Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Digital Appendices
2
Appendix A: Equilibrium solutions and expectations
3
Equilibrium solutions
4
Equations [1] to [7] can be solved at equilibrium, in terms of plant biomass, by assuming that
5
the rates of change are equal to 0. Provided that plants do not go extinct, plant uptake is
6
equal to the loss rate constants from the plant pool at equilibrium:
7
F ( N I* , PI* )  l B  r
8
The litter and soil organic pool sizes at equilibrium are:
9
rN B*
N 
m NL  l NL
[A2]
10
rN B*
P 
 m PL  l PL 
[A3]
11
The available pool sizes at equilibrium are:
12
N I* 
13
PI* 
14
The reactive P pool is equal to:
15
PR* 
16
The amount of plant biomass in the system at equilibrium depends on the limitation status
17
of the ecosystem.
[A1]
*
L
*
L
1
l NI
 l NL
[ S N  N B*  N ] where  N  l B  r 
 mNL  l NL



 l PL

1

[S P  N B*  P ] where  P  l B  r 
l PI
 mPL  l PL 
k1 PI* RS
k 2  k1 PI*
[A4]
[A5]
[A6]
1
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Nitrogen limitation
2
When N is limiting growth alone, plant biomass at equilibrium is equal to:
3
N B* 
4
where the available N pool is:
5
N I*  F 1 (l B  r )
6
For a real positive solution, all state variables are required to be greater than 0 and
7
therefore:
8
F ( N I ) is such that the maximum uptake rate constant is greater than the combined loss
9
rate constants
S N  l NI N I*
[A7]
N
lim
N I 
[A8]
F (N I ) > lB  r ;
10
*
available N input is greater than available loss S N  l NI N I ; and
11
available P input is greater than the rate of loss from the organic part of the system, that is,
12
can satisfy plant demand: S P  PB*  P
13
Phosphorus limitation
14
With P limitation of plant growth, conditions are the same as above with P replaced by N
15
and vice versa.
16
Nitrogen and phosphorus co-limitation
17
When N and P are co-limiting, we used the additive model shown in Equation [8].
18
Substituting [A4] and [A5] into Equation [8], and using the relationship defined by [A1], the
19
amount of N in plant biomass at equilibrium is given by:
[A9]
2
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
 1
  

 2
  S N  N

 
N

2
 

   4  N P


 N P
1
 S  P
1  S  N
N   N
   P
2  N
 P
2
where i 
3
In [A10], the term under the square root is always greater than 0, so the solution is always
4
real. The solution with the square root term added leads to negative available pool sizes (see
5
[A4] and [A5], and stability explanation), and thus the solution where the square root term is
6
subtracted gives the equilibrium amount of N in plant biomass provided:
7
U max  l B  r ,
8
A biological interpretation of the conditions in [A11] can be given. It is necessary for plant
9
persistence that the maximum uptake rate by the plant in the absence of limitation be
*
B

 S  P
    P
 P

liI K i
[A10]
U MAX
1
lB  r
N
SN

P
SP



1
[A11]
10
greater than the loss rate from the plant. The last condition can be understood by
11
considering the extreme case of limitation by only one nutrient as the other nutrient tends
12
to infinity at equilibrium. Thus, given Equation [8], as N I*   , F N I* , PI* 
13
Because this latter term is equal to l B  r , it can be shown that:
14
PI* 
15
This quantity is the level to which the limiting available nutrient (P in this case) would be
16
depleted by the plant in the absence of limitation by the other nutrient (in this case N).
17
Therefore, in the case of co-limitation, the last part of the conditions in [A11] requires that
18
the combined limiting available nutrient inputs be sufficiently greater than the loss of

KP

U max PI*
.
K P  PI*
[A12]
U max
1
lB  r
3
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
available nutrients, dependent on the levels to which they are depleted by the plant. This
2
depends on the strength of limitation; that is, in the absence of limitation by any one
3
nutrient, the condition simplifies to that given by single nutrient limitation (assuming in this
4
case, that uptake is represented by an explicit Michaelis-Menten type function, as opposed
5
to a general saturating function as shown previously).
6
Equilibrium stability
7
Nitrogen limitation
8
In the case of N limitation, plant biomass is determined by the limiting nutrient, and the
9
system simplifies to Equations [2], [4], and [6] with nutrient uptake denoted by F(NI). The
10
local stability of the equilibrium for N limitation can be assessed using standard methods
11
(Gurney and Nisbet 1998). The characteristic equation, det(J - λI) = 0, calculated from the
12
Jacobian matrix (J) is put in the form:
13
3  A12  A2   A3  0 .
14
The Jacobian matrix is:
15



  l NI   N B  F ( N I )  
 F (N I )
  N  

I

 

 F ( N I ) 

