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Adaptive economic and ecological forest management under risk
- Joseph Buongiorno^{1}Email author and
- Mo Zhou^{2}
https://doi.org/10.1186/s40663-015-0030-y
© Buongiorno and Zhou; licensee Springer. 2015
- Received: 26 September 2014
- Accepted: 10 February 2015
- Published: 24 February 2015
Abstract
Background
Forest managers must deal with inherently stochastic ecological and economic processes. The future growth of trees is uncertain, and so is their value. The randomness of low-impact, high frequency or rare catastrophic shocks in forest growth has significant implications in shaping the mix of tree species and the forest landscape. In addition, the fluctuations of wood prices influence greatly forest revenues.
Methods
Markov decision process models (MDPs) offer a rigorous and practical way of developing optimum management strategies, given these multiple sources of risk.
Results
Examples illustrate how such management guidelines are obtained with MDPs for combined ecological and economic objectives, including diversity of tree species and size, landscape diversity, old growth preservation, and carbon sequestration.
Conclusions
The findings illustrate the power of the MDP approach to deal with risk in forest resource management. They recognize that the future is best viewed in terms of probabilities. Given these probabilities, MDPs tie optimum adaptive actions strictly to the state of the forest and timber prices at decision time. The methods are theoretically rigorous, numerically efficient, and practical for field implementation.
Keywords
- Risk
- Management
- Economics
- Ecology
- Markov
- Optimization
Background
Forest planning and decision making models are often deterministic. They assume that the future is known, or they reduce uncertain variables to their expected values, with the hope that the results will work, at least “on average”. However, the future is highly uncertain^{a}, and it can only be described in terms of probabilities, be they “objective probabilities” derived from factual data, or “subjective probabilities” reflecting a personal belief (de Finetti 1937).
There is a large literature concerning the role of risk in forest ecosystems (see e.g. Kant and Alavalapati 2014, p. 307–369). But there are few applications of Markov decision process models (MDPs) in forestry, a paucity also observed in the general operations research literature (White 1993). Nevertheless, MDPs are powerful and general methods that deserve more attention in forest management. This was recognized early on in Lembersky and Johnson (1975), and Lembersky (1976) who dealt with applications to the management of forest plantations. Their pioneering work was followed by several applications to the uneven-aged management of various forest types for financial objectives (Kaya and Buongiorno 1987), ecological objectives (Zhou and Buongiorno 2006), and combined financial and ecological objectives (Lin and Buongiorno 1998; Rollin et al. 2005; Zhou et al. 2008a, b).
The purpose of this paper is to describe how stochastic simulation and MDPs can be used to deal with decision making under risk in forestry. The data and the specific applications described below deal with mixed loblolly pine (Pinus taeda L.)-hardwood stands in the southern United States. However, the methods are general and can be applied to a variety of forest ecosystems, ranging from even-aged monospecific plantations to mixed-species uneven-aged forests, and from single stands to large forest areas. The objective throughout is to develop simple adaptive management guidelines that can be readily applied in the field.
The paper is organized as follows. The next section presents the methods: stochastic forest growth models and their reduction to Markov chains which are then used for prediction and optimization. This is followed by the results which show first some of the consequences of (wrongly) predicting the evolution of forest ecosystems without taking into account the effects of random ecological, climatic, or economic shocks. Next are shown predictions of the long-term evolution of forest ecosystems with and without management and the implications for economic and ecological criteria. And, MDP results optimizing decisions adaptively for discounted or undiscounted economic and ecological objectives, and combinations thereof. The discussion and conclusion deal with limitations of the methods and potential improvements.
Methods
The guiding principle of the methods is to represent the evolution of a forest ecosystem subject to various impact variables, such as prices and catastrophic disturbances, with Markov chains: probability matrices describing the frequencies of transition of each variable between discrete states. After the transition probabilities between the system states have been determined, the Markov model is used to predict future system states subject to specific management policies. Furthermore, decision process models (MDPs), optimization techniques based on Markov chains, are used to determine the best management policy for a particular objective. A policy is a set of rules that prescribe a decision for each observed system state at decision time. Various adaptive strategies are obtained in this way depending on the objective function which may deal with discounted or undiscounted criteria and constraints that may limit the decision domain.
