growth dynamics of three tropical savanna grass species: an individual-module model

Ecological Modelling 154 (2002) 4560

Growth dynamics of three tropical savanna grass species: anindividual-module model

Miguel F. Acevedo a,*, Josep Raventos b

a Institute of Applied Sciences and Department of Geography, Uniersity of North Texas, P.O. Box 305279, Denton,TX 76203, USA

b Departamento de Ecologa, Uniersidad de Alicante, Alicante, Spain

Received 2 August 2001; received in revised form 13 February 2002; accepted 4 March 2002

Abstract

We model the dynamics of grass plant growth as a collection of the individual dynamic behavior of shoots andleaves. The model is inspired in data for plants of three species (Elyonurus adustus, Leptocoryphium lanatum andAndropogon semiberbis) of common grasses in the Venezuelan savannas that were sampled monthly for 1 year. Thesespecies represent different architecture and regeneration response to fire. Modules (shoots and leaves) were countedin each cell of a square grid in each one of several vertical levels. Module density per cell provides the horizontaldistribution within a level and is aggregated by level to obtain vertical distribution. Both distributions are simulatedby a dynamical model based on shoot emergence and mortality, elongation of shoots and leaves given by Richardsequation, plus a few simple geometric considerations. For quantitative comparisons of model results to data, thetransient and final values for vertical distribution plus two metrics of horizontal distribution at each level, werecalculated for the simulation results and the field data. Proportion of occupied cells and maximum distance to thecenter of growth were the two metrics selected to capture the dispersion and range of the horizontal distribution. Themodel results indicate predictable final vertical profiles (of proportional density plus the two metrics) similar to theprofiles of measured distributions for each species. A reasonable prediction of the transient behavior was alsoobtained but with larger deviations as evaluated by the root mean square error between model and data. Differencesin vertical and horizontal patterns of module density among species are explained by changing a set of parametervalues related to growth form and phenology. Thus, the model could be applied to generate plant functional typesfor analysis of savanna dynamics subject to fire. With modifications, the model is potentially applicable to other grassplants and other grassland ecosystems. 2002 Elsevier Science B.V. All rights reserved.

Keywords: Growth; Phenology; Shoots; Richards equation; Savanna; Grasses; Venezuela; Fire

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1. Introduction

Sessile organisms, and especially terrestrialplants, are discrete entities that live attached tosurfaces and interact mainly with their own or

* Corresponding author. Tel.: +1-940-565-2381; fax: +1-940-565-4297.

E-mail address: [email protected] (M.F. Acevedo).

0304-3800/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved.

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M.F. Aceedo, J. Raentos / Ecological Modelling 154 (2002) 456046

other species (Tilman et al., 1997). Growth andspatial arrangement of plant components are fun-damental processes underlying plant interactionwith their environment and other plants. There-fore, modelling plants functionally and geometri-cally as sets of interconnected organs can help topredict potential growth under changing condi-tions as required in theoretical agronomy (Espanaet al., 1998; Kaitaniemi et al., 1999; Lavorel et al.,1999). This perspective has been developed andextended to trees employing a variety of mod-elling approaches including Monte Carlo methodsas well as analytical and fractal geometry (DeReffye et al., 1995; De Reffye and Houllier, 1997;Berntson and Stoll, 1997). Also in forestry, effortshave included model formulations of tree archi-tectural analysis; for example, the pipe modeltheory (Chiba, 1998) and modular simulators(Chen and Reynolds, 1997).

Studies of herbaceous vegetation have focusedon demography of plant components (Harper etal., 1986) as well as descriptions of componentarrangement in horizontal and vertical dimensions(Raventos and Silva, 1988). Grassland modelsinclude a variety of approaches; for example,matrix population models to predict fire responseof a tall grass (Silva et al., 1991), individual-basedmodels of shortgrass communities emphasizingbelowground resources (Coffin and Lauenroth,1990; Coffin et al., 1993) and ecosystem processmodels (e.g. Coughenour and Chen, 1997).

Savannas are very important and productivetropical ecosystems characterized by co-domi-nance of herbaceous vegetation and less abundanttrees and shrubs. Aboveground vegetation coveris strongly influenced by climate, herbivores andfire (Walker, 1987; Skarpe 1996). Perennial herba-ceous vegetation can be analyzed using a fewfunctional types described by growth forms andfire response (Sarmiento and Monasterio, 1983;Skarpe, 1996).

We recognize that competition for belowgroundresources in grassland models has been considereda determinant factor in community composition,but in this paper we concentrate our efforts inaboveground processes. Competition for light alsodetermines important properties of plant growthand dynamics, such as shoot survivorship (Raven-

tos and Silva, 1988), and knowledge of verticaldistribution of plant canopy is important to un-derstand light availability at different vertical lev-els. In addition, aboveground distribution of plantcomponents affects fire intensity and propagationand in turn, fire regime is one of the main deter-minants of tropical savanna dynamics. Therefore,there is a need for models providing a betterunderstanding of how the dynamics of individualcomponents link to the architectural expression ofgrass plants in the vertical and horizontaldimensions.

