Visual models of plants interacting with their environment

Visual models of plants interacting with their environment
Visual Models of Plants Interacting with Their
Environment
Radomir Mech and Przemyslaw Prusinkiewicz
Department of Computer Science
University of Calgary
Calgary, Alberta, Canada T2N 1N4
e−mail: mech|pwp@cpsc.ucalgary.ca
Abstract
Interaction with the environment is a key factor affecting the development of plants and plant
ecosystems. In this paper we introduce a modeling framework that makes it possible to
simulate and visualize a wide range of interactions at the level of plant architecture. This
framework extends the formalism of Lindenmayer systems with constructs needed to model
bi−directional information exchange between plants and their environment. We illustrate the
proposed framework with models and simulations that capture the development of tree
branches limited by collisions, the colonizing growth of clonal plants competing for space in
favorable areas, the interaction between roots competing for water in the soil, and the
competition within and between trees for access to light. Computer animation and
visualization techniques make it possible to better understand the modeled processes and lead
to realistic images of plants within their environmental context.
Keywords: scientific visualization, realistic image synthesis, software design, L−system,
modeling, simulation, ecosystem, plant development, clonal plant, root, tree.
Reference
Radomir Mech and Przemyslaw Prusinkiewicz. Visual Models of Plants Interacting with Their Environment.
Proceedings of SIGGRAPH 96 (New Orleans, Louisiana, August 4−9, 1996). In Computer Graphics
Proceedings, Annual Conference Series, 1996, ACM SIGGRAPH, pp. 397−410.
Visual Models of Plants
Interacting with Their Environment
Radomı́r Měch and Przemyslaw Prusinkiewicz1
University of Calgary
ABSTRACT
Interaction with the environment is a key factor affecting the development of plants and plant ecosystems. In this paper we introduce a
modeling framework that makes it possible to simulate and visualize
a wide range of interactions at the level of plant architecture. This
framework extends the formalism of Lindenmayer systems with
constructs needed to model bi-directional information exchange between plants and their environment. We illustrate the proposed
framework with models and simulations that capture the development of tree branches limited by collisions, the colonizing growth of
clonal plants competing for space in favorable areas, the interaction
between roots competing for water in the soil, and the competition
within and between trees for access to light. Computer animation
and visualization techniques make it possible to better understand
the modeled processes and lead to realistic images of plants within
their environmental context.
CR categories: F.4.2 [Mathematical Logic and Formal Languages]: Grammars and Other Rewriting Systems: Parallel rewriting systems, I.3.7 [Computer Graphics]: Three-Dimensional
Graphics and Realism, I.6.3 [Simulation and Modeling]: Applications, J.3 [Life and Medical Sciences]: Biology.
Keywords: scientific visualization, realistic image synthesis, software design, L-system, modeling, simulation, ecosystem, plant development, clonal plant, root, tree.
1
INTRODUCTION
Computer modeling and visualization of plant development can be
traced back to 1962, when Ulam applied cellular automata to simulate the development of branching patterns, thought of as an abstract representation of plants [53]. Subsequently, Cohen presented
a more realistic model operating in continuous space [13], Linden1
Department of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N 1N4 (mech pwp@cpsc.ucalgary.ca)
j
mayer proposed the formalism of L-systems as a general framework
for plant modeling [38, 39], and Honda introduced the first computer
model of tree structures [32]. From these origins, plant modeling
emerged as a vibrant area of interdisciplinary research, attracting the
efforts of biologists, applied plant scientists, mathematicians, and
computer scientists. Computer graphics, in particular, contributed
a wide range of models and methods for synthesizing images of
plants. See [18, 48, 54] for recent reviews of the main results.
One aspect of plant structure and behavior neglected by most models
is the interaction between plants and their environment (including
other plants). Indeed, the incorporation of interactions has been
identified as one of the main outstanding problems in the domain of
plant modeling [48] (see also [15, 18, 50]). Its solution is needed to
construct predictive models suitable for applications ranging from
computer-assisted landscape and garden design to the determination
of crop and lumber yields in agriculture and forestry.
Using the information flow between a plant and its environment as
the classification key, we can distinguish three forms of interaction
and the associated models of plant-environment systems devised to
date:
1. The plant is affected by global properties of the environment,
such as day length controlling the initiation of flowering [23]
and daily minimum and maximum temperatures modulating the
growth rate [28].
2. The plant is affected by local properties of the environment, such
as the presence of obstacles controlling the spread of grass [2]
and directing the growth of tree roots [26], geometry of support
for climbing plants [2, 25], soil resistance and temperature in
various soil layers [16], and predefined geometry of surfaces to
which plant branches are pruned [45].
3. The plant interacts with the environment in an information feedback loop, where the environment affects the plant and the plant
reciprocally affects the environment. This type of interaction is
related to sighted [4] or exogenous [42] mechanisms controlling
plant development, in which parts of a plant influence the development of other parts of the same or a different plant through the
space in which they grow. Specific models capture:
; competition for space (including collision detection and access to light) between segments of essentially two-dimensional
schematic branching structures [4, 13, 21, 22, 33, 34, 36];
; competition between root tips for nutrients and water trans-
ported in soil [12, 37] (this mechanism is related to competition
between growing branches of corals and sponges for nutrients
diffusing in water [34]);
; competition
for light between three-dimensional shoots of
herbaceous plants [25] and branches of trees [9, 10, 11, 15,
33, 35, 52].
Models of exogenous phenomena require a comprehensive representation of both the developing plant and the environment. Consequently, they are the most difficult to formulate, implement, and
document. Programs addressed to the biological audience are often
limited to narrow groups of plants (for example, poplars [9] or trees
in the pine family [21]), and present the results in a rudimentary
graphical form. On the other hand, models addressed to the computer graphics audience use more advanced techniques for realistic
image synthesis, but put little emphasis on the faithful reproduction
of physiological mechanisms characteristic to specific plants.
In this paper we propose a general framework (defined as a modeling methodology supported by appropriate software) for modeling, simulating, and visualizing the development of plants that
bi-directionally interact with their environment. The usefulness of
modeling frameworks for simulation studies of models with complex (emergent) behavior is manifested by previous work in theoretical biology, artificial life, and computer graphics. Examples
include cellular automata [51], systems for simulating behavior of
cellular structures in discrete [1] and continuous [20] spaces, and
L-system-based frameworks for modeling plants [36, 46]. Frameworks may have the form of a general-purpose simulation program
that accepts models described in a suitable mini-language as input, e.g. [36, 46], or a set of library programs [27]. Compared to
special-purpose programs, they offer the following benefits:
At the conceptual level, they facilitate the design, specification,
documentation, and comparison of models.
At the level of model implementation, they make it possible to develop software that can be reused in various models. Specifically,
graphical capabilities needed to visualize the models become a
part of the modeling framework, and do not have to be reimplemented.
Finally, flexible conceptual and software frameworks facilitate
interactive experimentation with the models [46, Appendix A].
