Abstract: Plantations with fast-growing species play a crucial role in reducing global warming and
have great carbon capture potential. Therefore, determining optimal management strategies is a
challenge in the management of forest plantations to achieve the maximum carbon capture rate. The
objective of this work is to determine optimal rotation strategies that maximize carbon capture in
forest plantations. By evaluating an ecological optimal control problem, this work presents a method
that manages forest plantations by planning activities such as reforestation, felling, thinning, and
fire prevention. The mathematical model is governed by three ordinary differential equations: live
biomass, intrinsic growth, and burned area.
The characterization of the optimal control problem using
Pontryagin’s maximum principle is analyzed. The model solutions are approximated numerically by
the fourth-order Runge–Kutta method. To verify the efficiency of the model, parameters for three
scenarios were considered: a realistic one that represents current forestry activities based on previous
studies for the exotic species Pinus radiata D. Don, another pessimistic, which considers significant
losses in forest productivity; and a more optimistic scenario which assumes the creation of new forest
areas that contribute with carbon capture to prevent the increase in global temperature.
The model
predicts a higher volume of biomass for the optimistic scenario, with the consequent higher carbon
capture than in the other two scenarios. The optimal solution for the felling strategy suggests that, to
increase carbon capture, the rotation age should be prolonged and the felling rate decreased. The
model also confirms that reforestation should be carried out immediately after felling, applying
maximum reforestation effort in the optimistic and pessimistic scenarios. On the other hand, the
model indicates that the maximum prevention effort should be applied during the life cycle of the
plantation, which should be proportional to the biomass volume. Finally, the optimal solution for the
thinning strategy indicates that in all three scenarios, the maximum thinning effort should be applied
until the time when the fire prevention strategy begins.
Keywords: ecological model; biomass volume; carbon dioxide; optimal control; numerical simulation
1. Introduction
Carbon dioxide (CO2) is one of the main greenhouse gases (GHG) in the atmosphere.
Multiple human activities in most industrialized countries have contributed to the increase
in this gas and have exacerbated the negative effects of climate change. According to the
latest report of the Intergovernmental Panel on Climate Change (IPCC), climate change is
devastating today, in particular, because of the changes in the patterns of humidity, temperature, winds, snow, and ice, especially in coastal zones. These changes in climate conditions
could have negative impacts on human health, agriculture, and the economy [1–3]. Under
this worldwide situation, governments are making cooperative efforts agreements (e.g., the
Paris Agreement and the Kyoto Protocol) to create new forest areas to help prevent the global
average temperature rising more than 2 ◦C during the 21st century [4–6]. Forest ecosystems
cover approximately 4100 billion hectares of the Earth’s surface and have a huge potential for
Forests 2023, 14, 82. https://doi.org/10.3390/f14010082 https://www.mdpi.com/journal/forests
Forests 2023, 14, 82 2 of 17
carbon capture [7]. Of this total area, approximately 45% are exotic plantations whereas the
other 55% corresponds to native forests [8]. Because forest ecosystems can store the largest
amounts of carbon [9], it has been suggested that expanding forest areas and prolonging the
rotation age (i.e., the growth period required to derive maximum value from a stand of timber),
especially in exotic forest plantations [10], are key strategies to maximize carbon capture and
mitigate the negative effects of global climate change [11]. There is a large body of literature
where carbon capture is estimated [12,13]. In a temperate forest in Southern Europe, the
aboveground carbon capture in the species Eucalyptus nitens (Deane and Maiden), Eucalyptus
globulus Labill, and P. radiata, with rotation ages ranging from 10 to 35 years, was estimated to
be from 443 to 634 Tn C ha−1
[14]. The carbon sequestration with the same species established
in Chile was 212 Tn C ha−1
for P. radiata, 180 Tn C ha−1
for E. nitens, and 117 Tn C ha−1
for E. globulus (age of 20–24 years for P. radiata and 10–14 years for Eucalyptus) [15]. On the
other hand, in Panama the carbon stored in Tectona grandis E.L (Teca) plantations during 1 and
10 years was estimated to be 2.9 Tn C ha−1 and 40.7 Tn C ha−1
, respectively [16].
