1
Increased forest carbon storage with increased atmospheric CO2 despite nitrogen
2
limitation: A game-theoretic allocation model for trees in competition for nitrogen and
3
light
4
Authors: Ray Dybzinski*, Caroline E. Farrior, and Stephen W. Pacala
5
Appendix S1 – Detailed model description and analysis
6
This paper relies on a mathematical model of forest dynamics and competitive optimization
7
methods developed in a series of previous papers. The underlying model is a mechanistic
8
individual-based spatial forest simulator, the Perfect Plasticity Approximation (PPA), from
9
which we derive a series of integro-‐ partial differential equations that govern the size-structured
10
dynamics of each tree species at stand‐ level (Strigul et al., 2008). When calibrated with data on
11
individual tree vital rates, the equations predict observed successional dynamics (Purves et al.,
12
2008).
13
Two critical derivations of the PPA motivate our analysis. First, the lifetime reproductive
14
success (i.e. fitness) of an individual tree is calculated by summing over its growth rates and
15
mortality rates during its understory and canopy stages and by assuming it has a constant
16
fecundity per unit crown area while in the canopy. Its solution is:
𝐿𝑅𝑆 ≈ 𝛼𝐹𝑒
𝜇
−𝐷( 𝑈 )
𝐺𝑈
𝐺𝐶𝜃
Γ(𝜃 + 1),
𝜇 𝜃+1
(S1)
17
where LRS is lifetime reproductive success; 𝛼 and 𝜃 are constants that scale crown area by stem
18
diameter; D is the allometrically-related stem diameter at which a tree transitions from the
19
understory to the canopy; 𝜇 and G are the mortality and growth rates (for the understory,
20
subscript U, and canopy, subscript C); and Γ is the Gamma Function.
1
21
The second critical derivation of the PPA that motivates our analysis recognizes that for a
22
stand in dynamic equilibrium, each canopy tree will exactly replace itself in its lifetime, i.e.
23
̂ = 1. With this, we can solve for 𝐷
̂ , the allometrically-related stem diameter at which trees
𝐿𝑅𝑆
24
transition from the understory to the canopy (assuming they survive that long) in a stand in
25
dynamic equilibrium:
̂≈
𝐷
𝜇𝑈
𝐺𝐶𝜃
𝑙𝑛 (𝛼𝐹 𝜃+1 Γ(𝜃 + 1)).
𝐺𝑈
𝜇
(S2)
26
This shows that increasing the canopy stem growth rate, 𝐺𝐶 , increases the canopy height and thus
27
increases height-structured competition for light.
28
Although these more general features of the PPA model should not be forgotten, as they
29
provide the rigorous justification for scaling from individual-level strategies to stand-level
30
properties, our focus now narrows to describe how canopy growth rates depend on light,
31
nitrogen, CO2, and competition. The special case that we study here makes the canopy growth
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rate a function of the individual’s physiology and allocation strategy, as well as the physiology
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and allocation strategies of its neighbors in the context of light and nitrogen limitation. We focus
34
on canopy trees only because they dominate the land carbon sink. All other vital rates and
35
parameters are held constant.
36
Dybzinski et al. (2011) coupled these equations to a dynamic nitrogen model, disallowed
37
changes in litter chemistry that affect the decomposition rate of recalcitrant organic matter, and
38
focused on times scales over which inputs and losses of total ecosystem nitrogen are small.
39
These restrictions simplified the problem by making the rate of nitrogen mineralization
40
approximately constant. Dybzinski et al. (2011) then developed competitive optimization
41
methods to predict the winning foliage-wood-fine root allocation strategy as a function of the
2
42
nitrogen mineralization rate and showed that the predictions match patterns of biomass allocation
43
along natural productivity gradients (their Fig. 4).
44
A competitively optimal allocation strategy (i.e. the Evolutionarily Stable Strategy or
45
ESS) is found by introducing a rare “challenger” strategy into a monoculture of a “resident”
46
strategy, and then solving for the strategy that – as a resident – resists challenges by all other
47
strategies and that – as a challenger – can invade any nearby resident strategy (Geritz et al.,
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1998, McGill & Brown, 2007). Previous papers have shown that the most competitive
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challenger foliage-wood-fine root allocation strategy for trees while they are in the canopy is
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approximately the strategy that maximizes the stem growth rate of that challenger against a given
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resident in either early-successional or old growth forests (Purves et al., 2008, Dybzinski et al.,
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2011, Farrior et al., 2013a). This is not surprising because canopy trees that are overtopped
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suffer greatly reduced fitness and because fecundity increases with crown size, which increases
54
with stem size. Although lifetime reproductive success (i.e. fitness) is an increasing function of
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stem growth rate in the canopy, it is also a complicated function of investment in seed, the
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density-independent death rate in the canopy, and parameters governing understory growth and
57
survival (Strigul et al., 2008). However, allocation to fecundity can be neglected here because it
58
is a small component of the carbon and nitrogen budgets (Whittaker et al., 1979, Ladeau &
59
Clark, 2006, McCarthy et al., 2010).