Markov forest model
Definition of forest stand states according to basal area level by species and tree size
Pines (m ^{ 2 } · ha ^{ −1 } ) | Hardwoods (m ^{ 2 } · ha ^{ −1 } ) | |||||
---|---|---|---|---|---|---|
Small | Medium | Large | Small | Medium | Large | |
Low | ≤2.5 | ≤4.3 | ≤3.6 | ≤2.9 | ≤1.2 | ≤1.7 |
High | >2.5 | >4.3 | >3.6 | >2.9 | >1.2 | >1.7 |
State #13 | 0 | 0 | 1 | 1 | 0 | 0 |
In the following applications the parameters of model (1) are estimated from observations on permanent sample plots in the mixed loblolly pine (Pinus taeda L.)-hardwood forests of the Southern United States (Schulte et al. 1998)^{c}. The differences between the deterministic predictions of model (1), and the observations on the plots give observations on the random shocks ε _{ t } due to ice storms, wind, insect outbreaks, abnormal weather, etc.… that have affected forest growth during the observation period (Zhou and Buongiorno 2004).
To get the transition probabilities between stand states p(s’|s) model (1) is used to predict the future stand state, s’, of a random initial stand in state s, by bootstrapping a random shock ε _{ t } from the set of the observed shocks. This is repeated a sufficient number of times to obtain stable estimates of the transition probabilities p(s’|s) (Zhou 2005, p. 51).
Transition probabilities between stand states
State # at t | Stand compositiona ^{ 1 } | State # at t + 1 year (transition probability) |
---|---|---|
1 | 000,000 | 1(0.797), 2(0.027), 3(0.011), 5(0.055), 9(0.062), 17(0.019), 33(0.029) |
2 | 000,001 | 1(0.025), 2(0.782), 4(0.014), 6(0.064), 10(0.053), 18(0.027), 34(0.035) |
3 | 000,010 | 1(0.025), 3(0.78), 4(0.029), 7(0.049), 11(0.064), 19(0.025), 35(0.028) |
4 | 000,011 | 1(0.025), 4(0.817), 8(0.049), 12(0.059), 20(0.022), 36(0.027) |
5 | 000,100 | 1(0.036), 5(0.777), 6(0.022), 7(0.045), 13(0.063), 21(0.028), 37(0.030) |
6 | 000,101 | 1(0.025), 5(0.010), 6(0.794), 8(0.058), 14(0.053), 22(0.028), 38(0.032) |
7 | 000,110 | 1(0.025), 7(0.826), 8(0.027), 15(0.067), 23(0.027), 39(0.029) |
8 | 000,111 | 1(0.025), 8(0.857), 16(0.046), 24(0.031), 40(0.042) |
9 | 001,000 | 1(0.025), 9(0.842), 10(0.022), 13(0.060), 25(0.025), 41(0.026) |
10 | 001,001 | 1(0.025), 9(0.016), 10(0.843), 12(0.012), 14(0.052), 26(0.017), 42(0.035) |
11 | 001,010 | 1(0.025), 11(0.822), 12(0.047), 15(0.050), 27(0.023), 43(0.033) |
12 | 001,011 | 1(0.025), 12(0.893), 16(0.052), 44(0.030) |
13 | 001,100 | 1(0.025), 13(0.875), 14(0.018), 15(0.033), 29(0.018), 45(0.031) |
14 | 001,101 | 1(0.025), 10(0.013), 13(0.010), 14(0.851), 16(0.047), 30(0.023), 46(0.031) |
15 | 001,110 | 1(0.025), 15(0.850), 16(0.050), 31(0.026), 47(0.050) |
16 | 001,111 | 1(0.025), 12(0.013), 15(0.013), 16(0.917), 32(0.019), 48(0.013) |
17 | 010,000 | 1(0.067), 17(0.745), 18(0.020), 21(0.051), 25(0.083), 49(0.034) |
18 | 010,001 | 1(0.025), 2(0.033), 17(0.012), 18(0.739), 20(0.011), 22(0.054), 26(0.102), 50(0.024) |
19 | 010,010 | 1(0.025), 3(0.039), 19(0.731), 20(0.026), 23(0.047), 27(0.092), 51(0.030) |
20 | 010,011 | 1(0.025), 4(0.040), 20(0.750), 24(0.058), 28(0.095), 52(0.032) |
21 | 010,100 | 1(0.025), 5(0.027), 21(0.763), 22(0.023), 23(0.038), 29(0.099), 53(0.025) |
22 | 010,101 | 1(0.025), 6(0.036), 18(0.012), 22(0.800), 24(0.036), 30(0.066), 54(0.024) |
23 | 010,110 | 1(0.025), 7(0.037), 23(0.795), 24(0.030), 31(0.089), 55(0.025) |
24 | 010,111 | 1(0.025), 8(0.066), 24(0.745), 32(0.120), 56(0.013) |
25 | 011,000 | 1(0.025), 9(0.035), 25(0.828), 26(0.022), 29(0.061), 57(0.030) |
