Ecological Archives M079-016-A1

R. F. Grant, L. R. Hutyra, R. C. de Oliveira, J. W. Munger, S. R. Saleska, and S. C. Wofsy. 2009. Modeling the carbon balance of Amazonian rain forests: resolving ecological controls on net ecosystem productivity. Ecological Monographs 79:445–463.

Appendix A. Key governing equations for simulating net ecosystem productivity in ecosys: heterotrophic respiration.

Decomposition

DSi,j,l,C=D'Si,j,l,C Σn Mi,n,a,l,C   ftgl

   

[A1]

DZi,j,l,C = D'Zi,j,l,C Σn Mi,n,a,l,C  ftgl

   

[A2]

D'Si,j,l,C = {DSj,C[Si,j,l,C]}/{[Si,j,l,C] + KmD(1.0 +[Σn Mi,n,a,l,C]/KiD)}

substrate and water constraint on D

 

 

[A3]

D'Zi,j,l,C = {DZj,C[Zi,j,l,C]}/{[Zi,j,l,C] + KmD(1.0 +[Σn Mi,n,a,l,C]/KiD)}

  [A4]

ftgl = Tsl{e[B - Ha/(RTsl)]}/{1 + e[(Hdl - STsl)/( RTsl)] + e[(STsl - Hdh)/( RTsl)]}

Arrhenius function

 

[A5]

DSi,j,l,N,P = DSi,j,l,C(Si,j,l,N,P/Si,j,l,C)

N and P coupled with C during D

 

 

[A6]

DZi,j,l,N,P = DZi,j,l,C(Zi,j,l,N,P/Zi,j,l,C)

  [A7]

Yi,l,C = kts(aFs[Qi,l,C]b - Xi,l,C)

   

[A8]

Yi,l,N,P = Yi,l,C(Qi,l,N,P/Qi,l,C)

([Qi,l,C]b < Xi,l,C)

 

[A9]

Yi,l,N,P = Yi,l,C(Xi,l,N,P/Xi,l,C)

([Qi,l,C]b > Xi,l,C)  

[A10]

Microbial Growth

Rh = ΣiΣ nΣ lRhi,n,l

   

[A11]

Rhi,n,l = R'hn min{CNi,n,l,a/CNj, CPi,n,l,a/CPj}

Rh constrained by microbial N, P

 

[A12]

Rh'i,n,l = Mi,n,a,l,C {Rhi,n,l [Qi,l,C]}/{(KmQC +[Qi,l,C])}ftgl

Rh constrained by substrate C

 

[A13]

Rhi,n,l = Rh'i,n,l(UO2i,n,l/U'O2i,n,l)

Rh constrained by O2

 

[A14]

U'O2i,n,l = 2.67Rh'i,n,l

   

[A15]

UO2i,n,l = U'O2i,n,l[O2mi,n,l]/([O2mi,n,l] + KO2)

active uptake coupled with diffusion of O2

 

 

[A16a]

          = 4πn Mi,n,a,l,C DsO2l[rmrwl/(rwl rm)]([O2sl] -[O2mi,n,l]

  [A16b]

Rmi,n,j,l = RmMi,n,j,l,N ftml

   

[A17]

ftml = e[y(Tsl - 298.16)]

   

[A18]

Rgi,n,l = Rhi,n,l - Σ j Rmi,n,j,l

   

[A19]

Ui,n,l = Rgi,n,l(1 + ΔG/Em)

DOC uptake driven by Rg

 

[A20]

Ui,n,lN,P = Ui,n,lQi,l,N,P/Qi,l,C

   

[A21]

DMi,n,j,l,C= DMi,jMi,n,j,C ftg

first-order decay of microbial C, partial recycling of microbial N, P

 

 

[A22]

DMi,n,j,N,P = DMi,jMi,n,j,l,N,P ftgl  fdi,n,lN,P

  [A23]

δMi,n,j,l,Ct = FjUi,n,l - FjRhi,n,l DMi,n,j,l,C

[Rhi,n,l > Rmi,n,j,l]

 

[A24a]

δMi,n,j,l,Ct = FjUi,n,l - Rmi,n,j,l - DMi,n,j,l,C

[Rhi,n,l < Rmi,n,j,l]

 

[A24b]

δMCt = ΣiΣ nΣ jΣlδMi,n,j,l,Ct

   

[A24c]

