Rastetter et al.
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Running Head: A Stable Isotope Simulator
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A Stable Isotope Simulator that Can Be Coupled to
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Existing Mass-Balance Models
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Edward B. Rastetter1
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Bonnie L. Kwiatkowski1
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Robert B. McKane2
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1
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The Ecosystems Center
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Marine Biological Laboratory
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Woods Hole, MA 02543
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2
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US Environmental Protection Agency
Corvallis, OR 97333
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Contact:
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Edward B. Rastetter
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TEL.: 508-289-7483
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FAX.: 508-457-1548
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Email: erastett@mbl.edu
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Rastetter et al.
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Abstract
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We present software to simulate changes in the isotopic signatures of the stocks of an
3
element using the output from any parent model that specifies the stocks and flux rates of that
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element based on a mass-balance approach. This parent model can be a simple mass-balance
5
spreadsheet of the system. The isotopic simulations use a linear, donor-controlled approximation
6
of the fluxes in the parent model, which are updated for each time step. These approximations
7
are based on the output of the parent model, so no modifications to the parent model are
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required. However, all fluxes provided to the simulator must be gross fluxes and the user must
9
provide the initial isotopic signature for all stocks, the fractionation associated with each flux,
10
and the isotopic signature of any flux originating from outside the system.
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We illustrate the use of the simulator with two examples. The first is based on a model of
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the carbon and nitrogen mass balance in an eight-species food web. We examine the
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consequences of using the steady-state assumption implicit in multi-source mixing models often
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used to map food webs based on 13C and 15N. We also use the simulator to analyze a pulse-chase
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Long-Term Ecological Research site. We examine the constraints on net versus gross N
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mineralization that are necessary to match the observed changes in the isotopic signatures of the
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forest N stocks.
N labeling experiment based on a spreadsheet model of the nitrogen cycle at the Harvard Forest
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Key Words: Stable Isotopes, Isotope Simulation, Mass Balance Model, Food-web Model,
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Isotope Dynamics, 15N, 13C, Harvard Forest, Nitrogen Cycle
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Introduction
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Stable isotopes are now commonly used in biogeochemical, community, and
2
ecophysiological studies (Peterson and Fry 1987, Rundel et al. 1989, Ehleringer et al. 1993,
3
Lajtha and Michener 1994, Griffiths 1998). For example, 15N labeling has been used to assess
4
gross rates of N processing in soils (e.g., Stark and Hart 1997, Neill et al. 1999), to assess plant
5
uptake of organic N (e.g., Chapin et al. 1993), to track the fate of N inputs to ecosystems
6
(Nadelhoffer et al. 1999), and to examine niche partitioning (e.g., McKane et al. 1990, 2002).
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The natural abundances of C isotopes have been used to assess water relations in plants
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(Farquhar and Richards 1984, Farquhar, et al. 1989). The combined natural abundances of C and
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N isotopes are commonly used to map food webs (e.g., Peterson et al. 1986, 1997, Peterson and
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Howarth 1987, Schell and Zieman 1989, Kling et al. 1992, Hansson et al. 1997, Wollheim et
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al.1999, Saito et al. 2001, Phillips 2001, Phillips and Gregg 2001, 2003, Phillips and Koch 2002)
12
or to discern changes in the diet of individual species (e.g., Ben-David et al. 1997, Schell et al.
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1989). Clearly stable isotopes are playing an important and growing role in ecological studies.
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However, their incorporation into ecological models has lagged behind.
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Although isotope-based estimates of process rates and trophic connections have been
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used to parameterize dynamic simulation models (e.g., Saito et al. 2001), relatively few
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ecological models have been developed that simulate the dynamics of stable isotopes directly
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(e.g., van Dam and van Breemen 1995, Koopmans and van Dam 1998, Currie et al. 1999, Currie
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and Nadelhoffer 1999, Hobbie et al. 1999). A dynamic assessment of isotope signature could
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overcome the problems associated with the (often-implicit) assumption of steady state used in
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many isotope studies. For example, in many food-web studies, C and N isotopes are used to
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discern the relative importance of various prey species in the diet of a consumer based on a
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multiple-source mixing model. However, because of the underlying assumption of a near steady
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state in the application of these mixing models, quantifying the relative importance by this
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technique can be complicated if the diet of the consumer changes through time (O'Reilly et al.
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2002). This complication can be minimized by sampling tissues that turn over rapidly relative to
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the rate of change in the diet, but a more comprehensive assessment of the trophic interactions
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might be derived from a perspective that incorporates the dynamics of isotope turnover within
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the organisms. To incorporate these dynamics in an existing ecosystem model that does not
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already incorporate isotopes would require a major recoding of the model and would about
8
double its complexity (double the number of state variables).
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To avoid this recoding, we developed a Non-Equilibrium, Stable-Isotope Simulator
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(NESIS) that calculates dynamics in the isotopic signature of the stocks of an element using the
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output from any parent model that predicts the flux rates and stocks of that element based on a
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mass-balance approach. The software can also be used for radioisotopes if the half-life is long
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enough and simulation duration short enough to ignore losses caused by radioactive decay. The
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NESIS can be used for isotopes with shorter half-lives if the isotope is added in the simulations
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all at once rather than spread out through time; the results simply need to be corrected by
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removing the same proportion of isotope from all pools to account for the radioactive decay.
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Because the NESIS operates on the output of the parent model, no recoding is required.
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However, all fluxes provided to the NESIS must be gross fluxes (net fluxes allow only one-way
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isotope movement when the movement is actually in two directions). For models based on net
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fluxes, the model output can often be converted to gross fluxes with a few simple assumptions
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(e.g., an assumed ratio of net to gross fluxes). To apply the NESIS, the user must also provide
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the initial isotopic signature for all stocks, the fractionation associated with each flux, and the
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isotopic signature of any flux originating from outside the system.
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Strategy:
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The parent model generating the data to which the NESIS is applied will not, in general,
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be linear and it might use internal time steps that are shorter than the time steps of the output
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data. For example, if the model uses a fourth-order Runge-Kutta integrator, it requires four
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estimates of the derivative spaced across the output time step (Press et al. 1986), only one of
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which could be back-calculated from the model output. In addition, to maintain accuracy and
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reduce computation time, many integrators now use an adaptive step size that can take many
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intermediate steps for each output step when conditions in the simulation are changing rapidly
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and fewer steps when conditions are less dynamic (Press et al. 1986). Because the NESIS only
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has access to data at the output time steps, it has to estimate the flux rates between time steps.
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The approximation of the fluxes is based on the assumption of a linear, donor-controlled
13
representation of the fluxes with transfer coefficients (ai) estimated from the stocks and fluxes at
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both ends of the time step:
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1)
Fˆ i (t )  a i X j (t )
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where Fˆ i (t ) is flux i at time t as approximated in the linear, donor-controlled formulation, ai is
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the transfer coefficient for flux i, and Xj(t) is the amount of the element in the stock from which
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the flux originates (to maintain generality, fluxes are enumerated independently of the stocks,
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e.g., there might be several fluxes from stock j to stock k). The value of the ai is recalculated at
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each time step based on the average of the flux rate divided by the stock at the beginning and end
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of the time step as predicted by the parent model:
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Rastetter et al.
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2)
 F (t ) Fi (t  t ) 

