J. Vos, L.F.M. Marcelis, P.H.B. de Visser, P.C. Struik and J.B. Evers (eds.), Functional-
Structural Plant Modelling in Crop Production, 1-12.
© 2007 Springer. Printed in the Netherlands.
FUNCTIONAL-STRUCTURAL PLANT MODELLING IN
Adding a dimension
J. VOS, L.F.M. MARCELIS AND J.B. EVERS
Plant Sciences Group, Wageningen University and Research Centre, Wageningen,
Abstract. The role acquired by modelling in plant sciences includes integration of knowledge,
exploration of the behaviour of the plant system beyond the range of conditions covered experimentally
and decision support. The purpose of the model determines its structure. Initially process-based models
(PBM) were developed separately from structural (or: architectural or morphological) plant models
(SPM). Combining PBMs and SPM into functional-structural plant models (FSPM) or virtual plants has
become feasible. This adds a dimension to classical crop growth modelling. FSPM are particularly suited
to analyse problems in which the spatial structure of the system is an essential factor contributing to the
explanation of the behaviour of the system of study. Examples include intra-specific and interspecific
competition phenomena, analyses of mechanisms of physiological response to environmental signals that
affect allocation of carbon and nitrogen in the plant, and exploration of alternative, manipulated plant
architectures on production of fruits or flowers. Good modelling practice involves different steps in
model development. These steps are discussed and include the conceptual modelling, data collection,
model implementation, model verification and evaluation, sensitivity analysis and scenario studies.
A SHORT HISTORY OF (PLANT) MODELS
Ever since mankind settled and started to manipulate nature to the benefit of people,
i.e. started agricultural activities, people have been seeking to improve the
‘performance’ of the systems they managed. People learned by trial and error how
best to manipulate the environment. Gradually concepts developed, representing a
mental abstraction of reality and its functioning that could be communicated to
following generations. It would probably be difficult for us to comprehend the
perceptions on the relations between cause and effect that were manifest in eras
before the development of chemistry, physics and botany. Especially since the
Enlightenment the human activity called science, in which hypotheses on the nature
and behaviour of the real world were tested, has expanded enormously. Results from
2 J.VOS ET AL.
experiments were used to guide management of crops and animals. Concepts on
how ‘reality works’ were expressed as texts, or at best with mathematical equations.
For instance, the ‘laws’ of von Liebig, Liebscher and Mitscherlich, published in
1855, 1895 and 1925, respectively (De Wit 1992), on yield response to input of one
or two fertilizers can be regarded as early quantitative models in crop production.
Since the advent of computers the effects of larger numbers of factors can be
analysed in numerical calculations. Systems analysis began to be applied to
agricultural sciences, with computer-based models gaining a place as a tool to aid
researchers in expressing their ideas on ‘how reality works’. In this context a system
is defined as a limited part of reality, characterized by components, interrelations
between the components and an explicit boundary separating the system from the
outer world. Ideally the boundaries are chosen such that the outer world does affect
the behaviour of the system, but conversely the system does not or negligibly affect
the outer world. Driving forces from outside govern the behaviour of the system.
The use of a computer to study how such a model behaves is called simulation. Such
a model is a simplified representation of a system serving particular purposes. The
purposes of modelling include: (i) integration of knowledge (exceeding the capacity
of the human brain), (ii) quantitative testing of hypotheses, (iii) extrapolation of
effects of factors beyond the range of conditions covered experimentally, (iv)
revelation of knowledge gaps and ‘guiding’ research, and (v) to support practical
management decisions (input of resources, climate control in greenhouses, planning
Computer modelling has become a tool of historically unprecedented power to
enhance understanding of how physical and biological reality works and to explore
the possible behaviour of systems when implementing alternative hypotheses in
In agricultural sciences much emphasis has been on modelling the growth of crops
in relation to environmental conditions. We’ll refer to this class of models as
‘process-based models’ (PBMs). In PBMs, the components a crop commonly
include leaves, stems, roots and reproductive or storage organs (state variables), of
which attributes are expressed as quantities (e.g. weight, surface area, N content) per
unit area of soil surface (e.g. Van Ittersum et al. 2003). From the total leaf area per
unit surface area, i.e. the leaf area index (LAI), light interception and photosynthesis
can be computed. Various levels of sophistication are possible in the calculation of
the light-driven rate of dry-matter accumulation or photosynthesis and respiration
(Van Ittersum et al. 2003). Plant development (phenology) is commonly expressed
as a function of thermal time (i.e the accumulation of ‘degree time units’ above a
minimum temperature, e.g. degree days). Partitioning of carbohydrates among
component organs, i.e. the coordination of growth of components, is often
programmed as a more or less fixed function of the phenological stage (Marcelis et
al. 1998; Van Ittersum et al. 2003; cf. Marcelis and Heuvelink this volume).
