Soil Water Balance and Ecosystem Response to Climate Change

Author(s): Amilcare Porporato, Edoardo Daly and Ignacio Rodriguez‐Iturbe

Source: The American Naturalist , Vol. 164, No. 5 (November 2004), pp. 625-632
Published by: The University of Chicago Press for The American Society of Naturalists
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vol. 164, no. 5 the american naturalist november 2004

Soil Water Balance and Ecosystem Response to Climate Change

Amilcare Porporato,1,* Edoardo Daly,2,† and Ignacio Rodriguez-Iturbe3,‡

1. Department of Civil and Environmental Engineering, Duke empirical observations, simple and detailed models, and
University, Durham, North Carolina 27708; theoretical and numerical analyses. The importance of its
2. Department of Hydraulics, Transport, and Civil Infrastructures,
appropriate description is evident, and so are its impli-
Polytechnic of Turin, Corso Duca degli Abbruzzi 24, 10129 Turin,
Italy; cations for water resource availability, flood occurrence,
3. Princeton Environmental Institute and Department of Civil and biogeochemistry, and plant conditions (e.g., Noy Meir
Environmental Engineering, Princeton University, Princeton, New 1973; Stephenson 1990; Easterling et al. 2000; Allen and
Jersey 08544 Ingram 2002; Milly et al. 2002). Changes in rainfall regime
and hydrologic cycle due to increased concentration of
Submitted February 4, 2004; Accepted July 16, 2004;
greenhouse gases have already been detected and are pre-
Electronically published September 22, 2004
dicted to further increase (e.g., Easterling et al. 2000), and
it is thus crucial for the scientific community to concen-
trate its efforts on an improved understanding and pre-
diction of the ecological responses to such changes, em-
abstract: Some essential features of the terrestrial hydrologic cycle ploying proper combinations of experiments and
and ecosystem response are singled out by confronting empirical
theoretical analyses to overcome the inherent difficulty of
observations of the soil water balance of different ecosystems with
the results of a stochastic model of soil moisture dynamics. The
dealing with a complex nonlinear systems with essential
simplified framework analytically describes how hydroclimatic var- stochastic components (Clark et al. 2001).
iability (especially the frequency and amount of rainfall events) con- There is growing evidence that the predicted changes in
curs with soil and plant characteristics in producing the soil moisture rainfall regime due to climate change will reduce ecosystem
dynamics that in turn impact vegetation conditions. The results of net primary productivity and possibly induce shifts in
the model extend and help interpret the classical curve of Budyko, community composition (Knapp et al. 2002). Plant pro-
which relates evapotranspiration losses to a dryness index, describing
the partitioning of precipitation into evapotranspiration, runoff, and
ductivity and water stress as well as soil biogeochemistry
deep infiltration. They also provide a general classification of soil are strongly controlled by the pulsing and unpredictable
water balance of the world ecosystems based on two governing di- nature of soil moisture dynamics. Therefore, accounting
mensionless groups summarizing the climate, soil, and vegetation only for changes in mean responses to climatic variability
conditions. The subsequent analysis of the links among soil moisture is not sufficient for a realistic investigation of impact of
dynamics, plant water stress, and carbon assimilation offers an in- climate change on ecosystems, which instead must account
terpretation of recent manipulative field experiments on ecosystem
for the stochastic component of the hydrologic forcing
response to shifts in the rainfall regime, showing that plant carbon
assimilation crucially depends not only on the total rainfall during and its possible alterations in terms of frequency and
the growing season but also on the intermittency and magnitude of amount of rainfall events. Such alterations are responsible
the rainfall events. for modifying soil moisture dynamics and the temporal
structure (i.e., intensity, duration, and frequency) of pe-
Keywords: soil moisture, soil water balance, plant water stress, sto-
chastic processes, ecosystem response, climate change. riods of water stress and impaired plant assimilation (Por-
porato et al. 2001). In fact, soil moisture deficit induces
a reduction of plant water potential that, in turn, may
The terrestrial hydrologic cycle is an example of a manifold cause dehydration, turgor loss, xylem cavitation, stomatal
system whose understanding requires a synergistic use of closure, and reduction of photosynthesis (e.g., Nilsen and
Orcutt 1998). Even maintaining the same total rainfall, an
* Corresponding author; e-mail: amilcare@duke.edu. increase in the intensity of rainfall events, concomitant
†
E-mail: edoardo.daly@polito.it. with a reduction in their frequency, will affect soil moisture
‡
E-mail: irodrigu@princeton.edu. dynamics and plant conditions in a manner that depends
Am. Nat. 2004. Vol. 164, pp. 625–632. 䉷 2004 by The University of Chicago. on the soil and plant physiological characteristics at the
0003-0147/2004/16405-40302$15.00. All rights reserved. site.

