Optimal Rotation Age in Fast Growing Plantations: A Dynamical Optimization Problem

Abstract

Forest plantations are economically and environmentally relevant, as they play a key role in timber production and carbon capture. It is expected that the future climate change scenario affects forest growth and modify the rotation age for timber production. However, mathematical models on the effect of climate change on the rotation age for timber production remain still limited. We aim to determine the optimal rotation age that maximizes the net economic benefit of timber volume in a negative scenario from the climatic point of view. For this purpose, a bioeconomic optimal control problem was formulated from a system of Ordinary Differential Equations (ODEs) governed by the state variables live biomass volume, intrinsic growth rate, and area affected by fire. Then, four control variables were associated to the system, representing forest management activities, which are felling, thinning, reforestation, and fire prevention. The existence of optimal control solutions was demonstrated, and the solutions of the optimal control problem were also characterized using Pontryagin’s Maximum Principle. The solutions of the model were approximated numerically by the Forward–Backward Sweep method. To validate the model, two scenarios were considered: a realistic scenario that represents current forestry activities for the exotic species Pinus radiata D. Don, and a pessimistic scenario, which considers environmental conditions conducive to a higher occurrence of forest fires. The optimal solution that maximizes the net benefit of timber volume consists of a strategy that considers all four control variables simultaneously. For felling and thinning, regardless of the scenario considered, the optimal strategy is to spend on both activities depending on the amount of biomass in the field. Similarly, for reforestation, the optimal strategy is to spend as the forest is harvested. In the case of fire prevention, in the realistic scenario, the optimal strategy consists of reducing the expenses in fire prevention because the incidence of fires is lower, whereas in the pessimistic scenario, the opposite is true. It is concluded that the optimal rotation age that maximizes the net economic benefit of timber volume in P. radiata plantations is 24 and 19 years for the realistic and pessimistic scenarios, respectively. This corroborates that the presence of fires influences the determination of the optimal rotation age, and as a consequence, the net economic benefit.

Appendices

Appendix A. Computational Implementation

To continue with the numerical solution of the optimal control problem. First, we rewrite system (1) as follows.

$$\begin{aligned} \begin{array}{rlr} D'(t)&=g(t, D(t), U(t)) \end{array} \end{aligned}$$

(19)

where $D(t)=[B(t), r(t), I(t)]^\textrm{T}$ is the vector of solutions of the system for equations of state (1); and $U(t)=[F(t), T(t), R(t), S(t)]^\texttt {T}$ represents the control variables, g is the right-hand side vector of the equations of state. In addition, we rewrite the objective functional of Eq. (13), as

$$\begin{aligned} \begin{array}{rlr} J(F,T,R,S)=\int _{0}^{T_f}f[t, D(t), U(t)]dt. \end{array} \end{aligned}$$

Where f is the integrand of the objective functional (13). Following Eq. (16), we denote the adjoint variables ${\lambda }=(\lambda _1, \lambda _2, \lambda _3)^\textrm{T}$ that were defined in Eq. (14), We now rewrite the Hamiltonian function, as follows:

$$\begin{aligned} H= f[t, D(t), U(t)] + {\lambda }[g(t, D(t), U(t))] \end{aligned}$$

then the optimality condition is expressed

$$\begin{aligned} \frac{\partial H}{\partial U}=0, \quad \text {at} \quad U^*\Rightarrow f_U + {\lambda } g_U=0, \end{aligned}$$

the attached variables and cross-cutting conditions are also expressed, as follows

$$\begin{aligned} {\lambda }'=-\frac{\partial H}{\partial D}=-(f_D + {\lambda } g_U), \quad {\lambda }(T_f)=0. \end{aligned}$$

The Forward–Backward Sweep Method (FBSM) is a numerical technique for solving optimal control problems. FBSM is one of the indirect methods valid for several areas that is designed to numerically solve ODEs generated by Pontryagin’s Maximum Principle (Lenhart and Workman 2007). It is known that indirect methods present difficulties for complex and difficult to-analyze optimal control problems. For this, there are direct methods that allow obtaining an approximation of the optimal solution for complex problems (Bulirsch et al. 1994). However, it is interesting to know the convergence of the FBSM method. In Mattioli et al. (2022), they present a convergence result for an optimal control problem. The authors do not claim that the FBSM overcomes some difficulties of a more complex optimal control problem, but that the method is easy to apply to small problems and can provide a quick check of previous results of more efficient methods. For information on the convergence and stability of this method, see (Hackbusch 1978). The steps of the FBSM algorithm are then presented.