FUNCTIONAL-STRUCTURAL PLANT MODELLING 3
It is probably safe to state that PBMs have contributed enormously to our
understanding of differences in potential production levels as related to factors such
as latitude, planting time (defining the radiation regime), temperature, nutrients and
water and crop species. The models also have contributed to insight into the
significance of the distribution of photosynthesis-related parameters in the canopy to
the productivity of the plant stand (e.g. distributions of light-saturated rates of
photosynthesis, stomatal conductance and effects of mechanisms of stomatal
control). Such models are also suitable to explore effects of genetic variation in
‘global’ plant properties; ‘global’ meaning properties that pertain to the plant as a
whole and not to specific, single organs. For instance, the time of flowering of
cereals terminates the accumulation of vegetative mass and leaf area, both of which
determine the potential grain yield. Too early flowering reduces potential yield and
late flowering may also result in low yield when the suitable growing season has
ended before the crop matures (e.g. Yin et al. 1997).
Classical PBMs do not address feedbacks between processes at the level of an
individual organ (the ‘local level’) and the functioning of the plant or plant stand as
a whole (the ‘global level’). Since the 1980s (Field 1983; Hirose and Werger 1987),
awareness has grown in general plant ecology and crop ecology that the spatial
distribution of nitrogen (N) in the canopy and the associated photosynthetic capacity
is significant for the rate of photosynthesis of the stand. Optimization of allocation
of N in the plant became a subject of study (e.g. Anten 2005). Also in the domain of
physiology and modelling of trees it became apparent that the three-dimensional
(3D) structure of a tree strongly affects the processes involved, such as the
distribution of carbohydrates between autotrophic and heterotrophic tissues, the
interception of light, and the gas-exchange properties of the foliage (Sievänen et al.
2000 and references therein).
ARCHITECTURAL MODELS AND FUNCTIONAL-STRUCTURAL MODELS
In the literature the phrases architectural model, structural model, morphological
model and geometric model were used more or less interchangeably (e.g. Sievänen
et al. 2000); they refer to representations of the shape and orientation in space of the
components comprising a plant. An important line of work started when
Lindenmayer (1968a; 1968b) introduced a formalism for simulation of the
development of multicellular organisms, later named L-systems. That approach was
developed further and applied to plants, resulting in the publication of the book ‘The
Algorithmic Beauty of Plants’ (Prusinkiewicz and Lindenmayer 1990). Computer
graphics provided life-like visualizations of plants. However, in these models, plant
development is contained exclusively in the formalisms describing the shape,
position and properties of the (n+1)st element as functions of those of the nth
element. In other words: plants were treated as closed cybernetic systems;
development was regarded as autonomous and there was no interaction between
plant development and environment. Still, in these models provisions were already
included to simulate the transport of signals through the structure, mimicking
4 J.VOS ET AL.
physiological control (e.g., the Mycelis example in Prusinkiewicz and Lindenmayer
Functional-structural plant models, FSPMs (Sievänen et al. 2000; Godin and
Sinoquet 2005); or virtual plant models (Room et al. 1996; Hanan 1997) are the
terms used to refer to models explicitly describing the development over time of the
3D architecture or structure of plants as governed by physiological processes which,
in turn, are driven by environmental factors.