Microbial Nutrient Exchange

UNH4i,n,j,l = (Mi,n,j,l,C CNj - Mi,n,j,l,N)                                                  

UNH4 < 0

mineralization

[A25a]

UNH4i,n,j,l =min{(Mi,n,j,l,C CNj - Mi,n,j,l,N),

                     U’NH4 A i,n,j,l ([NH4+i,n,j,l]–[NH4+mn])/([NH4+i,n,j,l]–[NH4+mn] + KNH4)}

UNH4 > 0

immobilization

[A25b]

UNO3i,n,j,l = min{(Mi,n,j,l,C CNj - (Mi,n,j,l,N + UNH4i,n,j,l)) ,

                    U’NO3 A i,n,j,l ([NO3-i,n,j,l]–[NO3-mn])/([NO3-i,n,j,l]–[NO3-mn] + KNO3)}

UNO3 > 0

immobilization

[A25c]

UPO4i,n,j,l = (Mi,n,j,l,C CPj - Mi,n,j,l,P)                                                   

UPO4 < 0

mineralization

[A25d]

UPO4i,n,j,l =min{(Mi,n,j,l,C CPj - Mi,n,j,l,P),

                     U’PO4 A i,n,j,l ([H2PO4-i,n,j,l]–[H2PO4-mn])/([H2PO4-i,n,j,l]–[ H2PO4-mn] + KPO4)}

UPO4>0

immobilization

[A25e]

Φi,n=f,j,l = max{0, Mi,n=f,j,l,CCNj - Mi,n=f,j,l,N - max{0, Ui,n=f,j,l,N}}

N2 fixation driven by N deficit

 

[A26]

RΦi,n=f,j,l = EΦ Φi,n=f,j,l

   

[A27]

δMi,n,j,l,Nt = FjUi,n,l,N + UNH4i,n,j,l + UNO3i,n,j,l + Φi,n=f,j,l - DMi,n,j,l,N

δMi,n,j,l,Pt = FjUi,n,l,P + UPO4i,n,j,l - DMi,n,j,l,p

gains vs. losses of microbial N, P

 

[A28a]

[A28b]

Mi,n,a,l,C = Mi,n,j=labile,l,C + Mi,n,j=resistant,l,CFr/Fl

   

[A29]

Humification

HSi,j=lignin,l,C = DSi,j=lignin,l,C

decomposition products of litter added to POC depending on lignin

 

 

 

 

[A30]

HSi,j=lignin,l,N,P = DSi,j=lignin,l,N,P

  [A31]

HSi,j≠lignin,l,C = HSi,j=lignin,l,C Lhj

  [A32]

HSi,j≠lignin,l,N,P = HSi,j≠lignin,l,C Si,l,N,P/Si,l,C

  [A33]

HMi,n,j,l,C = DMi,n,j,l,C Fh

decomposition products of microbes added to humus depending on clay

 

 

[A34]

HMi,n,j,l,N,P = HMi,n,j,l,CMi,n,j,l,N,P/Mi,n,j,l,C

  [A35]
Root and Mycorrhizal Nutrient Uptake

UNH4i,r,l = {Uwi,r,l[NH4+l] + 2πLi,r,lDeNH4l ([NH4+l] – [NH4+i,r,l]) / ln(di,r,l /ri,r,l)}

            = U'NH4 Ai,r,l ([NH4+i,r,l] –[NH4+mn])/([NH4+i,r,l] –[NH4+mn] + KNH4) ftil

root uptake from mass flow + diffusion coupled with active uptake of NH4+, NO3- and H2PO4- , as for microbial uptake in [A25]

 

 

 

[A36a]

[A36b]

UNO3i,r,l = {Uwi,r,l[NO3-l] + 2πLi,r,l DeNO3l ([NO3-l] – [NO3-i,r,l]) / ln(di,r,l /ri,r,l)}

            = U'NO3 Ai,r,l ([NO3-i,r,l] –[NO3-mn] )/([NO3-i,r,l] –[NO3-mn] + KNO3) ftil

 

[A36c]

[A36d]

UPO4i,r,l = {Uwi,r,l[H2PO4-l] + 2πLi,r,lDePO4l ([H2PO4-l] – [H2PO4-i,r,l]) / ln(di,r,l /ri,r,l)}

            = U'PO4 Ai,r,l ([H2PO4-i,r,l] –[H2PO4-mn])/([H2PO4-i,r,l] –[H2PO4-mn] + KPO4) ftgl

