The formation of a tree leaf
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 359-377.

In this article, we build a mathematical model to understand the formation of a tree leaf. Our model is based on the idea that a leaf tends to maximize internal efficiency by developing an efficient transport system for transporting water and nutrients. The meaning of “the efficient transport system” may vary as the type of the tree leave varies. In this article, we will demonstrate that tree leaves have different shapes and venation patterns mainly because they have adopted different efficient transport systems. The efficient transport system of a tree leaf built here is a modified version of the optimal transport path, which was introduced by the author in [Comm. Cont. Math. 5 (2003) 251-279; Calc. Var. Partial Differ. Equ. 20 (2004) 283-299; Boundary regularity of optimal transport paths, Preprint] to study the phenomenon of ramifying structures in mass transportation. In the present paper, the cost functional on transport systems is controlled by two meaningful parameters. The first parameter describes the economy of scale which comes with transporting large quantities together, while the second parameter discourages the direction of outgoing veins at each node from differing much from the direction of the incoming vein. Under the same initial condition, efficient transport systems modeled by different parameters will provide tree leaves with different shapes and different venation patterns. Based on this model, we also provide some computer visualization of tree leaves, which resemble many known leaves including the maple and mulberry leaf. It demonstrates that optimal transportation plays a key role in the formation of tree leaves.

DOI : 10.1051/cocv:2007016
Classification : 92B05, 49Q20, 90B18
Mots clés : formation of a tree leaf, optimal transport system, leaf shape, leaf venation pattern
@article{COCV_2007__13_2_359_0,
     author = {Xia, Qinglan},
     title = {The formation of a tree leaf},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {359--377},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {2},
     year = {2007},
     doi = {10.1051/cocv:2007016},
     mrnumber = {2306641},
     zbl = {1114.92048},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2007016/}
}
TY  - JOUR
AU  - Xia, Qinglan
TI  - The formation of a tree leaf
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2007
SP  - 359
EP  - 377
VL  - 13
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2007016/
DO  - 10.1051/cocv:2007016
LA  - en
ID  - COCV_2007__13_2_359_0
ER  - 
%0 Journal Article
%A Xia, Qinglan
%T The formation of a tree leaf
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 359-377
%V 13
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2007016/
%R 10.1051/cocv:2007016
%G en
%F COCV_2007__13_2_359_0
Xia, Qinglan. The formation of a tree leaf. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 359-377. doi : 10.1051/cocv:2007016. http://archive.numdam.org/articles/10.1051/cocv:2007016/

[1] M. Bernot, V. Caselles and J.-M. Morel, Are there infinite irregation tree? J. Math. Fluid Mech. 8 (2006) 311-332. | Zbl

[2] A. Brancolini, G. Buttazzo and F. Santambrogio, Path functions over Wasserstein spaces. http://www.calcvar.sns.it/papers/brabutsan04/path.pdf

[3] T. De Pauw and R. Hardt, Size minimization and approximating problems. Calc. Var. Partial Differ. Equ. 17 (2003) 405-442. | Zbl

[4] Von C. Ettingshausen, Die Blatt-Skelete der Dikotyledonen. Wien: Staatsdruckerei, Wien (1861).

[5] E.N. Gilbert, Minimum cost communication networks. Bell System Tech. J. 46 (1967) 2209-2227.

[6] J.M. Harris, J.L. Hist and M.J. Mossinghoff, Combinatorics and graph theory. Springer-verlag (2000). | MR | Zbl

[7] L.J. Hickey, A revised classification of the architecture of dicotyledonous leaves, in Anatomy of the dicotyledons, 2nd edn., Vol. I, Systematic anatomy of the leaves and stem., C.R. Metcalfe, L. Chalk, Eds., Oxford, Clarendon Press (1979) 25-39.

[8] F. Maddalena, J.-M. Morel and S. Solimini, A variational model of irrigation patterns. Interfaces Free Bound. 5 (2003) 391-415. | Zbl

[9] R. Melville, Leaf venation patterns and the origin of angiosperms. Nature 224 (1969) 121-125.

[10] R. Melville, The terminology of leaves architecture. Taxon 25 (1976) 549-562.

[11] T. Nelson and N. Dengler, Leaf vascular pattern formation. Plant Cell 9 (1997) 1121-1135.

[12] Q. Xia, Optimal paths related to transport problems. Comm. Cont. Math. 5 (2003) 251-279. | Zbl

[13] Q. Xia, Interior regularity of optimal transport paths. Calc. Var. Partial Differ. Equ. 20 (2004) 283-299. | Zbl

[14] Q. Xia, Boundary regularity of optimal transport paths

Cité par Sources :