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Optimal thinning: a theoretical investigation on individual-tree level
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Paper I: In paper I, we asked how a tree should optimally allocate its resources to maximize its fitness. We let a subject tree grow in an environment shaded by nearby competing trees. The competitors were assumed to have reached maturity and had stopped growing, thus creating a static light environment for the subject tree to grow in. The light environment was modeled as a logistic function. For the growth model we used the pipe model as a foundation, linking tree width and leaf mass. This allowed us to construct a dynamic tree-growth model where the tree can allocate biomass from photosynthesis (net productivity) to either stem-height growth, crown-size growth, or reproduction (seed production). Using Pontryagin's maximum principle we derived necessary conditions for optimal biomass allocation, and on that built a heuristic allocation model. The heuristic model states that the tree should first invest into crown-size and then switch to tree height-growth, and lastly invest into crown-size before the growth investments stop and all investments are allocated to reproduction. To test our heuristic method, we used it to determine the growth in several different light environments. The results were then compared to the optimal growth trajectories. The optimal growth was determined by applying dynamic programming. Our less computationally demanding heuristic performed very well in comparison. We also found there exist a critical crown-size: if the subject tree possessed a larger crown-size, the tree would be unable to reach up to the canopy height.

Paper II: One of the most important aspects of modelling forest growth, and modelling growth of individual trees in general, is the competition between trees. A high level of competition pressure has a negative impact on the growth of individual trees. There are many ways of modelling competition, the most common one is by using a competition index. In this paper we tested 16 competition indices, in conjunction with a log-linear growth model, in terms of the mean squared error and the coefficient of determination. 5 competition indices are distance-independent (i.e. distance between the competitors are not taken into consideration) and 11 are distance-dependent. The data we used to fit our growth model, with accompanying competition index, was taken from an experimental site, in northern Sweden, of Norway spruce. The growth data for the Norway spruce comes from stands which were treated with one of two treatments, solid fertilization, liquid fertilization, or no treatment (control stand). We found that the distance-dependent indices perform better than the distance-independent. However, both the best distance-dependent and independent index performed overall well. We also found that the ranking of the indices was unaffected by the stand treatment, i.e. indices that work well for one treatment will work well for the others.

Paper III: In this paper we studied how spatial distribution and size selection affect the residual trees, after a thinning operation, in terms of merchantable wood production and stand economy. We constructed a spatially explicit individual-based forest-growth model and fitted and validated the model against empirical data for Norway spruce stands in northern Sweden. To determine the cost for the forest operation we employed empirical cost functions for harvesting and forwarding. The income from the harvested timber is calculated from volume-price lists. The thinnings were determined by three parameters: the spatial evenness of residual trees, the size selection of removed trees, and the basal area reduction. In order to find tree selections fulfilling these constraints we used the metropolis algorithm. We varied these three constrains and applied them for thinning of different initial configurations of Norway spruce stands. The initial configurations for the stands where collected from empirical data. We found that changing the spatial evenness and size selection improved the net wood production and net present value of the stand up to 8%. However, the magnitude of improvement was dependent on the initial configuration (the magnitude of improvement varied between 1.7%—8%).

