simulation modeling of plants and plant...

Simulation Modeling of Plants and PlantEcosystemsPrzemyslaw Prusinkiewicz

Reference

Przemyslaw Prusinkiewicz: Simulation Modeling of Plants and Plant Ecosystems. Communications of theACM vol. 43 no. 7, pp. 84−93.

84 July 2000/Vol. 43, No. 7 COMMUNICATIONS OF THE ACM

Przemyslaw Prusinkiewicz

The modeling of plants and plantecosystems has the captivating appeal of reproducingthe visual beauty of nature while providing insightsinto the way nature works. This interplay betweenart and science is rooted in history and can be tracedat least as far back as Leonardo da Vinci, whosenotes about plant architecture 500 years ago remainvalid today. The interdisciplinary character of plantmodeling research is echoed by the diversity of exist-ing and prospective applications of the models.

The complex architecture of plants consisting ofmany individual units, including branches, leaves,and flowers, is difficult to reproduce using traditionalmodeling techniques, which are better suited for arti-ficial rather than natural objects; the difficultyincreases when trying to model entireplant ecosystems. Simulation of plantdevelopment, based on botanical knowl-edge, offers a solution.

Simulation of natural phenomena isone of the most powerful methods forcreating realistic models, animations,and rendered images in computer

graphics. These simulations are often based inphysics, possibly involving the mechanics of objectssubject to forces in physically based modeling andanimations or the optics of light propagation anddistribution in radiative energy in rendering. Overthe last 15 years, steady progress has also been madein the simulation of the biological processes govern-ing plant development. This progress has led to bio-logically justified and visually realistic models ofplants and plant ecosystems.

Applications are divided into two broad categories:image synthesis and scientific visualization. In image-synthesis applications, the visual presentation of themodels is the main objective; consequently, the mod-eling methods for image synthesis emphasize the ease

of specifying and interactively manipulat-ing the models [6, 12]. In contrast, modelsdeveloped for scientific and scientific-visu-alization purposes emphasize the biologi-cal accuracy of the simulated processes andstructures [1, 8]. These models integrateour knowledge of plants and make it pos-sible to examine hypotheses when theC

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COMMUNICATIONS OF THE ACM July 2000/Vol. 43, No. 7 85

Simulation

Plants and PlantEcosystems

Modeling of

exact processes of plant growth and evolutionare not known by botanists. Here, I describethe state of the biologically motivated model-ing methodology, focusing on a dominanttechnique, called L-systems, a mathematicaltheory of plant development directly applica-ble to computer simulations.

Modular Architecture of PlantsThe diversity of plant forms in nature, ampli-fied by plants’ adaptation to their environ-ments, might leave the impression that it isimpossible to capture this variety in a unifiedand universally applicable way. Fortunately,botanists and ecologists have developed gen-eral concepts allowing us to consider not onlyindividual plants but entire plant ecosystemsfrom a common perspective. It is thereforepossible for computer scientists to developmultipurpose plant-modeling software that can beused to simulate different plant species and operate atdifferent levels of abstraction. The resulting modelsinclude “descriptive” models, which are limited to ageometric characterization of plant structure anddevelopment, and “functional-structural” models, orvirtual plants, which also capture the physiologicalprocesses governing plant growth [8].

The interwoven postulates of “modular plantarchitecture” and “a plant as a population of modules”are the key unifying concepts in all this software.Accordingly, a plant can be viewed as a modular struc-ture consisting of relatively independent, repetitiveunits, such as leaves, flowers, buds, and branch seg-ments. This view makes it possible to perform a num-ber of fundamental virtual modeling functions:

• Describe the development of a plant using asmall set of rules characterizing entire classes ofmodules, rather than individual modules;

• Characterize the physiological mechanisms con-trolling plant development in terms of informa-tion flow among the individual modules; and

• Apply similar modeling methodologies to indi-vidual plants (populations of modules) and entireplant ecosystems (populations of plants).

The concept of describing the development of apotentially unbound, or arbitrarily large, plantstructure using a finite set of rules applying to allobjects of the same type was introduced in 1968 bythe biologist Aristid Lindenmayer [5] and is for-mally expressed in the definition of L-systems, asLindenmayer defined it. An L-system model [10] isspecified by listing the following three components:

• The set of module types, or “alphabet,” of the L-system;

• The set of rewriting rules, or “productions,” thatcapture the behavior of individual modules overpredetermined time intervals; and

• The initial structure, or “axiom.”