Of course, there is no a priori limit to the physiological functionality that can or
should be included in FSPM. Photosynthetic carbon gain is among the prime
processes of interest in agricultural and horticultural applications of FSPM. Methods
are available to simulate the light distribution over any 3D object (e.g. Chelle this
volume). Given the distribution of photosynthetic properties in the canopy,
calculation of an instantaneous photosynthetic rate is rather straightforward (Müller
et al. this volume). It is more difficult to model how these photosynthetic properties
change with the development of the 3D structure and with change in environmental
conditions. In plant modelling, the partitioning of available carbohydrates among
competing centres of growth (sinks) has for long been an issue (Marcelis and
Heuvelink this volume). FSPM offers new ways to advance that issue (e.g.
contributions by Minchin, Allen et al., Drouet and Pagès, Kang and de Reffye,
Renton et al. to this volume). In principle FSPMs offer the possibility to model the
flow of material within the 3D structure and between the 3D structure and the
FSPMs are particularly suited to analyse problems in which the spatial structure
of the system is an essential factor contributing to the explanation of the behaviour
of the system of study. Examples include:
(i) Competition phenomena within species. Plants have divergent options to adapt
their architecture to the available space. Options include branching (dicots) or
tillering (Gramineae), and changes in leaf width, leaf length, leaf mass per unit
area, leaf angle, leaf longevity, and leaf:stem weight ratio. Some of these options
can be used simultaneously, whereas others come into play sequentially. In
particular, the spatial distribution of plant material in the canopy space is
decisive whether outgrowth or dormancy of tiller or branch buds occurs. For
instance, the number of tillers per plant declines for higher plant population
density in wheat (Evers et al. 2006). In turn, the growth or dormancy of a
particular bud of a branch or tiller affects the distribution of plant material in the
canopy space. FSPMs offer capacity to analyse such feedback between processes
at the local (i.e. organ) and the global level (i.e. the whole plant or canopy).
(ii) Competition phenomena between species. Plants of different species possess
different morphological options for occupying available space. Just as explained
under (i), FSPM can help to understand the competitive advantage of such
adaptive options in mixed plant communities (e.g. inter-crops, multiple crops,
crop–weed associations, grassland and natural vegetations) (Karley and Marshall
(iii)Exploration of alternative physiological hypotheses, explaining properties of the
structure. For instance, light has been postulated to govern the allocation of N in
plants (Drouet and Bonhomme 1999). Alternative hypotheses involve the
FUNCTIONAL-STRUCTURAL PLANT MODELLING 5
distribution of transpiration of water over the 3D structure and the associated
distribution of cytokinins (Pons et al. 2001). When coupled with microclimate
modules, FSPM can help to quantitatively explore, compare and expand such
hypotheses and derived postulates. The role of local assimilate production or
red:far-red ratio in the determination of the fate of tiller buds of wheat is another
example in this category (Evers et al. 2005; 2006).
(iv)Analyses of alternative canopy structures in production systems. In perennial
fruit trees and vines, and in ornamental crops (e.g. glasshouse roses) pruning is
applied to optimize the production of fruits or flowers over a number of years.
Such strategies are commonly empirically developed and FSPM offers
opportunities to strengthen their theoretical basis.
Models in applications (i) - (iv) are most useful if they are dynamic in nature, i.e.
simulate the changes in the system over time. However, there are also applications in
which the dynamic or functional aspect is less important than the adequate structural
representation of the system. Canopy structure helps to improve the interpretation of
remote-sensing data (Lewis this volume). Physiological functionality is not required in
such models. Also in analyses of interactions between insects, micro-organisms, plants
and environment, plant structure is important (e.g. Skirvin this volume). When only
snapshots of the structure at some points in time are needed for a particular study, the
digitization of real systems and their reconstruction in the computer is the most
efficient way to proceed (e.g. Sonohat et al. 2002; Drouet 2003; Kahlen this volume).