 

[A36e]

[A36f]

 

Definitions of Variables in Appendix A

Variable

Definition

Unit

Equation

Value

Reference

i

substrate-microbe complex: coarse woody litter, fine non-woody litter, POC, humus

       

j

kinetic component: labile, resistant, active

       

l

soil or litter layer

       

n

microbial functional type: heterotrophic (bacteria, fungi), autotrophic (nitrifiers, methanotrophs), diazotrophic, obligate aerobe, facultative anaerobes (denitrifiers), obligate anaerobes (methanogens)

       
           

A

microbial, root or mycorrhizal surface area

m2 m-2

[A25,A36]

   

a

total substrate + residue C = ([Si,j,C] +[Zi,j,C])

g C Mg-1

[A8]

   

B

parameter such that ftg = 1.0 at Tl = 298.15 K

 

[A5]

26.230

 

b

Freundlich exponent for sorption isotherm

 

[A8]

0.85

Grant et al. (1993a,b)

CN,Pi,n,a,l

ratio of Mi,n,a,N,P to Mi,n,a,C

g N or P g C-1

[A12]

   

CN,Pj

maximum ratio of Mi,n,j,N,P to Mi,n,j,C maintained by Mi,n,j,C

g N or P g C-1

[A12, A25, A26]

0.22 and 0.13 (N), 0.022 and 0.013 (P) for j = labile and  resistant, respectively

Grant et al. (1993a,b)

De NH4l

effective dispersivity-diffusivity of NH4+ during root uptake

m2 h-1

[A36]

   

De NO3l

effective dispersivity-diffusivity of NO3- during root uptake

m2 h-1

[A36]

   

De PO4l

effective dispersivity-diffusivity of H2PO4- during root uptake

m2 h-1

[A36]

   

DMi,j

specific decomposition rate of Mi,n,j at 30°C

g C g C-1 h-1

[A22, A23]

0.0125 and 0.00035 for j = labile and resistant, respectively

Grant et al. (1993a,b)

DMi,n,j,l,C

decomposition rate of Mi,n,j,l,C

g C m-2 h-1

[A22, A24, A34]

   

DMi,n,j,l,N,P

decomposition rate of Mi,n,j,l,N,P

g N or P m-2 h-1

[A23, A28]

   

DsO2l

aqueous dispersivity–diffusivity of O2 during microbial uptake in soil

m2 h-1

[A16]

   

DSi,j,l,C

decomposition rate of Si,j,l,C by ΣnMi,n,a,l 

g C m-2 h-1

[A1, A6, A30]

   

DSj,C

specific decomposition rate of Si,j,l,C by ΣnMi,n,a,l  at 25°C and saturating[Si,l,C]

g C g C-1 h-1

[A3]

1.0, 1.0, 0.15, and 0.025 for j = protein, carbohydrate, cellulose, and lignin

Grant et al. (1993a,b)

DSi,j, l,N,P

decomposition rate of Si,j,l,N,P by ΣnMi,n,a,l 

g P m-2 h-1

[A6, A31]

   

DZi,j,C

decomposition rate of Zi,j,l,C by ΣnMi,n,a,l 

g C m-2 h-1

[A29, A34]

   

DZi,j,N,P

decomposition rate of Zi,j,l,N,P by ΣnMi,n,a,l

g P m-2 h-1

[A7]

   

DZj,C

specific decomposition rate of Zi,j,l,C by ΣnMi,n,a,l  at 25°C and saturating[Zi,l,C]

g C g C-1 h-1

[A4]

0.25 and 0.05 for j = labile and resistant biomass

Grant et al. (1993a,b)

D'Si,j, l,C

specific decomposition rate of Si,j,l,C by ΣnMi,n,a,l  at 25°C

g C g C-1 h-1

[A1, A3]

   

D'Zi,j,l,C

specific decomposition rate of Zi,j,l,C by ΣnMi,n,a,l  at 25°C

g C g C-1 h-1

[A2, A4]

   

ΔG

energy yield of C oxidation and O2 reduction

kJ g C-1

[A20]

37.5

 

di,r,l

half distance between adjacent roots assumed equal to uptake path length

m

[A36]

Ls,z /Δz)-1/2

Grant (1998)

Em

energy requirement for growth of Mi,n,a,l 

kJ g C-1

[A20]