26 | 011,001 | 1(0.025), 10(0.039), 26(0.837), 28(0.016), 30(0.053), 58(0.030) |
27 | 011,010 | 1(0.025), 11(0.036), 27(0.816), 28(0.048), 31(0.045), 59(0.030) |
28 | 011,011 | 1(0.025), 12(0.040), 28(0.859), 32(0.045), 60(0.031) |
29 | 011,100 | 1(0.025), 13(0.045), 25(0.012), 29(0.825), 30(0.028), 31(0.041), 61(0.023) |
30 | 011,101 | 1(0.025), 14(0.031), 30(0.879), 32(0.048), 62(0.017) |
31 | 011,110 | 1(0.025), 15(0.030), 31(0.863), 32(0.050), 63(0.032) |
32 | 011,111 | 1(0.025), 16(0.062), 28(0.014), 32(0.872), 64(0.027) |
33 | 100,000 | 1(0.066), 33(0.738), 34(0.019), 35(0.013), 37(0.050), 41(0.054), 49(0.060) |
34 | 100,001 | 1(0.025), 2(0.045), 34(0.776), 38(0.050), 42(0.044), 50(0.060) |
35 | 100,010 | 1(0.025), 3(0.059), 35(0.703), 36(0.056), 39(0.051), 43(0.051), 51(0.055) |
36 | 100,011 | 1(0.025), 4(0.062), 36(0.702), 40(0.062), 44(0.055), 48(0.011), 52(0.084) |
37 | 100,100 | 1(0.025), 5/90.038), 37(0.772), 38(0.015), 39(0.040), 45(0.054), 53(0.056) |
38 | 100,101 | 1(0.025), 6(0.029), 38(0.812), 40(0.023), 46(0.052), 54(0.059) |
39 | 100,110 | 1(0.025), 7(0.065), 39(0.786), 40(0.020), 47(0.040), 55(0.065) |
40 | 100,111 | 1(0.025), 8(0.049), 40(0.845), 48(0.031), 56(0.049) |
41 | 101,000 | 1(0.025), 9(0.047), 41(0.777), 42(0.026), 43(0.011), 45(0.041), 57(0.072) |
42 | 101,001 | 1(0.025), 10(0.056), 41(0.019), 42(0.780), 46(0.052), 58(0.068) |
43 | 101,010 | 1(0.025), 11(0.060), 43(0.774), 44(0.027), 47(0.058), 59(0.056) |
44 | 101,011 | 1(0.025), 12(0.049), 44(0.822), 48(0.049), 60(0.038), 64(0.016) |
45 | 101,100 | 1(0.025), 13(0.058), 41(0.015), 45(0.789), 46(0.023), 47(0.035), 61(0.055) |
46 | 101,101 | 1(0.025), 14(0.034), 45(0.015), 46(0.810), 48(0.050), 62(0.065) |
47 | 101,110 | 1(0.025), 15(0.039), 47(0.850), 48(0.043), 63(0.043) |
48 | 101,111 | 1(0.025), 16(0.074), 48(0.800), 64(0.101) |
49 | 110,000 | 1(0.025), 17(0.039), 33(0.014), 49(0.752), 50(0.024), 53(0.049), 57(0.098) |
50 | 110,001 | 1(0.025), 18(0.047), 34(0.025), 50(0.768), 54(0.055), 58(0.079) |
51 | 110,010 | 1(0.025), 19(0.039), 35(0.018), 51(0.750), 52(0.035), 55(0.046), 59(0.087) |
52 | 110,011 | 1(0.025), 20(0.051), 28(0.010), 36(0.017), 51(0.014), 52(0.730), 56(0.031), 60(0.122) |
53 | 110,100 | 1(0.025), 21(0.035), 37(0.021), 53(0.775), 54(0.017), 55(0.031), 61(0.096) |
54 | 110,101 | 1(0.025), 22(0.032), 38(0.015), 53(0.017), 54(0.783), 56(0.042), 62(0.086) |
55 | 110,110 | 1(0.025), 23(0.044), 39(0.015), 55(0.783), 56(0.031), 63(0.102) |
56 | 110111 | 1(0.025), 24(0.061), 40(0.036), 52(0.012), 56(0.757), 64(0.109) |
57 | 111,000 | 1(0.025), 25(0.062), 41(0.018), 57(0.821), 58(0.023), 61(0.050) |
58 | 111,001 | 1(0.025), 26(0.036), 42(0.018), 57(0.016), 58(0.852), 62(0.052) |
59 | 111,010 | 1(0.025), 27(0.058), 43(0.016), 59(0.832), 60(0.033), 63(0.037) |
60 | 111,011 | 1(0.025), 28(0.025), 44(0.025), 60(0.860), 64(0.064) |
61 | 111,100 | 1(0.025), 29(0.062), 45(0.014), 61(0.835), 62(0.022), 63(0.043) |
62 | 111,101 | 1(0.025), 30(0.042), 46(0.019), 58(0.019), 61(0.019), 62(0.842), 64(0.034) |
63 | 111,110 | 1(0.025), 31(0.036), 47(0.025), 59(0.017), 63(0.846), 64(0.050) |
64 | 111,111 | 1(0.025), 32(0.068), 48(0.017), 64(0.890) |
Markov price model
Annual transition probability matrix between price levels
Price t + 1 | |||
---|---|---|---|
Low | Medium | High | |
Price t | < $84 · m^{−3} | [$84 · m^{−3}, $94 · m^{−3}] | > $94 · m^{−3} |
Low | 0.82 | 0.10 | 0.08 |
Medium | 0.11 | 0.78 | 0.11 |
High | 0.07 | 0.12 | 0.81 |