The terminology and structure of the L-system def-inition are evocative of formal language theory.Indeed, L-systems can be viewed as a type of formalgrammar, distinguished from the better knownChomsky grammars, which are widely used to for-mally characterize both natural and programminglanguages, by their parallel application of produc-tions. This parallelism reflects the simultaneousprogress of time in all parts of the modeled plant.

Figure 1 illustrates the concept of L-systemsthrough a simple model of a compound carrot-likeleaf. This L-system operates on two classes of mod-ules: terminal branch segments, called “apices,” andthe remaining segments, called “internodes.” Accord-ing to the first production of this L-system, an apexyields a branching structure consisting of an apexcontinuing the main axis, two lateral apices, and twointernodes. The second production states that, overthe same time interval, an internode doubles itslength. The figure shows a developmental sequencegenerated from a single apex. In each step, all apicesand internodes are transformed in parallel by theirrespective productions. The result is the intricatebranching structure of a compound leaf.

This developing leaf has a fractal structure; that is,individual branches of different orders are similar toeach other and to the whole leaf. This is an example


Figure 1. L-system productions and the resulting developmental sequence of a compound leaf

model. The apices are shown as thin lines, the internodes as thick lines [9].

of the often-observed self-similarity of plants andresults from the repetitive production of branches bythe apices, followed by the gradual elongation of theinternodes. The relationship between L-systems andfractals extends beyond plant models; L-systems canalso be used to generate classic space-filling curves andother linear fractals discovered and popularized byBenoit Mandelbrot [10]. On the other hand, L-sys-tems also capture structures and developmentalprocesses in plants that do not have a fractal character.

Information Flow in a Developing PlantThe relative independence of plant modules raisesthe question of how these modules communicateand interact with one another. It is convenient todistinguish three forms ofcommunication [9] amongplant modules:

Lineage. The transfer ofinformation from a par-ent module to its chil-dren. For instance, bothproductions in the com-pound leaf model in Fig-ure 1 represent this formof information transfer,since the type and parameters of the prede-cessor modules definethe successor structures completely.

Endogenous interaction.The interaction resulting from information trans-fer between adjacent elements of the branchingplant structure. The information could representwater, minerals, hormones, or the products ofphotosynthesis transported by plant tissues.

Exogenous interaction. The interaction resultingfrom information transfer through the physicalspace in which plants grow. Examples are thecompetition for space and light by branches of atree and the competition for water by roots.

This classification of communication forces amongplant modules is convenient in modeling practice,because different programming constructs areneeded to express different communication forms.They can be most clearly differentiated using sym-bolic representation of productions, called “brack-eted string notation” [5, 10]. The axes of abranching structure are represented by sequences ofletters indicating the types of the constituent mod-ules. Branches are enclosed in square brackets.

Bracketed string notation plays an important role inpractice, because it provides the foundation in pro-gramming languages for specifying plant models.

Lineage, conceptually the simplest form of infor-mation transfer, is expressed using context-free pro-ductions, in which the predecessor is a single module.For example, the productions governing developmentof the compound leaf in Figure 1 have the form:

A � I(1)[�A][�A]I(1)AI(x) � I(2x)

where A denotes the apex, I(x) denotes an internodeof length x, and the auxiliary symbols � and � indi-cate branching directions. In contrast, endogenous

interactions require the useof context-sensitive produc-tions. In the case of suchproductions, the successorof a module being replacedby a production dependsnot only on this moduleitself but also on its neigh-bors. For example, produc-tion S � I � S, where �separates the left context Sfrom the strict predecessorI, will replace an I with an Sonly if there already is an Sto the left of the I. Succes-sive applications of this pro-duction to the initial stringSIIIII will produce the

sequence of strings SSIIII, SSSIII, SSSSII, SSSSSI,SSSSSS. Thus, if S denotes an internode that hasbeen reached by a signal (such as one representing ahormone) and I is an internode that has not yet beenreached, the sequence can be interpreted as the prop-agation of a signal from the left to the right end of asequence of internodes. Similarly, the right-context-sensitive production I �S � S applied to the stringIIIIIS captures the flow of information from right toleft.