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The fact that trees change allocation dramatically when moving from the understory to
61
the canopy argues that we may separately optimize understory and canopy allocation. We focus
62
on the optimal canopy allocation in this paper, because we are interested in the size of the carbon
63
sink, which is dominated by canopy trees (Farrior et al., 2013a). Finally, we assume that the
64
density-independent death rate of canopy trees (i.e. from wind throw) is independent of the
3
65
foliage-wood-fine root allocation strategy. Although this is obviously debatable, we offer it as a
66
useful beginning because the death rate in the canopy is probably affected most by factors other
67
than those considered here, such as wood density and height-diameter allometry. As in
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Dybzinski et al. (2011), we consider three carbon pools: foliage, wood, and fine roots. However,
69
we have expanded the number of nitrogen pools from three to four. Foliage and wood N:C is
70
fixed at  (0.057 gN gC-1) and  (0.0024 gN gC-1, mean value in Kattge et al. 2011),
71
respectively. Unlike Dybzinski et al. (2011), we divide the nitrogen in fine roots into “structural”
72
and “metabolic” pools, with N:C ratios  (0.001934 gN gC-1, mean value in Kattge et al. 2011
73
divided by 10) and  (gN gC-1), respectively. The structural pool is primarily the nitrogen in
74
plant cell walls. The metabolic pool includes the nitrogen in proteins responsible for active
75
transport of nitrate and ammonium and the metabolism that supports and fuels this activity. In a
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nitrogen-limited forest, nitrogen uptake by fine roots is obviously a critical component of fitness.
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Trees should invest in the metabolic pool so as to maximize net nitrogen gains. We assume that
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the rate of uptake of nitrate and ammonium by fine roots is proportional both to fine root
79
biomass (R) and an increasing and concave-down function: f(). This function equals zero at  =
80
0 simply because the root has no active transport proteins if it has no nitrogen in its metabolic
81
pool. It is an increasing function of  because we assume that the density of active transport sites
82
on a root hair increases with , and it is a concave‐ down function for two reasons. First, the
83
probability that an active transport site will capture a nitrate or ammonium molecule once it
84
touches the surface of a root hair must saturate at one as the density of active transport sites
85
becomes large. Second, the density of transport sites on the surface of a root hair might saturate
86
as  becomes large. If the nitrogen mineralization rate is a constant and all mineralized nitrogen
87
is ultimately captured by canopy trees, then N gN m-2 yr-1 will be captured by a closed canopy
4
88
forest (see Appendix G in Dybzinski et al. (2011) for a full derivation from a nitrogen cycle
89
model).
90
91
Model structure
92
The nitrogen capture strategy of the resident strategy is given by its fine root mass per
93
unit crown area, Rr (gC m-2), and metabolic root stoichiometry r, where the subscript “r”
94
denotes a resident variable. The resident’s one-sided canopy leaf area index, Lr (m2 m-2) and
95
annual wood production rate Wr (gC m-2 yr-1) are the solutions of the following two equations,
96
the first of which closes a resident tree’s nitrogen budget and the second of which closes its
97
carbon budget:
98
N = (1 - p) lgMLr + ( rr + s ) tRr + wW r
(S3)
E ( Lr ) = aLr + bRr + cW r + F .
(S4)
and
99
In addition to the parameters and variables described above, p is the fraction of nitrogen that is
100
resorbed before senescence (0.5 unitless);  is the leaf turnover rate (1 yr-1); M is the leaf mass
101
per area (28 gC m-2); t is the turnover rate of fine roots (0.3 yr-1); E(Lr) is the canopy-level gross
102
photosynthetic rate as a function of Lr (gC m-2 yr-1); a is the annual cost of leaves per leaf layer
103
(57.37 gC m-2 yr-1); b is the annual cost of fine roots per root mass (1.6 gC gC-1 yr-1); c is the cost
104
of wood per new wood mass (1 gC gC-1); and F is the NPP associated with reproduction (34.6
105
gC m-2 yr-1). Included within a, b, and c are the carbon costs of respiration and construction. See
106
Dybzinski et al. (2011) for a detailed description of these parameters and their derivations (e.g. a
107
and b are composed of other physiological parameters such as respiration rates and turnover
108
times that we simply summarize here).