We adopt a view of the plant as composed by apopulation of modules (shoots, leaves and flower-ing shoots), developing in both spatial dimen-sions. In this paper we develop anindividual-based model formulated by shootemergence and mortality, shoot and leaf elonga-tion rates, plus simple geometrical rules in orderto analyze seasonal aboveground growth dynam-ics of perennial tropical grass plants in both hori-zontal and vertical dimensions. Predicted plantproperties are the distribution of modules in sev-eral height classes or levels and selected metrics ofhorizontal distribution at each height level. Ouraim was to identify species differences in modelparameter values with several aspects of plantfunctional types that could be used to modeltropical savannas (Sarmiento and Monasterio,1983; Smith et al., 1993; Scholes et al., 1997;Skarpe, 1996).

2. Study site and data collection methods

The field study was conducted in a seasonalsavanna near Barinas, Venezuela (8 38 N, 7012 W). Like many other savanna ecosystems, thisarea is subject to frequent burning. Mean annualtemperature is 27 C and mean annual rainfall is1700 mm, with a rainy season from May toNovember and a dry season from January toMarch. Burning usually takes place once a year,sometime during February through April, beforethe onset of the rainy season.

Among the dominant grass species at this site,we selected three perennial species with differentphenological traits (Fig. 1). (a) Elyonurus adustus,

M.F. Aceedo, J. Raentos / Ecological Modelling 154 (2002) 4560 47

a precocious grower with long and slender leaves,is a bunch grass that flowers after the annualburning at the end of the dry season. (b) Leptoco-ryphium lanatum, another bunch grass but withlong scleromorphic leaves, is an early species thatflowers in May, 1 month after the onset of rains.(c) Andropogon semiberbis, an erect and late flow-

ering grass with short soft leaves that blooms atthe end of the wet season. Growth, competition,demography and responses to fire of these specieshave been reported in several papers (Silva andAtaroff, 1985; Silva and Castro, 1989; Raventosand Silva, 1995; Silva and Raventos, 1999).

Three individuals (replicates) of each one of thethree species mentioned above were measured us-ing a metallic structure in which we could set ahorizontal frame of 225 (1515) square cells of55 cm in size at different heights (Fig. 2). For 1year, we performed monthly measurements usingthis structure from 10 to 100 cm above ground at10 cm intervals. A vertical level is defined as a 0.1m height interval starting at ground level andending at the maximum height of the plant. Themaximum number of levels achieved by a plantduring the year changed according to the replicateand the species; typical numbers by species arefive levels for E. adustus, seven for L. lanatum,and ten for A. semiberbis.

For each level and cell, module density wasobtained by counting all module intersectionswith the grid. This was repeated every monthstarting in September 1984 and ending in August1985, excluding July 1985. Fire occurred some-time after the January 1985 measurement, andthus the data for February 1985 indicated valuesof module density only in the first level. Detailson sampling method are described by Raventosand Silva (1988).

3. Model description

We postulate that an individual-based modelincluding shoot emergence and mortality, shootand leaf elongation and geometrical arrangementcan explain the dynamics of the horizontal andvertical distribution given by the data described inthe previous section. Each shoot and leaf is con-sidered to grow in length according to well-knowngrowth rules, such as the Richards differentialequation (Causton and Venus, 1981)

dl(t)dt= l(t)

gb

[1 (l(t)/L)b] (1)

where l(t) is shoot or leaf length (in m) at time t

Fig. 1. Different morphology of species studied. (E) E. adustus.(L) L. lanatum. (A) A. semiberbis. Adapted from Silva (1987).

Fig. 2. Frame used for field measurements showing the grid of225 (1515) square cells of 55 cm. This grid is set atdifferent heights to measure module density by vertical level.


Fig. 3. Geometric considerations to obtain height and projec-tion from length and elevation angles with respect to ground.(a) Shoot; (b) leaf.

h(t, 1)= l(t) sin 1 (2)

Similarly, the horizontal projection r(t, 1) orhorizontal distance of that point of the shoot isdetermined from its length using the cosine of thisangle

r(t, 1)= l(t) cos 1 (3)

The base of all shoots is assumed to be at thecenter of growth or root of the plant. The shootelevation angle was assumed to vary from 90 toa minimum elevation angle s in degrees (Fig. 3a).

