Annals of Botany 86: 1161±1167, 2000
doi:10.1006/anbo.2000.1286, available online at http://www.idealibrary.com on
Nitrogen States in Plant Ecosystems: A Viewpoint
M . G . R . CA N N E L L and J . H . M . T H O R N L E Y
Centre for Ecology and Hydrology (Edinburgh), Bush Estate, Penicuik, Midlothian EH26 0QB, UK
Received: 16 June 2000 Returned for revision: 8 August 2000
Accepted: 24 August 2000 Published electronically: 27 October 2000
Terrestrial ecosystems are considered to be in only two possible states: N unsaturated or N saturated. This view
lacks rigour and has led to three di€ering concepts of N saturation: (1) a continuum of changes in N states and
processes; (2) a threshold point, when N output increases; and (3) any equilibrium state, when N output equals N
input. A simple model of ecosystem N content, input and output is used to point out that, strictly, there are four
possible N states of ecosystems: (1) at equilibrium, but N unsaturated and N-limited, so that they will respond to
additional N input with increased N accumulation; (2) not at equilibrium and accumulating N, either in response to
additional N input, or more commonly in nature, when recovering from ®re, other disturbance or unfavourable
conditions; (3) at equilibrium but truly N saturated and not N-limited, so that any additional N input is matched
by equal N output; and (4) not at equilibrium and losing N because of disturbance or soil changes induced by N
addition itself or other factors. Most natural ecosystems are, for most of the time, in one of the two nonequilibrium states, especially in short-term N-addition experiments. It is not meaningful to regard them as being
# 2000 Annals of Botany Company
either N saturated or N-limited.
Key words: Nitrogen saturation, deposition, leaching, nitri®cation, plant ecosystems, forest.
I N T RO D U C T I O N
Nitrogen enrichment is a major global environmental
problem. Fossil fuel burning, fertilizer production, legume
growing, livestock farming and biomass burning are
releasing more reactive1 N into the global environment
than pre-industrial natural processes (Vitousek, 1994;
Galloway et al., 1995), and world energy and food demands
make it inevitable that this N release will continue. As a
result, reactive N is accumulating in the biosphere, oceans
and atmosphere, and is having powerful interlinked e€ects.
N accumulation in terrestrial ecosystems has attracted
particular attention because of changes that occur in plant
growth, species composition and carbon sequestration, and
because of the release of nitrate to ground-waters and N2O
into the atmosphere. Much of the research on N enrichment
of ecosystems has been guided by the notion that there is a
point of N saturation. Below this point, ecosystems are
thought to retain most of the N they receive, mostly in the
soil (Johnson, 1992; Kauppi et al., 1995; Nadelho€er et al.,
1999) but partly in vegetation, with the potential to increase
net primary productivity (NPP) and carbon sequestration
(Spieker et al., 1996; Houghton et al., 1998). Above this
point, ecosystems become `leaky', contributing to water
pollution, greenhouse forcing, acidi®cation and plant
nutrient imbalance, potentially depressing NPP and contributing to problems of forest decline (Gundersen, 1991;
Aber et al., 1989, 1998; Aber, 1992; Fenn et al., 1998).
Thus, ecosystems have tended to be de®ned as being
`unsaturated' with N if they retain most additional N, and
1
N2 is relatively unreactive. NOx , NHy and organic N are reactive.
NOx ˆ N combined with oxygen. NHy ˆ N combined with hydrogen.
Of the NOx , N2O is chemically unreactive, but is radiatively active; in
the stratosphere it contributes to ozone depletion.
0305-7364/00/121161+07 $35.00/00
`saturated' if an appreciable fraction of additional N is lost
as nitrate and N gases. The notion of a point of N
saturation, analogous to a bucket ®lling and over¯owing,
has been central to much of the thinking on `critical loads'
of N deposition (Skengton and Wilson, 1988; RoseÂn et al.,
1992; Bull et al., 1995).