5
109
Belowground competition for nitrogen is assumed to be well-mixed, where all roots have
110
access to the same resource pool, consistent with observational (Gilman, 1988, Stone & Kalisz,
111
1991, Casper et al., 2003), tracer (Gottlicher et al., 2008), and molecular-identification (Frank et
112
al., 2010, Jones et al., 2011) studies of root systems, which show extensive comingling of
113
individual root systems. This implies that to a first approximation, a challenger’s nitrogen uptake
114
per ground area is proportional to the nitrogen mineralization rate and its relative uptake capacity
115
(i.e. Rm f(m)/[Rr f(r)]) (Berendse & Elberse, 1990, Raynaud & Leadley, 2004, Craine et al.,
116
2005). Thus, a challenger strategy has a similar set of equations that closes its nitrogen and
117
carbon budgets that takes this into account:
N
118
Rm f ( rm )
Rr f ( rr )
= (1 - p) lgMLm + ( rm + s ) tRm + wW m
(S5)
and
E ( Lm ) = aLm + bRm + cW m + F ,
(S6)
119
where “m” subscripts challenger strategies. Note that the resident equations above (Eqs. S3, S4)
120
are simply a special case of the challenger equations (Eqs. S5, S6) when m = r.
121
We can solve Eq. (S5) for Lm,
Lm (W m ,Rm , r m ) =
122
æ R f (r )
ö
1
m
çç N m
- ( r m + s ) tRm - wW m ÷÷,
(1 - p) lgM è Rr f ( rr )
ø
(S7)
substitute the result into Eq. (S6), and set the result equal to zero:
é1
ê c E Lm [W m ,Rm , r m ]
ê
a
0 = ê- Lm [W m ,Rm , r m ]
ê c
ê b
F
ê- Rm - - W m
ë c
c
{
ù
}ú
ú
ú º G[W ,R , r ] .
m
m
m
ú
ú
ú
û
(S8)
6
123
We have defined a new function, G, that is an implicit function of Wm, Rm, and m and which
124
will become useful in a moment. Because G defines Wm as an implicit function of Rm, and m,
125
we may also write Wm as Wm(Rm, m).
To find the ESS strategy (denoted by “*”) we need to maximize Wm(Rm, m) with respect
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127
to Rm and m when evaluated at the resident strategy (Geritz et al., 1998, McGill & Brown,
128
2007),
dW m ( Rm , r m )
0=
dRm
0=
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R m =R r =R*
dW m ( Rm , r m )
drm
d 2W m ( Rm , rm )
and 0 >
dRm2
and 0 >
drm2
,
(S9)
r m = r r = r*
and verify that such a maximum is convergence-stable,
>
R m =R r =R*
d 2W m ( Rm , rm )
drr2
131
d 2W m ( Rm , rm )
r m = r r = r*
d 2W m ( Rm , rm )
dRr2
130
R m =R r =R*
>
r m = r r = r*
d 2W m ( Rm , rm )
dRm2
R m =R r =R*
.
d 2W m ( Rm , rm )
drm2
(S10)
r m = r r = r*
Using the implicit function theorem, we know that
dW m ( Rm , r m )
dRm
dG[W m ,Rm , r m ]
dRm
=dG[W m ,Rm , r m ]
dW m ( Rm , r m )
drm
dG[W m ,Rm , r m ]
drm
=.
dG[W m ,Rm , r m ]
(S11)
dW m ( Rm , rm )
and
(S12)
dW m ( Rm , rm )
7
132
The LHS of both Eq. (S11) and Eq. (S12) will be zero at the ESS according to Eq. (S9), which
133
implies that the numerators on the RHS of both Eq. (S11) and Eq. (S12) will also be zero at the
134
ESS, i.e.:
0=
135
dG[W m ,Rm , rm ]
dRm
(S13)
R m =R r =R*
r m = r r = r*
Wm =Wr =W*
and
0=
dG[W m ,Rm , rm ]
drm
.
R m =R r =R*
r m = r r = r*
(S14)
Wm =Wr =W*
136
First, we will use Eq. (S14) to solve for *, which we will then use together with Eq. S13 to
137
solve for R*.
138
139
140
Solution for *
Together, Eqs. (S14) and (S8) yield the solution for *:
{
}
æ dE L W ,R , r
ö
dLm [W m ,Rm , r m ]
m[ m
m
m]
ç
0=
- a÷
ç dLm [W m ,Rm , r m ]
÷
drm
è
ø
R m =R r =R*
.
(S15)
r m = r r = r* W =W =W*
m
r
141
The term in parentheses will be zero when the marginal return of gross photosynthesis on a
142
marginal increase in LAI exactly equals the total costs of that marginal increase. In other words,
143
this is the stopping point beyond which trees should not increase LAI and by which we define
144
nitrogen saturation (see Eq. 13 in Dybzinski et al. 2011). Since we are here interested in
145
solutions within the nitrogen-limited regime, where the term in parentheses will necessarily be
146
positive, the ESS solution is found by
8
0=
147
dLm [W m ,Rm , rm ]
drm
.
R m =R r =R*
r m = r r = r*
(S16)
Wm =Wr =W*
Using this together with Eq. (S7) yields
æ
ö
1
N df ( rm )
ç
0=
- tR*÷.