A shoot generates one leaf at the end of eachinter-node length D ; thus the number of nodes Nldetermines the number of leaves per shoot. For E.adustus and L. lanatum we assume only one leafper shoot, whereas for A. semiberbis we assumedseveral nodes per shoot. Leaf elevation angle 2 isassumed to be less than the corresponding shootelevation angle 1. The height at which a leaf isinitiated depends on shoot length l(t) and eleva-tion angle 1. Eqs. (2) and (3) provide the initialheight for a leaf as hini(t, 1)= l(t) sin 1and theprojection as rini(t, 1)= l(t) cos 1. In a similarmanner to the shoot elevation angle, a minimumleaf elevation angle of m (in degrees) is estab-lished as a parameter; but in addition, the maxi-mum angle is obtained usingf=m+ (1m)F which is parameterized by afraction F of the difference between the mini-mum angle and the shoot angle (Fig. 3b).

For species L. lanatum and A. semiberbis, theelevation angle was the only geometric consider-ation needed to account for length-to-height andhorizontal projection conversion. For E. adustus,however, there is a further net decrease of heightthat can be explained by leaf curvature. A powerfunction accounts for increasing curvature radiusc(2) with decreasing elevation angle 2

c(2)= (1c1)2m

90m

c2+c1 (4)

that varies from c(90)=1, for a radius of 1 m, toc(m)=c1. Therefore, c1 is the maximum horizon-tal extent and it would occur at the minimumangle. A radius of 1 was assumed to be equivalentto a nearly straight leaf. The radius of curvaturewas a parameter of a parabolic function to relate

(in months), g is an elongation rate coefficient (inmonth1), L is the maximum length (in m), and bis a coefficient (unitless) controlling the depen-dence of the elongation rate dl/dt on l(t). Thisequation is very flexible and its solution allows fora variety of sigmoid shapes by varying theparameter b. In particular, logistic elongation isobtained when b=1. Richards equation was ap-plied to shoot elongation with parameter valuesg1, b1, L1 and to leaf elongation with parametervalues g2, b2, L2.

A certain number Nf of shoots are convertedinto flowering shoots; these are assumed to beamong the first shoots that emerge during thegrowing season and continue to elongate up to amaximum length Lf following Eq. (1) with L=Lfand the same shoot parameters g1, b1.

Height h(t, 1) reached at time t by any point ofa shoot is a function of its length l(t) up to thattime and elevation angle 1 attained by the shootwith respect to ground. It can be calculated sim-ply using the sine of this angle


leaf height hf(t, 2) and horizontal projectionrf(t, 2) (both measured from the base given by hiniand rini) to a given point of the leaf (Fig. 3b).Total height and projection developed by the leafare hf(t, 2)+hini(t, 1) and rf(t, 2)+rini(t, 1).

A function relating height and horizontal pro-jection is inspired in the rate function of thelogistic equation

hf(t, 2)= tan(2)rf(t, 2)([1rf(t, 2)/c(2)] (5)

where tan(2) is the initial slope of the leaf.Height takes the value 0 when rf(t, 2) reaches theradius c(2). The slope of the leaf at any particu-lar time t is given by the derivative of thisfunction

dhf(t, 2)drf

= tan(2)[12rf(t, 2)/c(2)] (6)

and therefore the angle (t, 2) of the leaf at anyparticular point is the angle attaining this value oftangent

(t, 2)= tan1[tan(2)[12rf(t, 2)/c(2)]] (7)

In addition, we developed a circular trajectoryfor the leaf that allows for an explicit relationbetween height and projection. For this curvaturefunction, the radius c of curvature is the radius ofa circle tangent to the center of growth andcentered c units to the right of this point. In thiscase, the leaf follows an arc of the circumferencewith length lf(t). Height can be calculated fromthis length

hf(t, 2)=c(2) sin[lf(t)/c(2)] (8)

and thus the circle equation relates height tohorizontal projection, however, we used theparabolic form in this paper because it offersmore flexibility.

hf(t, 2)2=c(2)2 (rf(t, 2)c(2))2 (9)

For the horizontal distribution, an additionalazimuth angle was used and the coordinates ofany point of the leaf or shoot were calculatedusing spherical coordinates

x(t, , )=r(t, ) cos +xc (10)

y(t, , )=r(t, ) sin +yc (11)

where can be 1 or 2 according to whether weare using shoot or leaf respectively; similarly r canbe rf for leaves. The values xc, yc are coordinatesof the root of the plant, assumed to be at thecenter of the grid in the first level. The angle was measured counterclockwise with respect tothe positive side of the x axes (Fig. 4a).

A plant is composed of a collection of n shootsof different lengths (ls, s=1,, n) according tothe solution of Eq. (1). These shoots reach heightsand horizontal projections calculated from Eqs.(2) and (3). The elevation angle 1 for each shootwas chosen at random from a uniform distribu-tion from s to 90. Then, the leaf elevation anglewas selected at random from a uniform distribu-tion from the minimum m to the maximum f.The azimuth angle was selected at random from auniform distribution between 0 and 360.

