Bootstrap Simulation, Markov Decision Process Models, and Role of Discounting in the Valuation of Ecological Criteria in Uneven-Aged Forest Management

Abstract

This chapter presents general methods combining stochastic simulation and Markov decision process models, with a specific application to the issue of discounting ecological criteria. The literature argues for and against discounting ecological costs and benefits like standard financial investments. As part of this debate, this chapter investigates some purely ecological consequences of discounting ecological criteria. The methods are Markov decision process models with infinite time horizon and discounted or undiscounted ecological objectives, based on bootstrap simulations of stand growth. The data were from Douglas-fir/western hemlock forests in the U.S. Pacific Northwest, with the assumption of continuous-cover forestry. The ecological criteria examined here included the stand basal area per hectare, the tree species and size diversity measured with Shannon’s index, and the percentage of the forest in late seral stage. In maximizing expected tree species diversity, 18 out of 64 possible stand states would call for different decisions with discounting. For tree size diversity, the decisions differed for five stand states. For basal area all but two states called for the same decision. For late seral forest frequency, discounting led to different decisions for 13 stand states. However, for only a few stand states were the ecological criteria substantially different immediately after harvest. Thus, discounting would matter only for a few stand states, which in the study area had a low frequency. Given this initial condition, only the expected late seral forest frequency differed substantially after a decade with discounting. In the long run, the initial condition does not matter and discounting the criteria gave very similar values as not discounting.

Appendix

The matrix G and the vector r in the deterministic part of the growth model (8.2) have the following form (Liang et~al. 2005):

$$ \begin{aligned} & {\mathbf{G}} = \left[ {\begin{array}{*{20}{c}} {{{\mathbf{G}}_1}} & {} & {} & {} \\ {} & {{{\mathbf{G}}_2}} & {} & {} \\ {} & {} & {{{\mathbf{G}}_3}} & {} \\ {} & {} & {} & {{{\mathbf{G}}_4}} \\ \end{array} } \right],\ {{\mathbf{G}}_i} = \left[\arraycolsep4pt {\begin{array}{*{20}{c}} {{a_{i1}}} & {} & {} & {} & {} \\ {{b_{i1}}} & {{a_{i2}}} & {} & {} & {} \\ {} & \ddots & \ddots & {} & {} \\ {} & {} & {{b_{i,17}}} & {{a_{i,18}}} & {} \\ {} & {} & {} & {{b_{i,18}}} & {{a_{i,19}}} \\ \end{array} } \right] \\ & {\mathbf{r}} = \left[ {\begin{array}{*{20}{c}} {{{\mathbf{r}}_1}} \\ {{{\mathbf{r}}_2}} \\ {{{\mathbf{r}}_3}} \\ {{{\mathbf{r}}_4}} \\ \end{array} } \right],\quad {{\mathbf{r}}_i} = \left[ {\begin{array}{*{20}{c}} {{r_i}} \\ {0} \\ \vdots \\ {0} \\ \end{array} } \right] \\ \end{aligned} $$

where a _ij is the probability that a tree of species i and diameter class j stays alive and in the same diameter class between t and t + 1. i = 1 for Douglas fir, 2 for other shade tolerant species, 3 for western hemlock, and 4 for other shade-tolerant species. b _ij is the probability that a tree of species i and diameter class j stays alive and grows into diameter class j + 1, and r _i is the number of trees of species group i recruited in the smallest diameter class between t and t + 1, with a time period of 1 year. Recruitment is zero in the higher diameter classes. The b _ij probability is equal to the annual tree diameter growth, g _ij , divided by the width of the diameter class. Diameter growth is a function of tree diameter D _j (cm), stand basal area B (m²/ha), site productivity Q (m³/ha/year), tree species diversity H _s , and tree size diversity H _d .

$${\fontsize{9}{11}\selectfont{ \begin{aligned} & {g_{1j}} = 0.7860 + 0.0124{D_j} - 0.0001D_j^2 - 0.0107{B} + 0.0267Q + 0.0658{H_s} - 0.2426{H_d}\quad \\ & {g_{2j}} = 0.6104 - 0.0038{D_j} + 0.0001D_j^2 - 0.0080{B} + 0.0170Q + 0.0707{H_s} - 0.0693{H_d}\quad \\ & {g_{3j}} = 0.9026 + 0.0148{D_j} - 0.0001D_j^2 - 0.0107{B} + 0.0061Q - 0.0250{H_s} - 0.1750{H_d}\quad \\ & {g_{4j}} = 0.5851 + 0.0081{D_j} - 0.00003D_j^2 - 0.008{B} + 0.0178Q + 0.1285{H_s} - 0.1441{H_d}\quad \\ \end{aligned} }}$$