÷
(1 - p) lgM çè f ( r*) drm r m = r r = r*
ø
(S17)
148
A simple function for f() that has the right properties (increasing, concave-down, and
149
approaches zero at  = 0) is a power law: zu, where 0 < u < 1. Using this functional form and
150
the equation above,
R* r* =
u
N.
t
(S18)
151
This states that for competitively optimized trees, the total metabolic nitrogen in the fine root
152
system (R**) is a simple fraction (u/t) of the nitrogen mineralization rate (N). This is a pleasing
153
result because it provides a competitive optimization argument for the assumption in Dybzinski
154
et al. (2011) that a roughly constant fraction of nitrogen is allocated to foliage, with the
155
remainder going to other tissues in an unspecified way. As we describe below, the small wood
156
N:C () and structural fine root N:C () are a small perturbation on this result. In what follows,
157
we will use the result from f() = zu because it offers the simplest algebraic results and is
158
consistent with Dybzinski et al. (2011). Other functions that are tractable and yield similar results
159
include: f() = c1/(c2 + ) and f() = c1(1 – exp(c2)), where c1 and c2 are constants.
160
161
Using the power law functional form, it can be verified that * is a fitness maximum
according to Eq. (S9):
9
d 2 Lm [W m ,Rm , rm ]
0>
drm2
162
(S19)
R m =R r =R*
r m = r r = r*
Wm =Wr =W*
or
0>
1
Nu
( u -1) .
(1 - p) lgM r*2
(S20)
163
Everything outside the parentheses is necessarily positive, and because 0 < u < 1, the term in
164
parentheses is necessarily negative. Similarly, * is convergence-stable according to Eq. (S10):
æ d 2 L [W ,R , r ] d 2 L [W ,R , r ] ö
m
m
m
m
m
m
m
m
çç
÷÷
>
2
2
d
r
d
r
r
m
è
ø R m =R r =R*
165
(S21)
r m = r r = r* W =W =W*
m
r
or
u( u +1) > u( u -1) .
(S22)
166
Again, because 0 < u < 1, the LHS is necessarily positive and the RHS is necessarily negative,
167
making the inequality true.
168
169
Solution for R*
170
Together, Eqs. (S13) and (S8) yield the solution:
{
}
æ dE L W ,R , r
ö
dLm [W m ,Rm , r m ]
m[ m
m
m]
ç
0=
- a÷
-b
ç dLm [W m ,Rm , r m ]
÷
dRm
è
ø
R m =R r =R*
171
.
(S23)
r m = r r = r* W =W =W*
m
r
or, using Eq. (S7) for Lm and the result for R** (Eq. S18),
10
æ
ç
1 ç dE Lm [W m ,Rm , rm ]
R* =
b ç dLm [W m ,Rm , rm ]
ç
è
{
172
}
ö
־
ö
(1 - u) N
- a֍
- e R R*÷
֏ (1 - p) lgM
ø
÷
R m =R r =R*
r m = r r = r* W =W =W*
ø
m
r
(S24)
where
eR =
s
t
.
lM (1 - p)g
(S25)
173
Once a functional form for E(L) is chosen, this expression may be solved numerically. However,
174
note that R is the ratio of fine root structural N:C to total foliage N:C, which is a small number,
175
on the order of 0.1 or less, multiplied by the ratio of the turnover time of nitrogen in fine roots to
176
the turnover time of nitrogen in foliage, which is close to unity. We can find a close
177
approximation to the above expression by letting R = 0,
æ
ç
1 ç dE Lm [W m ,Rm , r m ]
R0* »
b ç dLm [W m ,Rm , r m ]
ç
è
{
}
ö
÷
(1 - u) N
,
- a÷
÷ (1 - p) lgM
÷
R m =R r =R*
r m = r r = r* W =W =W*
ø
m
r
(S26)
178
where we have subscripted R0* to indicate that it is the solution when R is approximated as zero.
179
The term outside the parentheses is necessarily positive, and the term inside the parentheses is
180
positive if the marginal return of gross photosynthesis on a marginal increase in LAI is greater
181
than the total costs of that marginal increase, i.e. if the tree is by definition nitrogen-limited (as
182
discussed above in reference to Eq. (S15)). It is here, incidentally, that the careful reader is
183
directed to note the goofy smiley face in Fig. 6m of the main text.
184
We verify that R0* is a fitness maximum by taking the second derivative of Eq. (S8) with
185
respect to Rm (assuming R = 0) and then evaluating the result at Rm = Rr = R0*. Using Eq. (S7)
186
for Lm (for which the second derivative with respect to Rm is zero), the result is
11
{
}
2
d 2 E Lm [W m ,Rm , rm ] æ dLm [W m ,Rm , rm ] ö
çç
÷÷
2
dRm
dLm [W m ,Rm , rm ] è
ø
.
R m =R r =R*
(S27)
r m = r r = r* W =W =W*
m
r
187
The term in parentheses is squared and thus necessarily positive. The term outside the
188
parentheses is the second derivative of the gross photosynthesis function, E(L) with respect to L.