PLATFORMS AND MODELLING TOOLS
Basically there are two ways to arrive at FSPMs (Sievänen et al. 2000): architectural
models could be expanded to accommodate ‘function’ and allow influence of
environmental factors, or PBMs should be expanded to accommodate the third
dimension. A step towards realization of the first option was the extension of the L-
system alphabet with communication symbols, which can exchange parameter values
with other models (M?ch and Prusinkiewicz 1996). Later improvements, i.e. the L+C
modelling language (Karwowski and Prusinkiewicz 2003), facilitated the development
of FSPM such as, for instance, L-Peach (Allen et al. this volume). L-systems and
associated programming environment (as presented in this volume by Prusinkiewicz et
al.) do not constitute the exclusive toolkit to make FSPMs. Alternatives described in
this volume include models programmed in C++ (Drouet and Pagès this volume),
models using the modelling language XL on the GroIMP platform (Kniemeyer et al.
this volume), programming in Matlab (Wernecke et al. this volume) and the Greenlab
methodology, which is implemented in several programming languages (Kang et al.
this volume). Hanan and Hearn (2003) showed an application linking L-Cotton, an
architectural model of cotton using L-systems, with a process-based crop model called
OZCOT. The latter is well calibrated, simulating growth and yield to several
environmental factors (e.g. water, nitrogen, temperature). It can be efficient and in the
interest of managing the models to keep such a growth model intact and have a
separate structural model that calculates the development of the architecture, based on
the daily growth rate provided by the crop growth model.
6 J.VOS ET AL.
Based on the work of Hallé and Oldeman (e.g. Hallé et al. 1978) the term
‘architectural model’ has acquired a special meaning in the botanical literature.
These authors drew attention to the various typifying patterns of branching and
flowering of plants, called architectural models, named after botanists. Examples of
such models include: Corner’s model, Leeuwenberg’s model and Rauh’s model. In
the context of FSPM, the terms architecture and architectural model have a slightly
different meaning. Godin (2000) defined plant architecture “… as any individual
description1 based on decomposition of the plant into components, specifying their
biological type and/or their shape, and/or their location/orientation in space and/or
the way these components are physically related one with another”. Important
implications of the definition are that a representation of plant architecture needs to
provide information in at least three areas:
plant composition, providing a description of the different types of components
the plant consists of;
geometrical properties, describing the shapes and relative spatial positions of
each of the components; and
plant topology, specifying which components are connected to each other,
implicitly containing information on the hierarchy among components of a
STEPS IN MODEL DEVELOPMENT
To be effective and communicable through publications, the modelling process has
to proceed in an orderly fashion. Descriptions of Good Modelling Practice (GMP)
have been articulated in water management (e.g. Scholten et al. 2001; Refsgaard and
Henriksen 2004). Scholten (pers. comm. 2006) provided the definition: “GMP are
practices to use models, shared and accepted by a substantial part of the professional
modelling community and consisting of explicit guidelines for quality assurance of
the modelling process”. It is wise to apply GMP in PBM (Van Oijen 2002) and
FSPM. Without pinning GMP down to strict rules the sections below deal with
features of GMP. In practice the development of a model often proceeds in a cyclic
series of activities, including development of concepts (mental work), modelling
(computer work) and experimentation. The steps in model development outlined
below are more or less in a logical and chronological order, but there are numerous
reasons to deviate from the sequence that is presented.
The conceptual model
Any modelling exercise starts off with the specification of the model’s purpose. In
research environments, modelling commonly serves purposes such as integrating
knowledge or the quantitative testing of hypotheses. Next, the system of interest
needs to be described. In horticultural and agricultural sciences the system of
interest is commonly a plant and very often a collection of interacting plants, i.e. a
FUNCTIONAL-STRUCTURAL PLANT MODELLING 7
row of plants or a homogeneous crop canopy. At this stage important decisions have
to be made on which aspects of function and structure the model needs to explain.