25

 

EΦ

energy requirement for non-symbiotic N2 fixation by heterotrophic diazotrophs (n = f)

g C g N-1

[A27]

5

Waring and Running (1998)

Fh

fraction of products from microbial decomposition that are humified (function of clay content)

 

[A34]

0.167 + 0.167*clay

 

Fl

fraction of microbial growth allocated to labile component Mi,n,l

 

[A24, A28, A29]

0.55

Grant et al. (1993a,b)

Fr

fraction of microbial growth allocated to resistant component Mi,n,r

 

[A24, A28, A29]

0.45

Grant et al. (1993a,b)

Fs

equilibrium ratio between Qi,l,C and Hi,l,C

 

[A8]

   

fdi,n,lN,P

fraction of N or P not recycled within Mi,n,j,l during decomposition

dimensionless

[A23]

   

ftgl

temperature function for microbial, root or mycorrhizal growth respiration

dimensionless

[A1, A2, A5, A13, A22, A23, A36]

   

ftml

temperature function for maintenance respiration

dimensionless

[A17, A18]

   

Φi,n=f,j,l

non-symbiotic N2 fixation by heterotrophic diazotrophs (n = f)

g N m-2 h-1

[A26,A27]

   

[H2PO4-]

concentration of H2PO4- in soil solution

g P m-3

[A25]

   

Ha

energy of activation

J mol-1

[A5,C10]

65×103

Addiscott (1983)

Hdh

energy of high temperature deactivation

J mol-1

[A5,C10]

225×103

 

Hdl

energy of low temperature deactivation

J mol-1

[A5,C10]

198×103

 

HMi,n,j,l,C

transfer of microbial C decomposition products to humus

g C m m-2 h-1

[A34, A35]

   

HMi,n,j,l,N,P

transfer of microbial N or P decomposition products to humus

g N or P m-2 h-1

[A35]

   

HSi,j,l,C

transfer of C hydrolysis products to particulate OM

g C m-2 h-1

[A30, A31, A32, A33]

   

HSi,j,l,N,P

transfer of N or P hydrolysis products to particulate OM

g N or P m-2 h-1

[A31, A33]

   

KNH4

M-M constant for NH4+ uptake at microbial, root or mycorrhizal surfaces

g N m-3

[A25,A36]

0.40

Barber and Silberbush (1984)

KNO3

M-M constant for NO3- uptake at microbial, root or mycorrhizal surfaces

g N m-3

[A25,A36]

0.35

Barber and Silberbush (1984)

KPO4

M-M constant for H2PO4-- uptake at microbial, root or mycorrhizal surfaces

g P m-3

[A25,A36]

0.125

Barber and Silberbush (1984)

KiD

inhibition constant for [Mi,n,a ] on Si,C , Zi,C

g C m-3

[A3, A4]

25

Grant et al. (1993a,b)

KmD

Michaelis–Menten constant for DSi,j,C

g C Mg-1

[A3, A4]

75

KmQC

Michaelis–Menten constant for R'hi,n on [Qi,C]

g C m-3

[A13]

36

KO2

Michaelis–Menten constant for reduction of O2s by microbes, roots and mycorrhizae

g O2 m-3

[A16]

0.032

Griffin (1972)

kts

equilibrium rate constant for sorption

h-1

[A8]

0.01

Grant et al. (1993a,b)

Lhj

ratio of nonlignin to lignin components in humified hydrolysis products

 

[A32]

0.10, 0.05, and 0.05 for j = protein, carbohydrate, and cellulose, respectively

Shulten and Schnitzer (1997)

Li,j,r

root length

m m-2

[A36]

   

Mi,n,j,l,C

microbial C

g C m-2

[A1, A2, A13, A16, A22, A24, A25, A29, A35]

   

Mi,n,j,l,N

microbial N

g N m-2

[A17,A26]

   

Mi,n,j,l,P

microbial P

g P m-2

[A23, A28, A25, A35]

   

Mi,n,a,l,C 

active microbial C from heterotrophic population n associated with (Si,j,l,C + Zi,j,l,C)

g C m-2

[A1, A2, A13, A16, A29]

   

[Mi,n,a,l,C ]

concentration of Mi,n,a  in soil water

g C m-3

[A3, A4]

   

[NH4+i,n,j,l]

concentration of NH4+ in soil solution or at microbial, root or mycorrizal surfaces

g N m-3

[A25,A36]