Our work with comprehensive process-based ecosystem
models, which couple the C, N and water cycles (Thornley
and Cannell, 1992, 1996; Cannell and Thornley, 1998;
Thornley, 1998) leads us to the conclusion that many of the
concepts of ecosystem N saturation met with in the
literature are confused and can be misleading. Thus,
research groups within the European NITREX project
came to di€erent conclusions on whether particular ecosystems were N saturated (Tietema et al., 1998).
The purpose of this paper is to o€er insights and sharply
de®ned concepts of N states of ecosystems that might be
helpful in clarifying ambiguous relationships and contribute to the discussion on N enrichment.
C U R R E N T CO N C E P T S O F N S AT U R AT I O N
N saturation is a state of a system and should therefore have
a rigorous de®nition. By analogy with a solution, it is the
state in which a system is fully charged with a substance.
Confusion has arisen because authors use di€erent de®nitions, none of which is satisfactory (Skengton and
Wilson, 1988). There are three main concepts.
N saturation as a continuum
As ecosystems become progressively N enriched they
undergo a series of changes. To begin with, N accumulates
# 2000 Annals of Botany Company
1162
Cannell and ThornleyÐN States of Ecosystems
in the soil and increasingly in the plants. A large fraction of
the mineral N is present as ammoniumÐwhich is retained
in the soil by cation exchange. Further ammonium-N
enrichment promotes the activity of nitrifying bacteria,
converting ammonium to nitrate (nitri®cation), releasing
gaseous N. Nitrate is a mobile anion, which is then readily
leached. By charge balance, loss of nitrate results in cation
removal (notably Ca2‡ and Mg2‡ ), an increase in the
concentration and mobility of hydrogen ions and inorganic
Al3‡ ions and potentially Al toxicity. The overall result is
increased N leaching, gaseous N loss (N2 , NO, N2O),
acidi®cation, possibly causing impairment of root and
mycorrhizal function and nutrient imbalances in the plants
(Gundersen, 1991; Aber et al., 1989, 1998; Aber, 1992).
Some authors speak of this series of processes as being an
`N saturation continuum' (Aber et al., 1989, 1998;
Peterjohn et al., 1996). This phrase confuses a state with a
process. The sequence of changes may be more correctly
termed an `N enrichment continuum'. It is best to reserve
the word `saturation' for a de®ned condition.
N saturation as a threshold or `critical load'
The search for a `critical load' of N, to aid policy makers,
has led to many attempts to de®ne N saturation in terms of
N leakage from ecosystems, on the assumption that there is
a threshold beyond which ecosystems lose a high fraction of
any additional N they receive. The notion is of a system
which absorbs a substantial fraction of additional N input
(N input exceeds N output), until some quite sharply
de®ned physiological or pathological change occurs which
damages the integrity of the system; thereafter N output
exceeds N input and the system can no longer retain the
quantity of N which was present before the change
occurred.
`Saturation' is then loosely de®ned as the point when an
ecosystem `loses its ability to retain N', and there is a
`marked increase in N leaching' or `N breakthrough' or
when `N leaching exceeds a threshold' or there is an abrupt
shift `from a closed to an open N cycle' (Skengton and
Wilson, 1988; Gundersen, 1991; Aber et al., 1998).
Again, in this literature, N saturation is not regarded as
a de®ned condition. Instead, it is a threshold or point
of discontinuity, beyond which there is an increase in
the fraction of additional N inputs which are lost as N
outputs.
N outputs equal to N inputs
AÊgren and Bosatta (1988) were the ®rst to suggest that
ecosystems are N saturated when they are in equilibrium, so
that N outputs equal N inputs. This is a rigorous de®nition,
which we elaborate below. But AÊgren and Bosatta (1988)
considered there to be only one equilibrium state, when the
amount of N in an ecosystem was the maximum that it
could contain in any given environment.
Nitrogen input
IN
System nitrogen
Nitrogen output
Nsys
ON
F I G . 1. Simple plant ecosystem model, with a single N pool and bulked
N input and output ¯uxes.