All shoots were initialized with a height equalto 0.1 m; that is to say, we assumed that theyreached level 1 at the beginning of the simulation.The length corresponding to this height was calcu-lated from Eq. (2) and the projection correspond-ing to this length was calculated from Eq. (3).This initial value of length was held constantduring a fixed time lag or latency T1 (given inmonths) to start growth to the second verticallevel. Shoots were then assumed to follow anelongation mechanism, prescribed by Eq. (1), withparameter values g1, b1 and L1. Once the latencyperiod is completed, this process unfolds rapidlyas shown by the fast increase in the proportion ofshoots in level 2. As shoots become longer, theirheight and projection change according to Eqs.(2) and (3). Leaf elongation proceeds followingEq. (1) but with different parameter values g2, b2,L2 and their height and projection are affected bycurvature.

As indicated by the data, module density de-creases due to mortality in October and Novem-ber, possibly due to the end of the rainy season.Thus, shoots were killed and excluded from thetally during the months established by the periodstarting at month T2 and ending with month T3.Shoots were killed randomly with probability i,i=1, 2, during month i of the mortality period.Also, shoot generation stops at the beginning ofthis period, i.e. month T2.


To simulate this model, we assumed that theplant produces a number n of shoots uniformlydistributed in age; therefore, we generated n/T2shoots per month at a constant rate. We simu-lated the model for 12 months in time steps ofdt=0.1 months. Therefore we generated n(dt/T2)shoots of height 0.1 m every simulation time stepdt. To obtain an integer number of generatedshoots per time step, n was made a multiple of T2.

For each shoot s, its length ls was updated by asmall increment dl every dt by numerically solvingEq. (1); then the resulting height hs and projectionrs were updated using Eqs. (2) and (3). For thispurpose we started with the short differential dland estimated

dh(t)=dl(t) sin[(t, 2)] (12)

from Eq. (2) and

dr(t)=dl(t) cos[(t, 2)] (13)

from Eq. (3). Recall that leaf angle (t, 2) iscalculated from Eq. (7).

As shoots and leaves elongate and increase inheight, they intersect the height levels k=1,, I

according to h(t, ) and cells according to thex(t, , ), y(t, , ) coordinates, and are countedto derive a simulated module density Nmk(i, j ) percell (i, j ) at each level k for each month m (Fig.4b). Here i=1,, 15 and j=1,, 15 startingwith the lower left corner of each level. Angle is1 and 2 for shoots and leaves, respectively. Wethen summed the number of shoots and leaves inall cells of each level k for each month m toobtain a simulated total module density Nmk forthe level and month. The Nmk(i, j ) are divided bythe total Nmk to obtain simulated proportionspmk(i, j ) in each cell. The densities Nmk(i, j ), theproportions and metrics of its spatial distribution(to be developed in the next section) are then usedto evaluate model performance when compared tothe field data.

4. Proportion by level and metrics of horizontaldistribution

To simplify the spatial distribution, we summa-rize each horizontal distribution by calculating a

Fig. 4. (a) Geometric considerations to obtain coordinates for a shoot or a leaf. (b) Cell intersections at month m and height levelk are used to obtain density Nmk(i, j ), occupancy Cmk and maximum distance Zmk.


Table 1List of parameters used in the model

Symbol (units) Explanation

(a) ConstantsShoot

Coefficient controlling dependence ofb1 (unitless)=20elongation rate on length

T2 (months)=9 Onset of mortality periodT3 (months)=10 End of mortality period

Minimum shoot elevation angles (degrees)=601 Probability of dying at month 1 of

the mortality period(probability)

=0.42 Probability of dying at month 2 of

(probability) the mortality period

=0.5

Leaf and flowering shootsCoefficient controlling dependence ofb2 (unitless)=0.1elongation rate on length

Nf (number)=6 Number of flowering shootsPower of curvature functionc2 (unitless)=1.0

(b) Species specificShootg1 (month

1) Elongation rate coefficientInter-node lengthD (m)

Nl (number) Number of leaves per shootLatency: months previous to start ofT1 (months)elongationNumber of shootsn (number)

Leaf and flowering shootsLeaf elongation rate coefficientg2 (month

1)Radius of curvaturec1 (m)Maximum elevation angle of theF (unitless)leaves, as fraction of the differencebetween minimum angle m andshoot elevation angleMinimum elevation angle of them (degrees)leavesMaximum length of leavesL2 (m)Maximum length of flowering shootsLf (m)

proportion of occupied cells. Each one of thesemetrics describes a different aspect of the horizon-tal distribution; namely, the range and dispersionof this distribution. For quantitative comparison,the metrics were calculated using the simulated aswell as the data distributions.