System states and transition probabilities
Decisions and immediate rewards
Decisions consist in moving instantly from one stand state to another by harvesting some of the trees. It is assumed that market prices are exogenous, i.e. the decisions have no effect on prices. A decision produces a stand state with less basal area, less carbon sequestered in the stand, and a different composition of tree species and size. It also generates revenues that depend on the volume of timber harvested and on the price at decision time^{e}. A policy consists of a set of decisions applied systematically to each stand-price state. If the policy is applied to an entire forest, decisions alter the distribution of stand states and thus the landscape diversity, and possibly the part of the forest that stays in an old-growth (late-seral) state.
The forest landscape diversity is also expressed similarly with f _{ s }, the fraction of the forest landscape in state s. Another useful criterion is the fraction of the forested landscape in “old growth” state. For some forest types, old growth or late-seral stands have been defined in previous studies (Hummel and Calkin 2005). Here, old-growth is defined as the most frequent states that develop in the steady state predicted with equation (3) in the absence of harvest (Zhou and Buongiorno 2006). With the data in Table 2, the five old-growth states are 001,011; 001,111; 011,111; 101,111; and 111,111, which all have high basal area in the largest softwoods and hardwood trees.
The data on the expected timber volume and ecological criteria by stand state are obtained during the simulations with model (1) that give the transition probabilities. The immediate ecological reward of a decision is the characteristic of the state induced by the decision, for example, its tree diversity. The immediate financial reward is equal to the change in volume obtained by harvesting the stand from the current state to another, multiplied by the price level at decision time. The carbon sequestered was estimated from the amount of growing stock left after harvest, assuming 1.24 t · m^{−3} of CO_{2}e for pulpwood and 1.57 t · m^{−3} for sawtimber (AFC 2014).
In sum, each immediate reward, monetary or ecological, is summarized by a vector V = [v _{ ik }] where v _{ ik } is the immediate reward when the stand-price system is in state i and the decision is k (which changes the stand state or leaves it intact).
Consequences of management policies
Predicting expected undiscounted rewards
Predicting expected discounted rewards
Optimizing management policies
In parallel with the predictions of the effects of specific management policies, the optimization of policies differs depending on whether the rewards are discounted or undiscounted.
Optimizing expected discounted rewards
Optimizing expected undiscounted rewards
The best decisions obtained with equation (17) are still deterministic (D’ _{ ik } = 0 or 1) and independent of the initial stand-price state. And, the maximum value of the objective function, such as the maximum expected undiscounted species diversity over an infinite time horizon, is also independent of the initial condition.
Multiple objectives
Policies that best meet multiple objectives simultaneously are obtained by modifying the objective function of models (14) and (16), or/and by adding constraints. In the results shown below the models were kept linear by expressing both the objective function and the constraints in undiscounted or discounted terms.