Using parameters, a model can capture not onlyqualitative aspects of information flow—the absenceor presence of a signal—but also quantitative mea-sures, such as concentrations of water, minerals, andthe products of photosynthesis transported by theplant [7]. Moreover, productions with a two-sidedcontext make it possible to express communicationmechanisms lacking a predefined direction, such asthe diffusion of substances from modules with higherconcentrations of a particular substance to moduleswith lower concentrations [10].


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The distribution of

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for light.

Plant models governed by lineage and endogenousinteraction can be viewed as closed cybernetic sys-tems, because control information comes from theplant models themselves. In contrast, models involv-ing exogenous interactions are “open” in the cyber-netic sense. The information exchange between aplant and its environment can be conceptualized as afeedback loop in which the environment affects theplant and the plant reciprocally affects the environ-ment. For example, a branch’s development maydepend on its local light environment, which in turndepends on the shadows cast by other branches. Sim-ilarly, the development of roots may depend on thedistribution of water in the soil, which is in turnaffected by the absorption of the water by the roots.

To communicate with the environment, plantmodels need input and output constructs. Within theframework of L-systems, these constructs are called“communication modules” [7]. Syntactically, a com-munication module is treated the same way as anyother symbol with associated parameters that mightappear in an L-system production. However, parame-ter values of a communications module are notmanipulated by L-system productions; instead, they

are sent to the environment (along with informationindicating the position of the communication mod-ule in space), processed by the environment, andeventually set to new values representing the environ-ment’s response.

Plant Development As Competition for SpaceThe three classes of control mechanisms—lineage,endogenous interaction, and exogenous interac-tion—make it possible to simulate an astoundingvariety of physiological processes and their resultingplant forms. They also reveal analogies betweenprocesses and structures taking place at differentspatial scales and levels of plant organization. Toillustrate this point, I focus on competition forspace—one of the most critical biological aspects ofplant development.

At a relatively small scale, competition for space ismanifested by the arrangement of individual organs,such as leaves and flowers. This arrangement oftentakes the form of spiral patterns in which initials ofthe individual organs, or “primordia,” are tightlypacked on their supporting surface, or the “recepta-


Figure 2. Models of green coneflower (left) and cacti (right) obtained using the collision-based algorithm [3].

cle.” These patterns are commonly referred to as“spiral phyllotaxis,” which means literally thearrangement of leaves on the stem and in relation toeach other; it is also commonly used to denote thearrangement of organs other than leaves as well. It isoften observed in nature that the angle between con-secutively issued primordia seen from the center ofthe receptacle approximates the golden angle ��137.5� [10], which results from the partition of the

full angle of 360� by the “golden mean” number, oneof the most important and extensively studied num-bers in mathematics. For a long time, mathemati-cians and botanists have been fascinated by thefrequent appearance of the golden angle and the reg-ularity of the emerging spiral patterns in manyplants and plant organs, including pine cones andsunflower heads.

Fully causal models have not yet led to realistic


Figure 3. Models of clover plants, with branches and leaves competing for space mediated by light. The plant grown in full light (top) has a different shape from

the one grown in shade (bottom) [4].

visualizations, but phyllotactic arrange-ments of plant organs have been repro-duced by models including descriptiveterms. For example, in the so-called colli-sion-based model of spiral phyllotaxis [3],the golden angle is assumed as given, butthe model uses an exogenous controlmechanism—collisions between consecu-tively issued primordia—to space primor-dia with respect to each other in the radialdirection. Figure 2 shows a flower head andan arrangement of cacti modeled using thisprinciple.

In the case of phyllotaxis, competitionfor space leads to the positioning of theindividual plant organs. In other cases,competition may determine whether ornot organs, such as branches, will beformed or maintained by the plant. Innature, the proximity of other organs andcompetition for space are often mediatedby light. For example, Figure 3 shows amodel of clover in two contrasting lightconditions. From biological studies, it isknown that clover (as well as other plants)is sensitive not only to the intensity oflight, which directly affects photosynthesis,but also by the ratio of the red and far-redcomponents of the light spectrum [4]. Thisratio is decreased in the light reflected byleaves and contributes to the perception ofshade by the plant’s light-sensitive organs.