The maximum distance metric was calculatedfor each level k and month m by finding themaximum value of the Euclidian distance of thetip of all modules to the center of growth atcoordinates xc and yc (see Eqs. (10) and (11)).That is to say,

Zmk=max{[(r(m, ) sin )2+ (r(m, ) cos )2]p1/2}(14)

where k is the height level and m is the month(Fig. 4b). Coordinates for the center xc and ychave the value (150.05m)/2=0.375 m. Angle is 1 or 2 according to whether module p is ashoot or leaf and the maximum is calculated overp=1,, Nmk that is all modules intersecting levelk. The projection r corresponds to rf for leaves.This maximum distance Zmk was normalized afterdividing by half of the diagonal of the grid (1.40.375=0.53 m) to obtain Z*mk and therefore allvalues of the metric are less than or equal to 1.

The number of occupied cells Cmk was tallied ateach level k for each month m. An occupied celli, j was detected by checking for nonzero values ofNmk(i, j ) (Fig. 4b). The number Cmk was thendivided by 225, the total number of cells in thegrid, to normalize it as C*mk and obtain values lessthan or equal to 1. Together with proportions Pmkthese two metrics Z*mk and C*mk characterize plantproperties to be predicted by the model.

5. Parameter estimation and simulation

Model parameters are summarized in Table 1;some of these parameters were considered to beconstant for all species (Table 1a) and some to bespecies-specific (Table 1b). An effort was made tohave as many constants as possible in order toreduce the parameters representing major differ-ences among species. To obtain the constants andparameter values (Table 1a and Table 2) we pro-ceeded as follows. Latency was first fixed accord-

proportional density by level and metrics charac-terizing the horizontal distribution by level. De-note by I the maximum number of levels for thespecies. First, the total in each level Nmk is dividedby the total of all levels k=1,, I for the monthm to obtain proportions Pmk by level. Second, weselected the following two metrics: maximum dis-tance with respect to the center of growth and


ing to the integer nearest to the average of thelatency at the first level observed in the threereplicates. Mortality period was assumed to last 2months beginning in October. The minimum andmaximum angle for shoots and leaves s, m andF were approximated from field observations.Number of shoots n was estimated from the aver-age number of shoots measured in the field at thefirst level for the three replicates at the monthprevious to the fire. The number Nl of leaves pershoot, the number Nf of flowering shoots and theinter-node length D were approximated based onobservations. L1 was calculated from Eq. (2) asthe shoot length required for a height of 0.1 m orthe height difference from level 1 to level 2, andestimated as DNl. The length of flowering shootsLf was estimated by the maximum height of theplants since flowering shoots are the ones thatreach the highest levels. Only E. adustus showssignificant leaf curvature, and therefore curvatureparameters for A. semiberbis and L. lanatum wereset at c1=c2=1. To facilitate estimation of cur-vature parameters for E. adustus we assumed thatthe power coefficient is c2=1 and adjusted c1 bysearching for a good fit to the data.

After these approximations were made, basedon direct field observations and assumptions, theremaining parameters related to shoot elongation(g1, b1), leaf elongation (g2, b2, L2), curvature c1for E. adustus and mortality rates (1, 2) wereadjusted by trial and error searching for a good fitto the data. The modeled rise of module densitypm2 at level 2 at month m right after the latency

was aimed to emulate the data by adjusting g1 andb1. We then found approximations of the otherparameters by attempting to match both transientPmk and final (January or month 12 after the fire,m=12) behavior of model and data for all levelswhile generating plants of the observed maximumheight.

To examine transients, the simulated propor-tions Pmk at each level k are plotted as a functionof time m beginning in February, the first monthafter the fire. The corresponding data are plottedbeginning in February 1985 (right after the fire)through August 1985 (except July 1985), thenfollowed from September to December 1984 andending in January 1985. Even though, the datacollection (September 1984June 1985) did notcorrespond to this chronology, we decided to plottransients this way to illustrate the typical post-fire dynamics of plant growth in this ecosystem.The individual 1 of species A. semiberbis died inthe course of the experiment yielding less data forparameter estimation.

Once a first approximation for all the parame-ters was obtained, we made multiple executions ofthe model attempting to mimic the measureddata. For this we used several graphical represen-tations of the data and the simulation results.First, the transient and final (month 12 after thefire) behavior of the horizontal distribution ateach level, were displayed as two-dimensionalgrids where density in each cell was coded withdifferent colors or shades. Second, the transientbehavior of the two horizontal distribution met-

Table 2Parameter values

D (m)g1 (month1) n (no.)T1 (months)Nl (no.)

(a) Shoot10 0.15 1 1 540E. adustus10 0.22 1L. lanatum 2 540

18020 5A. semiberbis 50.10

g2 (month1) c1 (m) F (unitless) m (degrees) L2 (m) Lf (m)

(b) Species-specific0.6 0.5 1.0E. adustus 30 0.50 0.45

L. lanatum 0.400.8 0.601.0 1.0 501.0 30 0.25 1.201.0A. semiberbis 0.3


Fig. 5. Horizontal patterns of module density at levels 1, 3 and5 for E. adustus. Each box corresponds to the 1515 grid.Gray cells are occupied cells and white cells are empty (zerodensity). Gray scale shows the density scaled in relation tomaximum value in the level.

