figure 1 indicates that there is a direct correlation between the quantity of timber that is reserved and the quantity of old-growth forest that can be provided to the murrelet.

[FIGURE 1 OMITTED]

Figure 2 graphs another thirty runs. This time the value of a stand is the core-area/edge-area (C/E) ratio. Again, the relationship between BFI of timber that is set aside and total murrelet habitat value (as measured by the sum of the core/edge ratios of the selected stands) is linear, at least over the range from low to intermediate BFI values. Where the slope of the curve flattens (about 200 BFI and total C/E = 33), further cost to PL starts to yield decreasing returns for the total core/edge objective.

[FIGURE 2 OMITTED]

Discussion and conclusion

Some general comments on the tradeoff curves in Figures 1 and 2 are warranted. The integer programming model is formulated and solved in terms of optimizing total nesting habitat value for a given constraint on applicant cost. The graphical interpretation of this is that for a set position on the horizontal axis, the model finds the highest point in the vertical direction that is a feasible solution. However, the knapsack model and graphical results can also be used indirectly to find, for a given level of habitat protection, what is the minimum applicant cost. For example, if negotiators could agree that a total C/E value of 33 would be sufficient habitat value conserved under the HCP, the knapsack model identifies the minimal amount of foregone timber harvest as about 200 BFI (Figure 2), and that particular set of stands would be revealed. It is important to realize that the knapsack model results and the graph can be interpreted in both of these ways, either from PL's viewpoint or from the biological viewpoint.

It is also important to realize the difference between optimal and suboptimal solutions in the figures. Points above the curve are infeasible -- no additional habitat protection can be attained for the amount of applicant cost. Points below the curve (which are numerous) are feasible but are suboptimal. Such points (not graphed) represent solutions that would require a higher applicant cost to reach a given level of murrelet protection; or, alternatively, that would reach a lower level of murrelet protection for a given applicant cost. The main value of the optimization exercise herein is that it identifies those solutions that represent "non-inferior" points (graphed in the figures). From the perspective of any non-inferior point, no gain can be made in either objective without sacrificing some of the other objective. Without the use of the integer programming model, it is easily possible that decisions would be considered using suboptimal alternatives existing below the optimal tradeoff curve.

The shape of the non-inferior tradeoff curves, particularly in Figure 1, shows a strong linear relationship between timber reserved and habitat reserved. If instead the tradeoff curve had a convex shape and was relatively steep for low values of BFI set aside and progressively flatter as foregone timber increased, that would show that there were some clearly superior sites that could attain relatively good murrelet protection, and that complementing those with additional sites would add relatively little to murrelet protection. From a negotiating viewpoint, this would likely be a desirable result, opening the door to a "win-win" type of solution for both sides. However, Figure 1 shows a nearly straight line, indicating that most of the groves as defined are good stands in terms of what they can add to marbled murrelet protection. Simply put, the more salable timber set aside, the more habitat available to the bird. It is not surprising to see this linear pattern, given that few if any candidate groves possess relatively low murrelet habitat quality along with high timber value, the type of data that would have prompted a convex shape in the tradeoff curve.

Turning to Figure 2, the tradeoff curve for the C/E ratio was relatively steep for low values of BFI set aside and progressively flatter as BFI increased beyond C/E of 33. Each unit of timber volume set aside for C/E values below 33 contributes more cumulative habitat value than timber volumes set aside above C/E of 33, suggesting that beyond C/E of 33, setting aside additional sites would add less to murrelet protection. However, C/E is probably a more limited gauge of marbled murrelet nesting habitat value because it is based more on the geometry of a stand rather than its composition. The volume of old-growth redwood is probably a superior habitat quality measure. During the negotiating process, government scientists developed a "stand productivity index" that combined old-growth volume and C/E to create a combined measure of stand habitat quality. This is a measure that can also be used in the knapsack model framework.

The knapsack model offers no guarantee that a specific amount of habitat that is set aside will be sufficient to ensure the murrelet's long-term survival in the Headwaters Forest area. Population viability analyses are required to attempt to answer that question. However, this relatively simple optimization approach does fit the data that were available during the HCP negotiations. The results presented here were not, in the end, employed in determining the results of the final Headwaters HCP agreement. Nevertheless, the model provides a methodology that can explicitly balance socio-economic and environmental goals and identify non-inferior solutions, rather than relying on subjective judgment. As HCPs proliferate and come under increasing scrutiny (Thomas 2001), there should be a continuing need for such modeling techniques. Methods that can incorporate not just habitat considerations but also economic factors should be increasingly relevant as controversies over "takings" escalate.

(Note: For the reader's convenience, the preliminary negotiated solution of March 1998 is plotted in Figures 1 and 2. However, stand boundaries had changed by the time of that agreement, and more changes occurred by the time of the final HCP agreement of March 1999.)

Acknowledgements

Data were provided by James Gaither of the California Resources Agency and Alisya Torregrosa of Thomas Reid Associates. The model development and analysis was supported by a postdoctoral fellowship at the National Center for Ecological Analysis and Synthesis at the University of California at Santa Barbara (UCSB), funded by the National Science Foundation (Grant No. DEB-94-21535), the State of California, and UCSB.

Literature cited

Hunter, J.E., K.N. Schmidt, H.B. Stauffer, S.L. Miller, C.J. Ralph, and L. Roberts. 1998. Status of the Marbled Murrelet in the Inner North Coast Ranges of California. Northwestern Naturalist 79:92-103.

McCarthy, S. 1993. A Seabird's Secret Life is Revealed--50 Miles Inland. Smithsonian 24:70-81.

Naslund, N. 1993. Why Do Marbled Murrelets Attend Old-Growth Forest Nesting Areas Year-Round? The Auk 110:594-602.

Thomas, G.A. 2001. Where Property Rights and Biodiversity Converge -- Part 2: The Role of Science. Endangered Species UPDATE 18:6-13.

Wagner, H. 1975. Principles of Operations Research, with Applications to Managerial Decisions, Second Edition. Prentice-Hall, Englewood Cliffs, New Jersey.

U.S. Fish and Wildlife Service (USFWS). 1992. Endangered and Threatened Wildlife and Plants: Determination of Threatened Status for the Washington, Oregon, and California Population of the Marbled Murrelet. Federal Register 57:45328-45337.