   

[NH4+mn]

concentration of NH4+ at microbial, root or mycorrizal surfaces below which UNH4 = 0

g N m-3

[A36]

0.0125

Barber and Silberbush (1984)

[NO3-i,n,j,l]

concentration of NH4+ in soil solution or at microbial, root or mycorrizal surfaces

g N m-3

[A25, A36]

   

[NO3-mn]

concentration of NO3- at microbial, root or mycorrizal surfaces below which UNO3 = 0

g N m-3

[A36]

0.03

Barber and Silberbush (1984)

[H2PO4-i,n,j,l]

concentration of H2PO4- in soil solution or at microbial, root or mycorrizal surfaces

g N m-3

[A25, A36]

   

[H2PO4-mn]

concentration of H2PO4- at microbial, root or mycorrizal surfaces below which UPO4 = 0

g N m-3

[A36]

0.002

Barber and Silberbush (1984)

[O2mi,n,l]

O2 concentration at heterotrophic microsites

g O2 m-3

[A16]

   

[O2sl]

O2 concentration in soil solution

g O2 m-3

[A16]

   

Qi,l,C

DOC from products of (DSi,j,l,C + DZi,j,l,C)

g C m-2

[A8, A21]

   

[Qi,l,C]

solution concentration of Qi,l,C

g C Mg-1

[A8, A13]

   

Qi,l,N,P

DON and DOP from products of (DSi,j,l,N,P + DZi,j,l,N,P)

g N or P m-2

[A9, A21]

   

R

gas constant

J mol-1 K-1

[A5,C10]

8.3143

 

RΦi,n=f,j,l

respiration for non-symbiotic N2 fixation by heterotrophic diazotrophs (n = f)

g C m-2 h-1

[A27]

   

Rgi,n,l

growth respiration of Mi,n,a,l  on Qi,l,C under nonlimiting O2 and nutrients

g C g C-1 h-1

[A19]

   

Rh

total heterotrophic respiration of all Mi,n,a,l  under ambient O2

g C m-2 h-1

[A11]

   

Rhi,n,l

heterotrophic respiration of Mi,n,a,l  under ambient O2, nutrients, temperature

g C m-2 h-1

[A11, A14, A19, A24]

   

Rhi,n,l

specific heterotrophic respiration of Mi,n,a,l  under nonlimiting [Qi,l,C], O2, and 25°C

g C g C-1 h-1

[A12, A13]

   

Rh'n

specific heterotrophic respiration of Mi,n,a,l  under nonlimiting [Qi,C], O2, nutrients, and 25°C

g C g C-1 h-1

[A12]

0.125

Shields et al. (1973)

Rh'i,n,l

heterotrophic respiration of Mi,n,a,l  on Qi,l,C under nonlimiting O2

g C m-2 h-1

[A13, A14, A15]

   

Rm

specific maintenance respiration at 25°C

g C g N-1 h-1

[A17]

0.0115

Barnes et al. (1998)

Rmi,n,j,l

maintenance respiration by Mi,n,j,l

g C m-2 h-1

[A17, A19, A24]

   

rwl

radius of rm + water film at current water content

m

[A16]

   

rm

radius of heterotrophic microsite

m

[A16]

2.5×10-6

 

ri,r,l

radius of root or mycorrhizae

m

[A36]

1.0×10-3, 5.0×10-6

 

rwl

thickness of water films

m

[A16]

   

S

change in entropy

J mol-1 K-1

[A5,C10]

710

Sharpe and DeMichelle (1977)

[Si,j,l,C]

concentration of Si,j,l,C in soil

g C Mg-1

[A3]

   

Si,j,l,C

mass of solid or sorbed organic C in soil

g C m-2

[A6, A32]

   

Si,j,l,N,P

mass of solid or sorbed organic N or P in soil

g P m-2

[A6, A32]

   

Tsl

soil temperature

K

[A5, A18]

   

Ui,n,l

uptake of Qi,l,C by ΣnMi,n,a,l under limiting nutrient availability

g C m-2 h-1

[A20, A21, A24]

   

Ui,n,N,P

uptake of Qi,l,N,P by ΣnMi,n,a,l under limiting nutrient availability

g P m-2 h-1

[A21, A28]

   

UNH4i,n,j,l

NH4+ uptake by microbes

g N m-2 h-1

[A25,A26,A28]

