umu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
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.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0001-9862-816x
International Institute for Applied Systems Analysis.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In forests, striving for light is matter of life or death for a tree, either by growing taller towards brighter conditions or by expanding the crown to capture more of the available light. Here we present a model to predict plasticity in optimal growth paths in terms of height and crown-size under a shading canopy, as determined by fitness maximization. Based on a mechanistic tree growth model, we determine the optimal growth path among all possible trajectories using a numerical method (dynamic programming) for a range of different light environments. We then test whether this path can be understood and approximated by optimization-principles. The key findings are: (i) Maximization of net productivity based on local light conditions in each time step does not lead to an optimal growth path and may make it impossible for the tree to reach the canopy. Instead, the tree must account for the full range of expected future light levels up to the top of the canopy. (ii) This complex problem can however be solved with remarkable accuracy based on just three allocation switching points. (iii) The results can explain stratification into canopy and sub-canopy species and why canopy species often get stuck at certain size under a shading canopy. The model can be used to enable plasticity in height versus diameter growth in individual based vegetation and forestry models.  

Keywords [en]
Allocation, Growth strategy, Life history, Optimal control, Tree
National Category
Ecology
Identifiers
URN: urn:nbn:se:umu:diva-156734OAI: oai:DiVA.org:umu-156734DiVA, id: diva2:1291696
Funder
Swedish Research Council Formas, 2012-1008Available from: 2019-02-26 Created: 2019-02-26 Last updated: 2019-03-19
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

Open Access in DiVA

No full text in DiVA

Authority records BETA

Fransson, PeterBrännström, Åke

Search in DiVA

By author/editor
Fransson, PeterBrännström, Åke
By organisation
Department of Mathematics and Mathematical Statistics
Ecology

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 64 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf