lorentzian model of roots for understory yellow birch and sugar maple saplings

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Journal of Theoretical Biology 246 (2007) 309–322

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Lorentzian model of roots for understory yellow birchand sugar maple saplings

Song Cheng�

Department of Biology, Groupe de Recherche en Ecologie Forestiere Interuniversitaire (GREFi), Concordia University,

7141 Sherebrooke Street West, Montreal, Quebec, Canada H4B 1R6

Received 26 June 2006; received in revised form 19 November 2006; accepted 18 December 2006

Available online 28 December 2006

Abstract

Total 66 small (o50m2), 24 medium (101–200m2) and 36 large (201–500m2) canopy gaps at the three sites of yellow birch (Betula

alleghaniensis Britton) and sugar maple (Acer saccharum Marsh) forests were established in southern Quebec, Canada. Half of the gaps

were covered by 8� 8m2 shading cloths to mimic a closed canopy. From these gaps, 46 understory yellow birch and 46 sugar maple

saplings with different tree ages and sizes were sampled. Single- and multi-variable linear and nonlinear models of root biomass and traits

(root surface area, volume, length and endings) were developed and examined. Lorentzian model as a multi-variable nonlinear model was

firstly applied to the simulations using both base diameter and height, and performed the best fit to total root biomass in both species

with the highest correlation coefficients (R2 ¼ 0:96 and 0.98) and smallest root mean squared deviations (RMSD ¼ 7.85 and 7.02) among

all the examined models. The model also accurately simulated small fine root (p2.0mm in diameter), coarse fine root (42.0–5.0mm) and

coarse root (45.0mm) biomass (R2¼ 0.87–0.99; RMSD ¼ 2.24–6.41), and the root traits (R2

¼ 0.71–0.99; RMSD ¼ 0.19–19.38). The

study showed yellow birch roots were longer, larger, had more endings (tips) and grew faster than sugar maple roots. The root traits were

largely distributed to small fine roots, sharply decreased from small fine roots to coarse fine roots, the fewest in coarse roots except for

root volume. When trees were large, coarse root biomass increased more rapidly than fine root biomass, but vise versa when the trees

were small.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Lorentzian model; Root biomass; Root traits; Sugar maple; Yellow birch

1. Introduction

Study on belowground growth system is more difficultthan aboveground growth system of tree. In most of thelimited studies, only part of the root system is taken intoaccount (Vercambre et al., 2003), due to the difficulties of(1) the opacity of soil, (2) the large volume of colonizedsoil, (3) the fragility of small fine roots (Vercambre et al.,2003). Thus, root models as an important tool have beendeveloping. Usually, single-variable linear and nonlinearmodels have been developed to estimate total root biomass.These models include linear, exponential, power and

e front matter r 2007 Elsevier Ltd. All rights reserved.

i.2006.12.026

87510881; fax: +86 27 87510251.

s: Wuhan Botanical Garden/Wuhan Institute of Botany,

ademy of Sciences, Wuhan 430074, PR China.

ess: [email protected].

logarithm models (Kurz et al., 1996; Kurz and Apps,1999; Vercambre et al., 2003; Li et al., 2003; Olson et al.,2003; Chen et al., 2004).Tree development includes the three phases of

establishment, rapid growth and maturation. Duringthe devolvement, the quantitative relationships amongthe various organ biomass of a tree are changeable(Chen et al., 2004). Root biomass has a strongly linearrelationship with height or diameter for small treesand a nonlinear relationship for large trees (Kolek andKozinika, 1992; Messier and Nikinmaa, 2000; Bolte et al.,2004). Therefore, the performance of the single-variablemodels varies in the different phases. The relationshipsalso change with different tree species (Li et al., 2003), anda changing environment (Chen et al., 2004). For example,the correlation coefficient (R2) is 0.799 for total rootbiomass of matured softwood species; 0.562 for matured

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ARTICLE IN PRESSS. Cheng / Journal of Theoretical Biology 246 (2007) 309–322310

hardwood species in the same forest (Li et al., 2003). Indrought soil, trees have a shorter stem with a deep andlarge root system. Thus, only height or diameter oraboveground biomass variable used for the estimationmay cause wide variations in precision, especially the treesgrowing in a complex changing environment. However, amulti-variable nonlinear model, which may accuratelysimulate root biomass of tree growing in a changingenvironment and the different phases of the development,has not been reported yet.

Very few empirical models have been developed toestimate root traits (Wu et al., 2005). The difficulties arefrom the complexity of root structure (e.g. branch degrees,angles, etc.), the root structural diversity in various species(e.g. drought-tolerant and -intolerant species), and thesensitivity of some root traits to changing abiotic and/orbiotic factors (e.g. mycorrhizal fungi increase or decreaseroot tips). Therefore, it is more difficult to accuratelyestimate root traits in various tree species growing in acomplex changing environment than root biomass (Salaset al., 2004; Dupuy et al., 2005).

The objectives of this study were to develop newmultiple-variable nonlinear models of root biomass andtraits of yellow birch and sugar maple saplings withdifferent ages (phases), tree sizes and light intensity; and toevaluate the quality of the single- and multi-variable linearand nonlinear models.

2. Materials and methods

2.1. Study site

The study was conducted in the Duchesnay Experi-mental Forest Station 461550N, 711400W, near QuebecCity, Canada in 1996, located on a moderate slopeat an elevation between 200 and 300m. The soil was amoder with a humo-ferric podzol underlain by well-drainedglacial till. The mean annual precipitation was 1200mmand the mean daily temperature ranged from 13 in Januaryto 28 1C in July (Environment Canada, 1982). In thestands, the overstory was dominated by sugar maple, beech(Fagus grandifolia Ehrh.) and yellow birch (60%, 20% and15% of merchantable volume, respectively) (Majcen andRichard, 1993). American yew (Taxus canadensis Marsh),mooseberry (Viburnum alnifolium Marsh), and stripedmaple (Acer pensylvanicum L.) grew in the understory.Pin cherry (Prunus pensylvanica L. f.), red-berried elder(Sambucus pubens Michx.) and raspberry (Rubus idaeus L.)were sometimes found in canopy gaps. The experimentaldistrict consisted of three sites in the station. Thesedistances among the sites were about 1 km from eachother with the almost same elevation, vegetationand weather conditions. Under all of the different sizedcanopy gaps, no significant differences in the soil tempera-ture, water, pH and nutrients were found (Cheng et al.,2005).