On the other hand, there are studies on the oil palm (Elaeis guineensis Jacq) which, due
to its high biomass production and expansion dynamics, plays an important role in carbon
capture [17]. By means of mathematical modelling the dynamics of both oil production and
carbon capture have been studied [18]. In [19], they formulated an optimal control problem
based on a system of ordinary differential equations that relate the dynamics of young and
mature trees and considers felling as a control variable. The authors concluded that palm
oil production and carbon capture increases with a controlled felling rate.
Notwithstanding, to increase CO2 capture the trees must remain for longer periods
in the field, which delays the rotation age [20,21]. However, in some situations, it is
risky to prolong the rotation age in order to increase carbon capture, since it increases the
probability of forest fires when there is more fuel in the field. More frequent forest fires
will increase CO2 levels in the atmosphere, causing extreme climate events and decreasing
relative humidity in many regions of the world [22]
. To model the probability of forest fire
occurrence some authors have used the Faustmann model generalized to the stochastic
Poisson process [23], whereas others have studied this phenomenon by using the Bellman
equation to determine the optimal rotation age in a forest stand that produces timber and
carbon benefits under fire risk [24]. The authors showed that higher fire risk will reduce the
optimal rotation age due to a lack of fire prevention and low carbon prices, while a higher
carbon price will increase the rotation age, thus obtaining a higher ecological benefit. It is
known that fires contribute to the increase of CO2 in the atmosphere.
In [25] they developed
a meteorological fire index to predict the risk of fire occurrence and help forest managers
take appropriate preventive measures. The authors determined that relative humidity is
a simple and feasible parameter to describe the occurrence of fires. Several mathematical
models have been developed to describe the dynamics of CO2 capture in reforestation
projects [26–28]. The atmospheric CO2 concentration decreases as the rate of reforestation
increases. Also in [29], they presented a study to model the greenhouse effect caused by
CO2 emissions through the optimal control theory. In the model, the authors addressed the
optimization of investments in reforestation and clean technologies associated with state
variables such as CO2 emissions, planted area, and Gross Domestic Product (GDP). They
concluded that it is more efficient to invest in reforestation than in clean technologies.
Because forested areas can contribute to climate change mitigation, it is necessary
to find optimal management strategies that maximize carbon capture. Strategies such as
large-scale reforestations are efficient in capturing huge amounts of carbon [30], whereas
the optimization of thinning, fire prevention, and harvesting strategies can also reduce CO2
emissions in forest plantation management [31]. In [32] they applied a thinning strategy
in Korean pine (Pinus koraiensis Sieb. et Zucc.) forest plantations and determined that the
optimal rotation age that maximizes wood production and carbon capture was at the age
of 86 years. In another study on oil palm [18], they applied the optimal control theory to
model the dynamics of biomass growth and intrinsic biomass growth as state variables
and considered felling as a control variable. The authors showed that the maximum oil
Forests 2023, 14, 82 3 of 17
production and carbon capture was reached at the age of 20 years. However, to our knowledge, no mathematical models have simultaneously modeled the relationship between
the living biomass, the intrinsic biomass growth, the burned area, the reforestation, the
felling and thinning, the fire prevention, and the relative humidity. Recently,
[33] modeled
the effects of the dynamics of living biomass, intrinsic growth, and burned area on carbon
capture in forest plantations. The authors showed that biomass decreases in each cycle
of regeneration because of forest fires, and suggested a strategy based on fire prevention
in order to obtain maximum carbon capture. In this context, the objective of the present
work is to determine optimal rotation strategies that maximize carbon sequestration in
forest plantations. Based on the optimal control theory, a mathematical model is proposed
to describe the dynamic relationship of carbon capture in forest plantations with control
strategies such as reforestation, felling, fire prevention, and thinning, which are associated
with state variables such as living biomass, intrinsic growth, and burned area. To verify the
efficiency of the model, three scenarios are considered: realistic, pessimistic, and optimistic,
using numerical methods to approximate its solution. In the case of the realistic scenario
we tested with data of the species P. radiata.