FSPM

FORMATH Vol. 12 (2013): 173–189

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Application of a Functional-Structural Plant Model (FSPM) to Optimize a Management Regime Surov´ y, P.∗ & Yoshimoto, A.∗ Keywords:

Functional-Structural Plant Model, optimization, Stone Pine, stand density

Abstract:

Functional Structural Plant Models (FSPM) facilitate decision making processes by rendering visualizations that permit decision makers to more easily evaluate multiple options or outcomes. This study simulates different stand management scenarios based on varying initial stand densities to demonstrate one potential application of FSPM as a decision making tool. We investigate the effects of several structural parameters on tree growth of Stone Pine (Pinus pinea L), and adjust them to model tree growth data collected in the field. Our simulation results show that individual trees develop more foliage when there is less competition, while greater competition (higher density) results in more foliage on a per hectare basis. The same relationship is observed for total biomass. Individual tree diameter growth is greater when there is less competition.

Received November 19, 2012; Accepted January 22, 2013 ∗ Institute of Statistical Mathematics, Japan

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1. Introduction Functional-Structural Plant Models (FSPM) are a newly developed class of Structural Plant Models (SPM) (Vos et al., 2009). Structural Plant Models were introduced to algorithmically describe complex structures such as tree crowns, tree roots, and other natural structures. An algorithmic description is useful, for example, when a process-based model is used to describe tree growth. Nikinmaa (1992, in Siev¨anen et al., 1999) and Givnish (1988) have shown that modeling 3-dimensional (3D) structure is essential for determining how carbohydrates are distributed between productive and non-productive structures in trees. Structure—or 3D crown development—also impacts the light interception and gas-exchange properties of trees (Landsberg and Gower, 1997, in Siev¨anen et al., 1999). Functional-Structural Plant Models result when these processes are embedded into Structural Plant Models (Kurth, 1994). Purely structural plant models can describe tree structure without considering any chronological change in tissue development, though the interaction between structure and the environment may be considered in the model. One example of such a case can be found in the GroIMP library (Kniemeyer and Kurth, 2008), as shown in Figure 1.

Figure 1. Reverse growth of a tree model.

Branch length corresponds to height above the ground (the higher the branch, the shorter its length), though the physiological develop-

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ment of a tree does not always directly correspond to the chronology of tree growth. The most challenging aspect of developing a functionalstructural plant model, or more generally a growth model, is to understand and describe growth elements and structural changes that follow realistic tree growth over time (such examples for spruce and beech can be found in the GroIMP library). For modelers, the biggest challenge is to understand and incorporate the characteristics (modules, organs) that have the greatest impact on the model’s final output. For users, computational time, which often exceeds a day, is one of the biggest impediments to employing FSPM, though more efficient approaches have been developed (Courn`ede et al., 2006). The model presented here is an improvement on a Stone Pine model previously developed by Surov´ y et al. (2012). In this paper we introduce a 3D version of that model with full sensitivity to neighboring environments. We focus on how green branches (those with leaves) correlate with non-green branches (those without leaves) and how this influences total production per hectare, as well as per tree, at different planting densities. There are two common planting regimes in forestry: (1) an initial high density planting followed by thinning, or (2) a low initial planting density. We demonstrate how the quality and quantity of future production varies depending on how key variables relating to tree productivity are managed over time. Similar studies for other species can be found in the literature (Siev¨anen et al., 2008). Other considerations include the effect of even or uneven aged stand management regimes on growth, as demonstrated by Courn`ede et al. (2009). Just as they simulated different thinning regimes in their study, our optimization framework could be applied to different management techniques in future studies.

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2. Materials and Methods We used previously digitized data collected from Stone Pine trees (Pinus pinea L.) in the Peg˜oes region of south Portugal. Trees at the study site are planted 4 meters apart in rows, which are also spaced at 4 meters. The complete digitization process and general growth evaluation procedures can be found in Surov´ y et al. (2011). Digitization was completed using a Polhemus Fastrak magnetic motion tracker device. The beginning and end points of each branch segment were recorded automatically, though diameter at each of these points was measured and entered manually. Additional information, such as branch age, was also recorded during the digitization process. 2.1 Data Evaluation and Variable Estimation Use of a 3D motion tracker to acquire data enables variables to be included for each branch segment (a process known as recording of topology, Danjon and Reubens, 1998). In such a case it is easy to register the age of branches and retrospectively display only branches of a certain age (as shown in Surov´ y et al., 2011). We compared a tree that has experienced high competition (8 competitors, Figure 2) to a tree that has experienced low competition (5 competitors, Figure 3). We studied the change in the ratio of branches with leaves (marked as green branches - G) to branches without leaves (marked as brown branches - B). Figure 2 shows the growth reconstruction of a tree in the high competition scenario. The G/B ratio decreases over time, with a value of 0.23 in the year of measurement. Figure 3 shows the growth reconstruction of a tree in the low competition scenario. The G/B ratio values are higher than those from the high competition scenario, with a final value of 0.27—approximately 20% greater.