2.2. Experimental design

Selective cuts in 1988–1989, 1993–1994 and 1994–1995created a gradient of gap sizes (ranging from o50 to500m2), sapling sizes and ages. The gaps were classified aseither small (o50m2), medium (101–200m2) or large(201–500m2). 66 small, 24 medium, and 36 large canopygaps were established at the three sites. In the experiment,half of the plots of each gap size at each site were coveredby 8� 8m2 shading cloths that reduced the light by 50% tomimic a closed canopy of the forests over 4 years beforeharvesting samples.

2.3. Light measurements

The light available to the saplings was measuredduring 4 homogenously overcast days during the monthof July, using the method of Messier and Puttonen (1995);three instantaneous light measurements were taken 5 cmabove the top of the saplings using a LI-189 lightradiometer (LI-COR, Lincoln, Nebraska, USA). At thesame time, a quantum sensor linked to a LI-1000datalogger (LI-COR, Lincoln, Nebraska, USA) recordedlight every minute in an adjacent open area. To calculatethe percentage of total overstory PPFD (photosyntheticphoton flux density) available, the values from above thetrees were divided by the reference values taken at the sametime in the open area. These punctual light measurementshave been shown to be highly correlated with the meanseasonal daily percent PPFD as measured under overcastand clear sky (Gendron et al., 1998). The light measure-ment confirmed all the sampled trees grew in different lightintensity.

2.4. Root sampling

In September 2000, 46 yellow birch and 46 sugar maplesaplings within these gaps were randomly harvested out of apopulation of around 600 individuals that had previouslybeen marked in August 1997. In the samples, their heightwas from 44.5 to 575 cm for yellow birch and 14.5 to 388 cmfor sugar maple; their age ranged from 4 to 18 years foryellow birch and 4 to 23 years for sugar maple whenharvesting; their relative light was from 0.34% to 31.7% foryellow birch, and 0.30% to 35% for sugar maple. Thesampled saplings were scattered among the three sites,canopies with and without the shading cloths. 15, 15 and 16of each species were from site 1, 2 and 3. In the study sites,19, 17 and 10 of each species were from the small, mediumand large gap sizes; 20 and 26 of yellow birch, 21 and 25 ofsugar maple were from non- and shading plots. The generalgrowth information of the samples was shown in Table 1and Figs. 1 and 2. The entire root system of each selectedtree was excavated as much as possible by cautiously diggingout by hand.

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Table 1

Analysis of variance for effects of species and shading (NS ¼ nonshading, SH ¼ shading) treatments and their interaction on root biomass (g), surface

area (dm2), volume (mm3), length (m) and endings (103�no.) in small fine roots, coarse fine roots and coarse roots of yellow birch and sugar maple

Size Traits Species (Sp) Shading (Sh) Interaction

Yellow birch Sugar maple NS SH Sp�Sh (p)

Total root biomass 28.40377.352 20.91677.274 38.268*77.690 11.05076.915 0.785

Small fine roots Area 11.68071.690 7.26771.692 12.460*71.768 6.46671.590 0.944

Volume 29.21474.608 20.09474.559 34.200*74.820 15.10874.334 0.864

Length 32.38874.528 21.39574.481 35.855*74.737 17.92874.260 0.971

Endings 18.775*72.657 5.12072.629 14.73272.780 9.16372.500 0.792

Biomass 9.67171.648 7.29671.631 11.935*71.723 5.03271.550 0.934

Coarse fine roots Area 2.604*70.380 0.82470.376 2.437*70.373 0.99170.336 0.397

Volume 17.051*72.533 5.84872.506 15.981*72.649 6.91872.382 0.597

Length 2.782*70.357 0.79770.353 2.504*70.373 1.07570.336 0.327

Endings 3.113*70.501 0.94870.496 2.943*70.524 1.11970.471 0.372

Biomass 5.22471.047 3.58271.036 6.560*71.095 2.24670.985 0.851

Coarse roots Area 1.05670.049 0.95670.044 1.62370.461 0.39070.415 0.421

Volume 12.78174.916 28.296*74.864 18.784*75.142 4.29474.624 0.530

Length 0.79170.372 0.74670.368 1.25670.389 0.28170.350 0.345

Endings 0.27370.187 0.19970.185 0.71170.195 0.26170.175 0.110

Biomass 13.50675.091 10.20775.038 19.842*75.325 3.77174.789 0.686

Bold and * are significant difference at 95%. p is the probability. The number of samples is 46 for yellow birch and 46 for sugar maple, and the same

numbers for other tables.

S. Cheng / Journal of Theoretical Biology 246 (2007) 309–322 311

2.5. Investigated traits

The root systems of the sampled trees were generallydivided into three classes: small fine roots (p2.0mm indiameter), coarse fine roots (42.0–5.0mm) and coarseroots (45mm) in the study. For each of the diameterclasses, the following root traits were measured: (1) surfacearea (dm2), (2) length (m), (3) number of endings (no.), and(4) volume (mm3) as well as (5) root biomass (g). Total rootbiomass (g) were calculated, tree height (cm), base diameter(cm) and tree age (year) were also recorded. To simulatethe distributions of the root traits within the root system,the small fine roots were further divided into the threesubclasses: 0.0–0.5,40.5–1.0,41.0–2.0mm in diameter bythe approach of Cheng et al. (2005).

2.6. Root measurements

After fully harvesting the whole root system of each tree,the roots were delivered to the field lab. and were washedfree of soil using a watering hose. Small roots weresupported on a 32-mesh sieve during washing. The rootswere sorted into the three diameter classes within five hoursafter the harvesting, using a digital caliper. Five freshsamples of the mid-diameter roots in each size categorywere selected to measure root surface area, volume, lengthand numbers of endings, using a McRhizo system (RegentInstruments Inc., Quebec City, Quebec, Canada). Forsmall trees (total root biomass o50 g), the used sub-samples were 70–85% of total roots, around 10–20% forlarge trees (total root biomass4200 g). For fine roots ofboth species, the endings were root tips. But for coarse

roots, the McRhizo system identified all small fibrousroots, herringbone or dichotomous architecture as the roottips. Thus, the endings were not precise in coarse roots. Ineach of the three root classes, the biomass was measuredusing a digital balance after drying in an oven at 70 1C for48 h; each of the traits (P) (root surface area or length orendings or volume) was calculated as the followingequation (Cheng et al., 2005):

P ¼TDW

SDW

� �VSP, (1)

where TDW was the total dry weight in that diameter class;SDW was the dry weight of the five mid-diameter samplesin the same class; VSP was the value of each root trait inthe sub-samples. To estimate the age of the tree, the ringson the basal disk of each tree were counted using amicroscope.

2.7. Data analyses and model development

Analysis of variance was used to examine effects ofspecies, shading treatments and their interaction on rootbiomass and traits in both species. Stepwise regressionanalysis was used to examine effects of height and basediameter on total root biomass. Single- and multi-variablelinear and nonlinear regression models were developed toinvestigate relationships between total root biomass, smalland coarse fine root biomass, coarse root biomass and roottraits, height and base diameter. All the variables weregraphically examined for the normality of their distribu-tion, using histograms, and for homogeneity of varianceusing scatter plots. All of the data were tested and satisfied

ARTICLE IN PRESS

Age (A) in years

3 9 12 15 18 21 24

Dia

met

er (

D)

in c

enti

met

ers

0

1

2

3

4

5

Hei

gh

t (H

) in

cen

tim

eter

s

0

200

400

600

800

HYB = -51.090 + 29.630A R2 = 0.69 ( p = 0.023)

HSM = -48.140 + 13.840A R2 = 0.53 ( p = 0.039)

DYB = -0.195 + 0.229A R2= 0.59 ( p = 0.031)

DSM = -0.083 + 0.107A R2= 0.46 ( p = 0.043)

6

a

b

Fig. 2. Linear regression models for the relationships of height (H) or base

diameter (D) in yellow birch (YB ¼K and solid line) and sugar maple

(SM ¼ J and dash line) with tree age (A).

Hei

gh

t (H

) in

cen

tim

eter

s

0

100

200

300

400

500

600

Relative light (L) in %

0 10 15 20 25 30 35

Dia

met

er (

D)

in c

enti

met

ers

0

1

2

3

4

HYB = 36.730 + 15.650L

R2 = 0.75 ( p = 0.011)

HSM = 39.070 + 8.892L

R2 = 0.74 ( p =0.015)

DYB = 0.497 + 0.120L

R2 = 0.70 ( p = 0.019)

DSM = 0.572 + 0.075L

R2 = 0.64 ( p = 0.028)

a

b

5

Fig. 1. Linear regression models for the relationships of height (H) or base

diameter (D) in yellow birch (YB ¼K and solid line) and sugar maple

(SM ¼ J and dash line) with light gradient (L). R2 is the correlation

coefficient (a ¼ 0.05). p is the probability. The number of samples is 46 for

yellow birch, 46 for sugar maple. The sampled numbers, R2 and p

definitions are as the same as for other figures.

S. Cheng / Journal of Theoretical Biology 246 (2007) 309–322312

the assumptions for the analyses. SigmaPlot (version 7.0)software (Aspire Software International, Leesburg, USA)and SPSS (version 10) statistical software (SPSS Inc.Chicago, USA) were used to perform the analyses; developand evaluate the root models. The following models weredeveloped in the study.

New multi-variables nonlinear models:

Lorentzian model:

M ¼a

½1þ ððD� bÞ=cÞ2�½1þ ððH � f Þ=gÞ2�, ð2Þ

Paraboloid model : M ¼ aþ bDþ cH þ fD2 þ gH2, (3)

Caussian model : M ¼ ae�0:5½ððD�bÞ=cÞ2þððH�f Þ=gÞ2�. (4)

Multi�variables linear model : M ¼ aþ bDþ cH, (5)

where M was the total root biomass, D was the basediameter, H was the height, a, b, c, f and g were thecoefficients of the equations; and e was the exponential.The above models simulated the relationship of total rootbiomass with both the diameter and height. The modelwith the highest correlation coefficient (R2) and smallestroot mean squared deviation (RMSD) was selected as thebest fit to the observed data (Kobayashi and Salam, 2000),then was further used to examine the relationships of smalland coarse fine root biomass, coarse root biomass, and theroot traits with D and H.Single-variable nonlinear models

Power model : MðiÞ ¼ aþ bMC , (6)

Exponential model : M ðiÞ ¼ aþ becM , (7)

Sigmoidal model : M ðiÞ ¼ aþb

1þ e�ððM�cÞ=f Þ, (8)

where M and e were defined as the same as the above, M(i)

was the biomass of coarse roots (CR) or coarse fine roots

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0 1 23456 78

Are

a (d

m2)

0

2

4

6

8

0 1 23456 78

Len

gth

(m

)

0

5

10

15

0 1 23456 78

En

din

gs

(10

3 x

no

.)

0

2

4

6

8

a

c

d

e

b

Small fine roots Coarse fine roots Coarse roots

Root diameter (X) in milimeters

0 1 2 3 4 5 6 7 8

Bio

mas

s (g

)

0

4

8

12

Vo

lum

e (m

m3)

0

15

30

45

Fig. 3. The distributions of root surface area (dm2), volume (mm3), length

(m), endings (103�no.) and biomass (g) within the root systems of yellow

birch (YB ¼ solid line) and sugar maple (SM ¼ dash line) saplings. The

small fine roots are divided into subclasses: 0.0–0.5, 40.5–1.0 and

41.0–2.0mm. The equations of all the curves are in Table 2.