S U G GE S T E D D E F I NE D N S TAT E S O F
E CO S Y S T E M S
Consider the simplest possible model of an ecosystem,
illustrated in Fig. 1. The total amount of N in soil plus
vegetation at any time is Nsys (kg N m ÿ2) with N inputs, IN
(kg N m ÿ2 y ÿ1; atmospheric deposition, N2 ®xation and N
addition) and N outputs, ON (kg N m ÿ2 y ÿ1; leaching,
gaseous losses and removals). This system can be in two
basic states: (1) at equilibrium, with IN ˆ ON and
* (the equilibrium N content), with no change
Nsys ˆ Nsys
in N content, or (2) in a non-equilibrium or transient state,
when it is gaining (IN 4 ON) or losing N (IN 5 ON).
The Appendix presents a formal analysis of the possible
relationships between IN , ON and Nsys in this simple model,
assuming that ON is a function of Nsys . A constant, k, and a
parameter, q (with no biological meaning) enable a variety
of dependencies to be de®ned. The input±output and
content relationships can behave like a bucket of water with
output proportional to the pressure head, an over¯owing
bucket, a bucket with only evaporation loss or a leaky
bucket (see Appendix). These possible behaviours are
referred to in the discussion below.
It is not meaningful to describe a non-equilibrium system
as being saturated, as in the `continuum' and `threshold'
de®nitions discussed above. Instead, we de®ne four possible
N states of ecosystems, according to whether they are at
equilibrium or not; N saturation is a special equilibrium
state. These states are illustrated conceptually in Fig. 2 for a
system which is subject to four step increases in N input.
Equilibrium, unsaturated (N-limited or colimited)
It is important to realize that ecosystems can be at
equilibrium with respect to N without being N saturated.
That is, ON can equal IN in systems that are N-limited (or,
more accurately, N-colimited), in the sense that they will
respond to an increase in IN with increased carbon ®xation,
which allows more N to accumulate in the soil and
vegetation. But as long as IN remains constant, the system
* .
will attain an equilibrium N content, denoted Nsys
In Fig. 2, the system is in this equilibrium, unsaturated,
N-limited state between time points P and Q, when it
responds to a step increase in IN , and again between times
R and S, when it responds to a further increase in IN .
In nature, a large proportion of terrestrial ecosystems are
N-limited, including most temperate and boreal forests,
temperate grasslands, tropical savannahs and almost all
areas of arctic and alpine tundra (Aerts and Chapin, 2000).
Some ecosystems may be near-equilibrium, averaged over
decadal timescales, if they are relatively undisturbed, but
none are likely to reach true equilibrium because of
variability and changes in climate and N deposition.
Cannell and ThornleyÐN States of Ecosystems
V
W
Non-equilibrium Equilibrium
(transient)
(saturated)
ON
Damaged
ecosystem
Equilibrium
(unsaturated)
Non-equilibrium
(transient)
Equilibrium
(unsaturated)
N input, IN, output ON
A System nitrogen input and output
P
Q
R
S
T
U
IN
IN
IN
ON
fraction of the N input is retained, so that IN 4 ON . The
same happens between times S and T.
Systems which are in the process of responding to an
increase in N input with increased carbon ®xation, and so
are accumulating N, are commonly regarded an `N-limited'.
This is not a meaningful use of the term because the system
is in a transient state. If systems are accumulating N, they
are already responding to the current N input and it will not
be apparent whether that level of N input is limiting or not
until the system comes to equilibrium.
At any time, the N content of a system in this nonequilibrium state, Nsys , can be de®ned relative to the
* , that it would reach with the
equilibrium N content, Nsys
given constant level of N input, by the fraction:
ON
fN * ˆ
B Total system nitrogen
Nsys* Nsat
Nsat
System N, Nsys
Nsys
Nsys
1163
Nsys
Nsys *
If it is assumed that N output is linearly dependent on N
content, which is approximately true in many instances
(Gundersen, 1991; Binkley and HoÈgberg, 1997), so that
ON ˆ kNsys (q ˆ 1 in the Appendix), it can be shown that
the current rate of N absorption, AN (kg N m ÿ2 y ÿ1), is
AN ˆ IN ÿ ON ˆ kNsys *…1 ÿ fN *†
Nsys*
Nsys*
Time
F I G . 2. Conceptual representation of the increase in total N in an
ecosystem (soil and vegetation), Nsys , in response to step increases in
annual N input (IN : continuous line) based on the model in Fig. 1 and
elaborated in the Appendix. N output ON is shown by the dashed line.