189
In the nitrogen-limited regime, the first derivative of E(L) with respect to L must be positive, i.e.
190
gross photosynthesis must increase with LAI. However, a reasonable function for E(L) should
191
saturate with increasing LAI because of self-shading, which means that the second derivative of
192
E(L) with respect to L must be negative. Thus the second derivative is necessarily negative,
193
indicating that R0* is a fitness maximum (Eq. S9).
194
We verify that R0* is convergence-stable according to (Eq. S10) (assuming R = 0) .
195
Again, using Eq. (S7) for Lm (for which the second derivative with respect to Rm is zero), the
196
condition for convergence stability becomes
2
2ù
2
é d 2 E {L } æ
dLm ö æ dE {Lm } ö d 2 Lm d E { Lm } æ dLm ö ú
m
ê
ç
÷
- a÷
ç
÷ +
ç
÷
2
2 >
dL2m è dRm ø úû
êë dLm è dRr ø çè dLm
ø dRr
R
,
m
=R r =R*
r m = r r = r*
(S28)
Wm =Wr =W*
197
where we have omitted the functional notation of Lm to save space. At the equilibrium point, Rm
198
= Rr = R0*, the first term on the LHS is equal to the RHS, which reduces the expression to
æ dE { L }
ö d2L
m
m
ç
- a÷÷
2
ç dL
m
è
ø dRr
> 0.
R m =R r =R*
(S29)
r m = r r = r* W =W =W*
m
r
199
As discussed in the analyses above, the term in the parentheses is necessarily positive in a
200
nitrogen-limited stand. It is easy to show that the second derivative of Lm (Eq. S7) with respect to
12
201
Rr is necessarily positive, and thus the condition for convergence stability is met. We verify this
202
for the exact model graphically in Fig. S2.
203
204
Solution for L*
205
We can revisit the nitrogen conservation equation for a challenger strategy in light of the solution
206
for R** (Eq. S18). At the ESS, Eq. (S7) becomes:
L* =
207
(1 - u) N
(1 - p) lgM
- e R R* -e WW* ,
(S30)
where
eW =
w
1
.
lM (1 - p)g
(S31)
208
Analogous to R, W is the ratio of wood N:C to (total) foliage N:C, which is also a small
209
number, on the order of 0.1 or less, multiplied by the inverse of the turnover time of nitrogen in
210
foliage, which is on the order of unity. If R and W are approximated as zero, then we recover,
211
with the change of a few symbols and parameters, the solution for L* found in Dybzinski et al.
212
(2011):
L0 * »
(1 - u) N ,
(1 - p) lgM
(S32)
213
which is their Eq. 14 for nitrogen-limited stands. Note that we have subscripted L0* to indicate
214
that it is the solution when R and W are approximated as zero.
215
216
ESS solutions using the simplified Farquhar model of photosynthesis used in Dybzinski et
217
al. (2011)
13
218
Dybzinski et al. (2011) use a simplified model of whole-crown photosynthesis in which
219
the top leaves of a canopy tree fix carbon at a light-saturated rate, light extinguishes through the
220
crown exponentially (Beer’s Law), and at some point in the crown leaves become light-limited
221
and fix carbon at a light-limited rate. See Dybzinski et al. (2011) for a derivation:
æ
E ( L) = ç A{CO 2 } + q
ç
è
[
]
ö
é
ìï f{CO } I üïù
2
-kL ÷ s
ê1+ lní
ú
ý - f{CO 2 }Ie
,
÷k
ïî A{CO 2} + q ïþúû
êë
ø
(S33)
222
where A(CO2) is the maximum net photosynthetic rate as a function of atmospheric CO2 (9.9•10-
223
5
224
with the mean of trees in Ainsworth and Long (2005)); q is the leaf respiration rate (9.9•10-6 gC
225
m-2 s-1), (CO2) is the quantum yield of photosynthesis (3.27•10-4 or 3.66•10-4 gC m-2 s-1 fPAR-1
226
for CO2 = 350ppm or 550ppm respectively, a 12% increase consistent with the mean in
227
Ainsworth and Long (2005) and where fPAR is the fraction of total photosynthetically active
228
radiation at a particular leaf), I is relative light at the top of the canopy (1 fPAR), k is the light
229
extinction coefficient (0.5 m2 m-2), and s scales per-second measurements to annual
230
measurements (2.26•106 s yr-1).