(CF) or small fine roots (SF), a, b, c and f were thecoefficients of the equations. Models 6–8 were developed totest the relationships of the three-sized root biomass withtotal root biomass, and their R2 and RMSD values wereused to determine the best-fit model:

Log normal model : P ¼ ae�b½lnðcX Þ�2 , (9)

Logarithm model : V ¼ aþ b ln X , (10)

Exponential growth model : M 0 ¼ aebX , (11)

where P was the root surface area, or length, or endings; V

was the root volume, M0 was the root biomass at a givenroot diameter, X was the root diameter. Models 9–11 wereused to simulate the distributions of root surface area,length, volume and endings with X within the root systemof each species.

The RMSD is one of the goodness-of-fit criterions toevaluate the predictive quality of a mode by quantifying themean difference between simulation and measurement(Kobayashi and Salam, 2000). It is defined as

RMSD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n

Xn

i¼1

ðx1 � y1

sÞ2. (12)

The smaller the RMSD is in comparison to measure-ments, the better the goodness-of-fit is (Kobayashi andSalam, 2000; Lobit et al., 2003).

3. Results and discussion

3.1. Differences in both species roots

Root traits vary in both species and different sized rootsof the same species. Generally, yellow birch roots werelonger and larger, had more endings than sugar mapleroots (Table 1, Fig. 3). These traits are asymmetricallydistributed within the root system (Eqs. in Table 2). Fig. 3showed the largest amounts of root surface area, lengthand endings were within the small fine roots. These peakvalues appeared between 0.8 and 1.5mm diameter roots.The amounts of the traits critically decreased from thesmall fine roots to the coarse fine roots. That indicate smallfine roots play a major role in soil resource uptake(McCully, 1990; Berntson, 1994; Dupuy et al., 2005). Oneof the reasons is the membranes of small fine roots and tipsare permeable to mineral nutrients and water, but thepermeability gradually loses with increasing the diameter,especially from coarse fine roots to coarse roots due to theformation of phellem on the root surface (McCully, 1990).Change in root absorptive ability is associated with abioticand biotic factors. Plant competition from other indivi-duals, as a biotic factor, reduces the fine roots of citrustrees (Notzold et al., 1998; Borowicz et al., 2005),sequentially reduces mineral nutrient content in the rootsand leaves (Borowicz et al., 2005). Abiotic factors alsochange the absorptive ability, for example, 20 1C soil

temperature is optimal temperature for root growth ofLactuca sativa L. cv. Panama. In the temperature, themineral content in the root and leaf tissue is higher thanbelow and above 20 1C (Tan et al., 2002). In the study, thegrowth environment for both tree species was thestatistically same. The differences of root traits imply bothspecies have the different ability of soil resource uptake(Comas et al., 2002; Vercambre et al., 2003; Comas andEissenstat, 2004). Yellow birch may have the strongerability than sugar maple.Root spatial heterogeneity in both species increases with

root development. By observation, around 90% of totalroots in small trees (total root biomass o50 g) in bothspecies grew within 20 cm depth from soil surface, wheremineral nutrients were rich. In large roots (e.g. 4200 g),approximating 65% of total roots for yellow birch and70% for sugar maple were within the layer of soil. As rootdevelopment, fine and coarse roots were expanded to deepfrom topsoil, forming a gradient of the vertical rootdistribution. More roots of yellow birch were distributed tothe deeper soil than sugar maple.Root growth varies in both species. Yellow birch roots

grow rapidly. The average age of yellow birch and sugarmaple samples was 8.9- and 11.5-year old. Aboveground,height and diameter of yellow birch grew more rapidly than

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Table 2

Single-variable nonlinear modes for simulating the distributions of root

surface area in dm2, volume in mm3, length in m, endings in 103�no. and

biomass (M0) in g with changing root diameter (X) in mm within the root

systems of yellow birch (YB) and sugar maple (SM)

Log normal model, logarithm model,

exponential growth model

R2 p

AareðYBÞ ¼ 5:365e�0:665½lnð0:803XYBÞ�2

(13) 0.99 o0.000

AareðSMÞ ¼ 3:924e�1:059½lnð0:850XSM Þ�2

(14) 0.92 o0.000

VolumeðYBÞ ¼ 10:360þ 4:149 lnX YB (15) 0.89 o0.000

VolumeðSMÞ ¼ 3:997e0:313XSM (16) 0.92 o0.000

LengthðYBÞ ¼ 15:680e�1:232½lnð0:868XYBÞ�2

(17) 0.97 o0.000

LengthðSMÞ ¼ 13:030e�2:032½lnð0:878XSM Þ�2

(18)

0.99 o0.000

EndingsðYBÞ ¼ 2:417e�0:843½lnð0:852XYBÞ�2

(19) 0.91 o0.000

EndingsðSMÞ ¼ 2:738e�0:882½lnð0:799XSM Þ�2

(20)

0.98 o0.000

M 0ðYBÞ ¼ 2:207e0:243XYB (21) 0.94 o0.000

M 0ðSMÞ ¼ 1:859e0:243XSM (22) 0.90 o0.000

R2 is the correlation coefficient (a ¼ 0.05). When p is o0.000, it means the

probably is between 0.000 and o0.0005. YB, SM and p are defined as the

same as for other tables.

05 0 1 00 1 50 3 0 0

Coar

se r

oot

bio

mas

s in

gra

ms

0

50

100

150

200

250

Total root biomass in grams

0 50 100 150 200 250 300

Fin

e r

oot bio

mas

s in

gra

ms

0

15

30

45

60 Small fine roots

Coarse fine roots

Yellow birch

b

a

Fig. 4. Sigmoidal model for the relationships of small fine root biomass (M(Spec

dash–dot line) and coarse root biomass (M(Species_CR) ¼ m and solid line) with t

The equations of the curves are in Table 5.