IN and ON coincide except over the ranges QR, ST and VW. Three
states occur: (1) PQ and RS, equilibrium unsaturated state with inputs
equal outputs, IN ˆ ON ; (2) QR and ST, non-equilibrium or transient
state, with N outputs rising to match the increased input; (3) TV,
equilibrium saturated state. At T, parameter q [eqn (A2)] is increased
from 1 to 100, so that the system behaves like a full bucket (Fig. A3).
At U the increased N input immediately appears as output, and the
system N content hardly changes. At point V, the N input is increased
again, it is assumed that this degrades the system internally, increasing
the parameter k ®ve-fold, and returning parameter q to unity. There is
a burst of increased N output, accompanied by a decrease in system N.
Widespread N limitation persists for many reasons,
including regular N loss during ®res or other disturbances,
the failure of N2 ®xers to reverse N de®ciencies and low
rates of decomposition of high C : N litter (Vitousek and
Howarth, 1991). On long time-scales, all of these systems
must approach the equilibrium, unsaturated N condition.
…1†
…2†
Thus, the three quantities which a€ect the N absorption of
an ecosystem during, say, times Q to R in Fig. 2, are: how
far the system is from equilibrium ( fN*), the N content of
the system at equilibrium (Nsys*), and the rate constant (k).
A large proportion of natural ecosystems are in this nonequilibrium state, accumulating N. They are commonly
regarded, misleadingly, as N-limited (see above). Thus, in
many natural ecosystems, N inputs from the atmosphere
and N2 ®xation seem to exceed N outputs to streamwaters
and the atmosphere (Vitousek and Howarth, 1991; Ineson
et al., 1991; Binkley and HoÈgberg, 1997). The commonest
reason for N (and C) accumulation is that ecosystems are
recovering from ®re or other natural disturbances over long
time-scales, or are responding to recently increased atmospheric N deposition. N accumulates very slowly in
ecosystems, basically because the N input±output ¯uxes
are small compared with the ecosystem cycling ¯uxes
(Cannell and Thornley, 1998). The time from disturbance
to N equilibrium often exceeds the return period of the
disturbance, so ecosystems tend to be constantly recovering.
After any disturbance of an equilibrium system, a
recovery time could be de®ned in terms of, say, the time
required for Nsys to move 90 % of the distance from its
value immediately following the disturbance towards its
new equilibrium value. However, even with the simple
model in Fig. 1, if q 6ˆ 1, a single meaningful recovery time
is not easily de®ned [(see eqns (A3) and (A5)], and such
de®nitions should be treated with caution.
Non-equilibrium, accumulating N
Equilibrium, saturated (not N-limited)
In Fig. 2, the system accumulates N between times Q and
R, following the step increase in IN at time Q. That is, a
*
N saturation should be de®ned as a special Nsys
equilibrium condition, when the system has N content
1164
Cannell and ThornleyÐN States of Ecosystems
Nsat, when a change in N input produces no further change
in N content. That is if:
*
@Nsys
ˆ 0;
@IN
Nsat ˆ N*sys
…3†
In Fig. 2, the system is N saturated at time T. This is
revealed by the fact that the step increase in N input at time
U produces no appreciable increase in N*sys (ˆNsat). Over
the time period U to V, ON ˆ IN ; the system is
unresponsive to added N and any increase in N input is
matched by an equal increase in N loss.
Clearly, in nature, this precise equilibrium condition is
likely to be relatively rare. It may be more useful to de®ne a
practical measure of N saturation in terms of, say, a 10 %
sensitivity of the system. That is, a system is e€ectively N
saturated if:
* =N*sys
DNsys
50 1
s…N*sys ; IN † ˆ
DIN =IN
…4†
s(y, x) denotes the sensitivity of the dependent variable y to
changes in the independent variable x. If say, s ˆ 0.5, then
* , so
a 10 % increase in N input causes a 5 % increase in Nsys
that s 5 0.1 means that a 10 % increase in N input causes
less than 1 % increase in N*sys .
