The formation of a tree leaf

Qinglan Xia

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 2, page 359-377
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Xia, Qinglan. "The formation of a tree leaf." ESAIM: Control, Optimisation and Calculus of Variations 13.2 (2007): 359-377. <http://eudml.org/doc/249988>.

@article{Xia2007,
abstract = { 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. },
author = {Xia, Qinglan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Formation of a tree leaf; optimal transport system; leaf shape; leaf venation pattern; selection principle; generation map; leaf venation pattern},
language = {eng},
month = {5},
number = {2},
pages = {359-377},
publisher = {EDP Sciences},
title = {The formation of a tree leaf},
url = {http://eudml.org/doc/249988},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Xia, Qinglan
TI - The formation of a tree leaf
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/5//
PB - EDP Sciences
VL - 13
IS - 2
SP - 359
EP - 377
AB - 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.
LA - eng
KW - Formation of a tree leaf; optimal transport system; leaf shape; leaf venation pattern; selection principle; generation map; leaf venation pattern
UR - http://eudml.org/doc/249988
ER -

References

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  12. Q. Xia, Optimal paths related to transport problems. Comm. Cont. Math.5 (2003) 251–279.  
  13. Q. Xia, Interior regularity of optimal transport paths. Calc. Var. Partial Differ. Equ.20 (2004) 283–299.  
  14. Q. Xia, Boundary regularity of optimal transport paths. http://math.ucdavis.edu/~qlxia/Research/boundary.pdf.  

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