Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T10:37:38.226Z Has data issue: false hasContentIssue false
  • English
  • Français

A Symbolic Method to Analyse Patterns in Plant Structure

Published online by Cambridge University Press:  09 July 2014

C. LOI ,
P.-H. COURNÈDE  and
J. FRANÇON
C. LOI
Affiliation:
Ecole Centrale Paris, Laboratory of Applied Mathematics and Systems, Châtenay Malabry, France (e-mail: paul-henry.cournede@ecp.fr)
P.-H. COURNÈDE
Affiliation:
Ecole Centrale Paris, Laboratory of Applied Mathematics and Systems, Châtenay Malabry, France (e-mail: paul-henry.cournede@ecp.fr)
J. FRANÇON
Affiliation:
University of Strasbourg, ICube Laboratory, Team ‘Informatique Géométrique et Graphique’, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Formal grammars such as L-systems have long been used to describe plant growth dynamics. In this article, they are used for a new purpose. The aim is to build a symbolic method that enables the computation of the stochastic distribution associated with the number of complex structures in plants whose organogenesis is driven by a multitype branching process. For that purpose, a new combinatorial framework is set in which plant structure is coded by a Dyck word. Moreover, organogenesis is represented by stochastic F0L-systems. In doing so, the problem is equivalent to determining the distribution of patterns in random words generated by a stochastic F0L-system. This method finds interesting applications in the parameter identification of stochastic models of plant development.