Some intensively managed grasslands and other agricultural systems may be close to N saturation, as indicated by a
lack of response to added N. Averaged over a number of
years, N output by o€-take, leaching and gaseous loss may
be close to N input.
The rate at which a perturbed system returns to equilibrium depends on Nsys , IN and the system parameters (k
and q, see Appendix). For example, if s ˆ 1, DIN/IN ˆ 0.1
and IN ˆ 10 kg N ha ÿ1 y ÿ1, then DIN ˆ 1 kg N ha ÿ1 y ÿ1
* /Nsys
* ˆ 001 [eqn (4)]. With Nsys
* ˆ 100 kg
and DNsys
* ˆ 1 kg N ha ÿ1, which could be provided
N ha ÿ1 y ÿ1, DNsys
in 1 year, depending on the values of k and q in eqn (A5). On
* ˆ 10 000 kg N ha ÿ1 y ÿ1,
the other hand, with Nsys
DN*sys ˆ 100 kg N ha ÿ1 which at best could be provided in
100 years. Thus, our de®nition of N saturation refers only to
`now' (t ˆ 0) and `equilibrium' (t ˆ 1) and does not give
information about the dynamic response.
Non-equilibrium, losing N
As mentioned above, in many natural ecosystems high N
inputs cause changes in the soil, such as loss of cations,
acidi®cation and Al toxicity, with impaired root and
mycorrhizal function, e€ectively `damaging' the system.
Consequently, these ecosystems may not sustain a period
with high N input at equilibrium, and so should not be
de®ned as saturated (Aber et al., 1989, 1998). They are in a
non-equilibrium state.
In Fig. 2, the increase in N input at time V is assumed to
`¯ip' the system to a new, `damaged' state (Murray, 1977;
Thornley and Johnson, 2000, pp. 148±150). The system is
altered so that the increased N input is not turned into
increased biomass and soil organic matter. (In the
Appendix model this is achieved by increasing parameter
k ®ve-fold, and returning parameter q to unity, simulating a
leaky bucket.) The system loses N; IN 5 ON until it reaches
a new, lower equilibrium N state, which may or may not be
N saturated.
There are, of course, reasons why ecosystems may lose N
other than being damaged by high N input. There may be a
temporary disturbance; environmental or management
changes may a€ect N ¯uxes directly, so that the old
equilibrium level of Nsys is no longer sustainable and the
system moves to a new lower equilibrium value; or changes
in the internal workings of the system may alter N relationships so that, for a given IN , Nsys can only be sustained at a
lower level. The latter changes may be pathological,
physiological or biochemical, and be triggered by environmental or management changes which do not directly
involve N.
DISCUSSION
The notion that ecosystems are in one of only two possible
states with respect to NÐunsaturated or saturatedÐhas
led to confusion in de®nitions and a lack of clarity. A more
rigorous approach would be to de®ne ecosystems as being
either at equilibrium with respect to N (IN ˆ ON) or not. If
they are at or near equilibrium, they may or not be N
saturated, depending on whether they will respond to
further N addition and accumulate more N. If they are not
at equilibrium, they are either accumulating or losing N.
There are, therefore, four possible N states: (1) equilibrium
unsaturated; (2) equilibrium saturated; (3) accumulating N;
and (4) losing N.
Currently, many researchers regard increased N output
(as nitrate or gases) in response to increased N input as
indicating `N saturation' (Peterjohn et al., 1996; Ohrui and
Mitchell, 1997; Aber et al., 1998; Berg and Verhoef, 1998).
By our de®nition, such a response would indicate that the
ecosystem is either near equilibrium, saturated or unsaturated, so that IN is approximately equal to ON , or that the
system has been altered or `damaged' in some way (e.g. with
increased nitri®cation or mycorrhizal impairment) and is in
a transient condition, losing N. In practice, it is rarely
possible to say which of these conditions pertain; all that
can be said is that they are losing an increased fraction of
added N. This is a more accurate statement than to say that
they are N saturated.