J= 
N B 
F N I   l B  r


N
I



0
r



16
Given that at equilibrium F ( N I )  l B  r , the coefficients of the characteristic equation are
17
shown to easily satisfy the Routh-Hurwitz conditions:
[A13]
4




 [A14]
0

 l NL  m NL 



m NL
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
A1  0
1
A3  0
[A15]
A1 A2  A3  0
2
Phosphorus limitation
3
In the case of P limitation, plant biomass is determined by the limiting nutrient, and the
4
system simplifies to Equations [1], [3], [5], and [7], with nutrient uptake denoted by F(PI),
5
and NB/α replaced by PB. Numerical simulations with random parameters and initial
6
conditions taken from uniform distributions with wide ranges showed that providing
7
conditions given in [A9] were met, with necessary replacements, solutions generally
8
converged to the expected equilibrium. Solutions did not converge when the reactive pool
9
could out-compete plants for available P. Where this occurred, plant extinction could
10
prevent the attainment of the predicted stable equilibrium. Extinction is associated with a
11
reactive pool that has a high capacity for adsorption that is, the maximum density of
12
sorption sites (RS) and the adsorption rate (k1) tend to be high, whilst there is a restricted
13
supply of available P that is, available P input (SP), mineralization of organic matter (mPL) and
14
desorption rate (k2) tend to be low. This means that the reactive pool is more of a sink for
15
the P that is available compared to the capacity of the plant to take it up. We used an explicit
16
saturating function to describe uptake, such that:
17
F ( PI ) 
18
Nitrogen and phosphorus co-limitation
19
Numerical simulations with random parameters taken from distributions with the ranges
20
shown in Table B2 showed that provided conditions given in [A11] were met, and the
21
solution to [A10] with the square root subtracted was used, simulations generally converged
22
to the expected equilibrium. Initial conditions were randomly chosen from within 10% of the
U max PI
K P  PI
[A16]
5
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
equilibrium value for the system in question. As with P limitation alone, reactive P could
2
occasionally out-compete available P leading to plant extinction even though persistence
3
conditions were met. This occurred with distributions as outlined above.
4
Equilibrium expectations
5
The expected direction of change in the available and reactive P pools at equilibrium can be
6
found by differentiating the pool sizes with respect to the management being applied.
7
Nitrogen fertilization
8
The effect of increased N input on equilibrium pool sizes depends on the limitation status of
9
the ecosystem.
10
Nitrogen limitation
11
From [A7], the differential of plant biomass N with respect to N addition is:
12
dN B*
1

0
dS N  N
13
From [A5], the differential of the available P pool is given by:
14
dPI*
 dN B*
 P
<0
dS N
l PI dS N
15
Phosphorus limitation
16
The differential of plant biomass P with respect to N addition is:
17
dPB*
0
dS N
18
Given [A18] there is no change in the available P pool at equilibrium following N addition.
[A17]
[A18]
[A19]
6
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Nitrogen and phosphorus co-limitation
2
From [A10], the differential of plant biomass N with respect to N addition is:
3



dN B*
1 

1
dS N 2 N 



 S N  N

 N
  S N  N

 
N




  0 [A20]

  N P  
 
 4 
 N P  

 S  P
    P
 P


 S  P
    P
 P


 


2



4
The differential of the available P pool is given by [A18] and is therefore necessarily less than
5
0.
6
Cutting
7
The effect of cutting on equilibrium pool sizes depends on the limitation status of the
8
ecosystem. In this section we prove the basis of an optimum level of cutting (lBopt) that can
9
exist for cutting to decrease available P under certain limitation conditions, and we also
10
describe how the case of co-limitation is necessarily between that for single nutrient
11
limitation alone. In the next section, we prove that available P pool decrease comes about
12
because the reduction in mineralization from the litter P pool at equilibrium is greater than
13
the reduction in uptake by the plant.
14
Nitrogen limitation
15
[A5] shows the equilibrium available P pool size.
16
Therefore:
17
dPI*
1

dl B
l PI
18
From [A7]:
 *  P N B*

 N B

 P 
l B
l B


[A21]
7
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
 N I* 

  N B*

l
NI 
*
N B
 l B 

l B
N
[A22]
2
N I*
 (l  r )
1
1

* B

*
l B
l B
F' NI
F ' N I*
3
[A23] is positive by definition, because plant uptake increases with the available N pool, and
4
therefore plant biomass always declines with cutting. [A23] shows that the available pool of
5
the limiting nutrient (N) will always increase with cutting.
 