that of sugar maple (Figs. 1 and 2). Belowground, thelarger and longer roots in yellow birch indicated thatyellow birch roots developed faster than sugar maple roots.The results agree with the view of Comas et al. (2002) thatis a fast-growing tree has longer, more fine roots than aslow-growing tree. The fast-growing ability of roots isassociated with plant physiological characteristics (Fieldand Mooney, 1983; Hunt et al., 1985; Sheriff et al., 1986;Kozlowski et al., 1991; Noland et al., 1997; Cavender-Bares et al., 2000; Rasse 2002). Compared to sugar mapleas a typical shade-tolerant species (Canham, 1989), yellowbirch, as a moderately shade-tolerant species (Barker,1949), has higher photosynthetic, transpiration and re-spiration rates (Walters and Reich, 2000). To satisfy itsphysiological requirements and support the rapid above-ground growth, understory yellow birch needs to take upmore soil resources by the rapid development of the rootsystem (Kinney and Lindroth, 1997).Increased total root biomass is heterogeneously

allocated to various sized roots. When both speciessaplings were small, increased total root biomasswas mainly invested to small fine roots for survivaland establishment of the trees, coarse roots developedslowly. When the trees grew large, the coarse rootbiomass increased the most rapidly, followed by smalland coarse fine root biomass (Fig. 4). For example,fine root biomass of yellow birch was around 40%, 30%and 18% of total root biomass of 50, 150 and 300 g.Fine roots are responsible for exploitation, whereas coarseroots are linked with soil exploration, and thereforeall sized roots continuously increase. For large trees,the coarse root biomass increased rapidly to establish a

05 0 1 00 1 5 0 50

Coar

se r

oot bio

mas

s in

gra

ms

0

50

100

150

200

250

300

Total root biomass in grams

0 50 100 150 200 250 300 350

Fin

e ro

ot bio

mas

s in

gra

ms

0

10

20

30

40

50

Sugar maple

Coarse fine roots

Small fineroots

c

d

ies_SF) ¼K and dash line), coarse fine root biomass (M(Species_CF) ¼ J and

otal root biomass (M) of yellow birch (YB) and sugar maple (SM) saplings.

ARTICLE IN PRESS

Table 3

Stepwise (forward) linear regression analyses for effects of base diameter

(D) in cm and tree height (H) in cm on total root biomass (M) in g of

yellow birch (YB) and sugar maple (SM) saplings

Equation R2

M(YB) ¼ �37.342+42.586 (DYB) p (o0.000) 0.76 (o0.000)

M(YB) ¼ �34.193+51.480 (DYB)�0.098 (HYB) p 0.78 (o0.000)

M(SM) ¼ �28.517+0.451 (HSM) p (o0.000) 0.53 (o0.000)

M(SM) ¼ �29.566+0.422 (HSM)+3.518 (DSM) p 0.53 (o0.000)

D, H and M are defined as the same as for other tables.

0

100

200

300

400

500

600

0

1

2

3

4

5

100

200

300

400

500

Tota

l ro

ot

bio

mas

s (M

) in

gra

ms

Diameter (D) in

centimeters

Height (H) in centimeters

0

100

200

300

400

500

0.0

0.5

1.0

1.52.0

2.53.0

3.5

50100

150200

250300

350

Tota

l ro

ot

bio

mas

s (M

) in

gra

ms

Diamete

r (D) i

n centim

eters

Height (H) in centimeters

Sugar maple

a

b

Yellow birch

Fig. 5. Lorentzian model for the relationships of total root biomass (M)

with base diameter (D) and height (H) in yellow birch (YB) and sugar

maple (SM) saplings. The equations are in Table 6.


strong root skeleton for physically supporting the above-ground growth and storing carbohydrates (e.g. starch,sugar, etc.) for further growth or acclimating to a changingenvironment. The results suggest that increase in theabsorptive ability of small fine roots depends on not onlyincreasing the root biomass but also changing someunknown biological metabolisms, which will be furtherstudied for biologists.

3.2. Lorentzian model of simulating total root biomass using

both diameter and height

Height and diameter have a different relationshipof total root biomass. Stepwise (forward) regression

Estimated total root biomass (E) in grams

0 50 100 150 200 250 300

Obse

rved

tota

l ro

ot

bio

mas

s (M

) in

gra

ms

0

50

100

150

200

250

300

Estimated total root biomass (E) in grams

0 50 100 150 200 250 300 350

Obse

rved

tota

l ro

ot

bio

mas

s (M

) in

gra

ms

0

50

100

150

200

250

300

350

a

b

Yellow birch

Sugar maple

MYB= -2.471 + 1.023EYB

R2= 0.96 ( p <0.000)

MSM = 0.451 + 0.997ESM

R2= 0.98 ( p <0.000)

Fig. 6. Linear regression models for the relationships between the

estimated (E) and observed total root biomass (M) of yellow birch (YB)

and sugar maple (SM) saplings.


analyses showed that total root biomass had a strongerrelationship with base diameter for yellow birch, butwith height for sugar maple (Table 3), similar resultshave been found by Chen et al. (2004). Therefore,the precision of the single-variable model changeswith different tree species. However, multi-variable linearmodel did not improve the precision for both species(Table 3).

Table 4

Multi- and single-variable nonlinear models for simulating total root biomass

birch (YB) and sugar maple (SM)