In the model in Figure 3 (described indetail in [4]), light determines the length ofinternodes, the size and elevation of leaves,and the delay between the emergence oflateral buds and their further developmentinto branches. The relationships betweenthe local light environment and plantgrowth have been modeled according toexperimental data. Light distribution wassimulated using the Monte-Carlo methodand stochastic sampling techniques. Themodel reveals the effect of self-shading andred/far-red ratio decrease on plant architec-ture; in the absence of these phenomena,patches of clover would become unrealisti-cally dense. Moreover, external light condi-tions affect the shapes of the patches,which are approximately circular in high-intensity light and become elongated inlow-intensity light. These changes corre-spond to observations of clover in nature.


Figure 4. Functional-structural models of a horse chestnuttree (a) and a stand of coniferous trees (b) simulating the

effects of photosynthesis. Light reaching leaves causes production of photosynthates, which are used for

maintaining leaves and branch segments and producing new branches. The surplus is transported toward the base

of the tree. The tree sheds branches using more photosynthates than they produce [7].


Figure 5. Two stages of a simulation of self-thinning. Initially, the field consists of a large number ofsmall plants (top); this number decreases as the plants grow and compete for space (bottom) [2].

In these models, the sites at which plant organsperceive environmental information—collision withother organs in phyllotaxis and local light intensity inclover organs—were assumed to be the same as thesites of the plant’s responses. But in nature, this co-location of the sites of perception and response oftenis not the case. For example, the model of a decidu-ous tree in Figure 4a combines open L-system con-structs for capturing interactions with theenvironment and context-sensitive productions forexpressing transportationand allocation of resourceswithin the plant [7]. (Thephysiological basis for thismodel was proposed by A.Takenaka [11].) Leaves (notshown so as to not blockthe tree’s architecture) pro-duce carbohydrates accord-ing to the amount of lightthey receive. These carbo-hydrates are used for main-taining the leavesthemselves, for thebranches supporting them,and possibly for the pro-duction of new branches.The surplus is transportedby the branch toward thebase of the tree. If theamount of resources needed to maintain thebranch exceeds the amount it produces, and this statepersists for some time, the tree sheds the branch.Thus, the distribution of branches in the tree crownis a result of branch competition for light.

Figure 4b shows a stand of coniferous trees, mod-eled according to the same principle. Six trees wereremoved from the stand to reveal the relationshipbetween the shape of the crown and the position ofthe remaining trees. The trees at the corner and onthe edge have asymmetric crowns; they retain theirbranches all the way down to the tree base whenexposed to light. The tree in the center of the standretains only its upper branches; lower branches wereshed due to insufficient light. This result conforms tobiological observations and could be used to estimatethe optimal distance for planting real trees.

The group of trees in Figure 4b was modeled usingthe same technique as if a single tree were being mod-eled. Extensions of this approach to ecosystems withhundreds or thousands of plants are conceptually pos-sible but present computationally challenging tasks.However, the time and space complexity of simula-tions can be reduced if they perform at two levels—

whole ecosystem and individual plant [2]. The distri-bution of plants and their general characteristics, suchas size and age, are determined first. Detailed plantmodels, conforming to these characteristics, are thensubstituted for the coarse models. Figure 5 shows anexample of an application of this technique. Themodel captures the essence of “self-thinning,” a funda-mental ecological process in which the number ofplants in a field decreases gradually as the individualplants grow in size and therefore compete for space.

The concept that a plant canbe viewed as a population ofits components bridges themodeling of individualplants with the modeling ofentire ecosystems; therefore,processes and structuresoccurring at a wide variety oflevels can be captured in aunified manner.

Conclusions and Further ResearchSimulation-based plantmodeling makes it possibleto capture the mechanismsof plant development in abiologically sound and visu-ally realistic manner. Theinterdisciplinary character

of plant modeling research is reflected in thediversity of existing and prospective applications ofthe models. Some models seek to increase our knowl-edge of plant development and apply this knowledgeto plant management in horticulture, agriculture,and forestry. Virtual plants are being introduced tostudy the effects of pruning, pinching, and spacingon plant development, analyze the effects of grazingon pasture plants, and investigate the effects of pesti-cides on insects foraging on plants [8]. Simulationmodels are also being used to analyze optimal plantdesign from an evolutionary perspective.