 
[A23]
6
7
Given [A22] and [A23], [A21] rearranges to:
8

l
dPI*
1  P  *   N

  NI *

N

1
B
dl B
l PI  N    P
 F' NI
9
For plant available P to decrease with cutting, it is necessary that the term enclosed within
 




[A24]
10
the outer brackets is greater than 0, which necessitates that:
11
 N   P  l B   P 
12
where  i 
13
σi refers to the rate constant of nutrient lost from the plant pool that is not recycled in the
14
system. [A25] therefore states that the recycling of the limiting nutrient has to be sufficiently
15
poor, whilst that of the non-limiting nutrient (P) is sufficiently good. Furthermore, the right
16
hand side of [A25] will be smaller where the strength of N limitation is strong, although the


l NI

*
* 
F
'
N
N
I
B 

 
[A25]
rl iL
, i = N or P.
liL  miL
8
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
system should also sustain a low rate of loss of available N. In sum the system has to have
2
good recycling properties for P with poor recycling properties for N.
3
4
[A25] allows derivation of the optimal cutting level ( l Bopt ) for declines in available P. Where
5
the condition in [A25] is met, the differential of available P is negative with respect to
6
increasing cutting rate. However, there comes a point at which the inequality in [A25]
7
becomes equal. This is because as cutting increases, the right hand side gets larger: N B* get
8
smaller, and l B increases. The left hand side of the inequality remains constant. The point at
9
which both sides are equal is the optimal cutting level, as shown in Figure A1. Beyond this
10
value, equilibrium available P will start increasing with cutting, given that the differential is
11
now positive, thus slowing down the depletion of the reactive P pool. Eventually, the loss
12
rate from the plant pool will be greater than the maximum uptake rate of the plant, and
13
extinction of the plant will occur (Condition [A9]). In such a scenario, the available pool sizes
14
at equilibrium will then simply be the ratio of input over available loss: i I* 
15
equal to N or P.
Si
, where i is
liI
16
17
Setting [A25] equal and rearranging in terms of l B , the optimal cutting rate to decrease
18
available P under N limited conditions is thus defined as:
19
l Bopt 
20
Phosphorus limitation
 
F ' N I* N B*
 N   P    P
l NI
[A26]
9
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
With P limitation of plant growth, available P will always increase with increases in cutting,
2
with similar reasoning to [A23]:
3
PI*
 (l  r )
1
1

* B

*
l B
l B
F ' PI
F ' PI*
4
Nitrogen and phosphorus co-limitation
5
The differential of available P with respect to cutting was too complicated to investigate by
6
inspection, as was the expression for the differential of plant biomass with respect to
7
cutting. We simulated 100,000 systems with parameters randomly picked from within
8
biologically reasonable ranges. Table A2 shows that when the condition in [A25] was
9
approximated for the co-limited case by this expression:
 
 
[A27]




l NI


  P  l B   P 
*
*

F N I , PI
*


N
B 

N I*


10
N
11
the differential was negative when the condition was satisfied. When the condition was not
12
satisfied, cutting would increase the available P pool size. There were no cases that did not
13
behave as expected.
14
The slope of the uptake function with respect to available N evaluated at equilibrium
15
F N I* , PI*
, was found by partially differentiating Equation [8]:
N I*
16
17






[A28]
U max K N
F N I* , PI*

2
N I*
 * KP KN

 NI 

  P *  N *  1 
I

  I
[A29]
[A28] was rearranged to calculate lBopt:
10
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
F  N I , PI 
NB
N I
 N   P    P

l NI
1
l Bopt
[A30]
2
Nitrogen limitation of growth alone: Available phosphorus at equilibrium decreases with
3
cutting because the reduction in mineralization is greater than the reduction in uptake
4
This section proves that the reduction in available P that occurs at or below the optimum
5
cutting rate is due to the decline in mineralization compared to the decline in uptake rate.
6
7
At equilibrium, plant uptake of P is equal to plant loss of P ([A1]). Therefore the differential
8
of plant uptake of P with respect to cutting:

N* 
d  F N I* B 
  N B* l B  r  N B*




dl B

  l B
 
9



[A31]
10
The differential of mineralization of P from the litter with respect to cutting:
11
d mPL PL*
rm PL
N B*

dl B
 mPL  l PL  l B
12
For the decrease in mineralization to be greater than the decrease in plant uptake, and given
13
that the change in biomass with respect to cutting is negative, it is necessary for:
14
rm PL
N B* N B* l B  r  N B*



 mPL  l PL  l B

  l B
15
Rearranging [A33] and substituting in the expression for
16
for available P to decrease with cutting:


[A32]



[A33]
11
N B*
([A22]), it can be shown that
l B
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1


l
N B*  N  1  NI *
 P
 F' NI
2
This is the same condition as the expression in the outer parentheses of [A24], for a decline
3
in available P with increases in cutting. Therefore the reduction in mineralization is greater
4
than the decline in plant uptake when available P decreases in the system with cutting at
5
equilibrium.
6
Dependence of the equilibrium reactive phosphorus pool on the available phosphorus
7
pool
8
The reactive P pool at equilibrium is given by [A6]. Taking the derivative of this with respect
9
to small increases in the available P pool at equilibrium gives:
 
[A34]
10
k 2 k1 RS
PR*

*
PI
k 2  k1 PI*
11
[A35] is necessarily greater than 0. This confirms that the reactive pool is in dynamic
12
equilibrium with the available P pool, and whichever direction the available pool changes,
13
the reactive pool mirrors. The only exception occurs where k1 PI*  k 2 ; in this case
14
PR*  RS , and the reactive pool should not change at equilibrium.

[A35]

2
12
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Appendix B: Investigating transient dynamics
2
Initial Conditions
3
Table B1 presents the initial conditions used to simulate the system. To maintain consistency
4
between runs we scaled initial conditions to be proportional to equilibrium pool sizes. As
5
explained in the text, the level of available nutrient was assumed to be variable in N
6
fertilization scenarios. However, given the restrictive constraints on when cutting was
7
expected to be successful we constrained the available nutrient values, to make it more
8
likely that the condition for cutting to lower available P would be satisfied. By using low
9
available N and high available P in comparison to their equilibrium values, we attempted to
10
make it more likely that they system was highly N limited. We calculated lBopt ([A30]) using
11
initial equilibrium conditions in the absence of management as scaled in Table A2.
12
Parameter Ranges
13
Empirical estimates of parameters within the model are rare. Where possible, ranges in
14
Table B2 were constrained as being an order of magnitude difference from available sources
15
as explained below. Although these values are not from former farmland, our aim was to
16
test generalities and thus absolute values are not crucial. In some instances, we could not
17
source estimates and used wide biologically reasonable ranges.
18
19
SN was from Vitousek (2004); SP was estimated from Chadwick and others (1999).
20
21
lNI, lNL, lPI, and lPL were calculated as the amount of nutrient lost (through leaching plus
22
denitrification in the case of available N) in a given year divided by pool sizes for Kokee in
23
Hawaii. Unavailable organic losses were assumed to be losses reported as dissolved organic
13
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
nutrient. Available losses were assumed to be losses reported as dissolved inorganic
2
nutrient. Calculations for the amount of loss were derived from Table 4 and Appendix B in
3
Hedin and others (2003). We assumed that the total amount of nutrient loss shown in Table
4
4 was made up of the percentages shown in Appendix B for soil solution concentrations of
5
dissolved organic P and dissolved inorganic P. Available N was derived from total N reported
6
in Crews and others (1995), assuming that 5% of the total was plant available (Sumner
7
2000), the remainder was assumed to be the organic N pool size in our model.
8
9
r was calculated as the amount of leaf litter fall divided by the leaf biomass pool size using
10
measurements from within Herbert and Fownes (1999); α was estimated as N leaf biomass
11
pool size divided by P leaf biomass pool size. Kinetic plant uptake parameters (Umax, KN, KP)
12
were sourced from O’Neill and others (1989).
13
14
mNL was calculated using data from Riley and Vitousek (1995). We divided their estimates for
15
flux by the organic N pool size estimated above. mPL was estimated using mineralization data
16
from Harrison (1982). He worked on Lake District soils and used 32P-RNA to assess relatively
17
labile litter P mineralization. He added 780ngP cm-3 labelled with 32P. 28ngP cm-3 day-1 was
18
mineralized giving a mineralization rate constant of 0.036 day-1. We assumed that this was
19
proportional for the year, but given that he only examined very fast turnover litter, we
20
assumed that the rate constant was an order of magnitude slower.
21
22
We could not find estimates for lB, k1, k2 and RS, and therefore used wide biologically
23
reasonable ranges for these parameters.
14
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Simulations: Notes on methodology and limitation scenarios
2
The text describes in general how we carried out simulations into management. In cutting
3
simulations, we checked whether the optimum level of cutting (calculated from initial
4
conditions) was positive, and whether it allowed plant persistence at equilibrium. We then
5
applied this initial optimum level throughout the management time period (50 years),
6
bearing in mind that the optimum level would change through time due to variation in the
7
state variables of the system (see below). We also checked whether systems eventually
8
converged to equilibrium with and without cutting. For N fertilizer simulations, we checked
9
whether plants could persist at equilibrium in the absence of fertilizer. Where persistence
10
was possible, we added 50kgN ha-1 y-1 for 50 years, and compared the reactive and available
11
pools at the end of the simulation in managed and unmanaged scenarios.
12
13
Tables 1 and 2 summarize the range of limitation scenarios. Table B3 gives all the possible
14
limitation scenarios for both N fertilization and cutting simulations. For cutting simulations,
15
category A (N limited throughout in both managed and unmanaged scenarios) was
16
subdivided into the following categories, taking into account how the optimum level of
17
cutting could change through the simulation:
18
19
A1: Applied level was below the calculated optimum in both managed and unmanaged
20
scenarios (top two rows of Table 1).
21
A2: Applied level was below the calculated optimum in managed scenarios, and was above
22
in unmanaged scenarios.
15
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
A3: Applied level was above the calculated optimum in managed scenarios, and was below
2
in unmanaged scenarios.
3
A4: Applied level was not always above nor below the calculated optimum in managed
4
scenarios, always above or below in unmanaged scenarios.
5
A5: Applied level was always above or below the calculated optimum in managed scenarios,
6
whereas it changed in the unmanaged scenarios.
7
A6: Applied level changed in relation to the calculated optimum in both managed and
8
unmanaged scenarios.
9
A7: Applied level was above the calculated optimum in managed scenarios, and was below
10
in unmanaged scenarios.
11
Parameter values for figures
12
Figure 2:
13
Parameter values for unperturbed system in a): SN: 18.5; SP: 0.006; lNI: 0.008; lNL: 0.0001; mNL:
14
0.001; lPI: 0.00002; lPL: 0.000001; mPL: 0.009; r: 0.011; c: 0; Umax: 9.5; KN: 64.6; KP: 6; α: 4.4; k1:
15
0.009; k2: 0.24; RS: 261.
16
Parameter values for unperturbed system in b): SN: 33; SP: 0.07; lNI: 0.021; lNL: 0.0017; mNL:
17
0.05; lPI: 0.00005; lPL: 0.00003; mPL: 0.009; r: 0.015; c: 0; Umax: 20.1; KN: 227.6; KP: 37.8; α: 8.8;
18
k1: 0.12; k2: 0.63; RS: 105
19
Figure 3:
20
Parameter values for unperturbed system in a): SN: 1.1; SP: 0.009; lNI: 0.064; lNL: 0.0002; mNL:
21
0.01; lPI: 0.0001; lPL: 0.000001; mPL: 0.0009; r: 0.026; c: 0.00001; Umax: 20.9; KN: 39.2; KP: 1.5;
22
α: 5.98; k1: 0.002; k2: 0.22; RS: 780.
16
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Parameter values for unperturbed system in b): SN: 2.7; SP: 0.02; lNI: 0.0018; lNL: 0.0017; mNL:
2
0.04; lPI: 0.00001; lPL: 0.00004; mPL: 0.001; r: 0.89; c: 0.00005; Umax: 32.2; KN: 11.54; KP: 45; α:
3
5.86; k1: 0.0224; k2: 0.296; RS: 1122.
17
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Appendix C: Robustness of model to assumptions
2
(1)
3
Our model does not include a potentially important loss from agricultural systems: that of
4
particulate reactive P (Jordan and Smith 1985; Heckrath and others 1995; Haygarth and
5
Condron 2004). Including this loss would lower the equilibrium reactive pool size but
6
otherwise should not alter the equilibrium insights. Where N fertilization lowers available P
7
the reactive P would also decrease as it buffers the available P pool. Given that the transient
8
dynamics mirrored our equilibrium expectations with N fertilization, there is no reason to
9
suspect that including this loss would substantially alter our conclusions. Furthermore,
Particulate loss
10
particulate loss from the reactive P pool may become a less important loss pathway as the
11
system moves from its former cropland status given that susceptibility to erosion should
12
decline as vegetative cover increases. However, organic P particulate loss may increase. This
13
would not alter the conclusions presented here.
14
15
(2)
N impact on P mineralization
16
Management may have other dynamic impacts on ecosystem P fluxes. In particular, N has
17
been shown to increase the activity of phosphatase enzymes (Olander and Vitousek 2000;
18
Treseder and Vitousek 2001; Pilkington and others 2005), potentially increasing the supply
19
and therefore availability of P, through organic matter mineralization. However, short term
20
N addition does not necessarily lead to increased root phosphatase enzyme activity,
21
probably dependent on the continuing status of N limitation within the system (Johnson and
22
others 1999). Therefore, P mineralization would increase where the system goes into P
23
limitation, in which case N fertilizer should not be added as already presented.
24
(3)
N impact on reactive P - available P exchange reaction
18
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
N fertilization often leads to acidification depending on the type of fertilizer applied
2
(Condron and Goh 1990; Eviner and others 2000; Crawley and others 2005). This can
3
influence parameters associated with the dynamic between the plant available and reactive
4
P pools and may therefore influence the success of management. Reactive P should still
5
decrease in size where plant growth is increased by N addition however. Furthermore, we
6
suggest that only non-acidifying fertilizers should be used, such as ammonium nitrate. At any
7
rate, acidification should be avoided as this tends to depress species diversity (Crawley and
8
others 2005), and is difficult to revert. The application of lime to counteract any acidification
9
should also be avoided as it may increase the reactive P pool by increasing the sorption
10
capacity of the soil, given the propensity of phosphate to bind to calcium (Tiessen and others
11
1984).
12
(4)
13
Cutting, and more likely N fertilization, could lead to changed N to P ratios either through
14
changed community composition or directly through a plastic response of plants, to
15
fertilization in particular (Tilman 1998; Gusewell 2004). This would be contrary to our
16
assumption of a fixed ratio. Our conclusions should not be changed by assuming a plastic
17
ratio: available and reactive P should still decline with N fertilization providing P uptake is
18
maintained. There are no obvious reasons to suspect that the rare and marginal success of
19
cutting would be altered with changed ratios although cutting may lead to the introduction
20
of species that demand more P and thus could lead to reduced available and reactive P pool
21
sizes. The converse may be equally likely. Mathematical results will likely change (Grover
22
2003). Community compositional change/direct change in ratio could have further impacts
23
upon litter quality, thereby influencing mineralization rates that may subsequently alter P
24
availability too (Hobbie 1992; Knops and others 2001).
25
(5)
Variable N to P ratios
Plant nutrient uptake limited by diffusion
19
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
We have assumed that uptake of nutrient by the plants is solely limited by nutrient
2
concentration, as presented by O’Neill (1989). However, the supply of nutrient to plant roots
3
may not depend on concentration and may rather depend on the diffusion of the nutrient
4
(Tinker and Nye 2000). This is unlikely to apply to N, given its high mobility (Tinker and Nye
5
2000; Raynaud and Leadley 2004), but P could be diffusion limited. We would argue that in
6
such cases, P would be limiting for plant growth, unlikely to be a management problem and
7
a constraint to species diversity.
20
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Literature cited
2
Chadwick, O. A., L. A. Derry, and others (1999). "Changing sources of nutrients during four
3
million years of ecosystem development." Nature 397: 491-497.
4
Condron, L. M. and K. M. Goh (1990). "Nature and availability of residual phosphorus in long-
5
term fertilized pasture soils in New Zealand." Journal of Agricultural Science,
6
Cambridge 114: 1-9.
7
8
9
Crawley, M. J., A. E. Johnston, and others (2005). "Determinants of species richness in the
Park Grass Experiment." American Naturalist 165: 179-192.
Crews, T. E., K. Kitayama, and others (1995). "Changes in soil-phosphorus fractions and
10
ecosystem dynamics across a long chronosequence in Hawaii." Ecology 76(5): 1407-
11
1424.
12
Eviner, V. T., F. S. Chapin, and others (2000). Nutrient manipulations in terrestrial
13
ecosystems. Methods in ecosystem science. O. E. Sala, R. B. Jackson, H. A. Mooney
14
and R. W. Howarth. New York, Springer-Verlag: 291-307.
15
Grover, J. P. (2003). "The impact of variable stoichiometry on predator-prey interactions: A
16
multinutrient approach." American Naturalist 162(1): 29-43.
17
Gurney, W. S. C. and R. M. Nisbet (1998). Ecological dynamics. Oxford, OUP.
18
Gusewell, S. (2004). "N:P ratios in terrestrial plants: variation and functional significance."
19
New Phytologist 164(2): 243-266.
20
Harrison, A. F. (1982). "32P-method to compare rates of mineralization of labile organic
21
phosphorus in woodland soils." Soil Biology and Biogeochemistry 14: 337-341.
21
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Haygarth, P. M. and L. M. Condron (2004). Background and elevated phosphorus release
2
from terrestrial environments. Phosphorus in environmental technology: Principles
3
and applications. E. Valsami-Jones. London, IWA Publishing: 79-92.
4
Heckrath, G., P. C. Brookes, and others (1995). "Phosphorus leaching from soils containing
5
different phosphorus concentrations in the Broadbalk Experiment." Journal of
6
Environmental Quality 24: 904-910.
7
8
9
10
11
12
13
Hedin, L. O., P. M. Vitousek, and others (2003). "Nutrient losses over four million years of
tropical forest development." Ecology 84(9): 2231-2255.
Herbert, D. A. and J. H. Fownes (1999). "Forest productivity and efficiency of resource use
across a chronosequence of tropical montane soils." Ecosystems 2: 242-254.
Hobbie, S. E. (1992). "Effects of plant-species on nutrient cycling." Trends in Ecology &
Evolution 7(10): 336-339.
Johnson, D., J. R. Leake, and others (1999). "The effects of quantity and duration of
14
simulated pollutant nitrogen deposition on root-surface phosphatase activities in
15
calcareous and acid grasslands: a bioassay approach." New Phytologist 141: 433 -
16
442.
17
Jordan, C. and R. V. Smith (1985). "Factors affecting leaching of nutrients from an intensively
18
managed grassland in County Antrim, Northern Ireland." Journal of Environmental
19
Management 20: 1-15.
20
Knops, J. M. H., D. Wedin, and others (2001). "Biodiversity and decomposition in
21
experimental grassland ecosystems." Oecologia 126(3): 429-433.
22
23
O'Neill, R. V., D. L. DeAngelis, and others (1989). "Multiple nutrient limitations in ecological
models." Ecological Modelling 46: 147-163.
22
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
2
3
Olander, L. P. and P. M. Vitousek (2000). "Regulation of soil phosphatase and chitinase
activity by N and P availability." Biogeochemistry 49: 175-190.
Pilkington, M. G., S. J. M. Caporn, and others (2005). "Effects of increased deposition of
4
atmospheric nitrogen on an upland Calluna moor: N and P transformations."
5
Environmental Pollution 135: 469-480.
6
7
8
9
Raynaud, X. and P. W. Leadley (2004). "Soil characteristics play a key role in modeling
nutrient competition in plant communities." Ecology 85(8): 2200-2214.
Riley, R. H. and P. M. Vitousek (1995). "Nutrient dynamics and nitrogen trace gas flux during
ecosystem development in montane rain forest." Ecology 76: 292-304.
10
Sumner, M. E., Ed. (2000). Handbook of Soil Science. New York, CRC Press.
11
Tiessen, H., J. W. B. Stewart, and others (1984). "Pathways of phosphorus transformations in
12
13
soils of differing pedogenesis." Soil Science Society of America Journal 48: 853-858.
Tilman, D. (1998). Species composition, species diversity, and ecosystem processes:
14
understanding the impacts of global change. Successes, limitations, and frontiers in
15
ecosystem sciences. M. L. Pace and P. M. Groffman. New York, Springer-Verlag: 452-
16
472.
17
18
Tinker, P. B. and P. H. Nye (2000). Solute movement in the rhizosphere. Oxford, Oxford
University Press.
19
Treseder, K. K. and P. M. Vitousek (2001). "Effects of soil nutrient availability on investment
20
in acquisition of N and P in Hawaiian rain forests." Ecology 82(4): 946-954.
21
22
Vitousek, P. M. (2004). Nutrient cycling and limitation: Hawai'i as a model system. Princeton,
Princeton University Press.
23
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
2
Figure A1
PI*
l Bopt
Increasing
3
24
lB
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Figure Legends
2
Figure A1: Schematic of change in available P pool (PI*) at equilibrium as cutting increases
3
when plant growth is N limited. In most cases, available P will increase given the reduction in
4
plant uptake due to decreased biomass (top solid black line). However, [A25] indicates that
5
in systems where recycling of P is sufficiently greater than recycling of N, and plant growth is
6
sufficiently N limited, available P may decline at equilibrium following increased losses
7
through cutting (bottom solid black line). The inequality in [A25] can not be satisfied for all
8
cutting rates and thus available P will increase at equilibrium beyond some optimal cutting
9
level ( l Bopt ) denoted by the dashed line. Too high a rate of cutting will make plant
10
persistence unfeasible, and available pools will then tend to the ratio of available input over
11
available loss.
12
25
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Table A1: Conditions for Available Phosphorus Decline at Equilibrium
Limitation Status
Management Type
Of System
N fertilization
Cutting
Available P declines:
N Limited

Available P declines
l NI
 F ' N I N B
 N   P  l B   P 



Available P declines†:
Co-limited
Available P declines
P limited
No change in available P
N
26


l NI
  P  l B   P 
 F  N I , PI 
NB

N I

Available P increases






Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Table A2: Evaluating the Co-limited Condition for Available P Reduction
Available P response
[A28] True
[A28] False
Increase
0
99381
Decrease
619
0
27
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Table B1: Initial Conditions for Simulations
State Variable
Initial Value
Plant Biomass
N B [0] 
Unavailable soil organic
and litter N
Unavailable soil organic
and litter P
Comment
N B*
100
In all simulation scenarios. Equilibrium
N L*
N L [0] 
100
In all simulation scenarios. Equilibrium
PL [0] 
value calculated with no management
value calculated with no management
In all simulation scenarios. Equilibrium
PL*
100
value calculated with no management
With N addition.
Plant available N
N I [0] 
*
I
N
 100 N I*
10
With cutting simulations, 1/10 equilibrium
value calculated without cutting
With N addition.
Plant available P
PI [0] 
PI*
 100 PI*
10
With cutting simulations, 1000 times
equilibrium value calculated without
cutting
Reactive P
RS
In all simulations
2
28
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Table B2: Parameter Dimensions, Definitions, and Ranges used for Simulations
Parameter
Dimension
Minimum
Maximum
Value
Value
Meaning
SN
kg ha-1 y-1
Available N input
0.7
70
l NI
y-1
Available N loss rate constant
0.001
0.1
lB
y-1
0.00001*
0.001*
r
y-1
0.01
1
0.00002
0.002
0.0009
0.09
Plant biomass loss rate
constant: cutting
Plant death rate constant to
unavailable organic matter
Unavailable N loss rate
l NL
y-1
constant
Mineralization of N rate
mNL
y-1
constant
SP
kg ha-1 y-1
Available P input
0.0009
0.09
l PI
y-1
Available P loss rate constant
0.000006
0.0006
l PL
y-1
0.000001
0.0001
mPL
y-1
0.0004
0.04

Dimensionless
1.3
130
RS
kg ha-1
100
10000
Unavailable P loss rate
constant
Mineralization of P rate
constant
Plant biomass N:P
Maximum density of reactive
P sorption sites
k1
ha kg-1 y-1
Adsorption rate constant
0.00001
1
k2
y-1
Desorption rate constant
0.00001
1
U max
y-1
0.9
90
4.5
450
Maximum uptake rate
constant
KN
kg ha-1
Half saturation constant for N
29
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
KP
kg ha-1
Half saturation constant for P
1
30
1.3
130
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Table B3: Limitation Scenarios for Transient Simulations
Ecosystem
Category
Managed
Unmanaged
A
N limitation throughout
N limitation throughout
B
N to P limitation
N limitation throughout
C
Multiple limitation
N limitation throughout
D
P limitation throughout
P limitation throughout
E*
P to N limitation
P limitation throughout
F*
Multiple limitation
P limitation throughout
G
N to P limitation
N to P limitation
H*
Multiple limitation
N to P limitation
I
P to N limitation
P to N limitation
J
Multiple limitation
P to N limitation
K
Multiple limitation
Multiple limitation
L*
N limitation throughout
N toP limitation
M
P limitation throughout
P to N limitation
N*
N limitation throughout
Multiple limitation
O
N to P limitation
Multiple limitation
P
P limitation throughout
Multiple limitation
Q
P to N limitation
Multiple limitation
2
31
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Table B4: Average Responses to Parameter Perturbations of +/- 10%.
10% increase
Parameter
Available P
Reactive P
10% decrease
Plant P
Available P
%
Reactive P
Plant P
%
SN
9.51
4.87
-0.26
-5.56
-4.16
0.14
l NI
-0.04
-0.05
0.25
0.04
0.05
-0.25
r
-0.8
-0.72
1.86
0.96
0.87
-1.9
l NL
0.01
0.001
0.06
-0.007
-0.002
-0.06
m NL
0.17
0.3
-3.35
-0.13
-0.44
3.38
SP
0.17
0.16
-0.03
-0.17
-0.16
0.03
l PI
0.001
0.001
0
-0.001
-0.002
0
l PL
-0.001
-0.001
0.001
0.0004
0.0005
-0.001
mPL
-0.25
-0.33
-0.83
0.26
0.37
0.74

-7.27
-7.52
1.71
13.5
9.58
-5.45
RS
-0.67
-0.38
0.46
0.73
0.38
-0.66
k1
-0.49
-0.46
0.38
0.53
0.5
-0.52
k2
0.22
0.82
0.34
-0.13
-0.83
-0.45
U max
0.69
0.61
-0.53
-0.78
-0.68
0.48
KN
-0.09
-0.06
0.006
0.09
0.06
-0.007
KP
-0.6
-0.54
0.4
0.66
0.6
-0.53
2
32
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
Table Legends
2
Table A1: Conditions for decline at equilibrium also give hypotheses for the transient effect,
3
following perturbations to the amount of nitrogen added to the system, or following cutting
4
and removal of plant biomass. Equations are defined within the main text to the Digital
5
Appendices.
6
†
7
However, simulations with a wide range of parameters showed that providing conditions
8
were as above, the differential of available phosphorus with respect to cutting was positive
9
or negative at equilibrium as expected (Table A2).
: In the case of the co-limited model, it was not possible to derive analytical conditions.
10
Table A2: Equilibrium results when evaluating the co-limited condition for a reduction in
11
available phosphorus following cutting ([A28]) using simulations taking parameters randomly
12
from biologically reasonable ranges. There were no cases where the inequality was satisfied
13
but available phosphorus increased, and vice versa, suggesting that the condition is a valid
14
approximation.
15
Table B1: No legend
16
Table B2: * In nitrogen addition simulations. The cutting value could be outside of this value
17
in the cutting simulations, depending on the other parameter values of the system, with lB
18
set to 0 in the no management scenario for cutting simulations.
19
Table B3: * These categories are not possible given N fertilization management.
20
Table B4: Table gives average sensitivity of responses pooled across systems shown in
21
Figures 2 and 3. The number shown is the percentage difference between the magnitude of
22
response to management in the original simulation and the response in the perturbed,
23
averaged across simulations. Qualitative dynamics were similar to those in the Figures, that
33
Perring, Edwards and de Mazancourt: Phosphorus removal from ecosystems
1
is, where management led to a decline in reactive P in the unperturbed scenario, this decline
2
was maintained with altered parameters, hence pooled responses were examined. A
3
positive result means that the percentage difference between pool sizes given management
4
was greater following perturbation to the parameter. This table does not indicate whether
5
pool sizes between managed and unmanaged increased or decreased following
6
management.
7
8
9
10
11
34