s u p p o r t i n g i n f o r m at i o n f o r

plant: A package for modelling forest trait ecology & evolution: Plant physiological model Daniel S. Falster, Richard G. FitzJohn and Mark Westoby Department of Biological Sciences, Macquarie University, Sydney, Australia

[email protected], [email protected]

The core job of the physiological model in p l a n t is to take a plant’s current size, light environment and physiological parameters as inputs, and return it’s growth, mortality and fecundity rates. In the default physiological model within p l a n t, these vital demographic rates are all derived from the rate at which living biomass is produced by the plant, which in turn is calculated based on well understood physiology (Fig. 1). Various physiological parameters influence demographic outcomes. Varying these parameters allows for speciesdifferences to be included, potentially via traits (see last section). Tables 1 and 3 gives units and definitions of all variables and parameters. Light intensity

Plant height leaf mass sapwood mass bark mass root mass

Mass budget

photosynthesis respiration of leaf, sapwood, bark & roots turnover of leaf,sapwood, bark & roots


Fecundity Reproduction

Net mass production Mass growth Allocation

(~ leaf area)

Height growth rate

Tissue costs

Fi g u r e 1 : Physiological model in p l a n t, giving demographic rates on the basis of its traits, size and light environment. as functions of net mass production. The dashed line to mortality indicates that although mortality rate is assumed to depend on mass production, no mass is actually allocated there.


l eaf photosynthesis

Let p( x, E) denote the gross rate of leaf photosynthesis per unit area at height within the canopy of a plant with traits x at light level E(z). We assume a relationship of the form p( x, E(z)) =

cp1 E(z) + cp2





s u p p o r t i n g i n f o r m at i o n

for the average of p across the year. The parameters cp1 , cp2 are derived from a detailed leaf-level model and correspond to maximum annual photosynthesis and light-level where photosynthesis is 50% of it’s maximum value. The average rate of leaf photosynthesis across the plant is then p¯ ( x, h, Ea ) =

Z h 0

p( x, E(z)) q(z, h),


where q(z, h) gives the density of leaf area at height z (Eq. 13).


s tandard model for mass production

The amount of biomass available for growth, db/dt, is given by the difference between income (total photosynthetic rate) and losses (respiration and turnover) within the plant (Mäkelä, 1997; Thornley & Cannell, 2000; Falster et al., 2011): db dt |{z}


net biomass production

 y al p¯ − mi ri − ∑ mi k i . ∑ |{z} |{z} i=l,b,s,r i=l,b,s,r | {z } {z } yield photosynthesis | turnover respiration


Here, m, r, and k refer to the mass, maintenance respiration rate, and turnover rate of different tissues, denoted by subscripts l=leaves, b=bark, s=sapwood and r=roots. A is the assimilation rate of CO2 per leaf area and y is yield: the fraction of assimilated carbon fixed in biomass (the remaining fraction being lost as growth respiration). The growth respiration component comes in addition to any costs of maintenance respiration. Gross photosynthetic production is proportional to leaf area, al = ml /φ, where φ is leaf mass per area. The total mass of living tissues ma = ml + mb + ms + mr .


h eight growth rate

The key measure of growth required by the demographic solver is rate of height growth for the plant, g( x, h, Ea ). To model growth in height requires that we translate mass production into height increment, accounting for the costs of building new tissues, allocation to reproduction, and architectural layout. Mathematically, height growth can be decomposed into a product of physiologically relevant terms (Falster et al., 2011): g( x, h, Ea ) =

dal dh dh dma db = × × × . dt dal dma db dt


The first term on the right of eq 4, dh/dal , is the growth in plant height per unit growth in total leaf area – accounting for the architectural strategy of the plant. Some species tend to leaf out more than grow tall, while other species emphasise vertical extension. The second term in eq 4, dal /dma , accounts for the marginal cost of deploying an additional unit of leaf area, including construction of the leaf itself and various support structures. As such, dal /dma can itself be expressed as a sum of construction costs per unit leaf area: dmb dal dms dmr −1 = φ+ + + . dma dal dal dal



s u p p o r t i n g i n f o r m at i o n


The third term in eq 4, dma /dmb , gives the fraction of net biomass production (eq. 3) that is allocated to growth rather than reproduction or storage. In the default physiological model we let the growth fraction decrease with height according to the function dma cr1 (h) = 1 − , db 1.0 + exp (cr2 (1.0 − h/hmat ))


where cr1 is the maximum possible allocation (0-1) and cr2 determines the sharpness of the transition.