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Figure 2. Growth reconstruction of a tree from a high competition scenario (G/B ratio values from left: 0.67, 0.50, 0.44, 0.38, and 0.23).

Figure 3. Growth reconstruction of a tree from a low competition scenario (G/B ratio values from left: 0.66, 0.54, 0.58, 0.54, and 0.27).

2.2 Model Construction We began by defining structural modules—buds, leaves, and branches—and functional components—branch rotation, bud competition, and mortality. As shown in Surov´ y et al. (2012), the easiest way to develop a model is to begin with a two-dimensional skeleton, which could be symmetric or one-sided. Figure 4 depicts the development of first order branches and their evolution over time. Branch bending and the subsequent development of reaction wood was thoroughly studied by Fourcaud and Lac (2003), and demonstrated by Fourcaud et al. (2003). These researchers utilized the finite element method (FEM) and propose a 3D multi-layer beam theory for realistically modeling branch shapes. More recently, Guillon et al. (2012a) and Guillon et al. (2012b) reformulated the question of

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growth biomechanics using rod theory. In our case we use a simplified approach that considers a linear increase of angles to adjust for the bending observed in the real data. This approach is similar to empirical modeling and it may be more suitable when 3D data is available. The example shows that new shoots begin with a vertical orientation, but the branch rotation module rotates them into a more horizontal orientation over time. At this stage, the branch rotation variable is adjusted in each year to mimic a real world growth form. Bud competition and mortality do not play an important role at this point in the simulation.

Figure 4. Growth of first order branches on a prototype skeleton (from left: 5 years, 10 years, and 15 years of age).

Figure 5 shows the growth of a prototype model with higher order branches. The angle at which branches sprout from the corresponding “parent” branch is obtained empirically from the measured data. Randomness or stochasticity may be assigned at this point to reflect a more realistic view of changes to the tree’s overall structure. Branching angles and the length of branch segments decrease with distance from the stem. Changes to the branches over time can clearly be seen in these examples, which appear to accurately model a real world growth scenario. The total biomass and G/B ratio can also be computed at each growth period. At this stage additional functional modules for bud competition and

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Figure 5. Growth of first order branches on a prototype skeleton with nth-order branches (from left: 5 years, 10 years, and 15 years of age).

mortality are deployed. The bud competition module evaluates the presence of other buds, or the structure, in the direction of potential growth. The mortality module eliminates those buds that have an obstacle in front of them, as well as those segments that touch the ground, branches with no green buds (dead buds), and so on. Because the mortality of brown branches is generally not decisive, more attention should be paid to bud competition settings. This rule for branching was once again derived from the measured data. To accurately reflect a realworld situation, the model must consider branch mortality; if it did not, the model would grossly exaggerate the number of branches. The field data demonstrates that most branch mortality occurs during a tree’s initial growth period, and our model reflects this observation. Several underlying mechanisms may be a cause for such mortality (thigmomorphogenesis, R/FR-ratio sensing by phytohormonal reactions, etc.); however, implementation of these mechanisms would require more data and are beyond the scope of this study. Finally, the principal branches are rotated around the stem. The rotation angles can be defined empirically or adjusted to a more realistic form by adding stochasticity. To reflect changes over time, we embedded a stem growth function

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into the model. The following Richard-Chapman function is commonly used to model trunk diameter:

[1]

y = a(1 − ebt )c

where y is diameter growth, t is time, and a, b, and c are coefficients. In this study we modified the function to respond to varying amounts of foliage in the crown.

[2]

y = a(1 − e(d+eS)t )c

where S is added as the total amount of foliage (defined by the total length of branches with leaves), and a, c, d, and e are coefficients. The coefficients were fitted using the least squares method and data from Surov´ y et al. (2011). 3. Demonstrative Results Our analysis provides both visual and physical results. Figure 6 shows a visualization of a tree growing under three different competition scenarios. From left, we can see that decreasing levels of competition result in increased branching and foliage. The tree in the far right image has grown within a 6 × 6 meter clearing, which implies nearly no competition. As the visualization shows, the crown is fully covered with leaves, suffering no disturbance. This kind of visualization may play an important role in understanding the visual quality of forest stands and gauging social aesthetic preferences. Our study’s physical results—in terms of productivity per tree and per hectare—allow us to investigate the interaction of photosynthetic organs in 3D space. Figure 7 shows the evolution of diameter under different initial stand density scenarios. Stand density has a negative