Results
Effects of disturbances on predicted stand growth
While the total basal area is qualitatively similar for both models, the predictions of the basal area of pine trees are very different. Ignoring the random shocks, pines totally disappear in about 300 years. Instead, when the random disturbances are taken into account the basal area of the pine trees is never less than 20 m^{2} · ha^{−1}, about a third of the total basal area. Forest scientists agree that stochastic disturbances “can be a major determinant of forest structure” (Oliver and Larson 1990). In the case of loblolly pine, recurrent fires are typical in the south of the United States. Lobllolly pines are more fire resistant than hardwoods. Fires also cause openings that favor pine regeneration. The simulations with stochastic shocks are thus more likely to predict correctly the long-term evolution of loblolly pine-hardwood stands than the deterministic version.
Effects of catastrophes on forest landscape
In addition to the high-frequency, low-impact disturbances, the forests are also subject to low-frequency catastrophes, such as hurricanes in the region investigated here. The long-term effects of catastrophic events on the forest landscape were predicted with the Markov model in Table 2, with and without the 0.025 probability of a catastrophe appearing in the first column. The fraction of the landscape that would be present in the long run (steady state) in each of the 64 possible states was predicted with equation (3).
Economic and ecological consequences of current management
Long-term effects of the current management on economic return and ecological criteria, compared with the outcome without management
Criteria | Unit | With current management | Without management |
---|---|---|---|
NPV of harvested wood | $.ha^{−1} | 1339 | – |
Sawtimber harvest | m^{3} · ha^{−1} · y^{−1} | 1.5 | – |
Pulpwood harvest | m^{3} · ha^{−1} · y^{−1} | 0.4 | – |
Basal area | m^{2} · ha^{−1} | 19 | 22 |
Species diversity^{a} | % | 101 | 100 |
Size diversity^{a} | % | 98 | 100 |
Landscape diversity^{a} | % | 103 | 100 |
Old growth fraction | % | 18 | 29 |
CO_{2}e sequestered | t.ha^{−1} | 203 | 244 |
Cutting cycle | year | 11 | – |
Continuing the current management indefinitely generated an NPV of $1,339 · ha^{−1} in timber revenues and an average expected production of 1.9 m^{3} · ha^{−1} · y^{−1} over the entire forest. Due to this harvest the expected average basal area was 2 m^{2} · ha^{−1} lower with the current management than it would be without human intervention.
Using the diversity indices without management as a basis of comparison, the current management increased tree species diversity slightly (1%). Like catastrophic events, management also increased landscape diversity, though to a lesser extent (3%). However, by harvesting large trees it decreased the expected tree size diversity by 2%.
The fraction of old-growth in the forest maintained by the current management was 11% lower than it would be in a natural forest, and the value of the average carbon sequestration was 42 t · ha^{−1} lower. All these outcomes stemmed from harvests in individual stands that occurred at average intervals of 7 years under the current management.
Optimizing financial objectives
The policy that maximized the expected NPV was obtained by solving model (14)–(15), with a uniform distribution of initial stand-price states (i.e. β _{j} = 1/192 ∀ j). This best policy was then used to obtain with equations (12)–(13) the expected NPV given the stand states distribution in the region of interest, as reflected by the plot data, and the current price level (“low” in 2014 according to the definitions in Table 3).