231
or 14.553•10-5 gC m-2 s-1 for CO2 = 350ppm or 550ppm respectively, a 47% increase consistent
With this functional form for E(L), we can solve Eq. (S24) for R* and, together with the
232
solution for L* (Eq. S30), use the carbon conservation equation (Eq. S4) to find W*. Together,
233
we find this system of equations that implicitly defines the ESS allocation strategy:
14
L* =
(1 - u) N
(1 - p) lgM
- e R R* -e WW*
éæ
ù
ö
é
ì f{CO } I ü
ïù÷
2
êç s A{CO } + q ê1+ lnï
ú
ú
í
ý - F + ae WW*
2
ç
÷
ê
ú.
k
ï
ï
ê
ú
A
CO
+
q
{
}
2
1
î
þûø
ë
ú
W* = êè
cê æ
ö
öú
é
s
(1 - u) kN ù÷æ (1 - u) kN
ê-çç I f{CO 2 } exp êke R R* +ke WW* - ke R R*÷ú
ú÷ç1+
(1 - p) lgM ûøè (1 - p) lgM
êë è k
ë
øúû
[
]
(S34)
ö
é
1æ
(1 - u) kN ù ö÷æ (1 - u) N
R* = çç Isf{CO 2 } exp ê ke R R* +ke WW* - e R R*÷
ú - a÷ç
bè
(1 - p) lgM û øè (1 - p) lgM
ë
ø
234
These are the equations that we solve numerically to produce the figures in the main text, after
235
converting to NPP values:
Foliage NPP = gML*
Wood NPP = W*
Fine Root NPP = tR *.
Fecundity NPP = F
(S35)
236
All NPP values are in units of gC m-2 yr-1. Total NPP sums these values, and fractional allocation
237
for a given organ is its NPP divided by total NPP. We use the parameter values indicated in the
238
text above, which are exactly as in Dybzinski et al. (2011) except where new parameters have
239
been introduced (we flagged those parameters by citing the sources and/or derivations of their
240
values above as they were introduced; all other parameter values are described in Dybzinski et al.
241
(2011)). We obtained numerical solutions using Mathematica’s FindRoot function (Wolfram
242
Research, 2008) with the approximation of R = W = 0 as starting values:
15
L0 * =
(1 - u) N
(1 - p) lgM
éæ
ù
ö
é
ì f{CO } I ü
ïù÷
2
êç s A{CO } + q ê1+ lnï
ú
ú
í
ý -F
2
ç
÷
ê
ú.
k
ï
ï
ê
ú
A
CO
+
q
{
}
2
1
î
þûø
ë
ú
W 0* = êè
cê æ
ö
é (1 - u) kN ù æ
s
(1 - u) kN öúú
ê-çç I f{CO 2 } exp ê÷
ú÷÷ç1+
êë è k
ë (1 - p) lgM ûøè (1 - p) lgM øúû
[
]
(S36)
é (1 - u) kN ù öæ (1 - u) N ö
1æ
R0 * = çç Isf{CO 2} exp ê÷
ú - a÷÷ç
bè
ë (1 - p) lgM û øè (1 - p) lgM ø
243
Note that, with the change of a few symbols and parameters, these approximations correspond
244
exactly to Eqs. (14), (16), and (15) in Dybzinski et al. (2011).
245
246
Closed-form approximations of carbon storage in wood using a Michaelis–Menten model of
247
whole canopy carbon gain
248
The simple Farquhar model of whole-crown photosynthesis (Eq. S33) is mechanistically-
249
based and yields reasonable closed-form organ ESS solutions when R = W = 0. However, it
250
does not permit closed-form solutions of absolute or relative carbon storage in living wood,
251
which are potentially useful expressions. Because Eq. (S33) saturates with increasing L due to
252
self-shading, it may be approximated by a phenomenological Michaelis-Menten model of whole-
253
crown carbon gain,
E ( L) =
hL
.
y+L
(S37)
254
This function saturates at h (gC m-2 yr-1) for large L, obtains half that value at L = y (m2 m-2), and
255
has an initial slope h/y at L = 0. With this functional form for E(L), we can solve Eq. (S24) for R*
256
and, together with the solution for L* (Eq. S30), use the carbon conservation equation (Eq. S4) to
257
find W*. Jumping straight to the approximation of R = W = 0, we find:
16
L0 * =
(1 - u) N
(1 - p) lgM
é
ù
ö2
1ê æ
(1 - u) N
W 0* = hç
÷ - Fú
c êë è (1 - u) N + (1 - p) ylgM ø
úû
.
(S38)
æ
öæ
ö-2
1 ç æ (1 - u) N
(1 - u) N ö
R0 * = hyç
+ y ÷ - a÷ç
֏ (1 - p) lgM ֿ
b çè è (1 - p) lgM
ø
ø
258
Because they are measurable physiological parameters, we understand how A(CO2) and
259
(CO2) depend on atmospheric CO2, but the same cannot be said for h and y. To gain that
260
insight, we fit the phenomenological Michaelis-Menten model to the simple Farquhar model at
261
both CO2 = 350 and CO2 = 550 over the range of LAI predicted by the full model (2.25 to 5.25
262
m2 m-2). The best fit is shown in Fig. S1 and gives
h=
h550 1966
=
= 1.11
h350 1776
y
3.34
y = 550 =
= 0.72
y 350 4.62
,
(S39)
263
where “550” and “350” subscripts values at atmospheric CO2 = 550ppm and 350ppm,
264
respectively, and  and  represent the ratios of h and y at elevated relative to ambient CO2.
