Introducing a Non-Stationary Matrix Model for Stand-Level Optimization, an Even-Aged Pine (Pinus Sylvestris L.) Stand in Finland

Abstract

Abstract:In general, matrix models are commonly applied to predict tree growth for size-structured tree populations, whereas empirical-statistical models are designed to predict tree growth based on a vast amount of field observations. From the theoretical point of view, matrix models can be considered to be more generic since their dependency on ad hoc growth conditions is far less prevalent than that of empirical-statistical models. On the other hand, the main pitfall of matrix models is their inability to include variation among the individuals within a size class, occasionally resulting in less accurate predictions of tree growth compared to empirical-statistical models. Thus, the relevant question is whether a matrix model can capture essential tree-growth dynamics/characteristics so that the model produces accurate stand projections which can further be applied in practical decision-making. Such a dynamic characteristic in our model is the basal area of trees, which causes nonlinearity in time. Therefore, our matrix model is a nonlinear model. The empirical data for models was based on 20 sample plots representing 8360 tree records. Further, according to the model, stand projections were produced for three Scots pine (Pinus sylvestris L.) sapling stands (age of 25 years, stand density fluctuating from 850 to 1400 stems ha −1 ). Then, (even-aged) stand management was optimized by applying sequential quadratic programming (SQP) among those growth predictions. The objective function of the optimization task was to maximize the net present value (NPV) of the ongoing rotation. The stands were located in Northern Ostrobothnia, Finland, on nutrient-poor soil type. The results indicated that initial stand density had an effect on optimal solutions-optimal stand management varied with respect to thinnings (timing and intensity) as well as to optimal rotation. Further, an increasing discount rate shortened considerably the optimal rotation period, and relaxing the minimum thinning removal to 30 m 3 ha −1 resulted in an increase both in number of thinnings and in the maximum net present value.

Carbon forestry compensation on estate level

“…Euros per ton. This inserted to Equation (11) together with a 3% discount rate, often applied in Forestry [29,30,31], results as an annual carbon rent of 0.75 Euros per ton of carbon dioxide.…”

Spruce forest stands in a stationary state

“…During such a period, a variety of disturbances may appear, interfering with the development of a stationary state. Obviously the recruitment of seedlings can be increased by artificial or seminatural regeneration (Busing 1994;Goodburn and Lorimer 1999;Pukkala 2006;Pyy et al 2017). Such actions easily lead to periodic forestry, instead of stationary forestry.…”

Demonstrating the Effect of Height Variation on Stand-Level Optimization with Diameter-Structured Matrix Model

“…y k i of the stand. The details about model was described by Pyy et al [13]. The matrix H is a K × K-matrix that describes how the trees move from a height class to another.…”

“…We assumed that H does not depend on time step k or diameter class i. The parameters are fitted on the data described in Section 2.2 below followed by other study [13]. The projected gradient method was used to solve the optimization problem (1) in accordance with other study [14].…”

“…Particularly in forestry size-structured models (e.g., interchangeably transition matrix model, matrix population model, matrix growth model or matrix model) are thriving with applications covering a wide range of areas [6,7]. For instance, transition matrix models have been applied in discovering biodiversity dynamics [8], in studying the impacts of climate change [9], in biomass carbon stock studies (e.g., [10]) and in assessing optimal forest management [11][12][13][14]. For optimal forest management, there are also other approaches widely applied for predicting tree growth-such as process-based [15][16][17][18] and statistical-empirical growth models [19][20][21].…”