Management policy that maximizes the net present value of harvested wood, depending on stand state and price level
Stand state # | Stand composition ^{ 1 } | Price | ||
---|---|---|---|---|
Low | Medium | High | ||
Best decision ^{ 2 } | ||||
1 | 000,000 | – | – | – |
2 | 000,001 | 1 | 1 | 1 |
3 | 000,010 | – | – | 1 |
4 | 000,011 | 3 | 3 | 1 |
5 | 000,100 | 1 | 1 | 1 |
6 | 000,101 | 1 | 1 | 1 |
7 | 000,110 | 3 | 3 | 1 |
8 | 000,111 | 3 | 3 | 1 |
9 | 001,000 | 1 | 1 | 1 |
10 | 001,001 | 1 | 1 | 1 |
11 | 001,010 | 3 | 3 | 1 |
12 | 001,011 | 3 | 3 | 1 |
13 | 001,100 | 1 | 1 | 1 |
14 | 001,101 | 1 | 1 | 1 |
15 | 001,110 | 3 | 3 | 1 |
16 | 001,111 | 3 | 3 | 1 |
17 | 010,000 | 1 | 1 | 1 |
18 | 010,001 | 1 | 1 | 1 |
19 | 010,010 | 3 | 3 | 1 |
20 | 010,011 | 3 | 3 | 1 |
21 | 010,100 | – | 1 | 1 |
22 | 010,101 | 21 | 1 | 1 |
23 | 010,110 | 3 | 3 | 1 |
24 | 010,111 | 3 | 3 | 1 |
25 | 011,000 | 1 | 1 | 1 |
26 | 011,001 | 1 | 1 | 1 |
27 | 011,010 | 3 | 3 | 1 |
28 | 011,011 | 3 | 3 | 1 |
29 | 011,100 | 21 | 1 | 1 |
30 | 011,101 | 21 | 1 | 1 |
31 | 011,110 | 3 | 3 | 1 |
32 | 011,111 | 3 | 3 | 1 |
33 | 100,000 | 1 | 1 | 1 |
34 | 100,001 | 1 | 1 | 1 |
35 | 100,010 | 3 | 3 | 1 |
36 | 100,011 | – | 3 | 1 |
37 | 100,100 | 1 | 1 | 1 |
38 | 100,101 | 1 | 1 | 1 |
39 | 100,110 | 3 | 3 | 1 |
40 | 100,111 | 36 | 3 | 1 |
41 | 101,000 | 1 | 1 | 1 |
42 | 101,001 | 1 | 1 | 1 |
43 | 101,010 | 3 | 3 | 1 |
44 | 101,011 | 36 | 3 | 1 |
45 | 101,100 | 1 | 1 | 1 |
46 | 101,101 | 1 | 1 | 1 |
47 | 101,110 | 3 | 3 | 1 |
48 | 101,111 | 36 | 3 | 1 |
49 | 110,000 | 1 | 1 | 1 |
50 | 110,001 | 1 | 1 | 1 |
51 | 110,010 | 3 | 3 | 1 |
52 | 110,011 | – | 3 | 1 |
53 | 110,100 | 21 | 1 | 1 |
54 | 110,101 | 21 | 1 | 1 |
55 | 110,110 | 3 | 3 | 1 |
56 | 110111 | 52 | 3 | 1 |
57 | 111,000 | 1 | 1 | 1 |
58 | 111,001 | 1 | 1 | 1 |
59 | 111,010 | 3 | 3 | 1 |
60 | 111,011 | 52 | 3 | 1 |
61 | 111,100 | 21 | 1 | 1 |
62 | 111,101 | 21 | 1 | 1 |
63 | 111,110 | 3 | 3 | 1 |
64 | 111,111 | 52 | 3 | 1 |
Long-term effects of maximizing economic returns (NPV) from harvested wood on NPV and ecological criteria, compared with the outcomes due to the current management
Criteria | Unit | Current management | Maximizing NPV |
---|---|---|---|
NPV of harvested wood | $ · ha^{−1} | 1339 | 5535 |
Sawtimber harvest | m^{3} · ha^{−1} · y^{−1} | 1.5 | 5.3 |
Pulpwood harvest | m^{3} · ha^{−1} · y^{−1} | 0.4 | 2.6 |
Basal area | m^{2} · ha^{−1} | 19 | 12 |
Species diversity^{a} | % | 101 | 96 |
Size diversity^{a} | % | 98 | 89 |
Landscape diversity^{a} | % | 103 | 31 |
Old growth fraction | % | 18 | 0 |
CO_{2}e sequestered | t · ha^{−1} | 203 | 98 |
Cutting cycle | year | 11 | 5 |
Optimizing ecological objectives
Long-term effects of maximizing expected tree species diversity or CO _{ 2 } e sequestered, compared with the outcome due to the current management
Criteria | Unit | Current management | Maximizing tree species diversity | Maximizing CO _{ 2 } e sequestered |
---|---|---|---|---|
NPV of harvested wood | $ · ha^{−1} | 1339 | 3363 | 119 |
Sawtimber harvest | m^{3} · ha^{−1} · y^{−1} | 1.5 | 4.3 | 0.1 |
Pulpwood harvest | m^{3} · ha^{−1} · y^{−1} | 0.4 | 1.1 | 0.1 |
Basal area | m^{2} · ha^{−1} | 19 | 17 | 22 |
Species diversity^{1} | % | 101 | 108 | 99 |
Size diversity^{1} | % | 98 | 96 | 100 |
Landscape diversity^{1} | % | 103 | 71 | 101 |
Old growth fraction | % | 18 | 13 | 15 |
CO_{2}e sequestered | t · ha^{−1} | 203 | 151 | 245 |
Cutting cycle | year | 11 | 10 | 86 |
Figure 3b shows how the tree species diversity index of the entire forest evolved from its initial level to the steady state with the policy that maximized expected tree species diversity. The diversity index is relative to the steady-state diversity of the unmanaged forest. With this policy it took approximately 30 years for the species diversity index to reach 90% of its maximum steady-state value of 108%.