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Figure 6. Visualization of a tree grown under different competition pressures (from left: 50cm, 3 meters, and 6 meters).

impact on the number of green branches, influencing output from the Richard-Chapman function. The values shown on the top side of Figure 7 represent cumulative growth, or the integral of the diameter growth function. These results show that in a high stand density scenario, the loss of photosynthetic organs (foliage) leads to decreased diameter growth. When density decreases, diameter growth improves, but only to a distance of 2 meters. Additional increases beyond 2 meters had no impact on diameter growth. Figure 8 shows total biomass production per hectare (top) and its evolution over time (bottom). Total biomass was greatest in the highest density scenario. As density decreased, total biomass also tended to decrease continuously over time. Figure 9 shows the evolution of green branches (expressed by their total length) per tree (top) and per hectare (bottom). These results clearly show that higher density results in fewer leaves per tree, but more leaves per hectare. That is, the highest density scenario results in the greatest number of leaves per hectare, even though each individual tree has fewer leaves.

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Figure 7. Diameter growth under different stand density scenarios final diameter after 14 years (top) and the evolution of diameter in different scenarios (bottom).

4. Discussion and Conclusions We demonstrated the use of a Functional-Structural Plant Model to simulate and evaluate different management scenarios that include varying planting densities.

We used the GroIMP 1.3 modeling

platform because it is easily deployed and accessible on the web (http://www.grogra.de, see the download section). There are other

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Figure 8. Total biomass production under different stand density scenarios - the final biomass per hectare after 14 years of growth (top) and the evolution of biomass under different density scenarios (bottom).

software packages and models that could be used, though most have been designed for different species. These include GreenLab (Kang et al., 2008), LIGNUM (Pertunnen et al., 1996), and modeling approaches described in Ford et al. 1990. Our intent, however, was to demonstrate

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Figure 9. Total biomass of green leaves under different density scenarios following 14 years of growth - per tree (top) and per hectare (bottom).

the integration of measured data and observed behavior from a specific species into the decision making process (via an FSPM model). From this perspective, a general modeling platform like GroIMP was the most practical approach. There are differing opinions among forestry professionals about ap-

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propriate planting densities. Some prefer to plant at high initial densities followed by thinning, while others prefer to plant fewer trees per hectare. Some suggest the practice of planting at lower densities can increase the number of photosynthetic organs per hectare, thereby increasing productivity and carbon assimilation (Groasis.com, 2012). Such a claim may be confirmed by using a 3D functional-structural model to study the behavior of buds and branches with leaves under different planting scenarios. We used measurements taken from Stone Pine trees (Pinus pinea L.) to adjust our model. In simulations we observed that the number of leaves (photosynthetic organs) per tree increased under less competitive conditions. However, this advantage was eliminated when extrapolated to an aggregate value on a per hectare basis. That is, greater competition resulted in more leaves per hectare, even though it reduced the number of leaves per tree. The same was observed in terms of the total biomass produced after 14 years of simulated growth. Our first scenario with half-meter distances among trees appeared to be the best in terms of production, though it should be noted that our model did not include a tree mortality module at the development stage. Tree survival may be a consideration for future research. Our results clearly show that decreasing density will not result in increased foliage or biomass production. Because our model is based on pine growth behavior and intercrown competition, the results cannot immediately be applied to other species. Diameter growth was influenced by competition, decreasing as competition increased. This relationship has been well documented since the historical works of Reineke (1933), and later Yoda et al. (1963) for plants. Similar observations can be found in yield tables that consider different planting densities, such as those of Smalley and Bailey (1974). Stand density’s influence on diameter growth is a common considera-

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tion in growth models (e.g., Calama and Montero, 2004, West, 1981, etc.). However, increasing competition does not result in linear changes in diameter growth. No improvement of diameter was observed after reducing competitors within a certain distance. We incorporated the amount of foliage into the Richard-Chapman growth function; however, the number of leaves may not be the best indicator of stem growth, or at least should not be used in a linear fashion. As other researchers have shown, crown area (horizontal crown projection) is highly correlated with stem sectional area (Tom´e et al., 2001). From this perspective, it may be better to combine horizontal area with the Leaf Area Index (LAI). Total light absorbance may be an even better growth predictor. Future research should focus on these hypotheses. Acknowledgement The first author would like to thank the Japanese Society for Promotion of Science for supporting the work presented here.

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