Our framework is intended both for purpose of image synthesis and
as a research and visualization tool for model studies in plant morphogenesis and ecology. These goals are addressed at the levels of
the simulation system and the modeling language design. The underlying paradigm of plant-environment interaction is described in
Section 2. The resulting design of the simulation software is outlined
in Section 3. The language for specifying plant models is presented
in Section 4. It extends the concept of environmentally-sensitive Lsystems [45] with constructs for bi-directional communication with
the environment. The following sections illustrate the proposed
framework with concrete models of plants interacting with their
environment. The examples include: the development of planar
branching systems controlled by the crowding of apices (Section 5),
the development of clonal plants controlled by both the crowding
of ramets and the quality of terrain (Section 6), the development
of roots controlled by the concentration of water transported in the
soil (Section 7), and the development of tree crowns affected by the
local distribution of light (Section 8) The paper concludes with an
evaluation of the results and a list of open problems (Section 9).
Plant
Environment
Reception
Response
Internal processes
Internal processes
Response
Reception
Figure 1: Conceptual model of plant and environment treated as
communicating concurrent processes
2
CONCEPTUAL MODEL
As described by Hart [30], every environmentally controlled phenomenon can be considered as a chain of causally linked events.
After a stimulus is perceived by the plant, information in some form
is transported through the plant body (unless the site of stimulus
perception coincides with the site of response), and the plant reacts. This reaction reciprocally affects the environment, causing
its modification that in turn affects the plant. For example, roots
growing in the soil can absorb or extract water (depending on the
water concentration in their vicinity). This initiates a flow of water
in the soil towards the depleted areas, which in turn affects further
growth of the roots [12, 24].
According to this description, the interaction of a plant with the
environment can be conceptualized as two concurrent processes
that communicate with each other, thus forming a feedback loop
of information flow (Figure 1). The plant process performs the
following functions:
reception of information about the environment in the form of
scalar or vector values representing the stimuli perceived by specific organs;
transport and processing of information inside the plant;
generation of the response in the form of growth changes (e.g.
development of new branches) and direct output of information
to the environment (e.g. uptake and excretion of substances by a
root tip).
Similarly, the environmental process includes mechanisms for
the:
perception of the plant’s actions;
simulation of internal processes in the environment (e.g. the
diffusion of substances or propagation of light);
presentation of the modified environment in a form perceivable
by the plant.
The design of a simulation system based on this conceptual model
is presented next.
3
SYSTEM DESIGN
The goal is to create a framework, in which a wide range of plant
structures and environments can be easily created, modified, and
used for experimentation. This requirement led us to the following
design decisions:
The plant and the environment should be modeled by separate
programs and run as two communicating processes. This design
is:
Plant
model
(L−system)
Plant
simulator
; compatible with the assumed conceptual model of plant-environment interaction (Figure 1);
; appropriate for interactive experimentation with the models;
in particular, changes in the plant program can be implemented
without affecting the environmental program, and vice versa;
Interface
plant−
environment
C
O
M
M
U
N
I
C
A
T
I
O
N
Environ−
mental data
Model of
the environment
Communication
specification
; consistent with the principles of structured design (modules
with clearly specified functions jointly contribute to the solution of a problem by communicating through a well defined
interface; information local to each module is hidden from
other modules);
C
O
M
M
U
N
I
C
A
T
I
O
N
Figure 2: Organization of the software for modeling plants interacting with their environment. Shaded rectangles indicate components
of the modeling framework, clear rectangles indicate programs and
data that must be created by a user specifying a new model of a plant
or environment. Shaded arrows indicate information exchanged in
a standardized format.
; extensible to distributed computing environments, where different components of a large ecosystem may be simulated
using separate computers.
The user should have control over the type and amount of information exchanged between the processes representing the plant
and the environment, so that all the needed but no superfluous
information is transferred.
Plant models should be specified in a language based on Lsystems, equipped with constructs for bi-directional communication between the plant and the environment. This decision has
the following rationale:
; A succinct description of the models in an interpreted lan-
guage facilitates experimentation involving modifications to
the models;
; L-systems capture two fundamental mechanisms that control
development, namely flow of information from a mother module to its offspring (cellular descent) and flow of information
between coexisting modules (endogenous interaction) [38].
The latter mechanism plays an essential role in transmitting
information from the site of stimulus perception to the site
of the response. Moreover, L-systems have been extended
to allow for input of information from the environment (see
Section 4);
; Modeling of plants using L-systems has reached a relatively
advanced state, manifested by models ranging from algae to
herbaceous plants and trees [43, 46].
Given the variety of processes that may take place in the environment, they should be modeled using special-purpose programs.
Generic aspects of modeling, not specific to particular models,
should be supported by the modeling system. This includes:
; an L-system-based plant modeling program, which interprets
L-systems supplied as its input and visualizes the results, and
; the support for communication and synchronization of processes simulating the modeled plant and the environment.
A system architecture stemming from this design is shown in Figure 2. We will describe it from the perspective of extensions to the
formalism of L-systems.
4
OPEN L-SYSTEMS
Historically, L-systems were conceived as closed cybernetic systems, incapable of simulating any form of communication between
the modeled plant and its environment. In the first step towards
the inclusion of environmental factors, Rozenberg defined table Lsystems, which allow for a change in the set of developmental rules
(the production set of the L-system) in response to a change in
the environment [31, 49]. Table L-systems were applied, for example, to capture the switch from the production of leaves to the
production of flowers by the apex of a plant due to a change in day
length [23]. Parametric L-systems [29, 46], introduced later, made
it possible to implement a variant of this technique, with the environment affecting the model in a quantitative rather than qualitative
manner. In a case study illustrating this possibility, weather data
containing daily minimum and maximum temperatures were used
to control the rate of growth in a bean model [28]. Environmentallysensitive L-systems [45] represented the next step in the inclusion of
environmental factors, in which local rather than global properties
of the environment affected the model. The new concept was the
introduction of query symbols, returning current position or orientation of the turtle in the underlying coordinate system. These
parameters could be passed as arguments to user-defined functions,
returning local properties of the environment at the queried location.
Environmentally-sensitive L-systems were illustrated by models of
topiary scenes. The environmental functions defined geometric
shapes, to which trees were pruned.
Open L-systems, introduced in this paper, augment the functionality
of environmentally-sensitive L-systems using a reserved symbol for
bilateral communication with the environment. In short, parameters
associated with an occurrence of the communication symbol can
be set by the environment and transferred to the plant model, or
set by the plant model and transferred to the environment. The
environment is no longer represented by a simple function, but
becomes an active process that may react to the information from the
plant. Thus, plants are modeled as open cybernetic systems, sending
information to and receiving information from the environment.
In order to describe open L-systems in more detail, we need to
recall the rudiments of L-systems with turtle interpretation. Our
presentation is reproduced from [45].
An L-system is a parallel rewriting system operating on branching
structures represented as bracketed strings of symbols with associated numerical parameters, called modules. Matching pairs of
square brackets enclose branches. Simulation begins with an initial string called the axiom, and proceeds in a sequence of discrete
derivation steps. In each step, rewriting rules or productions replace
all modules in the predecessor string by successor modules. The
applicability of a production depends on a predecessor’s context
(in context-sensitive L-systems), values of parameters (in productions guarded by conditions), and on random factors (in stochastic
L-systems). Typically, a production has the format:
id : lc < pred > rc : cond ! succ : prob
where id is the production identifier (label), lc, pred, and rc are
the left context, the strict predecessor, and the right context, cond is
the condition, succ is the successor, and prob is the probability of
production application. The strict predecessor and the successor are
the only mandatory fields. For example, the L-system given below
consists of axiom ! and three productions p1 , p2 , and p3 .