For those aspects the modeller should provide an explanation of the desired
functions as emergent from the behaviour of the relevant components. In other
words, these aspects need to be included in a ‘mechanistic way’. To keep the
complexity of the model within limits, it is wise to model those aspects of structure
and function in a descriptive manner that are of secondary importance in the context
of the purpose of modelling exercise. For instance, for the study of insect behaviour
in the 3D plant canopy space (Skirvin this volume) it is an adequate choice to model
the plants in a descriptive fashion.
In FSPM the conceptual model includes:
Recognition of the important components of which a plant consists.
Monocotyledonous and dicotyledonous plants, for instance, differ widely in
architecture. When emerging from seeds, cereals and grasses show a seminal
root system, composed of a defined number of main axes, and a crown root
system with four possible positions of roots on each main stem node (Klepper et
al. 1984). The main shoot develops a particular number of main stem leaves
defined by environmental cues inducing the terminal meristem to switch to the
initiation of floral structures. Buds in axils can grow out to form ‘side shoots’ i.e.
tillers, which in turn also produce a terminal inflorescence. Main stem and tillers
A potato plant, growing from a seed tuber, produces a variable number of main
shoots. Below ground, root axes emerge from the nodes. At some stage
belowground nodes also give rise to stolons, i.e. ‘lateral stems’ that stop
elongating in response to internal signals and start to swell to form a tuber. Buds
in the axils of the lowest main shoot leaves can grow out to produce a variable
number of basal lateral branches (Vos and Biemond 1992). Apical meristematic
activity is terminated with the formation of an inflorescence that forms flowers
and berries. Shoot growth can continue from buds in axils in the 2nd and 3rd leaf
below the inflorescence to give rise to apical laterals; several orders of apical
lateral branches can be produced sequentially till shoot production ceases.
These descriptions of cereals and potato include information on their
composition and topology and qualitative information on their changes over
time. For each plant species of interest such concepts are on the basis of
architectural modelling. This volume present examples of Gramineae (Fournier
et al.), chrysanthemum (de Visser et al.), faba bean (Ruiz-Ramos and Mínguez),
peach (Allen et al.) and cucumber (Kahlen).
A choice of the basal unit of plant modelling. Apical meristems produce cells,
cells differentiate into tissues and organs, organs form modules, e.g. phytomers
(or: metamers), phytomers form components such as a tiller or a branch,
components make plants and plants form a canopy. The phytomer or metamer
has been advanced as a convenient unit to describe vegetative composition. The
phytomer (Figure 1) consists of an internode with a bud at its bottom, and a node
at the top to which a leaf is attached. The leaf is composed of a sheath
(monocotyledonous plants) or petiole (broadleaf species) and a leaf blade. The
8 J.VOS ET AL.
botanical idea that a plant is a collection of basically similar units matches with
object-oriented modelling approaches. A conceptual FSP model should include
‘ideas’ on the timing of initiation of phytomers, the appearance and expansion of
its components (e.g. phyllochron, duration of leaf expansion), and the
coordination of initiation and expansion among components.
Figure 1. An example of a typical Gramineae phytomer, and its components internode, node,
tiller bud, leaf sheath and leaf blade
The physiological ‘functions’ to be included in the model, for instance:
photosynthesis, respiration, carbon allocation and sink–source interactions,
transport of water, nutrients and signals in the plant structure (cf. Section
The assumed relationship between environmental variables (particularly
temperature) and rate variables determining plant development (progression in
phenological stages) or growth processes.
The time step of relevance to the purpose of the model.
The construction of a diagram, e.g. Forrester diagram (Leffelaar 1999), showing
all the important components of the modelled system, their interrelations, the
flows of material and the flows of information, the external driving forces and
the processes they affect, using standardized symbols (Figure 2).
Decisions on the modelling platform and software to program the model. Also it
needs to be tested whether intended communications between different models
are technically feasible. It is subject to debate whether actual programming
activities still belong to the conceptual phase.