The expected recruitment of species i is represented by a Tobit model:

$$ {r_i} = \Phi ({{\mathbf{\beta }}_i}{{\mathbf{x}}_i}/{\sigma_i}){{\mathbf{\beta }}_i}{{\mathbf{x}}_i} + {\sigma_i}\phi ({{\mathbf{\beta }}_i}{{\mathbf{x}}_i}/{\sigma_i}) $$

with:

$$ \begin{array}{llll} & {{\mathbf{\beta }}_1}{{\mathbf{x}}_1} = - 21.9317 - 1.2996B + 0.0971{N_1} + 0.8007Q + 11.119{H_s} - 6.8020{H_d} \\ & {{\mathbf{\beta }}_2}{{\mathbf{x}}_2} = - 23.6333 - 0.8293B + 0.0975{N_2} + 0.2032Q + 8.4122{H_s} - 5.9733{H_d} \\ & {{\mathbf{\beta }}_3}{{\mathbf{x}}_3} = - 30.8842 - 0.9359B + 0.0926{N_3} + 0.6699Q + 14.693{H_s} - 9.8919{H_d} \\ & {{\mathbf{\beta }}_4}{{\mathbf{x}}_4} = - 34.5350 - 0.7512B + 0.0924{N_4} + 0.7375Q + 8.0361{H_s} - 2.2701{H_d} \end{array} $$

where N _i is the number of trees per hectare in species group i; Φ and ϕ are respectively the standard normal cumulative and density functions, and the standard deviations of the residuals are, σ ₁ = 23.5417, σ ₂ = 22.4354, σ ₃ = 27.3244, σ ₄ = 23.0297.

The probability of tree mortality per year, m _ij , is a species-dependent probit function of tree size and stand state:

$${\fontsize{8}{10}\selectfont{ \begin{aligned} & {m_1} = \frac{1}{{10.5}}\Phi ( - 2.1103 - 0.0356{D_j} + 0.0002D_j^2 + 0.0081B - 0.0200C + 0.0059{H_s} + 0.5110{H_d}){ } \\ & {m_2} = \frac{1}{{10.5}}\Phi ( - 1.4063 - 0.0204{D_j} + 0.0002D_j^2 + 0.0053B - 0.0147C + 0.0022{H_s} + 0.1411{H_d}){ } \\ & {m_3} = \frac{1}{{10.5}}\Phi ( - 3.1746 - 0.0416{D_j} + 0.0003D_j^2 + 0.0156B - 0.0230C + 0.3252{H_s} + 0.4192{H_d}){ } \\ & {m_4} = \frac{1}{{10.5}}\Phi ( - 1.5188 - 0.0093{D_j} - 0.0000D_j^2 - 0.0042B - 0.0303C - 0.2721{H_s} + 0.3528{H_d}){ } \end{aligned}}} $$

The probability that a tree stays alive and in the same size class from t to t + 1 is, then:

$$ {a_{ij}} = {1} - {b_{ij}} - {m_{ij}} $$

The expected single tree volume v _ij (m³) is a species-dependent function of tree and stand characteristics:

$$ {\fontsize{8.8}{10.8}\selectfont{\hspace*{-1pt}\begin{aligned} & {v_{1j}} = - 0.6116 - 0.0119{D_j} + 0.0012D_j^2 + 0.0132B + 0.0293Q - 0.1249{H_s} - 0.0330{H_d} \\ & {v_{2j}} = - 0.5487 + 0.0038{D_j} + 0.0009D_j^2 + 0.0044B + 0.0173Q + 0.0136{H_s} + 0.0358{H_d} \\ & {v_{3j}} = - 0.4621 - 0.0141{D_j} + 0.0013D_j^2 + 0.0072B{ + 0}{.0104}Q - 0.0659{H_s} + 0.0516{H_d} \\ & {v_{{4 }j}} = - 0.1401 - 0.0170{D_j} + 0.0011D_j^2{ + 0}{.0087}B{ + 0}{.0146}Q - {0}{.2471}{H_s} + 0.0125{H_d} \end{aligned}}} $$