6. Results and discussion

6.1. Spatial distribution

As explained before, we measured the monthlyhorizontal spatial distribution at all height levelsfor each of the three field replicates for the threespecies. For brevity, however, we will show onlyplots of one replicate for each species at the endof the simulation; i.e. January or 12 months afterthe fire (Figs. 57). All other spatial data andsimulation results including transient behavior areavailable following the links at http://emod.unt.edu.

Fig. 6. Horizontal pattern of module density at levels 1, 3, 5and 7 for L. lanatum. Boxes, columns and scales are the sameas in Fig. 5.

rics Z*mk, C*mk and the vertical distribution Pmkwere plotted in the manner described in the previ-ous paragraph.

Third, for each one of these three indices weused the root-mean-square (RMS) error betweenmodel and data. RMS error was calculated bylevel for all months of available data; thus, justone value summarizes the fit between the simu-lated and each replicate by level during the tran-sients. For all three indices (Pmk, Z*mk, C*mk) theseRMS errors together with a comparison of final(month 12 after the fire) values were used asbenchmark for the overall fit. Species-specificparameter values were then varied around theirfirst approximation searching for improvements inthis benchmark.


Fig. 7. Horizontal pattern module density at levels 1, 3, 5, 7and 9 for A. semiberbis. Boxes, columns and scales are thesame as in Fig. 5.

left hand side column whereas maps of replicate 3of the field data are shown in the right hand sidecolumn. Each panel represents the 1515 grid ata given level and the shade scale changes accord-ing to the maximum module density among alloccupied cells at that level. Lowest and highestvertical levels are at the bottom and top, respec-tively, of the figure.

All three species (Figs. 57) show commonfeatures in both simulated and observed patterns;namely, concentrated density at the bottom levelnear the center of growth and increased spread athigher levels. The model maintains a concentratedpattern at the lowest level, since we have assumeduniform shoot generation and no lateral displace-ment with respect to the center of growth.

For E. adustus, a bunch grass with curvedleaves, the match between observed and simulatedpattern is fairly good for lower and intermediatelevels (Fig. 5). The simulated pattern overesti-mates the occupied cells at the upper level. L.lanatum has a slender form with straight leaves;Fig. 6 shows that the model mimics the pattern atall levels, but overestimates the spread at level 5.Density is zero at the top level for the model aswell as for this replicate. L. lanatum has an archi-tectural form in between E. adustus and A.semiberbis. Fig. 7 shows that the model also mim-ics the architectural design of A. semiberbis, a tallgrass that produces more leaves per shoot thanthe other two species but with lower elevationangle. These leaves occupy many cells at interme-diate levels with a sharp decrease at higher levels.The simulated model captures these facts al-though it overestimates cell occupancy at level 5.

6.2. Transient behaior

The values of each metric for each monththroughout the simulation period are used as anindication of how well the model tracks the dataduring the simulation. For the sake of brevity, weonly show transient behavior for one metric: nor-malized cell occupancy C*mk. Transient behaviorfor other metrics is available to the interestedreader at http:emod.unt.edu.

Figs. 810 show the dynamics of normalizedcell occupancy for simulated (upper panel) and

Also for brevity, only odd numbered levels areshown (1, 3 and 5 for E. adustus, 1, 3, 5, and 7 forL. lanatum, and 1, 3, 5, 7, 9 for A. semiberbis).For comparison, simulated maps are shown in the


the average of the three field data replicates (lowerpanel) starting in February and ending in Janu-ary. As explained in the previous section, the fielddata was re-arranged to show this chronology.July is not shown because it was not measured.For clarity only levels 2, 3 and 5 are shown.Different latencies are evidenced for each speciesin the rise of values at level 2. For all species,occupancy shows an increasing pattern during theyear, for both model and data, and exhibits acorrection or sudden decrease for 2 months (Octo-berNovember) in its approach to the final Janu-ary value. This decreased occupancy is capturedin the model by shoot mortality during thesemonths (Figs. 810).

The simulated pattern fits the data relativelywell but with some exceptions. For E. adustuslevel 2 is overestimated for June, and levels 2 and3 are overestimated for AugustOctober. For L.lanatum level 3 is overestimated after August.

Fig. 9. Dynamics of occupied cells (normalized) at levels 2, 3,and 5 (numbered lines) for L. lanatum. Upper panel: simu-lated. Lower panel: average of field data.

Fig. 8. Dynamics of occupied cells (normalized) at levels 2, 3,and 5 (numbered lines) for E. adustus. Upper panel: simulated.Lower panel: average of field data.

However, we did not attempt to improve thesimulated pattern for each species by tuning allparameter values but rather to produce a goodcompromise among the species in order to keepsome parameters constant. This visual assessmentof fit during the transients will be made morequantitative when we discuss the RMS errors inthe following section.