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626 The American Naturalist

From the modeling viewpoint, the very large number neglected, thus excluding regions with marked topographic
of processes that make up the dynamics of the soil water effects.
balance and the extremely large degree of nonlinearity and We consider the relative soil moisture s (dimensionless),
space-time variability of hydrological and ecological phe- vertically averaged over the rooting zone of depth Zr (cm),
nomena (Clark et al. 2001; Porporato and Rodriguez- as the state variable describing the dynamics of the soil
Iturbe 2002) call for simplifying assumptions at different water balance. Accordingly, the total volume of soil water
levels. Whenever possible, the development of a low- per unit ground area at a given time t is snZr (cm), where
dimensional description in which the dominating deter- n is the vertically averaged soil porosity (volume of voids/
ministic (and possibly nonlinear) components are sepa- total volume of soil, i.e., dimensionless). Both n and Zr
rated from the high-dimensional (i.e., stochastic) are assumed to be time-invariant parameters. To model
environmental forcing is especially valuable. In particular, the soil water balance dynamics we assume that when s
simple models of soil moisture dynamics have been used exceeds a given threshold s 1, the rainfall in excess of the
to capture the essential features of the terrestrial water available storage capacity is immediately lost by runoff and
cycle and the resulting vegetation response (Eagleson 1978; deep percolation or drainage, LQ (cm day⫺1). The em-
Milly 1993; Rodriguez-Iturbe et al. 1999; Laio et al. 2001). pirical parameter s 1 depends on the type of soil and is
From the resulting analytical solutions, the role of the typically comprised between the so-called field capacity
controlling parameters clearly emerges, offering a theo- (i.e., the soil moisture level where drainage becomes neg-
retical framework whose generality surpasses that of more ligible) and complete saturation (s p 1). Notice that a
complicated models that require cumbersome numerical parameter similar to s 1 was also adopted in previous stud-
simulations. ies of the soil water balance (Milly 1993; Federer et al.
Here we follow a minimalistic approach to the modeling 2003). As discussed in Laio et al. (2001), the present ap-
of the soil-plant-atmosphere system by further simplifying proach to infiltration modeling is useful when the Dunne
a previous stochastic model of soil moisture dynamics or saturation-from-below mechanism of runoff formation
(Rodriguez-Iturbe et al. 1999; Laio et al. 2001) and cou- is dominant compared to the Hortonian runoff (i.e., rain-
pling it with a simple representation of the nonlinear link fall intensity exceeding the soil saturated hydraulic con-
between carbon assimilation and soil moisture at the daily ductivity); this is often the case for vegetated surfaces with
timescale. Our aim is to offer a very parsimonious yet negligible topography and absence of soil crusting.
realistic representation of soil water balance that captures Evapotranspiration, ET (cm day⫺1), is assumed to de-
its essential components: the water-holding capacity of the crease linearly from a maximum value (sometimes referred
soil, which is a function of soil and root characteristics to as potential evapotraspiration), ETmax, under well-
and is responsible for the threshold-like nonlinearity that watered conditions (s p s 1) to 0 at the wilting point
triggers deep infiltration and surface runoff; the soil-mois- (s p s w). The reduction of evapotranspiration with de-
ture dependence of evapotranspiration and photosynthe- creasing soil moisture is a well-established fact that can
sis; and the intermittency and unpredictability of rainfall, be ascribed to increased resistances to soil water transport
whose variability in terms of both frequency and depth of within the soil-plant-atmosphere continuum when soil
events is crucial not only for the soil water balance but water potential is reduced (e.g., increased soil-root resis-
also for the ecological processes (Noy Meir 1973; Rodri- tance, progressive cavitation in the xylem conduits, sto-
guez-Iturbe et al. 1999). matal closure). At a point scale in space, a marked non-
linearity is typically present in the evapotranspiration–soil
moisture relationship (Laio et al. 2001; Daly et al. 2004a);
Methods however, at larger scales (e.g., regional), the temporal var-
A Simple Stochastic Model for Soil Moisture Dynamics iability and special heterogeneity of hydrological processes
tends to significantly broaden the linear rise in the
We interpret the soil moisture dynamics at the daily time- evapotranspiration–soil moisture relationship (Wetzel and
scale, treating the soil as a reservoir with an effective stor- Chang 1987, 1988; Crow and Wood 2002). A similar ten-
age capacity that is intermittently filled by rainfall events dency to linearizing the soil water losses was also noticed
in the form of pulses of random depth. Soil water losses in a theoretical analysis of the mean soil moisture dynamics
occur via evapotranspiration, deep infiltration, and surface (Laio et al. 2002).
runoff (Milly 1993; Rodriguez-Iturbe et al. 1999). Both Rainfall input, R(t) (cm day⫺1), is modeled as a marked
the vertical and horizontal spatial variability are neglected, Poisson process with frequency l (day⫺1) and events car-
assuming that the propagation of the wetting front and rying a random depth of rainfall with exponential distri-
the soil moisture redistribution over the rooting zone are bution of mean a (cm). Such a model has been shown to
negligible at the daily timescale. Lateral water flow is also provide a simple and realistic representation of rainfall at