Lorentzian model

M ðYBÞ ¼729:800

½1þ ððDYB � 5:568Þ=0:998Þ2� � ½1þ ððHYB � 658:900Þ=419:000Þ2�(23

M ðSMÞ ¼5403:000

½1þ ððDSM � 3:852Þ=1:845Þ2� � ½1þ ððHSM � 462:700Þ=22:720Þ2�(24

Paraboloid model

M ðYBÞ ¼ 15:020� 29:910DYB � 0:020HYB þ 15:980D2YB þ 0:00002H2

YB (25

M ðSMÞ ¼ 19:220þ 26:230DSM � 0:794HSM � 5:514D2SM þ 0:003H2

SM (26

Caussian model

M ðYBÞ ¼ 790:900e�0:5½ððDYB�8:478Þ=2:484Þ2þððHYB�5:770�10

6Þ=3:723�108Þ2 � (27

M ðSMÞ ¼ 6:980� 107e�0:5½ððDSM�4:787Þ=2:933Þ2þððHSM�1906Þ=309:900Þ

2 � (28

Exponential model

M ðYBÞ ¼ 6:177e0:781DYB (29

M ðYBÞ ¼ 12:650e0:005HYB (30

M ðSMÞ ¼ 7:386e0:804DSM (31

M ðSMÞ ¼ 0:642e0:016HSM (32

Power model

M ðYBÞ ¼ 3:797D2:693YB (33

M ðYBÞ ¼ 0:013H1:487YB (34

M ðSMÞ ¼ 12:080D1:923SM (35

M ðSMÞ ¼ 6:216� 10�12H5:307SM (36

Logarithm modelM ðYBÞ ¼ 17:550þ 56:07 ln DYB (37

M ðYBÞ ¼ �20:680þ 48:060 ln HYB (38

M ðSMÞ ¼ 22:850þ 46:170 ln DSM (39

M ðSMÞ ¼ �84:180þ 24:290 ln HSM (40

RMSD is the root mean squared deviation and defined as the same as for oth

Lorentzian model, as a multi-variable nonlinear model,highly improved the precision using both base diameterand height variables, and did the best performance for bothspecies (Figs. 5 and 6) with the highest R2 and smallestRMSD values among the three multi-variable nonlinearmodels and the single-variable nonlinear models (Table 4),the linear models did the worst in the study (datanot shown). The new multi-variable nonlinear models

(M) in g using base diameter (D) in cm and/or height (H) in cm of yellow

R2 p RMSD

) 0.96 o0.000 7.85

) 0.98 o0.000 7.02

) 0.94 o0.000 37.38

) 0.84 o0.000 29.14

) 0.95 o0.000 35.66

) 0.96 o0.000 11.73

) 0.94 o0.000 36.87

) 0.38 0.021 42.75

) 0.36 0.025 43.42

) 0.95 o0.000 12.57

) 0.95 o0.000 35.31

) 0.44 0.019 40.12

) 0.43 0.170 40.61

) 0.94 o0.000 36.46

) 0.52 0.012 40.34

) 0.37 0.022 91.08

) 0.27 0.036 58.63

) 0.17 0.041 48.01

er tables.

ARTICLE IN PRESSS. Cheng / Journal of Theoretical Biology 246 (2007) 309–322 317

(Eqs. (23)–(28) in Table 4) performed better fit thanmost of the single-variable nonlinear models. Somesingle-variable nonlinear models also did as good asCaussian and Paraboloid models. For example, exponen-tial and power models estimated total root biomass ofyellow birch using diameter (Eqs. (29), (33) in Table 4) and

Table 5

Single-variable nonlinear models for simulating small fine (SF), coarse fin

M(Species_CR)) in g using total root biomass (M) in g of yellow birch (YB) and

Sigmoidal model: MðSpecies_SF ;CF ;CRÞ ¼ aþ b

1þe�ððM�cÞ=f Þ

Parameter a b cMðYB_SF Þ (41) �9.401 64.080 71.520

S.E. 4.676 6.592 5.904M ðYB_CF Þ (42) �14.650 45.720 41.100

S.E. 11.850 13.890 29.520M ðYB_CRÞ (43) �134.200 159.400 145.800

S.E. 101.1 4627.000 1164.000M ðSM_SF Þ (44) �3381.000 3926.00 �297.500

S.E. 1294.000 1295.000 262.900M ðSM_CF Þ (45) �48.440 88.100 �16.540

S.E. 58.310 59.990 105.700M ðSM_CRÞ (46) �15.670 313.900 196.800

S.E. 4.745 17.400 10.390

Power model: M ðSpecies_SF ;CF ;CRÞ ¼ aþ bMC

Parameter a b cM ðYB_SF Þ (47) �1.157 1.182 0.692

S.E. 1.066 0.345 0.054M ðYB_CF Þ (48) �0.765 0.771 0.664

S.E. 0.737 0.260 0.063M ðYB_CRÞ (49) 0.420 0.151 1.271

S.E. 0.149 0.039 0.051M ðSM_SF Þ (50) �5.208 5.453 0.398

S.E. 1.684 1.261 0.040M ðSM_CF Þ (51) �1.613 1.124 0.617

S.E. 0.820 0.366 0.046M ðSM_CRÞ (52) �0.281 0.060 1.435

S.E. 0.642 0.010 0.027

Exponential model: M ðSpecies_SF ;CF ;CRÞ ¼ aþ becM

Parameter a b cM ðYB_SF Þ (53) �1.378� 105 1.378� 105 2.561� 10

S.E. 2.112� 105 2.112� 105 0.001M ðYB_CF Þ (54) �6.584� 104 6.584� 104 4.504� 10

S.E. 6.488� 104 6.488� 104 0.001M ðYB_CRÞ (55) �137.100 136.500 0.003

S.E. 24.510 23.750 3.675� 10M ðSM_SF Þ (56) �2.391� 105 2.391� 105 3.229� 10

S.E. 2.698� 105 2.698� 105 1.965� 10M ðSM_CF Þ (57) �4083.000 4083.000 2.955� 10

S.E. 1.496� 105 1.496� 105 0.001M ðSM_CRÞ (58) �107.400 104.600 0.004

S.E. 16.690 16.030 3.126� 10

SF, CF, CR, MSpecies_SF, MSpecies_CF , MSpecies_CR and M are defined as the sa

using height for sugar maple (Eqs. (32), (36) in Table 4).However, the precision depended on tree species. TheLorentzian models can also describe root development wellin all various phases. For instance, small yellow birch had alinear relationship of total root biomass with basediameter, and the relationship initially changed to a curve

e (CF) and coarse root (CR) biomass (M(Species_SF), M(Species_CF) and

sugar maple (SM)

R2 p RMSD

f

44.460 0.96 o0.000 2.36

8.239

62.020 0.92 o0.000 2.48

22.260

305.500 0.99 o0.000 3.34

232.700

66.530 0.97 o0.000 1.96

19.730

88.400 0.97 o0.000 1.29

33.670

65.800 0.99 o0.000 1.45

8.349

0.95 o0.000 3.39

0.90 o0.000 2.26

0.99 o0.000 4.13

0.93 o0.000 2.75

0.95 o0.000 1.67

0.99 o0.000 1.46

�5 0.88 o0.000 5.38

�5 0.84 o0.000 3.13

0.99 o0.000 4.64

�4

�5 0.68 o0.000 7.47

�3

�5 0.86 o0.000 2.94

0.99 o0.000 3.28

�4

me as for other tables. S.E. is the standard error.