d iameter growth rate

In a similar way, basal area (ast ) increment can be expressed as the sum of increments in sapwood, bark & heartwood areas (as , ab , ah respectively): da da das dast = b+ + h. dt dt dt dt We now have an equation for basal area growth that contains many of the same elements as eq. 4:   da dal das dast dma db dah + b × = × × + . (7) dt dal dal dma db dt dt Diameter growth is then given by the geometric relationship between stem diameter (D) and ast : r dD π dast = . (8) dt ast dt


a functional -balance model for allocation

Here we describe an allometric model linking the various size dimensions of a plant required by most ecologically realistic vegetation models (i.e. =mass of leaves, mass of sapwood, mass of bark, mass of fine roots) to a plant’s height. This approach allows us to track only the plant’s height, while still accounting for the mass need to build leaves, roots, and stems. The growth rates of various tissues can then also be derived (Table 2). 5. 1

Leaf area

Based on empirically observed allometry, we assume an allometric log-log scaling relationship between the accumulated leaf area of a plant and its height: al = α1 h β 1 .


Note, scaling relationship reversed from (Falster et al., 2011). 5. 2

Vertical distribution of leaf area

We follow the model of Yokozawa & Hara (1995) describing the vertical distribution of leaf area within the crowns of individual plants. This model can account for a variety of canopy profiles through a single parameter η. Setting η = 1 results in a conical canopy, as seen in many conifers, while higher values, e.g. η = 12 , gives a top-weighted canopy profile similar 3


s u p p o r t i n g i n f o r m at i o n

to those seen among angiosperms. Let as (z) be the sapwood area at height z for a plant with top height h, as (= as (0)) be the sapwood area at the base of the plant, q(z, h) the probability density of leaf area at height z and Q(z, h) the cumulative fraction of a plant’s leaf above height z. Following Yokozawa & Hara (1995) we assume a relationship between as (z) and height such that   z  η 2 as ( z ) . (10) = 1− as h We also assume that each unit of sapwood area supports a fixed area of leaf (the pipe model Shinozaki et al., 1964), so that the total canopy area of a plant relates to basal sapwood area as : ml = θ as . (11) φ The pipe model is assumed to hold within individual plants, as well as across plants of different size. It then follows that   z  η 2 . (12) Q(z, h) = 1 − h Differentiating with respect to z then yields a solution for the probability density of leaf area as a function of height:  z  η   z  η −1 η q(z, h) = 2 1− . (13) h h h 5. 3

Mass of sapwood

Integrating as (z) gives a solution for the total mass of sapwood in the plant: ms = ρ

Z h 0

as (z) dz = ρ as h ηc ,


where ηc = 1 − 1+2 η + 1+12η (Yokozawa & Hara, 1995). Substituting from Eq. 11 into Eq. 14 then gives an expression for sapwood mass as a function leaf area and height: ms = ρ ηc θ al h. 5. 4


Bark mass

Bark and phloem tissue are modelled using an analogue of the pipe model, leading to a similar equation as that for sapwood mass (Eq. 15). Cross sectional-area of bark per unit leaf area is assumed to be a constant fraction b of sapwood area per unit leaf area such that mb = bms . 5. 5


Root mass

Also consistent with pipe-model assumption, we assume a fixed ratio of root mass per unit leaf area mr = α3 al . (17) Even though nitrogen and water uptake are not modelled explicitly, imposing a fixed ratio of root mass to leaf area ensures that approximate costs of root production are included in calculations of carbon budget.


s u p p o r t i n g i n f o r m at i o n



s eed production

The rate of seed production from the plant f ( x, h, Ea ) is a direct function of mass allocated to reproduction: dm (1 − a ) × db db dt , f ( x, h, Ea ) = (18) s + cacc where s is the mass of the seed and cacc is the cost per seed of accessories, such as fruits, dma gives the fraction of db that is allocated flowers and dispersal structures. The function db dt dma gives the fraction allocated to reproduction. to growth, while 1 − db



Instantaneous rates of plant mortality are taken as the sum of a growth independent and growth dependent terms (Falster et al., 2011; Moorcroft, Hurtt & Pacala, 2001): d( x, h, Ea ) = dI ( x, h) + dD ( x, h, Ea ).