a i  0.5 i

 X (t ) X (t  t ) 
j
j


3
4
where Fi (t ) and Fi (t  t ) are the flux rates at the beginning and end of the time step as
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predicted by the parent model, X j (t ) and X j (t  t ) are the stocks at the beginning and end of
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the time step as predicted by the parent model, and t is the size of the time step between
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predictions in the parent model.
8
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Transfer coefficients are then calculated for the heavy and light isotopes based on the ai,
the isotopic signature of the stock from which the flux originates, and the isotopic discrimination
10
associated with the flux. These calculations are based on the del () definition of isotopic
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abundance (Peterson and Fry 1987):
12
13
3)
 H



L
  1000
 1
 H

L std


 
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 L
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where H and L are the numbers of heavy and light isotopes in the sample and H
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of heavy to light isotope for a universally recognized standard (abbreviated as 0 below). Data
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that use the atom percent standard (i.e., A = 100 H/(H+L)) rather than the  standard should first
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be translated to  values before using the NESIS. The translation is easily made by calculating
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the isotope ratio as
std
is the ratio
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4)
H
L

A
100  A
2
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The first step in the calculation of transfer coefficients for the isotopes is to determine the
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ratios of the heavy to light isotope for the donor stock (Eq. 5) and for any flux originating outside
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the system (Eq. 6):
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7
5)
 j (t ) 
6)
Gi (t ) 
  j (t ) 

 0 1 
X Lj (t )
 1000 
X Hj (t )
8
9
G Hi (t )
  (t ) 
 0 1  Gi 
G Li (t )
1000 

10
11
where XHj(t) and XLj(t) are the amounts of the heavy and light isotope in stock j at time t,
12
 j (t ) is their ratio, 0 is that same ratio for a universally recognized standard (e.g., 0 =
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0.0112372 for 13C:12C, 0 = 0.0036765 for 15N:14N),  j (t ) is the "" representation of  j (t ) ,
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GHi(t) and GLi(t) are the heavy and light isotopes in flux Gi at time t, Gi (t ) is their ratio, and
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 Gi (t ) is the "" representation of Gi (t ) . Input fluxes originating from outside the system are
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assumed constant for the duration of the time step.
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Next the ratio of the isotopic ratio of the fluxes to isotopic ratio of the donor pool is
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calculated (ratio of the ratios) for the fluxes between stocks within the system and for fluxes
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leaving the system (see Appendix A for derivations of Eq. 7):
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7)
 i (t ) 
FHi (t )
0
fi
 1
FLi (t ) j (t )
 j (t ) 1000
2
3
 i (t ) is the ratio of heavy to light isotopes in flux i divided by the heavy to light isotopic ratio of
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the stock from which flux i originates,  j (t ) is the isotopic ratio of that stock, f i is the
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fractionation constant for flux i (by convention, a fractionation resulting in a lower  value in the
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product and a higher  value in the residual is positive, Peterson and Fry 1987), and FHi(t) and
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FLi(t) are the flux rates of the heavy and light isotopes in flux i.
8
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From the ratio of isotopes in the stocks (Eq. 5), the amount of each isotope in each stock
at time t can be calculated:
10
11
8)
X Hj (t ) 
9)
X Lj (t ) 
X j (t )  j (t )
1   j (t )
12
13
X j (t )
1   j (t )
14
15
From the ratio of isotopes in the inputs (Eq. 6), the flux rates of each isotope in each input from
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outside the system can also be calculated:
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10)
G Hi (t ) 
Gi (t ) Gi (t )
1  Gi (t )
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11)
G Li (t ) 
G i (t )
1  Gi (t )
2
3
Finally, the transfer coefficients for the linear, donor-controlled representation of the isotopic
4
fluxes can be calculated (see derivations in Appendix B):
5
6
12)
a Hi  a i
13)
a Li  a i
 i (t )1   j (t ) 
1   i (t ) j (t )
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8
1   (t )
j
1   i (t ) j (t )
9
10
where a Hi and a Li are the transfer coefficients for the heavy and light isotopes.
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From equations 8, 10, and 12 a full linear, donor-controlled model for the heavy isotope
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can be formulated and from equations 9, 11, and 13 the analogous model for the light isotope can
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be formulated. The models have the following form:
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15
14)
dX Hj t 
dt
  a Hi X Hk   a Hi X Hj   G Hi
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15)
dX Lj t 
dt
  a Li X Lk   a Li X Lj   G Li
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where the aHi XHk and aLi XLk are the flux rates of heavy and light isotope entering stock j from
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other stocks (k) within the model, the aHi XHj and aLi XLj are the flux rates of heavy and light
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isotope leaving stock j, and the GHi and GLi are the flux rates of heavy and light isotope entering
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stock j from outside the modeled system. The aHi, aLi, GHi, and GLi, are assumed constant
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between times t and t + t, where t is the output time step of the parent model. Equations 14
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and 15 can be integrated between time t and t  t to calculate the abundance of the two
7
isotopes in each stock at time t  t .
8
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As discussed above in relation to equation 1, the output time step of the parent model (t)
might be substantially longer than the time step used for the actual integration. There is
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therefore no guarantee that the integration of the isotope models will be stable using the output
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time step of the parent model. We therefore solve equations 14 and 15 by discretizing the
12
equation and iterate over many small time steps between t and t + t:
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14
16)
t 
dX  t

X      X  
n
dt n

15
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where  is time between t and t + t and n is the number of iterations. Within the limits set by
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the linear approximation of the derivative, the accuracy and stability of this method increases as
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the number of iterations increase. The solution will usually diverge if t/n times the sum of flux
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rates leaving a stock (i.e., second summation in Eqs. 14 and 15) is larger than the stock itself
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because the iterations can potentially step back and forth across zero. We therefore set the
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number of iterations based on the rate constant (ci) for the stock with the fastest turnover time.
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The rate constant for a stock is calculated as the sum of the ai for all of the fluxes leaving that
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stock and the stock with the fastest turnover time will have the largest rate constant. To increase
2
accuracy, we require n to be at least ten times larger than the minimum required to maintain
3
stability:
4
5
17)
n  trunc 10 max  c i  t  1
6
7
where "trunc" indicates the largest integer smaller than the number in brackets and ci is the rate
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constant for stock i. If n is less than ten, ten iterations are made.
9
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At the end of the time step, a new value for the isotopic signature of each stock is
calculated:
11
12
X Hj (t  t )
18)
 j (t  t ) 
19)
  j (t  t ) 
 j (t  t )  1000
 1

0


X Lj (t  t )
13
14
15
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The value of  j (t  t ) is saved and used for the next time step. To minimize the propagation
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of errors associated with the linear approximation, the values of X Hj (t  t ) and X Lj (t  t )
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are discarded and recalculated for the next time step based on  j (t  t ) and the size of stock j
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as predicted by the parent model. This correction is generally small for any single time step, but
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the errors can build up over long simulations. By recalculating the isotope abundances based on
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the isotope signatures and the mass as predicted by the parent model, the overall mass balance is
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maintained to the accuracy of the parent model. The isotopic signatures can nonetheless diverge.
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To check the accuracy of the isotopic simulation, the user should do a mass-balance assessment
3
of the isotope output through time (i.e., is the total amount of the heavy isotope in the system
4
equal to the initial amount, plus the cumulative inputs minus the cumulative losses). If the
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isotope simulation is not within desired accuracy limits, the step size of the parent model needs
6
to be shortened so that the NESIS can make better approximations of the flux rates through time.
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For convenience, isotopic signatures are also calculated for each of the model fluxes:
8
9
20)
  (t ) 
 Fi (t )  1000 i
 1
 0

10
11
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Example 1: C and N in a food web
To illustrate the NESIS, we built a typical food web model based on Lotka-Volterra
13
interactive terms (Arditi and Michalski 1996), but rather than simulating the numbers of
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individuals in each population, we simulated a mass balance for both C and N. The major
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purpose of this example is to illustrate the use of the NESIS to track changes in isotopic
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signatures in response to community dynamics (O'Reilly et al. 2002). We therefore built the
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model to reflect different types of trophic interactions rather than to represent any specific
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system (Fig. 1, Table 1). The model simulates an autotroph (Sp. 1), allochthonous organic
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matter (Sp. 2), a pure herbivore (Sp. 3), a herbivore/detritivore (sp. 4), a pure detritivore (sp. 5), a
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carnivore with three food sources (sp. 6), a carnivore with only one food source (sp. 7), and an
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omnivore feeding on the autotroph and the herbivore/detritivore (sp. 8). Allochthonous organic
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matter (AOM) was treated as a species, except that its supply rate was independent of its
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abundance (i.e., it was not self-propagating) and there was no associated respiration or N
2
excretion.
3
Parameters for the C components of the model were selected to be consistent with a
4
planktonic community and were set using four basic assumptions (Table 2): 1) The community
5
was at steady state for the purpose of parameter estimation. 2) For most trophic interactions,
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assimilation efficiency (ij) and respiration (rij) were set respectively to 25% and 15% of
7
consumption so that the trophic-transfer efficiency was 10% (= 25% - 15%; Wetzel 1975 p 485,
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Saito et al. 2001). However, to compensate for the high C:N ratio of the AOM, respiration
9
associated with the consumption of AOM was set to 22% of consumption (trophic-transfer
10
efficiency of 3%). 3) Based on reported annual production to biomass ratios (Wetzel 1975 p
11
478) and assuming a 6-month active season, we set a turnover time for all species of 10 days.
12
Thus at steady state, the C in the autotroph (Sp. 1) was ten times the daily net primary production
13
(NPP), AOM (sp. 2) was ten times the daily AOM inputs, and each of the other species was ten
14
times the daily assimilation minus respiration. Also, 10% of the C in each species was turned
15
over each day through all sources of mortality. 4) Finally we assumed that the mortality of the
16
species at the lower trophic levels was only due to consumption and that the consumption was
17
distributed evenly among all consumers of that species. For example, the autotroph (Sp. 1) is
18
consumed by the herbivore (Sp. 3), the herbivore/detritivore (Sp. 4), and the omnivore (Sp. 8).
19
Total consumption of the autotroph equaled 10% of its C mass per day and each of the three
20
consumers was responsible for one third of this total consumption.
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With these four assumptions all the steady state C stocks, fluxes, and associated
22
parameters can be calculated once the NPP and AOM inputs are specified. The global average
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NPP for fresh water is about 100 g C m-2 yr-1 (Schindler 1978); we assumed a 6 month active
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season and split this C input evenly between NPP by the autotroph and AOM C inputs (275 mg
2
C m-2 day-1 each). The NPP equation is based on the fraction of light absorbed by the autotroph
3
in a 10 m water column assuming only water, AOM, and the autotroph absorb substantial
4
amounts of light. The light extinction parameters (kw and kb) are based on data presented by
5
Yentsch (1980), but translated from a chlorophyll-based to a C-based model using a
6
C:chlorophyll ratio of 50 (Jorgensen et al. 1979 Table A15). The rate parameter was set to yield
7
an NPP rate of 275 mg C m-2 day-1 under the steady state conditions. The AOM inputs were
8
constant at 275 mg C m-2 day-1.
9
To reduce parameters, we assumed that the C:N of assimilated material was the same as
10
that of the prey species. Thus, to implement the N model, only the target C:N ratios of the
11
species and the rate constants for N excretion by the animal species (spp. 3-8) need to be
12
specified. For the autotroph and AOM, the C:N ratios always equal the target C:N ratios, which
13
were set to 5.68 g C g-1 N for the autotroph (Redfield 1958) and 50 g C g-1 N for the AOM (fine
14
particulate entering streams, Schlesinger 1991, p. 231). For the animal species, the target C:N
15
ratio was set to 7.5 g C g-1 N (Jorgensen et al. 1979 Tables A425 and A459). The excretion rate
16
parameters (i) regulate how fast N is excreted to maintain the C:N ratio in the biomass of the
17
animal species. We set all the excretion rate parameters to 10 day-1 to keep the C:N ratio in
18
biomass within 7% of the target C:N ratio for all species (only Sp. 7 deviated more than 3.5%
19
from the target, and then only during very rapid spurts of relative growth).
20
Although the model was parameterized to a steady state, that steady state is unstable.
21
Small perturbations to the populations result in the system approaching a limit cycle with a
22
period of about 280 days (Fig. 2). These internally driven dynamics provide an ideal example to
23
illustrate the use of the NESIS to track changes in isotopic signatures associated with changes in
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the diet of the species in the food web. We ran the model until it reached the stable limit cycle
2
then generated time-series data for two full cycles (560 days) and applied the NESIS to these
3
time series.
4
We applied the NESIS to the output of the food-web model twice, once to calculate
5
changes in the C isotope signatures and once to calculate changes in the N isotope signatures (see
6
Supplement for instructions on how to use the software). For the C calculations, we assumed
7
that the autotroph (Sp. 1) fixed C with a 13C of -30 o/oo, that the AOM (Sp. 2) had a 13C of -
8
13o/oo, and that there was no fractionation associated with any of the fluxes within the model.
9
Application to the N calculations was more complicated because few of the N fluxes are
10
specified explicitly in the food-web model. For example, the flux of N from species 1 to species
11
3 is represented in the model as the C flux divided by the C:N ratio of species 1. To correct this
12
problem, we used a spreadsheet to calculate the N fluxes from the C fluxes and the C:N ratios
13
before applying the NESIS. We assumed that both the autotroph and the AOM had N inputs
14
with a 15N of 3o/oo and that the only fractionation was due to excretion of N by the animal
15
species. By trial and error, we found that a fractionation of 4.2 o/oo for excretion yielded the
16
typical 3 o/oo positive shift in 15N in biomass between trophic levels (Peterson and Fry 1987).
17
To set the initial isotopic signatures for the species, we applied the NESIS to data for two limit
18
cycles and used the final isotope signatures as the initial values for the final 560-day simulation.
19
To assess accuracy of the isotope simulations, we did a mass-balance assessment for 13C and 15N
20
over the full 560-day simulation by checking if the amount of isotope in the whole food web at
21
the end of the simulation equaled the initial amount of isotope, plus all the inputs over the
22
duration of the simulation minus all the losses. The NESIS estimates accounted for 100.0042%
23
of the 13C and 99.994% of the 15N.
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Once the NESIS was applied to both the C and N output of the food-web model, we made
2
the standard plot of 15N versus 13C used to map trophic webs (Fig. 3). The results clearly show
3
the shift in isotopic signatures associated the changes in diet as the relative abundances of
4
various food sources change. For example, species 4 shifts its dominant consumption between
5
the algae (Sp.1) and the AOM (Sp. 2) depending on which food source is most abundant. The
6
13C signature therefore shifts toward -30o/oo when the algae are more dominant and toward -
7
13o/oo when the AOM is more dominant. In contrast, species 3 is an obligate feeder on the algae
8
and therefore has a constant 13C of -30o/oo.
9
The 15N signatures are somewhat more complicated. For the most part, there is a 3o/oo
10
increase in 15N with each step up the trophic web (Fig. 3). However, because of the high C:N
11
ratio of AOM in the model and because the animals in the model regulate their C:N ratios by
12
regulating N excretion, not by regulating respiration, the increase in 15N is only about 1o/oo
13
between AOM and its consumers (Sp. 4 and Sp. 5) . That is, even though the respiration
14
associated with AOM consumption was increased from 15% to 22%, the species consuming
15
AOM have to conserve more N to maintain their C:N ratios than the other species. Consequently
16
there is less N excretion and less fractionation. The isotope trajectories of species feeding on
17
both algae-derived and AOM-derived sources are therefore tilted toward lower 15N when the
18
dominant food source is AOM derived. There is an additional source of variation in the 15N
19
associated with the lag between N assimilation and N excretion. Thus, when there is a high N
20
assimilation per unit N already in the biomass, the N concentration increases and the
21
fractionation associated with N excretion has had little effect on the new N; the 15N is therefore
22
low (i.e., more like the food source). When there is a low N assimilation per unit N in the
23
biomass, the N concentration drops, fractionation associated with N excretion has a stronger
16
Rastetter et al.
1
influence, and the 15N is therefore higher. This effect of the lag between N assimilation and N
2
excretion interacts with differing isotopic signatures in the diet and results in a hysteresis in the
3
species trajectories on the 15N versus 13C plot (Fig. 3).
4
5
6
Example 2: 15N labeling at Harvard Forest
In our second example, the dynamics in isotopic signatures result from adding an
7
isotopically labeled tracer rather than from dynamic changes in the abundances of various
8
interacting components of the ecosystem. We used data derived from a 15N labeling experiment
9
in a fifty-year-old, predominantly oak forest at the Harvard Forest LTER site (Table 3;
10
Nadelhoffer et al. 1999, Currie et al. 1999). Although this site is aggrading slowly, we assumed
11
it was at steady state for our simulations, except that we allowed for a woody increment. The
12
parent model was built in a spreadsheet and represented amounts of N in leaves, wood, fine
13
roots, forest floor, mineral soil, and inorganic N pools in soil solution (Fig. 4). Consistent with
14
the assumption of steady state, the sum of all inputs to each of these components equaled the sum
15
of all outputs except for wood, which had an increment of 0.64 g N m-2 yr-1 (Nadelhoffer et al.
16
1999) and the commensurate discrepancy between inputs and outputs (Table 3). Only the last 4
17
years of wood were represented in the initial conditions for the model; earlier wood was assumed
18
to be segregated in heartwood and therefore did not interact with the simulated N cycle.
19
Because data were available for only net N mineralization, we had to make assumptions
20
about the gross rates of N processing within the soil. In the Currie et al. (1999) model of the
21
same data, they allowed N immobilization into both forest floor and mineral soil components.
22
For simplicity, we only allowed immobilization into the forest floor (high C:N ratio relative to
23
mineral soil). With the structure of our model and the joint constraints of a steady state and net
17
Rastetter et al.
1
N mineralization for the whole soil of 7.311 g N m-2 yr-1, we had two remaining degrees of
2
freedom for setting the four unknown fluxes within the soil.
3
To satisfy one of these degrees of freedom, we used the classic data from Staaf and Berg
4
(1982) in which litter cohorts had about 80% of the initial amount of N when only 20% of the C
5
remained. (Because the Staaf and Berg data were for Scots pine, and the plot we simulated was
6
mostly oak, we also used an assumption of 50% N remaining when 20% of the C remained.
7
However, we found that it made little difference to the major results of our simulations.) When
8
about 20% of the C in a litter cohort remains, it typically undergoes a transition from the
9
relatively fast early phases of decomposition with active N mineralization and immobilization to
10
the later phases with slow N mineralization (Melillo et al. 1989). Assuming that this shift in
11
decomposition dynamics corresponds roughly to the transition between forest floor and mineral
12
soil and that each year one litter cohort makes that transition with 80% of its initial N remaining,
13
then mass balance requires a net N mineralization from the forest floor equal to 20% of the
14
annual N input to the forest floor. If the whole soil is near steady state, then the mineral soil
15
must have an N mineralization rate equal to 80% of the annual N input to the soil to make up the
16
difference. Thus the ratio of net N mineralization between forest floor and mineral soil should
17
be 1:4.
18
The last degree of freedom can be satisfied by specifying the ratio of gross to net N
19
mineralization for the whole soil. This ratio is poorly quantified (Hart et al. 1994), however, we
20
found that it has important effects on the dynamics of 15N in our model. We tried various ratios
21
to try to match the 15N data reported in Nadelhoffer et al. (1999; since the publication of that
22
paper, an error was detected in the calculations of the % 15N recovered. The true values are
23
about twice those originally reported. Nadelhoffer Pers. Com.).
18
Rastetter et al.
1
We applied the NESIS to the spreadsheet model of the Harvard Forest N cycle assuming
2
an initial  15N of zero for all components and increased the  15N of N deposition to 5797 o/oo for
3
two years to simulate the 17 mg 15N m-2 yr-1 labeling. A mass-balance assessment of the
4
simulation indicated that the NESIS accounted for 99.966% of the label after a 20-year
5
simulation.
6
With a gross to net N mineralization ratio much below 20, the leaves and roots get
7
labeled during the two years of labeling but quickly lose that label immediately after labeling
8
stops. To have the label continue to increase in these tissues after the labeling stops, the label
9
has to be immobilized in the soil very rapidly and then re-released at a rate consistent with a
10
gross to net N mineralization ratio of about 21 (Fig. 5). This very fast immobilization of 15N is
11
consistent with the data of Berntson and Aber (2000), who found that 15N was immobilized in
12
the soil within about 45 minutes of labeling. Our simulations indicate that the turnover time for
13
this N in the forest floor is about 1 year. The label therefore gets re-released into the soil
14
solution fairly quickly and is available to be taken up into leaves and fine roots over several
15
years. However, even with this adjustment to the mineralization rates to match the general
16
pattern of 15N labeling in leaves and roots, discrepancies between the model and data remain. In
17
particular, the 15N data indicate that the rate of N uptake into roots may have been
18
underestimated (Fig. 5) suggesting that the rate of fine root production has probably been under
19
estimated.
20
21
22
23
Conclusions:
We presented a Non-Equilibrium Stable Isotope Simulator (NESIS) that estimates
changes in the isotopic signatures of the stocks and fluxes of an element using the output from
19
Rastetter et al.
1
existing models that predict the stocks and flux rates of that element based on a mass-balance
2
approach. The parent model can be based on a complex numerical simulation or on a simple
3
spreadsheet representation of the ecosystem element budget. We demonstrated the application of
4
the simulator on two models that differed in the nature of their isotope dynamics; in the first
5
model, the isotopic signatures of the model components varied because of changes in the diets of
6
species in an interactive food web. In the second model, most of the model components were in
7
steady state and the isotope dynamics were associated with a typical pulse-chase labeling
8
experiment. The NESIS should be widely applicable to studies of stable isotopes for several
9
elements, and probably for short-term studies of radioisotopes as long as the radioactive decay is
10
slow enough that the mass-balance assumption for the isotope is not severely violated. We
11
anticipate that use of the simulator will circumvent the need to develop new versions of existing
12
models so that they can be applied to isotope data. We also anticipate that the simulator will
13
facilitate analyses by non-modelers interested in applying isotope data to their spreadsheet mass-
14
balance calculations of ecosystem element cycles.
15
16
Acknowledgements:
17
The information in this document has been funded in part by the U.S. Environmental Protection
18
Agency (QT-RT-00-001667). It has been subjected to review by the National Health and
19
Environmental Effects Research Laboratory, Western Ecology Division and approved for
20
publication. Approval does not signify that the contents reflect the views of the Agency, nor
21
does mention of trade names or commercial products constitute endorsement or recommendation
22
for use. This work was also supported in part by grants from the National Science Foundation
20
Rastetter et al.
1
(DEB-0108960, OPP-9911681). We are grateful to Dr. Knute Nadelhoffer for his assistance
2
with the Harvard Forest data.
3
4
5
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4
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1
2
Table 1: Food web model
3
dC1
 PS  F13  F14  F18  L13  L14  L18
dt
dN1 PS
N

 F13  F14  F18  L13  L14  L18  1
dt
Q1
C1
4
dC 2
 A  F24  F25  L24  L25
dt
dN 2
A
N

 F24  F25  L24  L25  2
dt
Q2
C2
5
dC 3
 F13  F36  L36  R3
dt
dN 3
N
N
 F13 1  F36  L36  3  E3
dt
C1
C3
6
dC 4
 F14  F24  F46  F48  L46  L48  R4
dt
dN 4
N
N
N
 F14 1  F24 2  F46  F48  L46  L48  4  E 4
dt
C1
C2
C4
7
dC 5
 F25  F56  F57  L56  L57  R5
dt
dN 5
N
N
 F25 2  F56  F57  L56  L57  5  E 5
dt
C2
C5
8
dC 6
 F36  F46  F56  M 6  R6
dt
dN 6
N
N
N
N
 F36 3  F46 4  F56 5  M 6 6  E 6
dt
C3
C4
C5
C6
9
dC 7
 F57  M 7  R7
dt
dN 7
N
N
 F57 5  M 7 7  E 7
dt
C5
C7
10
dC 8
 F18  F48  M 8  R8
dt
dN 8
N
N
N
 F18 1  F48 4  M 8 8  E8
dt
C1
C4
C8
11
PS  p
12
R j   rij Fij summed over all species i consumed by species j

k LC1
1  e kw kb C1C2 z
k w  k L C1  C 2 



C 
E i  max 0,  i  N i  i  
Qi  


j
13
M i  mi C i
Fij   ij f ijC i C j
Lij  1   ij  f ijC i C j
14
15
Variables and parameters
16
A
allochthonous supply rate of suspended detritus (mg C m-2 day-1; assumed constant)
17
Ci
C in species i (mg C m-2)
18
Ei
N excretion by species i (mg N m-2 day-1)
27
Rastetter et al.
1
Fij
C assimilated from species i by species j (mg C m-2 day-1)
2
fij
rate parameter for consumption of species i by j (m2 mg-1 C day-1)
3
kb
light absorption parameters for biomass (m mg-1 C)
4
kw
light absorption parameter for water (m-1)
5
Lij
unassimilated C associated with flux Fij (mg C m-2 day-1)
6
Mi
mortality of species i (mg C m-2 day-1)
7
mi
mortality rate parameter for species i (day-1)
8
Ni
N in species i (mg N m-2)
9
PS
net primary production by species 1 (mg C m-2 day-1)
10
p
maximum photosynthetic rate (mg C m-2 day-1)
11
Qi
target C:N ratio in biomass of species i (mg C mg-1 N)
12
Ri
respiration by species i (mg C m-2 day-1)
13
rij
respiration rate parameter associated with flux Fij (fraction)
14
z
mixed layer depth (m)
15
i
excretion rate parameter for species i (day-1)
16
ij
assimilation efficiency for transfer from species i to j (fraction)
17
28
Rastetter et al.
1
Table 2: Food-web model parameter values
2
3
parameter
value
4
A
275 mg C m-2 day-1
5
p
559.375 mg C m-2 day-1
6
kw
0.03 m-1
7
kb
0.00032 m mg-1 C
8
z
10 m
9
mi
0.1 day-1 for i = 6, 7, & 8
10
rij
0.15 for ij = 13, 14, 18, 36, 46, 48, 56, & 57
11
0.22 for ij = 24 & 25
12
ij
0.25 for ij = 13, 14, 18, 24, 25, 36, 46, 48, 56, & 57
13
i
10 day-1 for i = 3, 4, 5, 6, 7, & 8
14
f13
3.6363610-4 m2 mg-1 C day-1
15
f14
2.5078410-4 m2 mg-1 C day-1
16
f18
3.3905510-4 m2 mg-1 C day-1
17
f24
3.7617610-4 m2 mg-1 C day-1
18
f25
1.2121210-3 m2 mg-1 C day-1
19
f36
5.5944110-3 m2 mg-1 C day-1
20
f46
2.7972010-3 m2 mg-1 C day-1
21
f48
5.0858210-4 m2 mg-1 C day-1
22
f56
2.7972010-3 m2 mg-1 C day-1
23
f57
2.4242410-2 m2 mg-1 C day-1
29
Rastetter et al.
1
Q1
5.68 mg C mg-1 N
2
Q2
50 mg C mg-1 N
3
Qi
7.5 mg C mg-1 N for i = 3, 4, 5, 6, 7, & 8
4
30
Rastetter et al.
1
Table 3: Data for N cycling at Harvard Forest
2
N Stock
g N m-2
Comments
3
Leaf N
6.03
Table 2 in Nadelhoffer et al. 1999
4
Wood N
6.8
Table 2 in Nadelhoffer et al. 1999,
5
1988-1992 wood only plus bark
6
Root N
7.11
Table 2 in Nadelhoffer et al. 1999
7
Organic Soil N
106.8
Table 2 in Nadelhoffer et al. 1999
8
Mineral Soil N
318.2
Table 2 in Nadelhoffer et al. 1999
9
Inorganic N
2.6
Figure 1 in Bormann et al. 1977
11
N Flux
g N m-2 yr-1
Comments
12
N deposition
0.8
Table 4 in Nadelhoffer et al. 1999
13
N fixation
0.387
Total losses plus woody increment
10
14
15
minus N deposition
Leaf production
5.825
Biomass production from Table 3 in
16
Currie et al. 1999 times N/biomass
17
from Table 2 in Nadelhoffer et al. 1999
18
Root production
2.427
Ditto
19
Wood production
1.900
Ditto, but used modeled data to include
20
belowground production
21
Total uptake
8.1
Table 3 in Currie et al. 1999
22
Retranslocation
2.052
Total production minus total uptake
23
from leaves
31
Rastetter et al.
1
Uptake to leaves
3.773
Production minus retranslocation
2
Uptake to wood
1.900
= Production
3
Woody increment
0.64
= Total N in 1988-1992 wood (exclusive of
4
bark) from Table 2 in Nadelhoffer et al.
5
1999 divided by 5 years
6
Uptake to roots
2.427
= Production
7
Leaf litter
3.773
= Production (steady state assumption)
8
Wood litter
1.260
= Production minus increment
9
Root litter
2.427
= Production (steady state assumption)
10
Whole-soil net N
7.311
Table 3 in Currie et al 1999 plus DIN losses
11
mineralization
from Table 7 in Currie et al. 1996
12
DON leaching
0.536
Table 7 in Currie et al. 1996
13
DIN leaching
0.011
Table 7 in Currie et al. 1996
14
Forest floor gross N
146.22
Assumes steady state for all soil
15
mineralization
components, whole-soil net N
16
mineralization of 7.311, ratio of forest
17
floor:mineral soil net N mineralization
18
= 1:4, ratio of whole-soil gross:net N
19
mineralization = 20.8. See text
20
Forest floor gross N
21
immobilization
22
Forest floor to mineral
23
144.758
Ditto
6.385
Ditto
soil flux
32
Rastetter et al.
1
2
Mineral soil N
5.849
Ditto
mineralization
3
33
Rastetter et al.
1
Figure Legends:
2
3
Figure 1: Trophic links in a food-web model used to illustrate the use of the NESIS.
4
5
Figure 2: Internally driven limit cycle in the abundance of eight species in a food-web model
6
used to illustrate the use of the NESIS.
7
8
Figure 3: Phase-plane plot of the 15N and 13C for eight species in a food-web model as
9
predicted by the NESIS.
10
11
Figure 4: The nitrogen cycle in an oak stand at Harvard Forest. Stocks are in g N m-2 and fluxes
12
are in g N m-2 yr-1. Data were derived as described in Table 3.
13
14
Figure 5: Percent recovery of a 15N label added to an oak forest at Harvard Forest as predicted
15
by the NESIS applied to data from the mass-balance model of the N cycle presented in Figure 4
16
and Table 3. Squares are data from Nadelhoffer et al (1999) for leaves (closed) and fine roots
17
(open). The Nadelhoffer et al. data have been corrected for a calculation error in the original
18
(Nadelhoffer Pers. Com.).
19
34