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A tree’s quest for light: optimal height and diameter growth under a shading canopy
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0003-2599-3611
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. International Institute for Applied Systems Analysis, Laxenburg, Austria.ORCID iD: 0000-0001-9862-816x
International Institute for Applied Systems Analysis.
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.

Place, publisher, year, edition, pages
2021. Vol. 41, no 1, p. 1-11
Keywords [en]
Allocation, Growth strategy, Life history, Optimal control, Tree
National Category
Ecology
Research subject
biology
Identifiers
URN: urn:nbn:se:umu:diva-156734DOI: 10.1093/treephys/tpaa110ISI: 000612199200001Scopus ID: 2-s2.0-85100069843OAI: oai:DiVA.org:umu-156734DiVA, id: diva2:1291696
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
In thesis
1. Optimal thinning: a theoretical investigation on individual-tree level
Open this publication in new window or tab >>Optimal thinning: a theoretical investigation on individual-tree level
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
Forest management, Simulation, Optimization, Spatially explicit model, Individual-based model
National Category
Computational Mathematics Forest Science
Identifiers
urn:nbn:se:umu:diva-156741 (URN)978-91-7855-031-9 (ISBN)
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

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Fransson, PeterBrännström, Åke

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