As mentioned above, most natural ecosystems are not at
equilibrium because of disturbance and changes in the
environment. Also, non-equilibrium conditions are likely to
be the most common, because the N dynamics of
ecosystems are slowÐit normally takes a minimum of
several decades for the N content of a plant±soil system to
reach a new equilibrium in response to a change in N input
(e.g. Rastetter et al., 1997; Cannell and Thornley, 1998;
Thornley and Cannell, 2000). Consequently, it may be
thought that the concept of N saturation as a nearequilibrium state is irrelevant, because it rarely exists. This
may be why the equilibrium de®nition of N saturation
proposed by AÊgren and Bosatta (1998) has received little
acceptance. Our viewpoint is that, to be rigorous, all
ecosystem N states should be de®ned relative to equilibrium
Cannell and ThornleyÐN States of Ecosystems
values to which they are heading in constant conditions. In
order to interpret the long-term consequences of N addition
and where the ecosystem is relative to equilibrium
saturation, it is necessary to know where you are along
the time-scale in Fig. 2.
N addition experiments can sometimes give a false
impression of system behaviour and long-term responses
to N because they always observe transient behaviour. For
instance, most `N saturation' experiments, such as those
conducted in the NITREX project, give the impression that
the ecosystem responds to added N like an over¯owing
bucket (with a high value of q in the Appendix analysis),
whereas, in reality the relationship between N input and
system N and C content may be more like a tall bucket with
variable loss (q ˆ 2 to 10 in the Appendix) (Thornley, 1998)
or like a leaky bucket, if the system is altered or damaged.
Rarely will the transient N output be equal to N input, as it
will be eventually at equilibrium.
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APPENDIX
Theoretical relationships between N input, N output and the
N content of ecosystems
There are a number of possible non-equilibrium and
equilibrium N states and relationships that can be derived
mathematically from the simple model illustrated in Fig. 1,
with the single reasonable assumption that ON is a function
of Nsys (Gundersen, 1991; Binkley and HoÈgberg, 1997).
Nsys is the only state variable and IN is regarded as the
driving variable, whose value is provided externally. In a
non-equilibrium state the quantity of N in the system, Nsys ,
changes with time:
dNsys
ˆ IN ÿ ON
dt
…A1†
The simplest assumption is that N output is linearly
dependent on N content, so that ON ˆ kNsys , where
1166
Cannell and ThornleyÐN States of Ecosystems
Output of N, ON
Input of N, IN (kg N m–2 y–1)
0.010
B
q=1
Equilibrium:
ON = IN = kNsys
Nsys variable
q = 100 (large)
Equilibrium:
ON = IN
Nsys = constant
C
D
q = 0.01 (small)
Equilibrium:
ON = constant
if IN < ON, Nsys = 0
if IN > ON, Nsys = ∞
q = 1, k high
Equilibrium:
ON = IN = kNsys
Nsys variable, but low
0.008
0.006
q = 100
0.004
10
2
1
0.002
0.5
0.1
0.01
0.000
0.0
0.4
0.8
1.2
1.6
2.0
Total system N, Nsys
Equilibrium system N, Nsys* (kg N m–2)
F I G . A1. N output, ON , as a simple function of N content of the
ecosystem, Nsys , as assumed for the simple model of Fig. 1 and in eqn
(A2). The allometric parameter q is assigned a range of values. Also
shown is the equilibrium system N, N*sys , with its dependence on N
input, IN , assuming equilibrium [ON ˆ IN , eqn (A5)]. Bucket
analogues of these responses are drawn in Fig. A3.
Total system N, Nsys (kg N m–2)
A
1.0
q = 10
2
0.9
1
0.8
0.5
0.7
0.1
0.6
0.5
0
200
400
600
Years
800
1000
F I G . A2. Dynamics of the simple model (Fig. 1). Solutions to eqn (A3)
with various values of the parameter q are drawn, showing how
sensitive the speed of approach to equilibrium is to the value of q.
k ˆ constant (y ÿ1). However, in order that we can explore
the consequences of non-linearity, we assume:
ON ˆ
kNqsys
k ˆ 0002 …kg N m ÿ2 †
1ÿq
y ÿ1 ; q ˆ 1
…A2†
Default values of the parameters are given in eqn (A2). If
q ˆ 1, the system is linear: doubling Nsys doubles the output
p
ON . If q ˆ 0.5, then doubling Nsys increases ON by 2. If
q ˆ 0, then the value of Nsys does not a€ect ON . The
responses of system N output to total system N are
illustrated in Fig. A1. By varying q, the system N output
can be made sensitive (high q) or insensitive (low q) to the
amount of N in the ecosystem.