   

UNH4i,r,l

NH4+ uptake by roots or mycorrhizae

g N m-2 h-1

[A36]

   

U'NH4

maximum UNH4 at 25 oC and non-limiting NH4+    

g N m-2 h-1

[A25,A36]

5.0×10-3

Barber and Silberbush (1984)

UNO3i,n,j,l

NO3- uptake by microbes

g N m-2 h-1

[A25,A26,A28]

   

UNO3i,r,l

NO3- uptake by roots or mycorrhizae

g N m-2 h-1

[A36]

   

U'NO3

maximum UNO3 at 25 oC and non-limiting NO3-    

g N m-2 h-1

[A25,A36]

5.0×10-3

Barber and Silberbush (1984)

UO2i,n

O2 uptake by Mi,n,a,l  under ambient O2

g m-2 h-1

[A14, A16]

   

U'O2i,n

O2 uptake by Mi,n,a,l  under nonlimiting O2

g m-2 h-1

[A14, A15, A16]

   

UPO4i,n,j,l

H2PO4- uptake by microbes

g N m-2 h-1

[A25,A26,A28]

   

UPO4i,r,l

H2PO4- uptake by roots or mycorrhizae

g N m-2 h-1

[A36]

   

U'PO4

maximum UPO4 at 25 oC and non-limiting H2PO4-    

g N m-2 h-1

[A25,A36]

5.0×10-3

Barber and Silberbush (1984)

Uwi,r,l

root water uptake

m3 m-2 h-1

[A36]

   

Xi,l,C

adsorbed C hydrolysis products

g C Mg-1

[A8, A10]

   

Xi,l,N,P

adsorbed N or P hydrolysis products

g P Mg-1

[A10]

   

y

selected to give a Q10 for ftm of 2.25

 

[A18]

0.081

 

Yi,l,C

sorption of C hydrolysis products

g C m-2 h-1

[A8, A9, A10]

   

Yi,l,N,P

sorption of N or P hydrolysis products

g P m-2 h-1

[A9, A10]

   

[Zi,j,l,C]

concentration of Zi,j,l,C in soil

g C Mg-1

[A4]

   

Zi,j,l,C

mass of microbial residue C in soil

g C m-2

[A7]

   

Zi,j,l,N,P

mass of microbial residue P in soil

g P m-2

[A7]

   

LITERATURE CITED

Addiscott, T. M. 1983. Kinetics and temperature relationships of mineralization and nitrification in Rothamsted soils with differing histories. Journal of Soil Science 34:343–353.

Barber,  S. A., and M. Silberbush. 1984.  Plant root morphology and nutrient uptake. Pages 65–87 in  S. A. Barber and D. R. Bouldin, editors. Roots, nutrient and water influx, and plant growth. Amer. Soc. Agron. Spec. Publ. no. 49. Madison, Wisconsin, USA.

Barnes, B. V., D. R. Zak, S. R. Denton, and S. H. Spurr. 1998. Forest Ecology Fourth Edition. Wiley and Sons. New York, New York, USA.

Grant, R. F. 1998. Simulation in ecosys of root growth response to contrasting soil water and nitrogen. Ecological Modelling 107:237–264.

Grant, R. F., N. G. Juma, and W. B. McGill. 1993a. Simulation of carbon and nitrogen transformations in soils. I. Mineralization. Soil Biology & Biochemistry 27:1317–1329.

Grant, R. F., N. G. Juma, and W.B. McGill. 1993b. Simulation of carbon and nitrogen transformations in soils. II. Microbial biomass and metabolic products. Soil Biology & Biochemistry 27:1331–1338.

Griffin, D. M. 1972. Ecology of soil fungi. Syracuse University Press, Syracuse, New York, USA.

Shields J. A., E. A. Paul, W. E. Lowe and D. Parkinson. 1973. Turnover of microbial tissue in soil under field conditions. Soil Biology & Biochemistry 5:753–764.

Sharpe, P. S. H., and D. W. DeMichelle. 1977. Reaction kinetics of poikilothermic development. Journal of Theoretical Biology 64:649–670.

Shulten, H. R., and M. Schnitzer. 1997. Chemical model structures for organic matter and soils. Soil Science 162:115–130.

Waring, R. H., and S. W. Running. 1998. Forest Ecosystems: Analysis at Multiple Scales. Second edition. Academic Press, London, UK.


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