!: A(1)B(3)A(5)
p1 : A(x) !A(x+1) : 0.4
p2 : A(x) !B(x–1) : 0.6
p3 : A(x) < B(y) > A(z) : y < 4 !B(x+z)[A(y)]
The stochastic productions p1 and p2 replace module A(x) by either A(x + 1) or B (x ; 1), with probabilities equal to 0.4 and
0.6, respectively. The context-sensitive production p3 replaces a
module B (y ) with left context A(x) and right context A(z ) by
module B (x + z ) supporting branch A(y ). The application of this
production is guarded by condition y < 4. Consequently, the first
derivation step may have the form:
A(1)B(3)A(5)
A(2)B(6)[A(3)]B(4)
)
It was assumed that, as a result of random choice, production p1
was applied to the module A(1), and production p2 to the module
A(5). Production p3 was applied to the module B (3), because it
occurred with the required left and right context, and the condition
3 < 4 was true.
In the L-systems presented as examples we also use several additional constructs (cf. [29, 44]):
Productions may include statements assigning values to local
variables. These statements are enclosed in curly braces and
separated by semicolons.
The L-systems may also include arrays. References to array
elements follow the syntax of C; for example, MaxLen[order].
The list of productions is ordered. In the deterministic case, the
first matching production applies. In the stochastic case, the set
of all matching productions is established, and one of them is
chosen according to the specified probabilities.
For details of the L-system syntax see [29, 43, 46].
→
U
→
+ −
→
L
&
\ H
/
^
Figure 3: Controlling the
turtle in three dimensions
In contrast to the parallel application of productions in each derivation step, the interpretation of the
resulting strings proceeds sequentially, with reserved modules acting as commands to a LOGO-style
turtle [46]. At any point of the
string, the turtle state is charac~ and
terized by a position vector P
interpret
derive
env. step
... A(a1,...,ak) ?E(x1,...,xm) B(b1,...,bn) ...
environment
... A(a1,...,ak) ?E(y1,...,ym) B(b1,...,bn) ...
Figure 4: Information flow during the simulation of a plant interacting with the environment, implemented using an open L-system
~ , U~ , and L~ , indithree mutually perpendicular orientation vectors H
cating the turtle’s heading, the up direction, and the direction to the
left (Figure 3). Module F causes the turtle to draw a line in the current direction. Modules +, , &, , = and rotate the turtle around
~ U~ , or L~ , as shown in Figure 3. The length
one of the vectors H;
of the line and the magnitude of the rotation angle can be given
globally or specified as parameters of individual modules. During
the interpretation of branches, the opening square bracket pushes
the current position and orientation of the turtle on a stack, and the
closing bracket restores the turtle to the position and orientation
popped from the stack. A special interpretation is reserved for the
module %, which cuts a branch by erasing all symbols in the string
from the point of its occurrence to the end of the branch [29]. The
meaning of many symbols depends on the context in which they
occur; for example, + and denote arithmetic operators as well as
modules that rotate the turtle.
;
^
n
;
The turtle interpretation of L-systems described above was designed to visualize models in a postprocessing step, with no effect
on the L-system operation. Position and orientation of the turtle
are important, however, while considering environmental phenomena, such as collisions with obstacles and exposure to light. The
environmentally-sensitive extension of L-systems makes these attributes accessible during the rewriting process [45]. The generated
string is interpreted after each derivation step, and turtle attributes
found during the interpretation are returned as parameters to reserved query modules. Syntactically, the query modules have the
form ?X (x; y; z ), where X = P; H; U; or L. Depending on the
actual symbol X , the values of parameters x, y , and z represent a
position or an orientation vector.
Open L-systems are a generalization of this concept. Communication modules of the form ?E (x1 ; : : : ; xm ) are used both to send
and receive environmental information represented by the values of
parameters x1 ; : : : ; xm (Figure 4). To this end, the string resulting
from a derivation step is scanned from left to right to determine
the state of the turtle associated with each symbol. This phase is
similar to the graphical interpretation of the string, except that the
results need not be visualized. Upon encountering a communication symbol, the plant process creates and sends a message to the
environment including all or a part of the following information:
the address (position in the string) of the communication module
(mandatory field needed to identify this module when a reply
comes from the environment),
values of parameters xi ,
the state of the turtle (coordinates of the position and orientation
vector, as well as some other attributes, such as current line
width),
the type and parameters of the module following the communication module in the string (not used in the examples discussed
in this paper).
The exact message format is defined in a communication specification file, shared between the programs modeling the plant and
the environment (Figure 2). Consequently, it is possible to include
only the information needed in a particular model in the messages
sent to the environment. Transfer of the last message corresponding
to the current scan of the string is signaled by a reserved end-oftransmission message, which may be used by the environmental
process to start its operation.
The messages output by the plant modeling program are transferred
to the process that simulates the environment using an interprocess
communication mechanism provided by the underlying operating
system (a pair of UNIX pipes or shared memory with access synchronized using semaphores, for example). The environment processes that information and returns the results to the plant model
using messages in the following format:
the address of the target communication module,
values of parameters yi carrying the output from the environment.
The plant process uses the received information to set parameter values in the communication modules (Figure 4). The use of addresses
makes it possible to send replies only to selected communication
modules. Details of the mapping of messages received by the plant
process to the parameters of the communication modules are defined
in the communication specification file.
After all replies generated by the environment have been received
(a fact indicated by an end-of-transmission message sent by the
environment), the resulting string may be interpreted and visualized,
and the next derivation step may be performed, initiating another
cycle of the simulation.
Note that, by preceding every symbol in the string with a communication module it is possible to pass complete information about the
model to the environment. Usually, however, only partial information about the state of a plant is needed as input to the environment.
Proper placement of communication modules in the model, combined with careful selection of the information to be exchanged,
provide a means for keeping the amount of transferred information
at a manageable level.
We will illustrate the operation of open L-systems within the context of complete models of plant-environment interactions, using
examples motivated by actual biological problems.
5
A MODEL OF BRANCH TIERS
Background. Apical meristems, located at the endpoints of
branches, are engines of plant development. The apices grow, contributing to the elongation of branch segments, and from time to time
divide, spawning the development of new branches. If all apices
divided periodically, the number of apices and branch segments
would increase exponentially. Observations of real branching structures show, however, that the increase in the number of segments
is less than exponential [8]. Honda and his collaborators modeled several hypothetical mechanisms that may control the extent of
branching in order to prevent overcrowding [7, 33] (see also [4]).
One of the models [33], supported by measurements and earlier
simulations of the tropical tree Terminalia catappa [19], assumes
an exogenous interaction mechanism. Terminalia branches form
horizontal tiers, and the model is limited to a single tier, treated
as a two-dimensional structure. In this case, the competition for
light effectively amounts to collision detection between the apices
and leaf clusters. We reproduce this model as the simplest example
illustrating the methodology proposed in this paper.
