Skip to main content

Efficient modelling of foliage distribution and crown dynamics in monolayer tree species

Abstract

In response to the computational limitations of individual leaf-based tree growth models, this article presents a new approach for the efficient characterisation of the spatial distribution of foliage in monolayered trees in terms of 2D foliage surfaces. Much like the recently introduced 3D leaf area density, this concept accommodates local crown plasticity, which is a common weak point in large-scale growth models. Recognizing phototropism as the predominant driver of spatial crown expansion, we define the local light gradient on foliage surfaces. We consider the partial differential equation describing the evolution of a curve expanding along the light gradient and present an explicit solution. The article concludes with an illustration of the incorporation of foliage surfaces in a simple tree growth model for European beech (Fagus sylvatica L.), and discusses perspectives for applications in functional-structural models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  • Bartelink HH (1997) Allometric relationships for biomass and leaf area of beech (Fagus sylvatica L). Ann For Sci 54:39–50

    Article  Google Scholar 

  • Beyer R (2016) Spatial leaf density-based modelling of teleonomic crown dynamics of crops and trees. PhD thesis, CentraleSupélec, France, Technische Universität München, Germany

  • Beyer R, Letort V, Cournède PH (2014) Modeling tree crown dynamics with 3d partial differential equations. Front Plant Sci 5

  • Beyer R, Letort V, Bayer D, Pretzsch H, Cournède P-H (2017a) Leaf density-based modelling of phototropic crown dynamics and long-term predictive application to European beech. Ecol Model 347:63–71

    Article  Google Scholar 

  • Beyer R, Bayer D, Letort V, Pretzsch H, Cournède P-H (2017b) Validation of a functional-structural tree model using terrestrial Lidar data. Ecol Model 357:55–57

    Article  Google Scholar 

  • Dewar RC (2010) Maximum entropy production and plant optimization theories. Philos Trans R Soc B Biol Sci 365:1429–1435

    CAS  Article  Google Scholar 

  • Farnsworth KD, Niklas KJ (1995) Theories of optimization, form and function in branching architecture in plants. Funct Ecol 9(3):355–363

    Article  Google Scholar 

  • Godin C (2000) Representing and encoding plant architecture: a review. Ann For Sci 57:413–438

    Article  Google Scholar 

  • Hallé F, Oldeman RAA, Tomlinson PB (1978) Tropical trees and forests. An architectural analysis. Springer, New York

    Book  Google Scholar 

  • Horn HS (1971) The adaptive geometry of trees. Princeton University Press, Princeton

    Google Scholar 

  • Honda H (1971) Description of the form of trees by the parameters of the tree-like body: Effects of the branching angle and the branch length on the shape of the tree-like body. J Theor Biol 31(2):331–338

    CAS  Article  PubMed  Google Scholar 

  • Kutschera U, Briggs WR (2016) Phototropic solar tracking in sunflower plants: an integrative perspective. Ann Bot 117(1):1–8

    Article  PubMed  Google Scholar 

  • Mäkelä A (1986) Implications of the pipe model theory on dry matter partitioning and height growth in trees. J Theor Biol 123:103–120

    Article  Google Scholar 

  • Martínez-Vilalta J, Vanderklein D, Mencuccini M (2007) Tree height and age-related decline in growth in Scots pine (Pinus sylvestris l.). Oecologia 150:529–544

    Article  PubMed  Google Scholar 

  • Medlyn BE (1998) Physiological basis of the light use efficiency model. Tree Physiol 18:167–176

    Article  PubMed  Google Scholar 

  • Mitchell KJ (1975) Dynamics and simulated yield of Douglas-fir. For Sci 21(4):a0001–z0001

    Google Scholar 

  • Monteith JL (1972) Solar radiation and productivity in tropical ecosystems. J Appl Ecol 2:747766

    Google Scholar 

  • Monteith JL (1977) Climate and the efficiency of crop production in Britain. Proc R Soc Lond B 281:277–294

    Google Scholar 

  • Niklas K (1994) Morphological evolution through complex domains of fitness. Proc Natl Acad Sci 91(15):6772–6779

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  • Pretzsch H (2014) Canopy space filling and tree crown morphology in mixed-species stands compared with monocultures. For Ecol Manag 327:251–264

    Article  Google Scholar 

  • Purves DW, Lichstein JW, Pacala SW (2007) Crown plasticity and competition for canopy space: a new spatially implicit model parameterized for 250 North American tree species. PLoS One 2(9):e870

    Article  PubMed  PubMed Central  Google Scholar 

  • Roloff A (2001) Baumkronen: Verständnis und praktische Bedeutung eines komplexen Naturphänomens. Ulmer, Stuttgart

  • Sievänen R, Nikinmaa E, Nygren P, Ozier-Lafontaine H, Perttunen J, Hakula H (2000) Components of a functional–structural tree model. Ann For Sci 57:399–412

    Article  Google Scholar 

  • Thornley JHM, Johnson IR (2000) Plant and crop modeling: a mathematical approach to plant and crop physiology. The Blackburn Press, Caldwell

    Google Scholar 

Download references

Acknowledgements

The author is grateful for support of this project by a doctoral scholarship from the Heinrich Böll Foundation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Robert Beyer.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Beyer, R. Efficient modelling of foliage distribution and crown dynamics in monolayer tree species. Theory Biosci. 136, 193–197 (2017). https://doi.org/10.1007/s12064-017-0249-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12064-017-0249-y

Keywords

  • Partial differential equations
  • Foliage surfaces
  • Leaf area density
  • Beer–Lambert’s law
  • Phototropism
  • Functional-structural tree modelling
  • Fagus sylvatica L.