Paper IV: With new technology and methods from remote sensing, such as LIDAR, becoming more prevalent in forestry, the ability to assess information on a detailed scale has become more available. Measurements for each individual tree can be more easily gathered on a larger scale. This type of data opens up for using individual-based model for practical precision forestry planning. In paper IV we used the individual-based model constructed in paper III to find the optimal harvesting time for each individual tree, such that the land expectation value is maximized. We employed a genetic algorithm to find a near optimal solution to our optimization. We optimized a number of initial Norway spruce stands (data obtained from field measurements). The optimal management strategy was to apply thinning from above. We also found that increasing the discount rate will decrease the time for final felling and increase basal area reduction for the optimal strategy. Decreasing relocation costs (the cost to bring machines to the stand) led to an increase in the number of optimal thinnings and postponed the first thinning. Our strategy was superior to both the unthinned strategy and a conventional thinning strategy, both in terms of land expectation value (>20% higher) and merchantable wood production.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2019. , p. 32
Series
Research report in mathematics, ISSN 1653-0810 ; 65/19
Keywords [en]
Forest management, Simulation, Optimization, Spatially explicit model, Individual-based model
National Category
Computational Mathematics Forest Science
Identifiers
URN: urn:nbn:se:umu:diva-156741ISBN: 978-91-7855-031-9 (print)OAI: oai:DiVA.org:umu-156741DiVA, id: diva2:1291736
Public defence
2019-03-29, MA121, MIT-huset, Umeå, 09:00 (English)
Opponent
Supervisors
Available from: 2019-03-08 Created: 2019-02-26 Last updated: 2019-03-04Bibliographically approved
List of papers
1. A tree’s quest for light: optimal height and diameter growth under a shading canopy
Open this publication in new window or tab >>A tree’s quest for light: optimal height and diameter growth under a shading canopy
2021 (English)In: Tree Physiology, ISSN 0829-318X, E-ISSN 1758-4469, Vol. 41, no 1, p. 1-11Article in journal (Refereed) Published
Abstract [en]

For trees in forests, striving for light is matter of life and death, either by growing taller toward brighter conditions or by expanding the crown to capture more of the available light. Here, we present a mechanistic model for the development path of stem height and crown size, accounting for light capture and growth, as well as mortality risk. We determine the optimal growth path among all possible trajectories using dynamic programming. The optimal growth path follows a sequence of distinct phases: (i) initial crown size expansion, (ii) stem height growth toward the canopy, (iii) final expansion of the crown in the canopy and (iv) seed production without further increase in size. The transition points between these phases can be optimized by maximizing fitness, defined as expected lifetime reproductive production. The results imply that to reach the canopy in an optimal way, trees must consider the full profile of expected increasing light levels toward the canopy. A shortsighted maximization of growth based on initial light conditions can result in arrested height growth, preventing the tree from reaching the canopy. The previous result can explain canopy stratification, and why canopy species often get stuck at a certain size under a shading canopy. The model explains why trees with lower wood density have a larger diameter at a given tree height and grow taller than trees with higher wood density. The model can be used to implement plasticity in height versus diameter growth in individual-based vegetation and forestry models.

Keywords
Allocation, Growth strategy, Life history, Optimal control, Tree
National Category
Ecology
Research subject
biology
Identifiers
urn:nbn:se:umu:diva-156734 (URN)10.1093/treephys/tpaa110 (DOI)000612199200001 ()2-s2.0-85100069843 (Scopus ID)
Funder
Swedish Research Council Formas, 2012-1008Wallenberg Foundations
Note

Originally included in thesis in manuscript form.

Available from: 2019-02-26 Created: 2019-02-26 Last updated: 2023-09-05Bibliographically approved
2. Comparing distance-independent and distance-dependent competition indices for Picea abies
Open this publication in new window or tab >>Comparing distance-independent and distance-dependent competition indices for Picea abies
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Competition between individual trees is important for forest growth. Modeling the underlying interactions is difficult and sometimes not possible. Consequently, simplified competition indices are often used as a proxy. Several studies have shown that including competition indices into growth models will improve accuracy. However, there is no consensus on which or what type of indices work well. Here, we compare several indices of two qualitatively different types: distance-independent and distance-dependent indices. These indices are incorporated into a log-linear growth model and tested against 5-year diameter growth data collected from three Norway spruce plots that each have different fertilization treatments. We find that although distance-dependent indices performed better, distance-independent indices can also perform well; R2adj = 0.66 vs. 0.62 for the best indices in each category. We find that Bella’s index to be the best distance-dependent index (the index depends on distance between competitors and their diameters) and Reineke’s index to be the best distance-independent index (this index depends on the mean quadratic diameter and density in the stand). We also find that the performance ranking of indices is consistent across three different fertilization treatments.

Keywords
Competition index, Modelling, Picea abies, Diameter growth
National Category
Forest Science
Identifiers
urn:nbn:se:umu:diva-156737 (URN)
Funder
Swedish Research Council Formas, 2012-1008
Available from: 2019-02-26 Created: 2019-02-26 Last updated: 2019-03-19
3. Model-based investigation on the effects of spatial evenness, and size selection in thinning of Picea abies stands
Open this publication in new window or tab >>Model-based investigation on the effects of spatial evenness, and size selection in thinning of Picea abies stands
Show others...
2019 (English)In: Scandinavian Journal of Forest Research, ISSN 0282-7581, E-ISSN 1651-1891, Vol. 34, no 3, p. 189-199Article in journal (Refereed) Published
Abstract [en]

Size and spatial distribution of trees are important for forest stand growth, but the extent to which itmatters in thinning operations, in terms of wood production and stand economy, has rarely beendocumented. Here we investigate how the choice of spatial evenness and tree-size distribution ofresidual trees impacts wood production and stand economy. A spatially explicit individual-basedgrowth model was used, in conjunction with empirical cost functions for harvesting andforwarding, to calculate net production and net present value for different thinning operations inNorway spruce stands in Northern Sweden. The in silico thinning operations were defined by threevariables: (1) spatial evenness after thinning, (2) tree size preference for harvesting, and (3) basalarea reduction. We found that thinning that increases spatial evenness increases net productionand net present value by around 2.0%, compared to the worst case. When changing the spatialevenness in conjunction with size preference we could observe an improvement of the netproduction and net present value up to 8.0%. The magnitude of impact differed greatly betweenthe stands (from 1.7% to 8.0%) and was highest in the stand with the lowest stem density.

Place, publisher, year, edition, pages
Taylor & Francis, 2019
Keywords
thinning, individual-based model, spatial distribution, forest management, simulation
National Category
Forest Science
Identifiers
urn:nbn:se:umu:diva-156738 (URN)10.1080/02827581.2019.1577914 (DOI)000459142200003 ()2-s2.0-85061783652 (Scopus ID)
Funder
Swedish Research Council Formas, 2012-1008
Available from: 2019-02-26 Created: 2019-02-26 Last updated: 2023-03-24Bibliographically approved
4. A simulation-based approach to a near optimal thinning strategy: allowing for individual harvesting times for individual trees
Open this publication in new window or tab >>A simulation-based approach to a near optimal thinning strategy: allowing for individual harvesting times for individual trees
Show others...
2020 (English)In: Canadian Journal of Forest Research, ISSN 0045-5067, E-ISSN 1208-6037, Vol. 50, no 3, p. 320-331Article in journal (Refereed) Published
Abstract [en]

As various methods for precision inventories, such as LiDAR, are becoming increasingly common in forestry, individual-tree level planning is becoming more viable. Here, we present a method for finding the optimal thinning times for individual trees from an economic perspective. The method utilizes an individual tree-based forest growth model that has been fitted to Norway spruce (Picea abies (L.) Karst.) stands in northern Sweden. We find that the optimal management strategy is to thin from above, i.e. harvesting trees that are larger than average. We compare our optimal strategy with a conventional management strategy and find that it results in approximately 20% higher land expectation value. Furthermore, we find that increasing the discount rate will, for the optimal strategy, reduce the final harvest age and increase the basal area reduction. Decreasing the cost to initiate a thinning (e.g., machinery-related transportation costs) increases the number of thinnings and delays the first thinning.

National Category
Forest Science
Identifiers
urn:nbn:se:umu:diva-156740 (URN)10.1139/cjfr-2019-0053 (DOI)000516761900008 ()2-s2.0-85080097125 (Scopus ID)
Funder
Swedish Research Council Formas, 2012-1008
Note

Originally included in thesis in manuscript form 

Available from: 2019-02-26 Created: 2019-02-26 Last updated: 2023-03-23Bibliographically approved

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