Communication specification. The plant communicates with the
environment using communication modules of the form ?E (x).
Messages sent to the environment include the turtle position and the
value of parameter x, interpreted as the vigor of the corresponding
apex. On this basis, the environmental process determines the fate
of each apex. A parameter value of x = 0 returned to the plant
indicates that the development of the corresponding branch will be
terminated. A value of x = 1 allows for further branching.
The model of the environment. The environmental process considers each apex or non-terminal node of the developing tier as the
center of a circular leaf cluster, and maintains a list of all clusters
present. New clusters are added in response to messages received
from the plant. All clusters have the same radius , specified in the
environmental data file (cf. Figure 2). In order to determine the fate
of the apices, the environment compares apex positions with leaf
cluster positions, and authorizes an apex to grow if it does not fall
into an existing leaf cluster, or if it falls into a cluster surrounding
an apex with a smaller vigor value.
The plant model. The plant model is expressed as an open Lsystem. The values of constants are taken from [33].
#define r1 0.94
/* contraction ratio and vigor 1 */
#define r2 0.87
/* contraction ratio and vigor 2 */
/* branching angle 1 */
#define 1 24.4
#define 2 36.9
/* branching angle 2 */
/* divergence angle */
#define 138.5
: –(90)[F(1)?E(1)A(1)]+( )[F(1)/?E(1)A(1)]
+( )[F(1)?E(1)A(1)]+( )[F(1)/?E(1)A(1)]
+( )[F(1)?E(1)A(1)]
!
'
'
'
'
'
p1 : ?E(x) < A(v) : x == 1 !
[+(2 )F(v*r2 )?E(r2 )A(v*r2 )] –(1 )F(v*r1 )/?E(r1 )A(v*r1 )
p2 : ?E(x) ! "
The axiom ! specifies the initial structure as a whorl of five branch
segments F . The divergence angle ' between consecutive segments
is equal to 138:5 . Each segment is terminated by a communication
symbol ?E followed by an apex A. In addition, two branches
include module =, which changes the directions at which subsequent
branches will be issued (left vs. right) by rotating the apex 180
around the segment axis.
Production p1 describes the operation of the apices. If the value of
parameter x returned by a communication module ?E is not 1, the
associated apex will remain inactive (do nothing). Otherwise the
apex will produce a pair of new branch segments at angles 1 and 2
with respect to the mother segment. Constants r1 and r2 determine
the lengths of the daughter segments as fractions of the length of
their mother segment. The values r1 and r2 are also passed to the
process simulating the environment using communication modules
?E . Communication modules created in the previous derivation
step are no longer needed and are removed by production p2 .
The environment responds by setting x to the intensity of incoming light, ranging from 0 (no light) to 1 (full light).
Figure 5: Competition for space between two tiers of branches
simulated using the Honda model
Simulation. Figure 5 illustrates the competition for space between
two tiers developing next to each other. The extent of branching
in each tier is limited by collisions between its apices and its own
or the neighbor’s leaf clusters. The limited growth of each structure in the direction of its neighbor illustrates the phenomenon of
morphological plasticity, or adaptation of the form of plants to their
environment [17].
6
A MODEL OF FORAGING IN CLONAL PLANTS
Background. Foraging (propagation) patterns in clonal plants provide another excellent example of response to crowding. A clonal
plant spreads by means of horizontal stem segments (spacers),
which form a branching structure that grows along the ground and
connects individual plants (ramets) [3]. Each ramet consists of a
leaf supported by an upright stem and one or more buds, which may
give rise to further spacers and ramets. Their gradual death, after
a certain amount of time, causes gradual separation of the whole
structure (the clone) into independent parts.
Following the surface of the soil, clonal plants can be captured using
models operating in two dimensions [5], and in that respect resemble Terminalia tiers. We propose a model of a hypothetical plant
that responds to favorable environmental conditions (high local intensity of light) by more extensive branching and reduced size of
leaves (allowing for more dense packing of ramets). It has been
inspired by a computer model of clover outlined by Bell [4], the
analysis of responses of clonal plants to the environment presented
by Dong [17], and the computer models and descriptions of vegetative multiplication of plants involving the death of intervening
connections by Room [47].
Communication specification. The plant sends messages to the environment that include turtle position and two parameters associated
with the communications symbol, ?E (type;x). The first parameter is equal to 0, 1, or 2, and determines the type of exchanged
information as follows:
The message ?E (0; x) represents a request for the light intensity
(irradiance [14]) at the position of the communication module.
The message ?E (1; x) notifies the environment about the creation
of a ramet with a leaf of radius x at the position of the communication module. No output is generated by the environment in
response to this message.
The message ?E (2; x) notifies the environment about the death of
a ramet with a leaf of radius x at the position of the communication module. Again, no output is generated by the environment.
The model of the environment. The purpose of the environment
process is to determine light intensity at the locations requested
by the plant. The ground is divided into patches (specified as a
raster image using a paint program), with different light intensities
assigned to each patch. In the absence of shading, these intensities
are returned by the environmental process in response to messages
of type 0. To consider shading, the environment keeps track of the
set of ramets, adding new ramets in response to a messages of type
1, and deleting dead ramets in response to messages of type 2. If
a sampling point falls in an area occupied by a ramet, the returned
light intensity is equal to 0 (leaves are assumed to be opaque, and
located above the sampling points).
The plant model. The essential features of the plant model are
specified by the following open L-system.
#define 45
#define MinLight 0.1
#define MaxAge 20
#define Len 2.0
#define ProbB (x)
#define ProbR (x)
#define Radius(x)
: A(1)?E(0,0)
!
p1:
/* branching angle */
/* light intensity threshold */
/* lifetime of ramets and spacers */
/* length of spacers */
(0.12+x*0.42)
(0.03+x*0.54)
(sqrt(15–x*5)/ )
A(dir) > ?E(0,x) : x >= MinLight
R(x)B(x,dir)F(Len,0)A(–dir)?E(0,0)
2 : A(dir) > ?E(0,x) : x < MinLight
!
p
!"
! [+(*dir)F(Len,0)A(–dir)?E(0,0)] : ProbB (x)
! " : 1–ProbB (x)
p3:
p4:
B(x,dir)
B(x,dir)
p5:
p6:
R(x)
R(x)
p7:
p8:
@o(radius,age): age < MaxAge
@o(radius,age): age == MaxAge
! [@o(Radius(x),0)?E(1,Radius(x))] : ProbR (x)
! " : 1–ProbR (x)
! @o(radius,age+1)
! ?E(2,radius)
p9: F(len,age): age < MaxAge ! F(len,age+1)
p10: F(len,age): age == MaxAge ! f(len)
p11: ?E(type,x) ! "
The initial structure specified by the axiom ! consists of an apex A
followed by the communication module ?E . If the intensity of light
x reaching an apex is insufficient (below the threshold MinLight),
the apex dies (production p2 ). Otherwise, the apex creates a ramet
initial R (i.e., a module that will yield a ramet), a branch initial
B , a spacer F , and a new apex A terminated by communication
module ?E (production p1 ). The parameter dir, valued either 1 or
-1, controls the direction of branching. Parameters of the spacer
module specify its length and age.