The growth independent rate is taken to be a constant, independent of plant performance, but potentially varying with species traits. The growth dependent part is assumed to decline exponentially with the rate of mass production per unit leaf area, i.e. dD ( x, h, Ea ) = cd2 exp(−cd3 X ),


where X = db/dt/al . This relationship allows for plants to increase mortality as growth rate approaches zero, while also allowing for differences among species in the parameters cd2 and cd3 . We also require a function § g ( x 0 , h0 , Ea0 ) for survival through germination. For the demographic model to behave smoothly, SG ( x 0 , h0 , Ea0 )/g( x, h0 , Ea0 ) should approach zero as g( x, h0 , Ea0 ) → 0. Following (Falster et al., 2011), we use the function S G ( x 0 , h 0 , Ea0 ) =

1 X 2 + 1.0


where X = cs0 al and cs0 is a constant. The behaviour of Eq. 21 is consistent with Eq. 20, db/dt as both cause survival to decline with mass production.


hyper- parameterisation of physiological model via t raits

The plant physiological model includes default values for all parameters needed (see Table 3). Species are known to vary considerably in many of these parameters, such as φ, ρ, c p1 , s; so by varying parameters one can account for different ecologies. When altering one parameter in the model, however, one must also consider whether there are trade-offs linking parameters. p l a n t allows for hyper-parameterisation of the plant physiological model via plant functional traits, i.e. simultaneous variation in multiple parameters because of assumed trade-off. In the default physiological model we implement the following relationships. For more details see make_FFW16_hyperpar.R. 5


8. 1

s u p p o r t i n g i n f o r m at i o n

Leaf mass per area

The trait leaf mass per area (φ) directly influences growth by changing dal /dma . In addition we link φ to the rate of leaf turnover, based on widely observed scaling relationship from Wright et al. (2004):  − B4 φ . k l = k l0 φ0 We normalise the relationship around a global mean l m a of φ0 – this allows us to vary k l0 and B4 without displacing the relationship from the observed mean. We also vary mass based leaf respiration rate so that it stays constant per unit leaf area and varies with l m a, as is empirically observed Wright et al. (2004): c Rl = 8. 2

c Rl0 . φ

Wood density

The trait wood density (ρ) directly influences growth by changing dal /dma . In addition we link ρ to the rate of growth independent mortality:  cd0 = d00

ρ ρ0

−d1 .

In addition, ρ is assumed to influence the rate of sapwood turnover,  − B5 ρ , k s = k s0 ρ0 and the rate of sapwood respiration per unit mass: c Rs =

4012.0 . ρ

As with φ, we normalise the mortality and turnover relationships around a global mean value of ρ0 . 8. 3

Seed mass

Effects of the trait seed mass (s) are naturally embedded in the equation determining fecundity (Eq. 18) and the initial height of seedlings (XXX). In addition, we let the accessory cost per seed be a linear function of seed size: c acc = 3.0 ∗ s, following observed relationships (Henery & Westoby, 2001). 8. 4

Nitrogen per leaf area

Photosynthesis per unit leaf area and respiration rates per unit leaf mass (or area) are assumed to vary with leaf nitrogen per unit area ν: c p1 = c PN ν, c Rl = c RN ν.


s u p p o r t i n g i n f o r m at i o n




Ta b l e 1 : Key variables in physiological model. Subscripts i = l, s, b, r, a refer to leaves, sapwood, roots, total of these vegetative tissues. Similarly ai denotes areas, of leaves (i = l) and of cross-sections of total stem and sapwood (i = st, ss) respectively. Symbol h b mi ai y

Unit m kg kg m2 kg kg−1

p, p¯ ri

kg yr−1 m−2 kg yr−1 − 1 kg yr−1


Description height of plant biomass originating from parent plant mass retained on plant in tissue i surface-area or cross-sectional-area of tissue i Yield; ratio of carbon fixed in mass per carbon assimilated photosynthetic rate per unit area respiration per unit tissue mass turnover rate for tissue



s u p p o r t i n g i n f o r m at i o n

Ta b l e 2: Equations for an allometric growth model, based on functional balance assumptions. The key assumptions of the physiological model are listed in a), under "function". From these assumptions we derive allocation functions for both tissue areas and masses (b). We can also express growth rate for each tissue as a function of growth rate in leaf area. Variable Function a) Assumed relationship to leaf area β1


h = α1 a

sapwoood area

as = θ al

bark area

ab = b θ al



Growth rate

d h = β α a β 1 −1 1 1 l d al d as =θ d al d ab = bθ d al

d h = d h d al dt d al dt da da d as = s l dt d al dt d ab da da = b l dt d al dt

dml =φ d al  dms = θ ρ ηc h + al dh d al d al  dmb = b θ ρ ηc h + al dh d al d al dmr = α3 d al

dml dml d al = dt d al dt dms dms d al = dt d al dt dmb dmb d al = dt d al dt dmr d al dmr = dt d al dt