Combining eqns (A1) and (A2), therefore:
dNsys
ˆ IN ÿ kNqsys
dt
…A3†
F I G . A3. Bucket analogy of the simple model of Fig. 1. A, the out¯ow
is proportional to the pressure head giving a linear dependence of N
output on N content. B, The full bucket simply over¯ows so that any
increases in input are immediately re¯ected in increased output. C,
There may be a constant loss rate by `surface evaporation', but if the
input is greater than this, the bucket continues to ®ll ad in®nitum. D,
The system is `damaged' so that the output is increased and the system
moves to a lower equilibrium with less N.
For linearity with q ˆ 1, integration gives the solution:
* ÿ ‰Nsys
* ÿ Nsys …t ˆ 0†Š e ÿkt
Nsys ˆ Nsys
…A4†
* denotes
Nsys(t ˆ 0) is the value of Nsys at time t ˆ 0. Nsys
the equilibrium value of Nsys . This equation describes an
* ), as in the
exponential approach to the asymptote (Nsys
monomolecular growth function (Thornley and Johnson,
2000, pp. 76±77).
Solutions of eqn (A3) are illustrated in Fig. A2. The
rate at which an ecosystem approaches equilibrium
depends on parameter q, that is, the dependence of N
outputs, ON , on ecosystem N content, Nsys . A low value
of q, which could represent soil organic matter pools which
are deep within the soil, highly protected, or strongly
substrate-limited, will lead to slow rates of equilibration.
In equilibrium, IN ˆ ON , and, with eqn (A2), therefore:
r
* ˆ q IN
…A5†
Nsys
k
Cannell and ThornleyÐN States of Ecosystems
* on N input I is also illustrated in
The dependence of Nsys
N
Fig. A1 if the ordinate is read as IN (because IN ˆ ON in
equilibrium) and the abscissa is read as N*sys . It can be seen
that the equilibrium N of the system, N*sys , is highly
dependent on IN over a narrow range of IN about 0.002 for
low q, e.g. q ˆ 0.1, is linearly dependent for q ˆ 1, and has
low dependency on IN for high q, e.g. q ˆ 100, with
N*sys ˆ 1 for any value of IN (ˆ ON).
A bucket analogy can give a helpful physical picture of
the system (Fig. A3).
Tall bucket, variable loss. Referring to Fig. A1, the q ˆ 1,
moderate k curve resembles a tall leaky bucket of water so
the output (ON) is driven by the pressure head (Nsys) giving
linearity, as shown in Fig. A3A.
Over¯owing bucket. The q ˆ 100 curve of Fig. A1
resembles a bucket of water without leaks or evaporation
from the upper surfaceÐeventually it becomes full to, in
this case, about 1 kg N m ÿ2 whatever the N input
(Fig. A3B). This system is not able to absorb increased
1167
inputs to any useful extent, and increased inputs appear
very quickly in the outputs.
Tall bucket, constant loss. The q ˆ 0.01 curve of Fig. A1
resembles a tall bucket of water with a constant crosssection, without leaks and with evaporation at a constant
rate from the upper surface (Fig. A3C)Ðthe output rate is
0.002 kg N m ÿ2 y-1, whatever the system N level, to which
it is insensitive. If the input rate is less than the constant
output rate, then the bucket empties. If the input rate is
greater than the constant output rate, then the bucket goes
on ®lling up forever, and there is no equilibrium solution.
This system can absorb increased inputs very easily with
little change in output.
Leaky bucket, damaged ecosystem. Fig. A3D represents a
damaged ecosystem which is unable to process the N input
which is being imposed, the consequence of this being that
the ecosystem ¯ips into a state where the output constant k
is much increased, and the amount of N stored in the
ecosystem is much diminished.