A branch initial B may create a lateral branch with its own apex
A and communication module ?E (production p3 ), or it may die
and disappear from the system (production p4 ). The probability of
survival is an increasing linear function ProbB of the light intensity
x that has reached the mother apex A in the previous derivation
step. A similar stochastic mechanism describes the production of
a ramet by the ramet initial R (productions p5 and p6 ), with the
probability of ramet formation controlled by an increasing linear
function ProbR . The ramet is represented as a circle @o; its radius
is a decreasing function Radius of the light intensity x. As in the
case of spacers, the second parameter of a ramet indicates its age,
initially set to 0. The environment is notified about the creation of
the ramet using a communication module ?E .
The subsequent productions describe the aging of spacers (p7 ) and
ramets (p9 ). Upon reaching the maximum age MaxAge, a ramet is
removed from the system and a message notifying the environment
about this fact is sent by the plant (p8 ). The death of the spacers
is simulated by replacing spacer modules F with invisible line segments f of the same length. This replacement maintains the relative
position of the remaining elements of the structure. Finally, production p11 removes communication modules after they have performed
their tasks.
Simulations.
Division of the
ground into patches used in the sim0.2
0.6
ulations is shown in Figure 6. Arabic numerals indicate the intensity
III
II
0.0
of incoming light, and Roman numerals identify each patch. The deV
1.0
velopment of a clonal plant assum0.2
ing this division is illustrated in FigIV
I
ure 7. As an extension of the basic
Figure 6: Division of
model discussed above, the length
the ground into patches
of the spacers and the magnitude of
the branching angle have been varied using random functions with a normal distribution. Ramets have
been represented as trifoliate leaves.
The development begins with a single ramet located in relatively
good (light intensity 0.6) patch II at the top right corner of the growth
area (Figure 7, step 9 of the simulation). The plant propagates
through the unfavorable patch III without producing many branches
and leaves (step 26), and reaches the best patch I at the bottom left
corner (step 39). After quickly spreading over this patch (step 51),
the plant searches for further favorable areas (step 62). The first
attempt to reach patch II fails (step 82). The plant tries again, and
this time succeeds (steps 101 and 116). Light conditions in patch II
are not sufficient, however, to sustain the continuous presence of the
plant (step 134). The colony disappears (step 153) until the patch is
reached again by a new wave of propagation (steps 161 and 182).
The sustained occupation of patch I and the repetitive invasion of
patch II represent an emerging behavior of the model, difficult to
predict without running simulations. Variants of this model, including other branching architectures, responses to the environment,
and layouts of patches in the environment, would make it possible
to analyze different foraging strategies of clonal plants. A further
extension could replace the empirical assumptions regarding plant
responses with a more detailed simulation of plant physiology (for
example, including production of photosynthates and their transport and partition between ramets). Such physiological models
could provide insight into the extent to which the foraging patterns
optimize plants’ access to resources [17].
Figure 7: Development of a hypothetical clonal plant simulated
using an extension of L-system 3. The individual images represent
structures generated in 9, 26, 39, 51, 62, and 82 derivation steps
(top), followed by structures generated in 101, 116, 134, 153, 161,
and 182 steps (bottom).
7
A MODEL OF ROOT DEVELOPMENT
Background. The development of roots provides many examples
of complex interactions with the environment, which involve mechanical properties, chemical reactions, and transport mechanisms
in the soil. In particular, the main root and the rootlets absorb water
from the soil, locally changing its concentration (volume of water
per unit volume of soil) and causing water motion from water-rich to
depleted regions [24]. The tips of the roots, in turn, follow the gradient of water concentration [12], thus adapting to the environment
modified by their own activities.
Below we present a simplified implementation of the model of root
development originally proposed by Clausnitzer and Hopmans [12].
We assume a more rudimentary mechanism of water transport,
namely diffusion in a uniform medium, as suggested by Liddell
and Hansen [37]. The underlying model of root architecture is similar to that proposed by Diggle [16]. For simplicity, we focus on
model operation in two-dimensions.
Communication specification. The plant interacts with the environment using communication modules ?E (c; ) located at the
apices of the root system. A message sent to the environment in~ , the heading vector H~ , the value of
cludes the turtle position P
parameter c representing the requested (optimal) water uptake, and
the value of parameter representing the tendency of the apex to
follow the gradient of water concentration. A message returned to
the plant specifies the amount of water actually received by the apex
as the value of parameter c, and the angle biasing direction of further
growth as the value of .
The model of the environment.
θin∇C
The environment maintains a field
→
C of water concentrations, repreT
sented as an array of the amounts
∇C
of water in square sampling areas.
θout
Water is transported by diffusion,
→
simulated numerically using finite
H
differencing [41]. The environFigure 8: Definition of
ment responds to a request for wathe biasing angle out
ter from an apex located in an area
(i; j ) by granting the lesser of the
values requested and available at that location. The amount of water
in the sampled area is then decreased by the amount received by the
apex. The environment also calculates a linear combination T~ of
~ and the gradient of water concentration
the turtle heading vector H
C (estimated numerically from the water concentrations in the
sampled area and its neighbors), and returns an angle between the
~ (Figure 8). This angle is used by the plant model
vectors T~ and H
to bias turtle heading in the direction of high water concentration.
Figure 9: A two-dimensional model of a root interacting with water
in soil. Background colors represent concentrations of water diffusing in soil (blue: high, black: low). The initial and boundary values
have been set using a paint program.
r
The root model. The open L-system representing the root model
makes use of arrays that specify parameters for each branching order
(main axis, its daughter axes, etc.). The parameter values are loosely
based on those reported by Clausnitzer and Hopmans [12].
#define N 3
Define: array
Req[N] = 0.1, 0.4, 0.05 ,
MinReq[N] = 0.01, 0.06, 0.01 ,
ElRate[N] = 0.55, 0.25, 0.55 ,
MaxLen[N] = 200, 5, 0.8 ,
Sens[N] = 10, 0, 0 ,
Dev[N] = 30, 75, 75 ,
Del[N–1] = 30, 60 ,
BrAngle[N–1] = 90, 90 ,
BrSpace[N–1] = 1, 0.5
f
f
f
f
f
g
!:
g
f
f
f
g
f
f
g
g
g
g
g
g
g
/* max. branching order + 1 */
/* requested nutrient intake */
/* minimum nutrient intake */
/* maximum elongation rate */
/* maximum branch length */
/* sensitivity to gradient */
/* deviation in heading */
/* delay in branch growth */
/* branching angle */
/* distance between branches */
Figure 10: A three-dimensional extension of the root model. Water
concentration is visualized by semi-transparent iso-surfaces [55]
surrounding the roots. As a result of competition for water, the
roots grow away from each other. The divergence between their
main axes depends on the spread of the rootlets, which grow faster
on the left then on the right.