b) Derived equation for mass of tissue


leaf mass

ml = φ al

sapwood mass

ms = θ ρ ηc al h

bark mass

mb = b θ ρ ηc al h

root mass

mr = α3 al

s u p p o r t i n g i n f o r m at i o n


Ta b l e 3 : Model parameters Description Plant construction Crown-shape parameter Leaf mass per area Wood density Leaf area per sapwood area Root mass per leaf area Height / leaf area scaling Height / leaf area scaling Production Leaf photosynthesis per area Saturation of leaf photosynthesis per area Yield, = fraction of carbon fixed converted into mass Ratio of bark area to sapwood area Leaf respiration per mass Fine-root respiration per mass Sapwood respiration per mass Turnover rate for leaves Turnover rate for sapwood Turnover rate for bark Turnover rate for fine roots Fecundity Seed mass Accessory cost per seed Height at maturation Maximum allocation to reproduction Parameter determining rate of change in r ( x, ml ) around hm Mortality Survival probability during dispersal Parameter influencing survival through germination Baseline rate for intrinsic mortality Baseline rate for growth-related mortality Risk coefficient for dry-mass production per unit leaf area in growth-related mortality





η φ ρ θ α3 α1 β1

kgm−2 kgm− 3 kgm−2 m−1 -

12 0.1978791 608 4669 0.07 5.44 0.306

eta lma rho theta a3 a1 B1

c p1 c p2 y

kgyr−1 m−2 0-1 0-1

150.36 0.19 0.7

c_p1 c_p2 Y

b rl rr rs kl ks kb kr

kgyr−1 kg−1 kgyr−1 kg−1 kgyr−1 kg−1 yr−1 yr−1 yr−1 yr−1

0.17 198.4545 217 6.598684 0.4565855 0.2 0.2 1

b c_Rl c_Rr c_Rs k_l k_s k_b k_r

s cacc hmat cr1 cr2

kg kg m 0-1 -

0.000038 0.000114 16.59587 1 50

mass_seed c_acc hmat c_r1 c_r2

SD cs0

0-1 kgm−2 yr−1

0.25 0.1

Pi_0 c_s0

cd0 cd2 cd3

yr−1 yr−1 yrm−2 kg−1

0.01 5.5 20

c_d0 c_d2 c_d3



s u p p o r t i n g i n f o r m at i o n

r e f erences Falster, D.S., Brännström, Å., Dieckmann, U. & Westoby, M. (2011) Influence of four major plant traits on average height, leaf-area cover, net primary productivity, and biomass density in single-species forests: a theoretical investigation. Journal of Ecology, 99, 148–164. Henery, M. & Westoby, M. (2001) Seed mass and seed nutrient content as predictors of seed output variation between species. Oikos, 92, 479–490. Mäkelä, A. (1997) A carbon balance model of growth and self-pruning in trees based on structural relationships. Forest Science, 43, 7–24. Moorcroft, P.R., Hurtt, G.C. & Pacala, S.W. (2001) A method for scaling vegetation dynamics: the Ecosystem Demography model (ED). Ecological Monographs, 71, 557–586. Shinozaki, K., Yoda, K., Hozumi, K. & Kira, T. (1964) A quantitative analysis of plant form the pipe model theory. I. Basic analyses. Japanese Journal of Ecology, 14, 97–105. Thornley, J.H.M. & Cannell, M.G.R. (2000) Modelling the components of plant respiration: representation and realism. Annals of Botany, 85, 55–67. Wright, I.J., Reich, P.B., Westoby, M., Ackerly, D., Baruch, Z., Bongers, F., Cavender-Bares, J., Chapin, F., Cornelissen, J., Diemer, M., Flexas, J., Garnier, E., Groom, P., Gulias, J., Hikosaka, K., Lamont, B., Lee, T., Lee, W., Lusk, C., Midgley, J., Navas, M.L., Niinemets, Ü., Oleksyn, J., Osada, N., Poorter, H., Poot, P., Prior, L., Pyankov, V., Roumet, C., Thomas, S., Tjoelker, M., Veneklaas, E. & Villar, R. (2004) The world-wide leaf economics spectrum. Nature, 428, 821–827. Yokozawa, M. & Hara, T. (1995) Foliage profile, size structure and stem diameter plant height relationship in crowded plant-populations. Annals of Botany, 76, 271–285.


plant: A package for modelling forest trait ecology ... - GitHub

Department of Biological Sciences, Macquarie University, Sydney, Australia .... of leaf area, including construction of the leaf itself and various support structures.

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