265
Because the carbon residence time of wood is approximately two orders of magnitude
266
greater than its residence time in either foliage or fine roots, the carbon storage in living forest
267
biomass is dominated by wood. Thus, we can approximate storagedifference (Eq. 3) and storageratio
268
(Eq. 4) by focusing on approximate ESS wood allocation, W0* at atmospheric CO2 = 550ppm
269
and 350ppm.
270
First, storagedifference (Eq. 3) is approximately
é
ö2 æ
ö2 ù
1 ê æ
(1 - u) N
(1 - u) N
(W * -W 0,350*) = mc h êhç (1 - u) N + yy(1 - p) lgM ÷ - ç (1 - u) N + (1 - p) ylgM ÷ úú,
m 0,550
ø è
ø û
ë è
1
(S40)
17
271
where  is the canopy tree mortality rate (0.013 yr-1) and L0* is an increasing function of N (Eq.
272
S38).
273
274
By taking the derivative of the above expression with respect to N, we can show that
storagedifference will increase with N if the following condition is true:
é (1 - u) N + y (1 - p) lgM ù3
hy ê
ú > 1.
ë (1 - u) N + yy (1 - p) lgM û
(S41)
275
The numerator and denominator are necessarily positive, which simplifies the analysis. At the
276
limit of N  0, the condition becomes
h
> 1,
y2
(S42)
277
which is true given the numbers above (Eq. S39). At the opposite limit of N  , the condition
278
becomes
hy >1,
(S43)
279
which is false given the numbers above (Eq. S39). Together, this suggests that storagedifference
280
will increase with N at low N and decrease with N at high N, but it does not suggest how these
281
mathematical limits correspond to biologically “low” and “high” N. Numerically, the exact
282
solution shows that storagedifference increases with N across the range of N that supports closed-
283
canopy, nitrogen-limited forests (Fig. 4f).
284
storageratio (Eq. 4) is approximately
é
ù
ö2
mê æ
(1 - u) N
hhç
÷ - Fú
c êë è (1 - u) N + yy (1 - p) lgM ø
úû
mW 0,550*
.
=
é
ù
mW 0,350*
ö2
mê æ
(1 - u) N
hç
÷ - Fú
c êë è (1 - u) N + y (1 - p) lgM ø
úû
(S44)
18
285
We can simplify this expression by neglecting the fecundity term, F, which is much smaller than
286
the positive term:
2
W 0,550* æ (1 - u) N + y (1 - p) lgM ö
» hç
÷.
W 0,350* è (1 - u) N + yy (1 - p) lgM ø
(S45)
287
By taking the derivative of the above expression with respect to N, we can show that storageratio
288
will increase with N if the following condition is true:
y >1.
(S46)
289
Given the value determined in Eq. (S39), this condition is not true. Thus, storageratio is expected
290
to decrease with increasing N, as confirmed numerically for the exact model (Fig. 5).
291
292
293
Down-regulation of carbon fixation
Finally, the nitrogen and carbon conservation equations (Eqs. S3, S4, S5, S6) assume that
294
trees use all of the nitrogen and carbon that they can capture to make tissues. It is easy to show
295
that no strategy that down-regulates is competitively optimal in a nitrogen-limited ecosystem.
296
That is, strategies that release or voluntarily forgo capture of any carbon or nitrogen rather than
297
making tissue of it are never competitively optimal. For example, we can introduce , a carbon
298
gain “down-regulation factor,” in front of the challenger’s LAI function (Eq. S7), allowing it to
299
decrease both its carbon capture and the costs associated with that carbon capture
æ R f (r )
ö
d
m
m
ç
Lm (W m ,Rm , rm ) =
N
- ( rm + s ) tRm - wW m ÷÷.
(1 - p) lgM çè Rr f ( rr )
ø
(S47)
300
The parameter  varies from zero (complete shutdown of carbon gain) to one (no down-
301
regulation). The derivative of W (Eq. S8) with respect to  is
19
{
}
æ dE L W ,R , r
ö
m[ m
m
m]
ç
0<
- a÷Lm [W m ,Rm , rm ] .
ç dLm [W m ,Rm , r m ]
÷
è
ø
(S48)
302
As we saw above, the term in parentheses will be zero when the marginal return of gross
303
photosynthesis on a marginal increase in LAI exactly equals the total costs of that marginal
304
increase. Since we are here interested in solutions within the nitrogen-limited regime, the term in
305
parentheses will necessarily be positive. The term outside the parentheses is simply LAI, which
306
is also necessarily positive. Together, the whole expression is necessarily positive in a nitrogen-
307
limited stand. Thus, the competitively optimal down-regulation factor takes on the boundary
308
value of one (i.e. no down-regulation); * = 1. In contrast, models that down-regulate assume
309
that trees voluntarily forgo capture of some of either one or the other, so as to keep in
310
stoichiometric balance. Our analysis allows competitively optimal strategies to keep in
311
stoichiometric balance both by shifting allocation among organs (which differ in their mean
312
stoichiometry) and by adjusting tissue stoichiometry itself.