ARTICLE IN PRESS

Table 6

Lorentzian model for simulating biomass of small fine roots, coarse fine roots and coarse roots (M(Species_SF), M(Species_CF) and M(Species_CR)) in g using

base diameter (D) in cm and height (H) in cm of yellow birch (YB) and sugar maple (SM)

Lorentzian model: M ðSpecies_SF ;CF ;CRÞ ¼a

½1þððD�bÞ=cÞ2 ��½1þððH�f Þ=gÞ2 �

Parameter a b c f g R2 p RMSDM ðYB_SFÞ (59) 156.600 5.283 1.489 831.300 401.200 0.92 o0.000 3.64

S.E. 131.700 8.570 0.295 545.000 136.400M ðYB_CF Þ (60) 40.490 4.812 1.299 469.500 352.500 0.87 o0.000 2.60

S.E. 13.530 7.540 0.401 429.300 102.200M ðYB_CRÞ (61) 1555.000 5.825 0.522 666.700 378.300 0.97 o0.000 6.41

S.E. 1476.000 1.052 0.492 521.000e 254.600M ðSM_SF Þ (62) 53.99 3.490 1.474 463.600 256.300 0.91 o0.000 3.12

S.E. 15.300 0.143 0.266 214.000 165.480M ðSM_CF Þ (63) 3757.000 3.625 1.371 563.200 20.400 0.91 o0.000 2.24

S.E. 800.400 3.042 0.437 283.200 18.000M ðSM_CRÞ (64) 9120.000 3.693 1.009 429.600 9.560 0.99 o0.000 2.83

S.E. 12990.000 1.970 0.233 298.300 4.497

S.E. is the standard error.

Are

a (d

m2)

5

10

15

20

25

30

Volu

me

(mm

3)

30

60

90

120

150

Len

gth

(m

)

30

60

90

120

0 5 10 15 20 25 30 35

0

3

0

6

9

12

0

20

0

40

60

80

0

2

0

4

6

8

10

12

2

4

6

8

10

12

30

60

90

120

150

2

4

6

8

Area = 1.968 + 1.078MSF

R2 = 0.95 (p < 0.000)

Volume = 3.413 + 2.816MSF

R2 = 0.96 (p < 0.000)

Area = 0.016 + 0.554MCF - 0.004MCF2

R2 = 0.90 ( p < 0.000)

Volume = 0.318 + 3.645MCF - 0.025MCF2

R2 = 0.91 (p < 0.000)

Length = 0.158 + 0.589MCF - 0.006MCF2

R2 = 0.92 ( p < 0.000)

Area = 0.327 + 0.054MCR

R2 = 0.96 ( p < 0.000)

Volume = 2.737 + 0.742MCR

R2 = 0.97 (p < 0.000)

Length=0.285+0.034MCR

R2=0.92( p<0.000)

Length =9.540 + 2.496MSF

R2 = 0.86 ( p < 0.000)

0 10 20 30 40 50

Endin

gs

(10

3 x

no.)

0

30

60

90

120 Endings = 3.213 + 2.045MSF

R2 = 0.73 ( p < 0.000)

0 5 10 15 20 25 30 35

4

8

12

16

20 Endings = - 0.157 + 0.687MCF - 0.003MCF2

R2 = 0.85 (p < 0.000)

Coarse root biomass (MCR) in gramsCoarse fine root biomass (MCF) in gramsSmall fine root biomass (MSF) in grams

0 30 60 90 120 150 180 210

1

2

3

4

5

Endings = 0.082 + 0.021MCR

R2 = 0.79 ( p < 0.000)

a

b

c

d

e

f

g

h

i

j

k

l

Fig. 7. Linear and quadratic regression models for the relationships of root surface area (dm2), volume (mm3), length (m), endings (103�no.) with small

fine (SF), coarse fine (CF) and coarse root (CR) biomass (MSF, MCF and MCR) of yellow birch saplings.


ARTICLE IN PRESSS. Cheng / Journal of Theoretical Biology 246 (2007) 309–322 319

when the diameter increased above around �1.5 cm(Fig. 5a).

Based on the R2 and RMSD values, Sigmoidal, powerand exponential models obtained the accurate simulationsof the three-sized root biomass using total root biomass(Table 5). Among the single-variable models, the Sigmoidalmodels performed the best fit in the similar studies(Vercambre et al., 2003; Olson et al., 2003; Chen et al.,2004). However, total root biomass is more difficultlyavailable than height and diameter variables, especially inthe field. Compared to the RMSD and S.E. values(Table 6), the quality of the Lorentzian models wassimilar with that of the Sigmoidal and power models.The Lorentzian models are more convenient to be appliedto the simulations.

Both species were sampled from various ages,tree sizes and light intensity created by canopy gapsand shading cloths. Each of the abiotic and bioticfactors significantly affected both species growth(Table 1, Figs. 1 and 2). As a result, the belowgroundspatial heterogeneity is large and the root development ishighly variable (Li et al., 2003). Generally, the single-variable nonlinear and linear models of roots have a greaterror in many cases. The high-quality Lorentzian model is

Are

a (d

m2)

5

10

15

20

25

Volu

me

(mm

3)

30

60

90

120

Len

gth

(m

)

30

60

90

120

Area = 0.917 + 0.847MSF

R2 = 0.96 (p<0.000)

Volume = 0.583+2.608MSF

R2 = 0.97( p < 0.000)

Length = 1.715 + 2.631MSF

R2 = 0.97( p < 0.000)

0 10 20 30 40

Endin

gs

(10

3 x

no.)

0

10

20

30 Endings = 0.153 + 0.663MSF

R2 = 0.98 ( p < 0.000)

2

4

6

8

20

40

60

2

4

6

8

0 10 2

20

40

60

80

100

Area = 0.105 + 0.

R2 = 0.95 ( p < 0.0

Volume = 0.624 +

R2 = 0.97 ( p < 0.0

Length = 0.117 +

R2 = 0.95 ( p < 0.0

Endings = 0.838 +

R2 = 0.95 (p < 0.

Coarse fine root bioSmall fine root biomass (MSF) in grams

a

b

c

d

e

f

g

h

Fig. 8. Linear regression model for the relationships of root surface area (dm2)

fine (CF) and coarse root (CR) biomass (MSF, MCF and MCR) of sugar maple

applied to estimate the root biomass both widely andprecisely.

3.3. Lorentzian model of simulating root traits using both

diameter and height

The three-sized root biomass were more stronglyassociated with their root traits than height anddiameter in both species based on the R2 values only(Figs. 7 and 8, Tables 7 and 8). However, each ofthe root traits in the three-sized roots had a signi-ficant Lorentzian’s relationship of both height anddiameter in yellow birch and sugar maple (Tables 7and 8). The RMSD values of the Lorentzian models(Tables 7 and 8) were similar with that of the linearand quadratic (Figs. 7e–h) models (the RMSD valuesnot shown for Figs. 7 and 8). It indicates the qualityof the Lorentzian models is similar with that of thesemodels.Summarily, both height and diameter have the strongest

Lorentzian’s relationship of total root biomass, followedby the three-sized root biomass and root traits in bothspecies. There might be some possible reasons for thevariations: (1) the changing environment more strongly

0 30 40

5

10

15

20

25

50

100

150

200

250

5

10

15

20

0 50 100 150 200 250 300

20

40

60

80

216MCF

00)

1.573MCF

00)

0.204MCF

00)

2.598MCF

000)

Endings = 4.390 + 0.304MCR

R2 = 0.58 ( p < 0.000)

Area = 0.045 + 0.091MCR

R2 = 0.99 ( p < 0.000)

Volume = 0.349 + 0.997MCR

R2 = 0.99 ( p < 0.000)

Length = 0.014 + 0.078MCR

R2 = 0.99 ( p < 0.000)

Coarse root biomass (MCR) in gramsmass (MCF) in grams

i

j

k

l

, volume (mm3), length (m), endings (103�no.) with small fine (SF), coarse

saplings.

ARTICLE IN PRESS

Table 7

Lorentzian model for simulating root surface area, volume, length, endings in small fine (SF), coarse fine (CF) and coarse roots (CR) using base diameter

(D) in cm and height (H) in cm of YB

Lorentzian model R2 p RMSD

AreaðYB_SF Þ ¼4108:000

½1þ ððDYB � 5:909Þ=1:775Þ2� � ½1þ ððHYB � 1263:000Þ=132:100Þ2�(65) 0.84 o0.000 4.55

VolumeðYB_SF Þ ¼623:500


LengthðYB_SF Þ ¼10050:000


EngdingsðYB_SF Þ ¼44950:000


AreaðYB_CF Þ ¼1064:000


VolumeðYB_CF Þ ¼493:100


LengthðYB_CF Þ ¼27:180


EngdingsðYB_CF Þ ¼2710:000


AreaðYB_CRÞ ¼1209:000


VolumeðYB_CRÞ ¼4385:100


LengthðYB_CRÞ ¼5914:000


EngdingsðYB_CRÞ ¼2140:000



influenced the root traits and fine root biomass than thetotal root biomass; (2) some of the finest roots lost in soilwhen being harvested.

In the study, only part of conventional or typical modelsof roots was tested and discussed. It was difficult to test allof them. Some conventional models may also simulate rootbiomass and traits well.

4. Conclusion

Lorentizen model is a high quality of multi-variablenonlinear model that can precisely simulate root biomassand traits for understory yellow birch and sugar maplesaplings.

Soil environment also influences the root develop-ment of tree. For example, tree has a superficial rootsystem in high soil texture or high water table with low soil

aeration. It is suggested for future research: whether theLorenztian models will perform well for overstory yellowbirch and sugar maple; or other tree species, includingsoftwood species, when they grow in different soilenvironments.

Acknowledgments

This study was supported by Concordia UniversityGraduate Fellowship and NSERC PCS A to S. Cheng, andNSERC strategic grant to Christian Messier (PI). The authoris also grateful to Christian Messier and Paul Widden forproviding the research opportunity, and anonymous viewersfor their constructive comments on the paper, and JoelCoburn, Sylvain Delagrange, Marie-Helene Croisetiere, JulieM. Richard, Nathalie Bourdonnais-Spear, Jocelyn Poissant,

ARTICLE IN PRESS

Table 8

Lorentzian model for simulating root surface area, volume, length, endings in small fine (SF), coarse fine (CF) and coarse roots (CR) using base diameter

(D) in cm and height (H) in cm of SM

Lorentzian model R2 p RMSD

AreaðSM_SF Þ ¼47:950

½1þ ððDSM � 3:366Þ=1:449Þ2� � ½1þ ððHSM � 459:100Þ=268:900Þ2�(77) 0.93 o0.000 2.35

VolumeðSM_SF Þ ¼138:400


LengthðSM_SFÞ ¼143:500


EndingsðSM_SF Þ ¼35:430

½1þ ððDSM � 3:359Þ=1:394Þ2� � ½1þ ððHSM � 429:200=240:9ÞÞ2�(80) 0.86 o0.000 1.68

AreaðSM_CF Þ ¼2449:000


VolumeðSM_CF Þ ¼15340:000


LengthðSM_CF Þ ¼2187:000


EndingsðSM_CF Þ ¼24890:000


AreaðSM_CRÞ ¼2247:000


VolumeðSM_CRÞ ¼27570:000


LengthðSM_CRÞ ¼1690:000


EndingsðSM_CRÞ ¼20310:000

½1þ ððDSM � 3:137Þ=0:195Þ2� � ½1þ ððHSM þ 608:000Þ=11:120Þ2�(88) 0.89 o0.000 7.69


Mario Bonneau, Alexandre Piboule, Johanne Campbell andRebecca Tittler for assistance in the field.

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