Keywords

Primary 05ASecondary 92B60J
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 5: Honouring the Memory of Philippe Flajolet - Part 1 , September 2014 , pp. 842 - 860
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Athreya, K. and Ney, P. (2004) Branching Processes, Dover.Google Scholar
[2]Barthélémy, D. and Caraglio, Y. (2007) Plant architecture: A dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny. Ann. of Botany 99 375–407.Google Scholar
[3]Bell, A. (1991) Plant Form: An Illustrated Guide to Flowering Plant Morphology, Oxford University Press.Google Scholar
[4]Capasso, V. and Morale, D. (2009) Stochastic modelling of tumour-induced angiogenesis. J. Math. Biol. 58 219–233.CrossRefGoogle ScholarPubMed
[5]Costes, E., Smith, C., Renton, M., Guédon, Y., Prusinkiewicz, P. and Godin, C. (2008) MAppleT: Simulation of apple tree development using mixed stochastic and biomechanical models. Funct. Plant Biol. 35 936–950.Google Scholar
[6]Cournède, P.-H., Kang, M. Z., Mathieu, A., Barczi, J.-F., Yan, H. P., Hu, B. G. and de Reffye, P. (2006) Structural factorization of plants to compute their functional and architectural growth. Simulation 82 427–438.Google Scholar
[7]Diao, J., de Reffye, P., Lei, X., Guo, H. and Letort, V. (2012) Simulation of the topological development of young eucalyptus using a stochastic model and sampling measurement strategy. Computers and Electronics in Agriculture 80 105–114.Google Scholar
[8]Eichhorst, P. and Savitch, W. J. (1980) Growth functions of stochastic Lindenmayer systems. Inform. Control 45 217–228.Google Scholar
[9]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[10]Guédon, Y., Barthélémy, D., Caraglio, Y. and Costes, E. (2001) Pattern analysis in branching and axillary flowering sequences. J. Theoret. Biol. 212 481–520.Google Scholar
[11]Hallé, F., Oldeman, R. A. A. and Tomlinson, P. B. (1978) Tropical Trees and Forests: An Architectural Analysis, Springer.CrossRefGoogle Scholar
[12]Harris, T. E. (1963) The Theory of Branching Processes, Springer.Google Scholar
[13]Hemmerling, R., Kniemeyer, O., Lanwert, D., Buck-Sorlin, G. and Kurth, W. (2008) The rule based language XL and the modeling environment GroIMP illustrated with simulated tree competition. Funct. Plant Biol. 35 739–750.Google Scholar
[14]Henkel, A., Müller, J. and Pötzsche, C. (2012) Modeling the spread of phytophthora. J. Math. Biol. 65 1359–1385.Google Scholar
[15]Kang, M. Z., Cournède, P.-H., de Reffye, P., Auclair, D. and Hu, B. G. (2008) Analytical study of a stochastic plant growth model: Application to the GreenLab model. Math. Comput. Simul. 78 57–75.Google Scholar
[16]Karwowski, R. and Prusinkiewicz, P. (2003) Design and implementation of the L+C modeling language. Electron. Notes Theoret. Comput. Sci. 86 1–19.Google Scholar
[17]Kniemeyer, O., Buck-Sorlin, G. and Kurth, W. (2003) Representation of genotype and phenotype in a coherent framework based on extended L-systems. In Advances in Artificial Life: Proceedings of the 7th European Conference on Artificial Life, ECAL (Banzhaf, W., Christaller, T., Dittrich, D., Kim, J. T., and Ziegler, J., eds), Vol. 2801 of Lecture Notes in Artificial Intelligence, Springer, pp. 625–634.Google Scholar
[18]Kniemeyer, O., Buck-Sorlin, G. and Kurth, W. (2007) GroIMP as a platform for functional–structural modelling for plants. In Functional–Structural Plant Modelling in Crop Production (Vos, J.et al., eds), Springer, pp. 43–52.Google Scholar
[19]Knuth, D. E. (1997) The Art of Computer Programming, third edition, Addison-Wesley.Google Scholar
[20]Kurth, W. (1994) Growth Grammar Interpreter GROGRA 2.4: A Software Tool for the 3-Dimensional Interpretation of Stochastic, Sensitive Growth Grammars in the Context of Plant Modelling, Berichte des Forschungszentrums Waldökosysteme der Universität Göttingen.Google Scholar
[21]Kurth, W. (1994) Morphological models of plant growth: Possibilities and ecological relevance. Ecological Modelling 75/76 299–308.Google Scholar
[22]Lindenmayer, A. (1968) Mathematical models for cellular interactions in development I: Filaments with one-sided inputs. J. Theoret. Biol. 18 280–289.Google Scholar
[23]Loi, C. and Cournède, P.-H. (2008) Generating functions of stochastic L-systems and application to models of plant development. In Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings Vol. AI, pp. 325–338.Google Scholar
[24]Loi, C., Cournède, P.-H. and Françon, J. (2010) A symbolic method to compute the probability distribution of the number of pattern occurrences in random texts generated by stochastic 0l-systems. In 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms: AofA'10, DMTCS Proceedings Vol. AM, pp. 473–488.Google Scholar
[25]Lopez, G., Favreau, R. R., Smith, C., Costes, E., Prusinkiewicz, P. and DeJong, T. M. (2008) Integrating simulation of architectural development and source-sink behaviour of peach trees by incorporating Markov chains and physiological organ function submodels into L-PEACH. Funct. Plant Biol. 35 761–771.CrossRefGoogle ScholarPubMed
[26]Mathieu, A., Cournède, P.-H., Letort, V., Barthélémy, D. and de Reffye, P. (2009) A dynamic model of plant growth with interactions between development and functional mechanisms to study plant structural plasticity related to trophic competition. Ann. of Botany 103 1173–1186.Google Scholar
[27]Mech, R. and Prusinkiewicz, P. (1996) Visual models of plants interacting with their environment. In ACM SIGGRAPH'96 Proceedings, pp. 397–410Google Scholar
[28]Mode, C. J. (1971) Multitype Branching Processes: Theory and Applications, American Elsevier.Google Scholar
[29]Moré, J. J. (2006) The Levenberg–Marquardt algorithm: Implementation and theory. In Numerical Analysis: Dundee 1977, Vol. 630 of Lecture Notes in Mathematics, Springer, pp. 105–116.Google Scholar
[30]Pevzner, P. A., Borodovski, M. and Mironov, A. (1989) Linguistics of nucleotide sequences I: The significance of deviations from mean statistical characteristics and prediction of the frequencies of occurrence of words. J. Biomol. Struct. Dyn. 6 1013–1026.Google Scholar
[31]Pfeifer, N., Gorte, B. and Winterhalder, D. (2004) Automatic reconstruction of single trees from terrestrial laser scanner data. In Proceedings of 20th ISPRS Congress.Google Scholar
[32]Prusinkiewicz, P. and Lindenmayer, A. (1990) The Algorithmic Beauty of Plants, Springer.Google Scholar
[33]Régnier, M. and Szpankowski, W. (1998) On pattern frequency occurrences in a Markovian sequence. Algorithmica 22 631–649.Google Scholar
[34]Riordan, J. (2002) An Introduction to Combinatorial Analysis, Dover.Google Scholar
[35]Rozenberg, G. and Salomaa, A. (1980) The Mathematical Theory of L-Systems, Academic Press.Google Scholar
[36]Smith, A. R. (1984) Plants, fractals and formal languages. In Proceedings of the 11th Annual Conference on Computer Graphics and Interactive Techniques, ACM, pp. 1–10.Google Scholar
[37]Xu, H., Gossett, N. and Chen, B. (2007) Knowledge and heuristic-based modeling of laser-scanned trees. ACM Trans. Graphics 26 #19.Google Scholar