Maximizing carbon sequestration
The amount of CO_{2}e sequestered in the trees living biomass in steady state could be increased by 42 t · ha^{−1} compared to the current management (last column of Table 7), but in doing so, wood production would be very low (0.2 m^{3} · ha^{−1} · y^{−1}). In fact, the level of CO_{2}e sequestered in this way would be almost the same as that obtained by letting the forest grow naturally without any harvest (Table 4, last column). Indeed, the hands-off option maybe superior as it kept 29% of the landscape in old-growth state against only 15% when maximizing CO_{2}e sequestration, at the small opportunity cost of $119 · ha^{−1} in foregone NPV from timber revenues^{j}. The other criteria were similar to those obtained by natural forest growth.
Constrained financial and ecological objectives
Discussion and conclusions
In a 1999 interview, G.B. Dantzig, the inventor of linear programming, remarked that “all problems that are solved under deterministic means have that fundamental weakness—they don’t properly take uncertainty into account” (Dantzig 1999). Forestry planning is no different. Risk, uncertainty, and stochastic behavior are critical parts of how forest ecosystems work. As illustrated above, simulations of mixed loblolly pine-hardwood stands reveal how ignoring random disturbances due to biological or catastrophic shocks in forest growth models can lead to wrong ecological predictions, such as the disappearance of pines in this context. To these natural sources of risk must be added the high risk that stems from price fluctuations which complicates decision making with financial objectives.
The methods suggested in this paper to handle risk in forest decision making follow the modeling approach outlined by Holling et al. (1986). The first step is to “bring the world to the laboratory” with possibly complex and non-linear stochastic models, and then simplify the models to allow for efficient optimization with Markov decision process models. MDPs are attractive for their simplicity while keeping the essence of planning problems under risk^{l}. They recognize that the future is unknown and that it may be described only in terms of probabilities. And, they lead to best adaptive management policies whereby decisions depend entirely on the systems state, i.e. the state of knowledge of the decision maker at decision time.
The fact that with MDP models the future state of a forest ecosystem depends only on its current state should not be viewed as a shortcoming, but recognition of fact. Predictions can rely only on current knowledge. This current knowledge may include past behavior of the system (such as past tree growth) if necessary. Thus, Markov models are not necessarily “memory less” and apparent failure of Markov models to correctly represent forest growth (Binkley 1980; Roberts and Hruska 1986; Johnson et al. 1991) may be due to incomplete description of the current ecosystem state rather than to a shortcoming of the Markov model. Similarly, Markov models of price changes are very general, embracing random walk, rational expectations, autoregressive, and “any stochastic model in which the price is conditional on previous prices” (Taylor 1984). In sum, Markov chains allow the simplification of complex, multi-dimensional stochastic processes and make their optimization easier or at all possible (Holling et al. 1986; Insley and Rollins 2005).
There exists considerable potential and flexibility to enrich the models presented in this paper as they are relatively small compared with the size of the problems that can be solved with current linear programming software. Still, to be successful in forest ecosystem management the MDPs should adhere to the general principles of parsimony and simplicity. To this end, the number of system states must be kept as small as possible, with a few state variables (tree species and size categories, prices) and a few levels of these variables.
In the same spirit of simplicity, multiple objectives such as optimizing financial returns subject to ecological constraints, of vice-versa, have been treated here with linear models. This requires expressing both the objective function and the constraints in either discounted or undiscounted criteria. While discounting financial returns from forestry has long been a standard procedure (Faustmann 1849), it is less so for ecological criteria. Nevertheless, it is plausible to give more weight to the present ecological characteristics of a forest ecosystem than to their future values. Indeed, it seems preferable to maintain desirable current states (such as old-growth stands) rather than lose them, even if they could be restored later (albeit after a long delay). It has also been argued on theoretical grounds that “the future benefits provided by a generic public good--environmental quality--should be discounted at a rate that is close to the market rate of return for risk-free financial assets” (Howarth 2009), a principle that has been applied for example in discounting future carbon sequestered in forests (Boscolo et al. 1997)^{m}.
The methods presented here can be expanded in many directions, in particular to deal with the new carbon markets and the highly stochastic prices for CO_{2}e sequestered in forests. The treatment is parallel to the stochastic prices of timber. This would increase substantially the size of the problem, but still keep it well within the capabilities of current linear programming software.
Other issues may require deeper modifications, to deal with non-stationary processes. For example, “climate change can affect forests by altering the frequency, intensity, duration, and timing of fire, drought, introduced species, insect and pathogens outbreaks, hurricanes, windstorms, ice storms, or landslides” (Dale et al. 2001). This implies a change in the transition probabilities over time. Although non-stationary problems can in principle be converted to stationary ones by a reformulation (Bertsekas 1995, p. 167), efficient numerical methods are still elusive (Ghate and Smith 2013). Nevertheless, the consequences of changes in transition probabilities on the steady-state criteria can be readily explored with the methods described in this paper. In sum, given the conceptual generality and the well-developed theory of MDPs, coupled with the powerful solution techniques available, the MDP approach is well suited to deal with risk in forest ecosystem management, and to develop practical adaptive management policies with both economic and ecological objectives.
Endnotes
^{a}In this paper the terms “risk” or “uncertainty” are synonymous. Although Knight (1921) defines uncertainty as the absence of probabilities, it may be argued that probabilities, objective (data based) or subjective (opinion based) always exist to some degree.
^{b}While modern linear programming software places almost no limit (computationally) on the number of states, it is best for all practical purposes (identifying states of nature, doing the computation, and laying out the recommendations) to use the minimum number of states needed for sufficient accuracy and realism of applications.
^{c}For other applications, any stochastic model of forest growth can be used to compute the transition probabilities used in the MDP approach.
^{d}If the price series is not stationary, the price data are first de-trended, and the trend is added to the discount rate in present value calculations.
^{e}In the particular application described here, prices vary by tree size and species, and all the prices change in parallel with the probabilistic changes of the price index described in Table 3.
^{f}That is: y _{ ik } = z _{0ik } + dz _{1ik } + d ^{2} z _{2ik } + d ^{3} z _{3ik } + … where z _{ tik } is the probability of state i and decision k at time t and d = 1/(1 + ρ) is the discount factor (Hillier and Lieberman 2005, p. 921).
^{g}As applied to forestry, the MDP described by equations (10) and (11) is a generalization of (Faustmann 1849) formula recognizing that future stand states and prices are known only as probability distributions. The classical, deterministic Faustmann formula is a special case in which the transition probabilities are 0 or 1 (Buongiorno 2001).
^{h}According to Timber Mart South (2014), the average price of softwood sawtimber stumpage in the South of the United States in the second quarter of 2014 was $25 per short ton or approximately $19 · m^{−3}, placing it in the low range according to the definitions in Table 3.
^{i}Previous results also show that the optimum adaptive policy derived here is superior in terms of NPV to an optimum fixed policy that converts stands to a chosen state at fixed intervals (Zhou et al. 2008b).
^{j}The reduction in old-growth fraction from 29% to 15% was due to the best decision calling for a harvest when the stand was in the old-growth state #32 (011,111). Although the harvest occurred only every 86 years on average, this was equivalent to a low frequency natural catastrophe reducing substantially the fraction of the landscape in old-growth state.
^{k}In Figure 4, the old growth fraction is the right-hand side of equation (18) with O* replaced by its annual constant perpetual equivalent, ρO */(1 + ρ).
^{l}See Buongiorno and Gilless (2003, p. 337–371) for an introduction to Markov and MDP models in forestry.
^{m}Alternatively, models (10)–(11) and (12–13) can be extended to deal with discounted objective functions and undiscounted constraints, or the reverse, by introducing non-linear constraints (e.g. Rollin et al. 2005; Zhou 2005), but at the cost of the attendant numerical difficulties.
Declarations
Acknowledgements
The preparation of this paper was supported in part by the USDA Forest Service, Southern Research Station, through a cooperative research agreement with Joseph Buongiorno, directed by Jeff Prestemon. It was also supported in part by the USDA McIntire-Stennis fund WVA00105. We are also grateful to the organizers of the workshop on Risk and Uncertainty in Ecosystems Dynamics, Beijing Forestry University, October 13–17, 2014, for motivating the study.
Authors’ Affiliations
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