A(0,0,0)?E(Req[0],Sens[0])
p1: A(n,s,b) > ?E(c,) : (s > MaxLen[n]) || (c < MinReq[n]) ! "
p2: A(n,s,b) > ?E(c,) :
(n >= N–1) || (b < BrSpace[n]) fh=c/Req[n]*ElRate[n];g
! +(nran(,Dev[n]))F(h) A(n,s+h,b+h)?E(Req[n],Sens[n])
p3: A(n,s,b) > ?E(c,) :
(n < N–1) && (b >= BrSpace[n]) fh=c/Req[n]*ElRate[n];g
! +(nran(,Dev[n]))B(n,0)F(h)
/(180)A(n,s+h,h)?E(Req[n],Sens[n])
p4: B(n,t) : t < Del[n] ! B(n,t+1)
p5: B(n,t) : t >= Del[n]
! [+(BrAngle[n])A(n+1,0,0)?E(Req[n+1],Sens[n+1])]
p6: ?E(c,) ! "
The development starts with an apex A followed by a communication module ?E . The parameters of the apex represent the branch
order (0 for the main axis, 1 for its daughter axes, etc.), current axis
length, and distance (along the axis) to the nearest branching point.
Productions p1 to p3 describe possible fates of the apex as described
below.
If the length s of a branch axis exceeds a predefined maximum value
MaxLen[n] characteristic to the branch order n, or the amount
of water c received by the apex is below the required minimum
MinReq[n], the apex dies, terminating the growth of the axis (production p1 ).
If the branch order n is equal to the maximum value assumed in the
model (N 1), or the distance b to the closest branching point on
the axis is less than the threshold value BrSpace[n], the apex adds
a new segment F to the axis (production p2 ). The length h of F
is the product of the nominal growth increment ElRate[n] and the
ratio of the amount of water received by the apex c to the amount
requested Req [n]. The new segment is rotated with respect to its
predecessor by an angle nran(; Dev [n]), where nran is a random
function with a normal distribution. The mean value , returned by
the environment, biases the direction of growth towards regions of
;
higher water concentration. The standard deviation Dev [n] characterizes the tendency of the root apex to change direction due to
various factors not included explicitly in the model.
If the branch order n is less than the maximum value assumed in the
model (N 1), and the distance b to the closest branching point on
the axis is equal to or exceeds the threshold value BrSpace[n], the
apex creates a new branch initial B (production p3 ). Other aspects
of apex behavior are the same as those described by production p2 .
;
After the delay of Del[n] steps (production p4 ), the branch initial B
is transformed into an apex A followed by the communication module ?E (production p5 ), giving rise to a new root branch. Production
p6 removes communication modules that are no longer needed.
Simulations. A sample two-dimensional structure obtained using
the described model is shown in Figure 9. The apex of the main axis
follows the gradient of water concentration, with small deviations
due to random factors. The apices of higher-order axes are not
sensitive to the gradient and change direction at random, with a
larger standard deviation. The absorption of water by the root and
the rootlets decreases water concentration in their neighborhood;
an effect that is not fully compensated by water diffusion from the
water-rich areas. Low water concentration stops the development
of some rootlets before they have reached their potential full length.
Figure 10 presents a three-dimensional extension of the previous
model. As a result of competition for water, the main axes of the
roots diverge from each other (left). If their rootlets grow more
slowly, the area of influence of each root system is smaller and
the main axes are closer to each other (right). This behavior is
an emergent property of interactions between the root modules,
mediated by the environment.
8
MODELS OF TREES CONTROLLED BY LIGHT
Background. Light is one of the most important factors affecting the development of plants. In the essentially two-dimensional
structures discussed in Section 5, competition for light could be
considered in a manner similar to collision detection between leaves
and apices. In contrast, competition for light in three-dimensional
structures must be viewed as long-range interaction. Specifically,
shadows cast by one branch may affect other branches at significant
distances.
The first simulations of plant development that take the local light
environment into account are due to Greene [25]. He considered
the entire sky hemisphere as a source of illumination and computed
the amount of light reaching specific points of the structure by
casting rays towards a number of points on the hemisphere. Another
approach was implemented by Kanamaru et al. [35], who computed
the amount of light reaching a given sampling point by considering
it a center of projection, from which all leaf clusters in a tree were
projected on a surrounding hemisphere. The degree to which the
hemisphere was covered by the projected clusters indicated the
amount of light received by the sampling point. In both cases,
the models of plants responded to the amount and the direction
of light by simulating heliotropism, which biased the direction of
growth towards the vector of the highest intensity of incoming light.
Subsequently, Chiba et al. extended the models by Kanamaru et
al. using more involved tree models that included a mechanism
simulating the flow of hypothetical endogenous information within
the tree [10, 11]. A biologically better justified model, formulated
in terms of production and use of photosynthates by a tree, was
proposed by Takenaka [52]. The amount of light reaching leaf
clusters was calculated by sampling a sky hemisphere, as in the work
by Greene. Below we reproduce the main features of the Takenaka’s
model using the formalism of open L-systems. Depending on the
underlying tree architecture, it can be applied to synthesize images
of deciduous and coniferous trees. We focus on a deciduous tree,
which requires a slightly smaller number of productions.
Communication specification. The plant interacts with the environment using communication modules ?E (r). A message sent by
~ , which represents the center of a
the plant includes turtle position P
spherical leaf cluster, and the value of parameter r, which represents
the cluster’s radius. The environment responds by setting r to the
flux [14] of light from the sky hemisphere, reaching the cluster.
The model of the environment. Once all messages describing
the current distribution of leaves on a tree have been received, the
environmental process computes the extent of the tree in the x, y ,
and z directions, encompasses the tree in a tight grid (32 32
32 voxels in our simulations), and allocates leaf clusters to voxels
to speed up further computations. The environmental process then
estimates the light flux Φ from the sky hemisphere reaching each
cluster (shadows cast by the branches are ignored). To this end,
the hemisphere is represented by a set of directional light sources
S (9 in the simulations). The flux densities (radiosities) B of the
sources approximate the non-uniform distribution of light from the
sky (cf. [52]). For each leaf cluster Li and each light source S , the
environment determines the set of leaf clusters Lj that may shade
Li . This is achieved by casting a ray from the center of Li in the
direction of S and testing for intersections with other clusters (the
grid accelerates this process). In order not to miss any clusters that
may partially occlude Li , the radius of each cluster Lj is increased
by the maximum value of cluster radius rmax .
To calculate the flux reaching cluster Li , this cluster and all clusters
Lj that may shade it according to the described tests are projected
on a plane P perpendicular to the direction of light from the source
S . The impact of a cluster Lj on the flux Φ reaching cluster Li is
then computed according to the formula:
Φ = (Ai Aij )B + Aij B
where Ai is the area of the projection of Li on P , Aij is the area
of the intersection between projections of Li and Lj , and is the
light transmittance through leaf cluster Lj (equal to 0.25 in the
simulations). If several clusters Lj shade Li , their influences are
multiplied. The total flux reaching cluster Li is calculated as the
sum of the fluxes received from each light source S .
;
The plant model. In addition to the communication module ?E ,
the plant model includes the following types of modules:
Apex A(vig; del). Parameter vig represents vigor, which determines the length of branch segments (internodes) and the diameter of leaf clusters produced by the apex. Parameter del is used
to introduce a delay, needed for propagating products of photosynthesis through the tree structure between consecutive stages
of development (years).
Leaf L(vig; p; age; del). Parameters denote the leaf radius vig ,
the amount of photosynthates produced in unit time according
to the leaf’s exposure to light p, the number of years for which
a leaf has appeared at a given location age, and the delay del,
which plays the same role as in the apices.
