The matrix growth model can be integrated into an optimization model consisting of a function that represents the forest management objective(s) given a set of constraints. For example, the objective of the landowner can be maximizing the financial returns from timber harvesting, while maintaining a sustained supply of timber [
19,
40]. Those management objectives can also imply optimizing simultaneously the provision of timber and other goods and services, such as carbon sequestration [
21,
22], tree diversity [
20], or the simultaneous provision of water and carbon services, e.g., for even-aged forest [
41].
In this paper, we explored the individual and joint optimization of water and timber yields, in uneven-aged forest stands, while ensuring a sustainable forest management and conservation (that is land use change, and total clear-cutting is not allowed). We considered the direct impact of different restriction levels to timber harvesting, and the effect of these restrictions on harvesting decisions, and consequently on forest state (structure) and total LAI, and indirectly on water provisioning service (hereinafter water yield).
We determined the optimal path of harvesting decisions to get a forest structure that maximizes timber or joint timber and water benefits, while satisfying a set of constrains on harvesting and forest growth. Optimal harvesting decisions and resulting forest structure depend on the initial conditions at each forest grid cell (based on virtual forest inventories). As indicated before (
Section 2.2), we used the output of the ecohydrological model FORHYCS [
37]), which simulates stand dynamics in a spatially distributed way), the forest growth model (depending on total stand LAI at each grid cell and growing period
), and restrictions imposed to the maximum share of the trees that can be harvested).
2.3.1. The Harvesting/Thinning Decision Model
The optimal harvesting/thinning decision model is implemented as a finite-horizon discrete-time optimal-control problem, where the state vector reports the number of trees ( of species in group i and size class j at each period of time, which in turn is discrete and indexed by t∈ {0,1,2,…,T}. T represents the evaluated time horizon (20 years and 40 years here). Our model further considers a 10-year harvesting frequency or harvesting cycle (S).
The objective function of the decision problem is framed as the maximization of the finite time horizon net present discounted value (NPV) of the net benefits delivered by uneven-aged forest stands, when both the economic value of the (standing) timber stock at the beginning and the end of the decision period were taken into account [
42]. The decision problem is framed from a landowners perspective assuming that water benefits are internalised though payments to the landowner.
In the simplest case, the objective function is to maximize the NPV from timber revenues over a finite time horizon. When dealing with infinite time horizon problems, analysts usually maximize the land expectation value (LEV), which represents the perpetual income stream produced by periodic crops, such as tress, starting with bare land. This latter implies that LEV is estimated by subtracting the initial timber stock value from the NPV of an infinite sequence of forest rotations [
43].
In order to deal with a finite time horizon forest optimization problem that is consistent with the optimal solutions for an infinite chain of forest harvesting cycles, we followed Samuelson’s [
2] recommendation to correct Fisher’s [
44] false solution based on a single forest cycle. Fisher argued that providing positive real interest rates, it would be optimal to cut a forest stand when its growth rate equals the rate of interest. Samuelson [
2] claimed that Fisher’s solution was wrong, as this solution is only correct when a forest owner does not continue with the forest activity after harvesting. Frequently, a forest owner maybe interested in replanting the forest stand, and then cutting and replanting it again in an infinite sequence of rotations. The correct solution of the optimal forest rotation problem considering this infinite sequence of rotations was proposed by [
1]. Maximizing the net present value (NPV) for just one planting cycle will give a longer rotation period that when an infinite sequence of rotations were considered, implying is that case that cutting more rapidly growing young trees would increase the NPV of the forest compared to a rotation period that follows Fisher recommendations.
Our forest benefit function includes a
proxy of the market land rental value. We approximated this market land rental, by quantifying the vector of the expected timber revenues (
=
) for each tree of species
i and size class
j, given the survival and growth probabilities defined by the uneven forest growth model presented in
Section 2.2, and a timber price vector (
v =
) that represents the standing value of trees (i.e., timber prices minus variable harvesting costs for each species group and size class). We used the survival and up-growth probabilities to estimate the maximum harvesting feasible, which corresponds to the harvesting rates that equal net timber growth over a growing cycle. The harvesting probabilities were estimated for each one of the size classes, and those indicate the probability (
) an individual tree of a species in group
i to be felled as it reaches a size class
j.
For the estimation of the
values, we assumed that all living trees will be eventually felled, and therefore the aggregated harvesting probabilities over a
T growing cycle totals one:
The economic problem is to find the optimal harvesting path (
) and tree stock (
) in a harvesting cycle
T that maximize the NPV of the net forest benefits over a finite time horizon, given constrains on: (i) forest (biological) growth (Equation (
12)), the feasible harvest (Equation (
13)), (iii) non-negativity constrains (Equations (
14) and (
15), and the initial forest state, defined as the initial distribution of trees by species and size class (
y at each forest grid cell (indexed by
):
subject to:
where FC is the fixed harvesting costs,
is the stock value of the standing trees at the beginning of the evaluated period,
y a vector representing the stock of trees at the end of the evaluated period in the
forest unit,
is the discount function (
= 1/(1 + r), being
r the discount rate).
To estimate the market land rental value after the cutting cycle T, we considered a subsequent growing cycle of a -years duration. It is assumed that identical chains of the growing cycle are periodically repeated. The term () is used to approximate the stock value of the expected timber revenues as the sum of an infinite geometric series (where < 1).
The coefficient stands for the maximum harvesting intensity allowed, (where ), which is used to avoid complete deforestation solutions. We considered, in that case that harvesting/thinning interventions cannot be higher than a share of the tree stock, at every harvesting period evaluated.
For the joint maximization of timber and water benefits, we added a second term to estimate the value of felled trees and final tree stock, defined by the price vector v = , and the expected benefits associated with water yield. Those price vectors account for the potential water benefits due to a decrease in the number of trees and their associated LAI, which result in an increase in water yield, as shown later.
The objective function in the presence of payments for increasing water yields can be written as follows:
where
=
, being
the expected value of each individual tree of species
i and size class
j in terms of their contribution to water benefits. As a decrease in tree density, hence in total LAI, favors water yields in this model, no additional water provisioning services is associated with to the stock of trees.
2.3.2. Forest State Dynamics
The optimal stock (forest structure) at each time period is defined by the state vector (
y) at each period of time
t. Changes in the state variable along the evaluated time periods rely upon the ingrowth, mortality, and up-growth functions, which in turn depend on the total LAI at the beginning of each
growing period, and on the optimal harvesting decisions. Forest growth for each species in group
i is estimated using Equation (
1) for each one of the
growing periods comprised in one harvesting cycle
T. LAI is recalculated after each
growing period, and up-growth, mortality and ingrowth probabilities reappraised in accordance, considering Equations (
6), (
7) and (
9), and integrated again into Equation (
1).
The decision maker must decide upon the optimal number of trees to be felled by species and diametric class at each harvesting cycle S. Harvesting is the control variable that models the decision mechanism. The control variable enters in the state motion equation through a single control vector (h) that influences the forest state.
The model is solved in Matlab R2016a for different finite time horizon periods (), using linear programming techniques, and assuming that harvesting takes place at the beginning of each harvesting period S, every 10-years, which in turn comprise two consecutive growing periods of 5 years.