Other models focus on image-synthesis applica-tions. Such models have been included in computeranimations and games, manipulated in virtual real-ity installations, incorporated into animations andmultimedia presentations for educational purposes,used to visualize the development of extinct plants,incorporated into computer-assisted landscape andgarden designs, and applied to the visual-impactanalysis of forest harvesting. Realistic plant modelshave also been proposed for use in plant registries(holding descriptions of different plant varietiesdeveloped by breeders) and for advertising orna-


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Conceptually, the

most exciting open

problem is how to

incorporate genetic

mechanisms into the

simulation models

of plants.

mental plant varieties to potential customers.Experience with simulation-based plant modeling

also reveals problems requiring further research. Con-struction of well-calibrated models of individual plantspecies is a tedious process, because it requires exten-sive experimental data and may involve examinationof alternative models when the exact nature of theprocesses in plants is not well understood. Develop-ment of a methodology that would accelerate con-struction of well-calibrated models is thereforeimportant to biologists and applied plant scientists.

Conceptually, the most exciting open problem ishow to incorporate genetic mechanisms into the sim-ulation models of plants. This research question couldlead to new insights into the fundamental mecha-nisms of development—from genes all the way to thefinal forms of the plants around us—and furtheradvance our understanding of nature through the syn-ergy of biology, computer science, and art.

References1. Bouchon, J., de Reffye, P., and Barthélémy, D., Eds. Modélisation et

Simulation de l’Architecture des Végétaux. INRA Editions, Paris, 1997.2. Deussen, O., Hanrahan, P., Lintermann, B., Mech, M., Pharr, M., and

Prusinkiewicz, P. Realistic modeling and rendering of plant ecosystems.In Proceedings of SIGGRAPH’98 (Orlando, Fla., July 19–24). ACMPress, New York, 1998, 275–286.

3. Fowler, D., Prusinkiewicz, P., and Battjes, J. A collision-based model ofspiral phyllotaxis. In Proceedings of SIGGRAPH’92 (Chicago, July26–31), ACM Press, New York, 1992, 361–368; see also Comput.Graph. 26, 2 (July 1992).

4. Gautier, H., Mech, R., Prusinkiewicz, P., and Varlet-Grancher, C. 3Darchitectural modeling of aerial photomorphogenesis in white clover(Trifolium repens L.) using L-systems. Annals of Botan. 85, 3 (Mar.2000), 359–370.

5. Lindenmayer, A. Mathematical models for cellular interaction in devel-opment, Parts I and II. J. Theoret. Biol. 18 (1968), 280–315.

6. Lintermann, B. and Deussen, O. Interactive modeling of plants. IEEEComput. Graph. Appl. 19, 1 (Jan.–Feb. 1999), 56–65.

7. Mech, R. and Prusinkiewicz, P. Visual models of plants interacting withtheir environment. In Proceedings of SIGGRAPH’96 (New Orleans, La.,Aug. 4–9). ACM Press, New York, 1996, 397–410.

8. Prusinkiewicz, P. Modeling of spatial structure and development ofplants: A review. Scientif. Horticult. 74 (1998), 113–149.

9. Prusinkiewicz, P., Hammel, M., Hanan, J., and Mech, R. Visual mod-els of plant development. In Handbook of Formal languages, Vol. III:Beyond Words, G. Rozenberg and A. Salomaa, Eds., Springer-Verlag,Berlin, 1997, 535–597.

10. Prusinkiewicz, P. and Lindenmayer, A. The Algorithmic Beauty ofPlants. Springer-Verlag, New York, 1990 (with J. Hanan, F. Fracchia,D. Fowler, M. de Boer, and L. Mercer).

11. Takenaka, A. A simulation model of tree architecture developmentbased on growth response to local light environment. J. Plant Res. 107(1994), 321–330.

12. Weber, J. and Penn, J. Creation and rendering of realistic trees. In Pro-ceedings of SIGGRAPH’95 (Los Angeles, Aug. 6–11). ACM Press, NewYork, 1995, 119–128.

Przemyslaw Prusinkiewicz ([email protected]) is a professor in the Department of Computer Science at the University ofCalgary in Calgary, Alberta, Canada.

The work described here was supported in part by the Natural Sciences and

Engineering Research Council of Canada.

© 2000 ACM 0002-0782/00/0700 $5.00

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