Internode
F (vig).
Consistent with the turtle interpretation, the
parameter vig indicates the internode length.
8192
Branch width symbol !(w; p; n), also used to carry the endogenous information flow. The parameters determine: the width of
the following internode w, the amount of photosynthates reaching the symbol’s location p, and the number of terminal branch
segments above this location n.
f
! ' f
16
20
24
The flux r also determines the fate of the apex, captured by productions p6 to p8 . If the amount of photosynthates r LP LM
transported from the nearby leaf exceeds a threshold value PB , the
apex produces two new branches (p6 ). The second parameter in
the first branch symbol ! is set to PB , to subtract the amount
of photosynthates used for branching from the amount that will be
transported further down. The length of branch segments vig is
reduced with respect to the mother segment by a predefined factor
V D, reflecting a gradual decrease in the vigor of apices with age.
The branch width modules ! following the first apex A are introduced to provide context required by productions p10 and p11 , as in
the axiom.
;
;
g
?E(r) < A(vig,del) : r*LP–LM>PB vig=vig*VD;
/( )[+( 2 )!(W,–PB,1)F(vig)L(vig,LP,0,0)A(vig,0)
[!(0,0,0)]!(W,0,1)]
–( 1 )!(W,0,1)F(vig)L(vig,LP,0,0)/A(vig,0)
7 : ?E(r) < A(vig,del) : r*LP–LM > PG vig=vig*VD;
/( )–( 0 )[!(0,0,0)]
!(W,–PG,1)F(vig)L(vig,LP,0,0)A(vig,0)
A(vig,0)
8 : ?E(r) < A(vig,del) : r*LP–LM <= PG
9 : ?E(r)
p
12
!
! ' 8
exceeding the amount LM needed to maintain a cluster, it remains
in the structure (p4 ). Otherwise it becomes a liability to the tree and
dies (p5 ). Another condition to production p5 prevents a leaf from
occupying the same location for more than LS years.
!
!
4
Figure 11: The number of terminal branch segments resulting from
unrestricted bifurcation of apices (continuous line), compared to the
number of segments generated in a simulation (isolated points)
!
p6:
8
year
A(vig,del) : del<Del
A(vig,del+1)
L(vig,p,age,del) : (age<LS)&&(del<Del–1)
L(vig,p,age,del+1)
L(vig,p,age,del) : (age<LS)&&(del==Del–1)
L(vig,p,age,del+1)?E(vig*0.5)
4 : L(vig,p,age,del) > ?E(r) : (age<LS) && (r*LP>=LM)
&& (del == Del)
L(vig,LP*r–LM,age+1,0)
5 : L(vig,p,age,del) > ?E(r) : ((age == LS)||(r*LP<=LM))
&& (del == Del)
L(0,0,LS,0)
p
32
0
!
p
512
128
0
'
#define
137.5
/* divergence angle */
#define 0 5
/* direction change - no branching */
#define 1 20
/* branching angle - main axis */
#define 2 32
/* branching angle - lateral axis */
#define W 0.02
/* initial branch width */
#define VD 0.95
/* apex vigor decrement */
#define Del 30
/* delay */
#define LS 5
/* how long a leaf stays */
#define LP 8
/* full photosynthate production */
#define LM 2
/* leaf maintenance */
#define PB 0.8
/* photosynthates needed for branching */
#define PG 0.4
/* photosynthates needed for growth */
#define BM 0.32
/* branch maintenance coefficient */
#define BE 1.5
/* branch maintenance exponent */
#define Nmin 25
/* threshold for shedding */
Consider: ?E[]!L
/* for context matching */
: !(W,1,1)F(2)L(1,LP,0,0)A(1,0)[!(0,0,0)]!(W,0,1)
!
2048
2
The corresponding L-system is given below.
p1:
p2:
p3:
number of terminals
32768
g
p
!
p
!"
p10: !(w0 ,p0 ,n0 ) > L(vig,pL ,age,del) [!(w1 ,p1 ,n1 )]!(w2 ,p2 ,n2 ) :
fw=(w1 ˆ2+w2 ˆ2)ˆ0.5; p=p1 +p2 +pL –BM*(w/W)ˆBE;g
(p>0) || (n1 +n2 >=Nmin ) ! !(w,p,n1 +n2 )
p11: !(w0 ,p0 ,n0 ) > L(vig,pL ,age,del) [!(w1 ,p1 ,n1 )]!(w2 ,p2 ,n2 )
! !(w0 ,0,0)L%
The simulation starts with a structure consisting of a branch segment
F , supporting a leaf L and an apex A (axiom !). The first branch
width symbol ! defines the segment width. Two additional symbols
! following the apex create “virtual branches," needed to provide
proper context for productions p10 and p11 . The tree grows in stages,
with the delay of Del + 1 derivation steps between consecutive
stages introduced by production p1 for the apices and p2 for the
leaves. Immediately before each new growth stage, communication
symbols are introduced to inform the environment about the location
and size of the leaf clusters (p3 ). If the flux r returned by the
environment results in the production of photosynthates r LP
;
If the amount of photosynthates r LP LM transported from the
leaf is insufficient to produce new branches, but above the threshold
PG, the apex adds a new segment F to the current branch axis
without creating a lateral branch (p7 ). Again, a virtual branch
containing the branch width symbol ! is being added to provide
context for productions p10 and p11 .
If the amount of photosynthates is below PG, the apex remains dormant (p8 ). Communication modules no longer needed are removed
from the structure (p9 ).
Production p10 captures the endogenous information flow from
leaves and terminal branch segments to the base of the tree. First, it
determines the radius w of the mother branch segment as a function
of the radii w1 and w2 of the supported branches:
p
w = w12 + w22 :
Thus, a cross section of the mother segment has an area equal to the
sum of cross sections of the supported segments, as postulated in
the literature [40, 46]. Next, production p10 calculates the flow p of
photosynthates into the mother segment. It is defined as the sum of
the flows pL , p1 and p2 received from the associated leaf L and from
both daughter branches, decreased by the amount BM (w=W )BE
representing the cost of maintaining the mother segment. Finally,
production p1 0 calculates the number of terminal branch segments
n supported by the mother segment as the sum of the numbers of
terminal segments supported by the daughter branches, n1 and n2 .
Production p10 takes effect if the flow p is positive (the branch is
not a liability to the tree), or if the number n of supported terminals
Figure 12: A tree model with branches competing for access to
light, shown without the leaves
Figure 14: A model of deciduous trees competing for light. The
trees are shown in the position of growth (top) and moved apart
(bottom) to reveal the adaptation of crown geometry to the presence
of the neighbor tree.
of apices in unfavorable local light conditions, and shedding of
branches which do not receive enough light to contribute to the
whole tree. Both phenomena limit the extent of branching, thus
controlling the density of the crown. This property of the model
is supported by the simulation results shown in Figure 11. If the
growth was unlimited (production 6 was always chosen over 7
and 8 ), the number of terminal branch segments would double
every year. Due to the competition for light, however, the number
of terminal segments observed in an actual simulation increases
more slowly. For related statistics using a different tree architecture
see [52].
p
Figure 13: A climbing plant growing on the tree from the previous
figure
N
is above a threshold min . If these conditions are not satisfied,
production 11 removes (sheds) the branch from the tree using the
cut symbol %.
p
Simulations. The competition for light between tree branches is
manifested by two phenomena: reduced branching or dormancy
p
p
A tree image synthesized using an extension of the presented model
is shown in Figure 12. The key additional feature is a gradual
reduction of the branching angle of a young branch whose sister
branch has been shed. As the result, the remaining branch assumes
the role of the leading shoot, following the general growth direction
of its supporting segment. Branch segments are represented as
texture-mapped generalized cylinders, smoothly connected at the
branching points (cf. [6]). The bark texture was created using a
paint program.