313
20
314
References
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
Berendse F, Elberse WT (1990) Competition and nutrient availability. In: Perspectives on plant
competition. (eds Grace JB, Tilman D). San Diego, Academic Press.
Casper BB, Schenk HJ, Jackson RB (2003) Defining a plant's belowground zone of influence.
Ecology, 84, 2313-2321.
Craine JM, Fargione J, Sugita S (2005) Supply pre-emption, not concentration reduction, is the
mechanism of competition for nutrients. New Phytologist, 166, 933-940.
Dybzinski R, Farrior C, Wolf A, Reich PB, Pacala SW (2011) Evolutionarily Stable Strategy
Carbon Allocation to Foliage, Wood, and Fine Roots in Trees Competing for Light and
Nitrogen: An Analytically Tractable, Individual-Based Model and Quantitative
Comparisons to Data. American Naturalist, 177, 153-166.
Farrior CE, Dybzinski R, Levin SA, Pacala SW (2013) Competition for Water and Light in
Closed-Canopy Forests: A Tractable Model of Carbon Allocation with Implications for
Carbon Sinks. American Naturalist, 181, 314-330.
Frank DA, Pontes AW, Maine EM, Caruana J, Raina R, Raina S, Fridley JD (2010) Grassland
root communities: species distributions and how they are linked to aboveground
abundance. Ecology, 91, 3201-3209.
Geritz SaH, Kisdi E, Meszena G, Metz JaJ (1998) Evolutionarily singular strategies and the
adaptive growth and branching of the evolutionary tree. Evolutionary Ecology, 12, 35-57.
Gilman EF (1988) Tree Root Spread in Relation to Branch Dripline and Harvestable Root Ball.
Hortscience, 23, 351-353.
Gottlicher SG, Taylor AFS, Grip H, Betson NR, Valinger E, Hogberg MN, Hogberg P (2008)
The lateral spread of tree root systems in boreal forests: Estimates based on N-15 uptake
and distribution of sporocarps of ectomycorrhizal fungi. Forest Ecology and
Management, 255, 75-81.
Jones FA, Erickson DL, Bernal MA et al. (2011) The Roots of Diversity: Below Ground Species
Richness and Rooting Distributions in a Tropical Forest Revealed by DNA Barcodes and
Inverse Modeling. PLoS ONE, 6.
Ladeau SL, Clark JS (2006) Elevated CO2 and tree fecundity: the role of tree size, interannual
variability, and population heterogeneity. Global Change Biology, 12, 822-833.
Luyssaert S, Inglima I, Jung M et al. (2007) CO2 balance of boreal, temperate, and tropical
forests derived from a global database. Global Change Biology, 13, 2509-2537.
Mccarthy HR, Oren R, Johnsen KH et al. (2010) Re-assessment of plant carbon dynamics at the
Duke free-air CO2 enrichment site: interactions of atmospheric [CO2] with nitrogen and
water availability over stand development. New Phytologist, 185, 514-528.
Mcgill BJ, Brown JS (2007) Evolutionary game theory and adaptive dynamics of continuous
traits. Annual Review of Ecology Evolution and Systematics, 38, 403-435.
Purves DW, Lichstein JW, Strigul N, Pacala SW (2008) Predicting and understanding forest
dynamics using a simple tractable model. Proceedings of the National Academy of
Sciences, 105, 17018-17022.
Raynaud X, Leadley PW (2004) Soil characteristics play a key role in modeling nutrient
competition in plant communities. Ecology, 85, 2200-2214.
Stone EL, Kalisz PJ (1991) On the Maximum Extent of Tree Roots. Forest Ecology and
Management, 46, 59-102.
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359
360
Figure S1. Model fit of the simple Farquhar model of whole-crown carbon gain (black, Eq. S33
361
and parameter values in text) by the phenomenological Michaelis-Menten model (gray, Eq. S37),
362
yielding parameter values for h, y, , and  (Eq. S39).
363
22
364
365
Figure S2. Pairwise invasion plots of the exact model (Eq. S34) at three different nitrogen
366
mineralization rates and two different CO2 concentrations. Black indicates areas where a
367
challenger (vertical axis) would be successful against a resident (horizontal axis). Note the
368
different scale in e & f. Like the approximate solutions, the exact solutions are convergence-
369
stable: against residents below the ESS, challengers with relatively greater fine root NPP
370
succeed, and against residents above the ESS, challengers with relatively smaller fine root NPP
371
succeed.
372
373
23