6.3. Final alues and transient RMS error for allmetrics

Figs. 1116 summarize the comparison ofmodel versus data for the final values as well asthe errors made during the transients for Pmk andthe two metrics Z*mk, C*mk. The left hand sidepanels of each one of these figures correspond toa comparison by level between the metric orproperty for the three replicates (three white bars)and the model (black bar) at the end of the


simulation. The right hand side column of thesefigures corresponds to the RMS error during theentire simulation period for simulated versus allreplicates (white bars). These errors are absoluteand not relative to the value at this level.

Fig. 12. Final simulated (black bar) and data replicates (threewhite bars) by level for E. adustus and transient RMS errorbetween the simulated and each one of the three field datareplicates (all white bars). (a, b) Occupied cells; (c, d) maxi-mum distance from the root.

Fig. 10. Dynamics of occupied cells (normalized) at levels 2, 3,and 5 (numbered lines) for A. semiberbis. Upper panel: simu-lated. Lower panel: average of field data.

Fig. 13. Final vertical distribution for L. lanatum. (a) Simu-lated (black bar) and data replicates (three white bars). (b)Transient RMS errors between the simulated and each one ofthe three field data replicates (all white bars).

Fig. 11. Final vertical distribution for E. adustus. (a) Simulated(black bar) and data replicates (three white bars). (b) TransientRMS errors between the simulated and each one of the threefield data replicates (all white bars).

Fig. 11a shows that for E. adustus, a bunchgrass, there are proportionally many more shootsat the first level with fast decrease of density withheight. The simulated model fits this vertical pat-


tern except for underestimation at the first leveland overestimation at higher levels. The RMSerrors (Fig. 11b) show good agreement betweendata and model with errors that decrease with

Fig. 16. Final simulated (black bar) and data replicates (threewhite bars) by level for A. semiberbis and transient RMS errorbetween the simulated and each one of the three field datareplicates (all white bars). (a,b) Occupied cells; (c,d) maximumdistance from the root.

Fig. 14. Final simulated (black bar) and data replicates (threewhite bars) by level for L. lanatum and transient RMS errorbetween the simulated and each one of the three field datareplicates (all white bars). (a,b) Occupied cells; (c,d) maximumdistance from the root.

height. The largest errors occur at the first level.The resulting cell occupancy with height repro-duces the data fairly well (Fig. 12a) with smallRMS errors at all levels (Fig. 12b).

Together, Fig. 11a and Fig. 12a indicate that atlevel 1 there are more modules but these arepacked more tightly on fewer cells; whereas atlevels 2 and 3 there are fewer modules but theseare spread in a greater number of occupied cells.For this species, this fact is due to two differentfactors. One is the increased spread with increas-ing height due to the elevation angle and theother, is due to the curvature of the leaf, thatallows a leaf to intercept the same level twice ondifferent cells.

At the end of the year, the vertical pattern ofmaximum distance from the root at differentheights (Fig. 12c) is similar to that shown by celloccupancy, lower at the base with higher values atother levels. Again, this indicates greater spread at

Fig. 15. Final vertical distribution for A. semiberbis. (a) Simu-lated (black bar) and data replicates (three white bars). (b)Transient RMS errors between the simulated and each one ofthe three field data replicates (all white bars).


intermediate and higher levels. In this case themodel underestimates distance at the lower levels.The transient RMS errors (Fig. 12d) are higherthan the ones obtained for cell occupancy andproportional density (Fig. 11b and Fig. 12b).

L. lanatum is a slender species and this is repre-sented by a more gradual decrease of final pro-portional density with height, showing a fairlygood agreement between data and model (Fig.13a). The transient RMS error decreases withheight (Fig. 13b). Due to the fact that leaves andshoots of this species grow straight, with no ap-preciable curvature, increased cell occupancy atintermediate levels is predicted by increasedspread due to the leaf elevation angle. Transienterrors are small at all levels with larger values atintermediate levels (Fig. 14b). Similarly to E.adustus, the simulated profile underestimates themaximum distance to the root at the basal levels(Fig. 14c), showing large transient errors (Fig.14d).

A. semiberbis is a tall, slender grass with addi-tional production of leaves at intermediate levels.This fact is reflected at the end of the year, bothfor data and model, by an increase in propor-tional density and cell occupancy from the basallevel to the second level and a subsequent de-

crease with height towards the upper levels (Fig.15a and c). This species shows, similarly to theother two species, a larger transient error forproportional density at the lowest level and gener-ally low error at all levels for cell occupancy (Fig.16a). The final maximum distance for the data(Fig. 16c) shows a nearly constant value withheight except for the bottom level, demonstratinga different architecture of this species when com-pared to the other two species (Fig. 12c and Fig.14c). The maximum distance is underestimated bythe model at the bottom and top levels (Fig. 16c)and shows slightly higher transient errors towardsthe top (Fig. 16d).

