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The Leaves of a Tree, Hong Kong Baptist University,
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2024-03-11
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2012美国大学生数学建模特等奖论文集,数学建模,英文版,中文版
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Team # 14297 Page 1 of 45
Summary
Leaf shape plasticity has long been a heated topic in the field of morphology and
taxonomy. Although numerous researchers have been involved in the study of how the
shape formation is controlled by genes and environmental stress, little has been done
to address the interactions between different plant organs. In this paper, we approach
this issue in a creative way: starting from isolated parts (leaves), building connections
(leaf-leaf, leaf-branch) and drawing the big picture of the entire system (tree).
The mathematical model introduced in this paper describes morphological,
physiological and partially ecological properties of the vegetations. First and foremost,
relative errors of the numerical outputs are limited within 7.5% and all the values
predicted are within the right order of magnitude. Moreover, biological intuition is
presented in the consistent manner. For instance, the dynamic positive feedback is
applied to describe leaf growth pattern and transportation process. It is also worth
noticing that a much simple and flexible algorithm is achieved by executing ingenious
geometric strategy.
Interestingly findings were generated in the procedures of solving the problems.
The seemly unrelated bio-phenomena could be explained when morphological,
physiological and partially ecological interactions were taken into account. To
illustrate, as we conducted the leaf mass estimation, it could give us a rough picture of
leaf-leaf, leaf-branch, and leaf-tree relations.
Team # 14297 Page 2 of 45
Dear editor,
The topic of this paper is about mathematical modeling of tree leaves as well as how
leaf traits impose influence on the tree system.
In this paper, we addressed the morphological, physiological and partially ecological
properties of the vegetations by applying a series of modeling process to related issues.
The performance of our model in the sensitivity test satisfied our expectation.
Different from the previous research, we approached this issue in a creative way:
starting from isolated parts (leaves), building connections (leaf-leaf, leaf-branch) and
drawing the big picture of the entire system (tree).
Interestingly, the seemly unrelated bio-phenomena could be explained when
morphological, physiological and partially ecological interactions were taken into
account. Properties of the individual leaves are closely related to the characteristics of
the entire tree system.
We are looking forward to your comments and revision. Hopefully, our paper could be
published in this Journal.
Your sincerely
Team # 14297 Page 3 of 45
Introduction
The mechanism of plant shape plasticity has long been a topic of numerous
researches in the field of morphometrics. And leaves have been always favored as the
subject because of its two-dimensionality and relative simplicity. The change of leaf
shapes are generally regarded as both a genetically determination (Howell, 1998) and
results of structural optimization to environmental stress (Hemsley and Poole, 2004).
However, little attention is drawn to the interactions between different plant organs
(e.g. leaf, branch) and the influence from the system as an entity. To fill this gap, we
construct a mathematical model to interpret such in-leaf communications, and
perform our analysis on simple leaves.
As is shown in some literatures (Stern et al., 2008), simple leaves are classified into
different shape catalogs (Figure 1), such as flabellate leaves (e.g. ginkgo), palmate
leaves (e.g. maple), and cordate leaves (e.g. Hibiscus tiliaceus). The major criteria of
classification are the shape apex, base, and margin of leaves.
Figure 1 Shape of Leaves
Leaf venation is also a critical component of leaf types. There are three major
patterns of major vein organization: pinnate, palmate, and parallel, as are shown in
Figure 2.
Team # 14297 Page 4 of 45
Figure 2 Leaf Venation
Besides the diversification in leaf shapes and venation, arrangement of leaves on a
stem (phyllotaxy) are generally in three distinct patterns (Figure 3), which are defined
by the number of leaves attached to a single node. In most species, leaves are attached
alternately or in a spiral along a stem, with one leaf per node. This is called an
alternate arrangement. If two leaves are attached at each node, they provide an
opposite arrangement. Leaves are whorled when three or more occur at a node. A
special case in opposite arrangement is called decussate, if successive pairs are
orientated at 90
o
to each other.
Figure 3 Leaf Arrangement
Review of previous researches shows that most models developed are based on
statistical results (Greig-Smith, 1983). And so far, no model is built to indicate the
effect of plant organ interactions (leaf-leaf, leaf-branch) on leaf shapes. The
mathematical model presented by us demonstrates the logic of plant-environment and
in-plant communications at physiology, inter-organ, and tree system levels. We first
Team # 14297 Page 5 of 45
start from isolated parts (leaves), build connections (leaf-leaf, leaf-branch) and end
with the big picture of the entire system (tree).
Problem A requires mathematical modeling of tree leaves as well as how leaf traits
impose influence on the tree system. It is suggested that shape factor of individual
leaves to be analyzed via the following aspects.
Influencing elements of leaf shapes;
Inter-leaf relations (i.e. overlapping between leaves)
Branch-leaf relations
Tree-leaf relations (i.e. estimating mass of leaves)
The model used should be able to capture not only the characteristics of single
leaves but also their interactions with other plant organs and eventually the tree
system. We have to start from isolated parts (leaves), build connections (leaf-leaf,
leaf-branch) and end with the big picture of the entire system (tree).
We divide the modeling section into four subsections.
Section 2.1: Descriptive model of leaf traits;
Section 2.2: Model of cell growth & physiological activities;
(photosynthesis, transportation, transpiration)
Section 2.3: Model of leaf arrangement & branching structure;
Section 2.4: Model of the tree characteristics.
(ATP governing equation, Driving Force governing equation)
The first two subsections of modeling are for analysis at leaf level. To quantify the
geometric characteristics of various leaves, shapes are described by functions and
curves are placed in Polar Coordinates, so that differences can be visualized.
Parameter selection can then easily adjust the leaf shapes, thus providing more
flexibility in deriving the bio-features such as size and perimeter. As for the shape
influencing factors, processes of growth and photosynthesis are assumed to determine
the effective blade area of leaves, while plant transport system and transpiration are
closely related with leaf venation. These physiological activities are modeled in
Section 2.2.
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