Step 1.:

Make an initial guess for U on the interval.

Step 2.:

Using the initial condition $D1 = D(0) = [B(0), r(0), I(0)]^{\textrm{T}}=[B_0,r_0,I_0]^{\textrm{T}}$ and the values of U, solve D forward in time according to its differential equation in the optimality system (19).

Step 3.:

Using the transversality condition ${\lambda }_{N+1} = (\lambda _1(T_f),\lambda _2(T_f),\lambda _3(T_f))= (0,0,0)$, where $N+1$ is the step number and the values of U and D, solve ${\lambda }$ backward in time according to its differential equation in the optimality system.

Step 4.:

Update U by introducing the new values of D and ${\lambda }$ in the optimal control characterization.

Step 5.:

Check convergence. If the values of the variables in this iteration and in the last iteration are very close, the current values are output as solutions. If the values are not close, go back to Step 2.

Appendix B. Proof of Condition $(A_4)$

To show that the objective functional is concave in U, we define

$$\begin{aligned} \mathcal {Z}(t,X,U)=[ p_1FB+p_2TB-c_1F^2 - c_2T^2 - c_3R^2 - c_4S^2]. \end{aligned}$$

The definition of the convex set is as follows

$$\begin{aligned} (1-q)\mathcal {Z}(t,X,u) + q\mathcal {Z}(t,X,s)\le \mathcal {Z}(t,X,(1-q)u + qs); \end{aligned}$$

(20)

where u, s are two vectors and $q\in (0,1)$. First, we will develop the right-hand side of Equation (20), such that

$$\begin{aligned} \begin{aligned} (1-q)\mathcal {Z}(t,X,u) + q\mathcal {Z}(t,X,s)&=p_1Bu_1+p_2Bu_2 - c_1u_1^2-c_2u_2^2-c_3u_3^2-c_4u_4^2\\&\quad - qp_1Bu_1- qp_2Bu_2 + qc_1u_1^2+qc_2u_2^2\\&\quad +qc_3u_3^2+qc_4u_4^2 +qp_1Bs_1+qp_2Bs_2\\&\quad - qc_1s_1^2-qc_2s_2^2-qc_3s_3^2-qc_4s_4^2, \end{aligned} \end{aligned}$$

(21)

now we will develop the left-hand side of Eq. (20), we obtain that

$$\begin{aligned} \begin{aligned} \mathcal {Z}(t,X,(1-q)u + qs)&=p_1Bu_1 -p_1qBu_1 + p_1qBs_1 + p_2Bu_2 - p_2qBu_2 +p_2qBs_2 \\&\quad -[c_1u_1^2-c_1qu_1^2 + c_1[(q^2-q)u_1^2 + 2q(1-q)u_1s_1 + q^2s_1^2]]\\&\quad -[c_2u_2^2-c_2qu_2^2 + c_2[(q^2-q)u_2^2 + 2q(1-q)u_2s_2 + q^2s_2^2]]\\&\quad -[c_3u_3^2-c_3qu_3^2 + c_3[(q^2-q)u_3^2 + 2q(1-q)u_3s_3 + q^2s_3^2]]\\&\quad -[c_4u_4^2-c_4qu_4^2 + c_4[(q^2-q)u_4^2 + 2q(1-q)u_4s_4 + q^2s_4^2]]\\ \end{aligned} \end{aligned}$$

(22)

replacing Eqs. (22) and (21) in Eq. (20), it is obtained that

$$\begin{aligned} \begin{aligned}&(1-q)\mathcal {Z}(t,X,u) + q\mathcal {Z}(t,X,s)-\mathcal {Z}(t,X,(1-q)u + qs)\\&\quad =- \sum _{i=1}^4c_i\left[ \sqrt{q(1-q)}u_i-\sqrt{q(1-q)}s_i\right] ^2< 0, \end{aligned} \end{aligned}$$

(23)

furthermore, as the objective functional (13) there is a discount factor $e^{-\delta t}$ and it is always positive. Therefore, from Eq. (23) it is justified that the objective functional is is concave in U, which determines the demonstration of $(A_4)$.