The deliverables of the conceptual modelling phase include at least a list of
parameters that are needed to construct a functional-structural model. Also it needs
to be specified how parameter estimates can be obtained, either from literature,
FUNCTIONAL-STRUCTURAL PLANT MODELLING 9
existing data or from dedicated experiments. Protocols need to be made specifying
how unknown parameters will be measured in experiments.
flow of material or
energy and its direction
state variable, or integral
of a rate (flow)
sink and source quantities of
auxiliary or intermediate
variable in the flow of
information or material
constant or parameter
flow of information and
Figure 2. Symbols – as defined by Forrester (Forrester 1961; Leffelaar 1999) – used in the
construction of relational diagrams
Experimentation, collection and analysis of data
A thorough account of experimentation, collection and analysis of data is provided
in the contribution of van der Heijden et al. to this volume. Therefore the issue is not
treated here. Very specific to FSPM is the collection of geometric attributes of
components and plant topology (see section "Plant architecture").
Parameterization, model verification, calibration and sensitivity analysis
Once data requirements are satisfied, the modeller can proceed to finish
programming the model. Once technically running, the next step is to verify whether
the model assembly really represents the functioning of the system. That process is
called model or model code verification, i.e. the question at stake is “is the model
built right?” Verification is achieved by analysing the structure and results of the
model for their consistency. Mass balances and dimensions, for instance, need to be
In the technical sciences, the term ‘model calibration’ is used to refer to the
procedure of adjustment of parameter values of a model to reproduce the response of
reality within the desired range of accuracy. When using models as a research tool,
calibration should be applied judiciously, for instance to improve the performance of
the functions of the model that are necessary but of secondary relevance in the
context of the objectives of the study.
The objective of sensitivity analysis is to explore whether the model results are
critically dependent on the values of particular parameters. This is commonly done
by analysing the relations between variation in input parameters and output
10 J.VOS ET AL.
variables. Plots can be made of relative change in output versus relative change in
input (e.g. stepwise increment from -20 to 20% of the initial value). The slope
represents the elasticity; the smaller its value, preferably <1, the less sensitive the
model reacts to the value of the input parameter (Figure 3). When varying values of
input parameters, only those within the biologically relevant range should be used,
nor should associations be violated between variables that are dictated by the
biological reality. The deliverables of this stage include a technically sound model
with insight into sensitivity to variation of input-parameter values. Results of the
sensitivity analysis may inspire new experimentation or adjustment of the model
relative change in parameter value (%)
relative change in output value (%)
Figure 3. A typical figure associated with a sensitivity analysis; the x-axis represents the
relative change in the input parameter, and the y-axis represents the relative change in
output. The slope of the regression line is the elasticity
Model validation, scenario studies and uncertainty analysis
Model validation seeks to answer the question “was the right model built?” The
answer to that question is commonly obtained by comparing model results with data
from the real system. These data should not have been used during model
development. Such tests of the performance of the model gain in value if
independent data include data from conditions that differ from the ones for which
the model was derived, for example from a different agro-ecological zone.
Validation of a model under a wide range of conditions using independent data sets
is perhaps not practised as widely as desirable, while it is of utmost importance if we
want to have reliable models.
FUNCTIONAL-STRUCTURAL PLANT MODELLING 11
If the model passed the testing phase successfully one can proceed to conduct
scenario studies, basically to answer ‘what if?’ questions. For instance: what if
conditions change? what if plant properties change? Such studies yield insight into
the quantitative significance of various processes and parameters and can help to
explore divergent management options. This process of ‘deriving insight into reality
through analyses of model output for different conditions’ is also simply called
‘simulation’ (Refsgaard and Henriksen 2004).
Functional-structural plant models (FSPM) seek to integrate plant structure with
plant functioning, i.e. the flow of material and energy through the system as
dependent on the genotype and as driven by the environment. FSPMs are
particularly suited to analyse problems in which the spatial structure of the system is
an essential factor contributing to the explanation of the behaviour of the system of
study. In that sense, FSPM adds a dimension to conventional crop growth models.
1 Alternative: ‘description of individuals’, stressing that the individual plant is the unit of measurement
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