7. Conclusions

The results discussed in the previous sectionhave direct connection to the morphology andphenology of the three species studied (Fig. 17).E. adustus and L. lanatum have similar architec-ture, with shoot emergence and leafing at thebasal level, shorter plants, higher module density,longer leaves and early or precocious growth afterthe fire. A. semiberbis is a taller plant with addi-tional leafing above the basal level, more leavesper shoot, lower module density, shorter leavesand late growth after the fire.

Parameters values obtained for E. adustus andL. lanatum are similar except for higher minimumleaf elevation angle m and shorter radius ofcurvature c1 for L. lanatum. Lower elevation angleand radius of curvature of E. adustus mimic abunch grass with curved leaves, whereas higherelevation angles and no curvature of L. lanatumreflect a taller grass with straighter leaves. Mostparameter values obtained for A. semiberbis aredifferent from both E. adustus and L. lanatum.Higher number of leaves Nl per shoot, fewershoots n, lower maximum leaf elevation angles F,shorter leaf length L2 and longer flowering shootsLf reflect striking architectural differences for A.semiberbis (Figs. 1 and 17). Different values oflatency were required for the three species, short-est for E. adustus, longer for L. lanatum andlongest for A. semiberbis. These latencies reflectdifferential response to fire of these three species,

Fig. 17. Different morphology and phenology of species stud-ied. (E) E. adustus. (L) L. lanatum. (A) A. semiberbis. Majorparameters that vary when changing species are emphasized.


since E. adustus is precocious, L. lanatum is earlyand A. semiberbis is a late species (Fig. 17). Addi-tional differences in temporal response are cap-tured by differences in coefficients for leafelongation rate g2 between E. adustus and L.lanatum and for shoot elongation rate g1 for A.semiberbis.

This model is able to simulate very well thefinal (i.e. 12 months after the fire) distributionsand reasonably well the transient behavior. Quan-titative evaluation of distributions is greatly aidedby metrics of horizontal distribution (cell occu-pancy and maximum distance to the root). Thesemetrics provide a measure of dispersion and rangeof module distribution on the grid by level. Tran-sient behavior was evaluated calculating the RMSerrors for all the properties studied. The errorsindicated a reasonable prediction of the dynamics,especially for the proportional density and celloccupancy for all species. However, the modeledtransient behavior presents larger differences withrespect to the data than the final distribution.This is due to the inherently more difficult task ofpredicting a measure of the horizontal distribu-tion at all times during plant development.

Our goal in this paper has been to test the ideathat a generic and simple individual componentbased model would approximately reproduce ob-served plant growth for all species by changingparameter values. Thus we did not attempt tooptimize the fit of the model to data by tuning allparameters but rather sought to obtain reasonablematch to data while keeping some parametersconstant across species. Because the model com-bines descriptors for architectural design of theplants with their fire response, it can be applied toincorporate plant growth forms (Sarmiento andMonasterio, 1983) in analysis of savanna dynam-ics subject to repetitive fire events and has poten-tial applications to define and model plantfunctional types in these ecosystems (Smith et al.,1993; Skarpe, 1996; Scholes et al., 1997). Thesetypes could then be used to predict ecosystemdynamics by other more aggregated and compre-hensive models.

In this paper, we implicitly modeled modulecompetition by introducing a mortality factor atselected months. Explicit modeling of inter-spe-

cific competition utilizing an extended data setwith combinations of these three species is subjectof our current research efforts on this problem.This paper provides a simple and generic mecha-nistic model of the three-dimensional above-ground shoot density of tropical grass plantssubject to frequent fire disturbance. The model istightly coupled to field data. It shows that con-trasting plant phenology and architecture can becaptured by different parameter values of thesame simple generic model. Thus, the model ispotentially applicable to other grass plants andother grassland ecosystems.

Acknowledgements

We thank Juan Silva, ICAE, Universidad deLos Andes, for useful discussions and comments.Luis Nieto, also of ICAE, provided invaluablehelp while working in the field, and Maria JesusGras, Universidad de Alicante for assistance withphotographs. We wish to thank the Vicerrec-torado of the Universidad de Alicante, for provid-ing travel support for MFA and funding for JRduring his sabbatical year. We also wish to thankthe University of North Texas for providing adevelopment leave of absence to MFA. Furthersupport has been received from NSFs Division ofInternational Programs (Grant INT-0104728).

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Growth dynamics of three tropical savanna grass species: an individual-module modelIntroductionStudy site and data collection methodsModel descriptionProportion by level and metrics of horizontal distributionParameter estimation and simulationResults and discussionSpatial distributionTransient behaviorFinal values and transient RMS error for all metrics

ConclusionsAcknowledgementsReferences

growth dynamics of three tropical savanna grass species: an individual-module model

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