As an illustration of the flexibility of the modeling framework presented in this paper, Figure 13 shows the effect of seeding a hypothetical climbing plant near the same tree. The plant follows the
surface of the tree trunk and branches, and avoids excessively dense
Figure 16: Relationship between tree form and its position in a
stand.
Figure 15: A model of coniferous trees competing for light. The
trees are shown in the position of growth (top) and moved apart
(bottom).
colonization of any particular area. Thus, the model integrates several environmentally-controlled phenomena: the competition of tree
branches for light, the following of surfaces by a climbing plant, and
the prevention of crowding as discussed in Section 6. Leaves were
modeled using cubic patches (cf. [46]).
In the simulations shown in Figure 14 two trees described by the
same set of rules (younger specimens of the tree from Figure 12)
compete for light from the sky hemisphere. Moving the trees apart
after they have grown reveals the adaptation of their crowns to the
presence of the neighbor tree. This simulation illustrates both the
necessity and the possibility of incorporating the adaptive behavior
into tree models used for landscape design purposes.
The same phenomenon applies to coniferous trees, as illustrated
in Figure 15. The tree model is similar to the original model by
Takenaka [52] and can be viewed as consisting of approximately
horizontal tiers (as discussed in Section 5) produced in sequence
by the apex of the tree stem. The lower tiers are created first and
therefore potentially can spread more widely then the younger tiers
higher up (the phase effect [46]). This pattern of development is
affected by the presence of the neighboring tree: the competition
for light prevents the crowns from growing into each other.
The trees in Figure 15 retain branches that do not receive enough
light. In contrast, the trees in the stand presented in Figure 16 shed
branches that do not contribute photosynthates to the entire tree,
using the same mechanism as described for the deciduous trees.
The resulting simulation reveals essential differences between the
shape of the tree crown in the middle of a stand, at the edge, or
at the corner. In particular, the tree in the middle retains only
the upper part of its crown. In lumber industry, the loss of lower
branches is usually a desirable phenomenon, as it reduces knots
in the wood and the amount of cleaning that trees require before
transport. Simulations may assist in choosing an optimal distance
for planting trees, where self-pruning is maximized, yet there is
sufficient space between trees too allow for unimpeded growth of
trunks in height and diameter.
9
CONCLUSIONS
In this paper, we introduced a framework for the modeling and visualization of plants interacting with their environment. The essential
elements of this framework are:
a system design, in which the plant and the environment are
treated as two separate processes, communicating using a standard interface, and
the language of open L-systems, used to specify plant models
that can exchange information with the environment.
We demonstrated the operation of this framework by implementing
models that capture collisions between branches, the propagation of
clonal plants, the development of roots in soil, and the development
of tree crowns competing for light. We found that the proposed
framework makes it possible to easily create and modify models
spanning a wide range of plant structures and environmental processes. Simulations of the presented phenomena were fast enough
to allow interactive experimentation with the models (Table 1).
There are many research topics that may be addressed using the
simulation and visualization capabilities of the proposed framework.
They include, for instance:
Fundamental analysis of the role of different forms of information flow in plant morphogenesis (in particular, the relationship
Fig.
5
7
9
10
12
15
Number of
branch
leaf
segments clusters
138
140
786
229
4194
34b
37228
448b
22462
19195
13502
3448
Derivation
steps yrs
5
182
186
301
744
194
5
NA
NA
NA
24
15
Timea
sim.
render.
1s
50 s
67 s
15 min
22 min
4 min
1s
2s
3s
70 s
13 sc
8 sd
a Simulation and rendering using OpenGL on a 200MHz/64MB Indigo2 Extreme
b active apices
c without generalized cylinders and texture mapping
d branching structure without needles
Table 1: Numbers of primitives and simulation/rendering times for
generating and visualizing selected models
between endogenous and exogenous flow). This is a continuation
of the research pioneered by Bell [4] and Honda et al. [7, 33].
Development of a comprehensive plant model describing the
cycling of nutrients from the soil through the roots and branches
to the leaves, then back to the soil in the form of substances
released by fallen leaves.
Development of models of specific plants for research, crop and
forest management, and for landscape design purposes. The
models may include environmental phenomena not discussed in
this paper, such as the global distribution of radiative energy in
the tree crowns, which affects the amount of light reaching the
leaves and the local temperature of plant organs.
The presented framework itself is also open to further research. To
begin, the precise functional specification of the environment, implied by the design of the modeling framework, is suitable for a
formal analysis of algorithms that capture various environmental
processes. This analysis may highlight tradeoffs between time,
memory, and communication complexity, and lead to programs
matching the needs of the model to available system resources in an
optimal manner.
A deeper understanding of the spectrum of processes taking place in
the environment may lead to the design of a mini-language for environment specification. Analogous to the language of L-systems for
plant specification, this mini-language would simplify the modeling
of various environments, relieving the modeler from the burden of
low-level programming in a general-purpose language. Fleischer
and Barr’s work on the specification of environments supporting
collisions and reaction-diffusion processes [20] is an inspiring step
in this direction.
Complexity issues are not limited to the environment, but also arise
in plant models. They become particularly relevant as the scope of
modeling increases from individual plants to groups of plants and,
eventually, entire plant communities. This raises the problem of
selecting the proper level of abstraction for designing plant models,
including careful selection of physiological processes incorporated
into the model and the spatial resolution of the resulting structures.
The complexity of the modeling task can be also addressed at the
level of system design, by assigning various components of the
model (individual plants and aspects of the environment) to different
components of a distributed computing system. The communication
structure should then be redesigned to accommodate information
transfers between numerous processes within the system.
In summary, we believe that the proposed modeling methodology
and its extensions will prove useful in many applications of plant
modeling, from research in plant development and ecology to landscape design and realistic image synthesis.
Acknowledgements
We would like to thank Johannes Battjes, Campbell Davidson, Art
Diggle, Heinjo During, Michael Guzy, Naoyoshi Kanamaru, Bruno
Moulia, Zbigniew Prusinkiewicz, Bill Remphrey, David Reid, and
Peter Room for discussions and pointers to the literature relevant to this paper. We would also like to thank Bruno Andrieu,
Mark Hammel, Jim Hanan, Lynn Mercer, Chris Prusinkiewicz, Peter Room, and the anonymous referees for helpful comments on
the manuscript. Most images were rendered using the ray tracer
rayshade by Craig Kolb. This research was sponsored by grants
from the Natural Sciences and Engineering Research Council of
Canada.
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