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 Water Balance and Ecosystem Response 627

the daily timescale for different hydroclimatic regimes important climate, soil, and vegetation parameters in con-
(Milly 1993; Rodriguez-Iturbe et al. 1999). It is particularly trolling soil moisture dynamics.
useful to explicitly and efficiently account not only for Following Rodriguez-Iturbe et al. (1999), the master
changes in mean rainfall rates but also for changes in equation of the probability density function (PDF) of x
frequency and amount of rainfall events. Thus for a typical can be obtained and solved analytically for steady-
growing season of duration Tseas, the total average rainfall state conditions. The result is a truncated gamma distri-
amount is alTseas. bution; that is,
According to the modeling scheme describe above, the
soil moisture balance equation can thus be written as N (l/h)⫺1 ⫺gx
p(x) p x e (2)
h
ds
nZ r p R(t) ⫺ ET[s(t)] ⫺ LQ[s(t), t]. (1)
dt
for 0 ! x ≤ 1. The normalization constant is N, and
Because of the forcing term R(t), equation (1) is a sto-
chastic differential equation that requires a solution in hg l/h
Np , (3)
probabilistic terms (see “Normalization and Probabilistic G(l/h) ⫺ G(l/h, g)
Steady State Solution”). Details about the implications of
some of the modeling assumptions and their possible gen- where G(7) is the gamma function and G(7, 7) is the in-
eralization can be found in Laio et al. (2001a). Applications complete gamma function (Abramowitz and Stegun 1964).
to natural ecosystems of this more complete model and Because of the truncation, the shape of the soil moisture
its possible extensions to cases where seasonal trends in PDF depends on the scale parameter g, as well as the shape
rainfall and evapotranspiration are important can be found parameter l/h.
in Laio et al. (2001b, 2002) and Porporato et al. (2003). The mean effective relative soil moisture, AxS, can also
A critical discussion of the implications of neglecting the be obtained analytically,
soil moisture vertical distribution (i.e., propagation of the
wetting front, hydraulic lift, etc.) at the daily timescale can
1
be found in the literature (Laio et al. 2001a; Guswa et al. GxH p (l ⫺ Ne⫺g) , (4)
2002; Federer et al. 2003). We only note here that the hg
errors introduced with such a simplification tend to be
negligible compared to the uncertainties in the external and so can the normalized water balance,
hydroclimatic forcing.
GETH GLQH GLQH
1p ⫹ p D I GxH ⫹ , (5)
Normalization and Probabilistic Steady State Solution GRH GRH GRH

Defining x p (s ⫺ s w)/(s 1 ⫺ s w) as the “effective” relative

where AETS p AxS ETmax . Equation (5), with the aid of
soil moisture and w0 p (s 1 ⫺ s w)nZ r as the maximum soil
equation (4), describes the partitioning of the rainfall input
water storage available to plants, the governing quantities
into evapotranspiration and deep infiltration plus runoff
of the process are w0, a, l, and ETmax. According to di-
as a function of the governing parameters of the climate,
mensional analysis, these quantities can be grouped into
soil, and vegetation system.
two dimensionless numbers as g p w0 /a and l/h p
(lw0 )/ETmax or g p w0 /a and D I p (gh)/l p ETmax / ARS,
where DI is Budyko’s dryness index, h is the normalized
evapotranspiration loss under well-watered conditions, Results
ETmax /w0, and ARS is the mean rainfall rate, ARS p al
(Milly 2001). Physically, this means that the terrestrial wa- The model proposed here is a minimalist representation
ter balance is governed by the ratio between the soil storage of soil moisture dynamics. In the following applications
capacity and the mean rainfall input per event and either we show that it provides a realistic description of the ter-
the dryness index, that is, the ratio between the maximum restrial water balance under a wide range of conditions.
evapotranspiration and the mean rainfall rate, or l/h, Moreover, when suitably combined with a threshold de-
which is the ratio between the rate of occurrence of rainfall fining incipient plant water stress, it also offers a useful
events and the maximum evapotranspiration rate. Such framework to link hydrologic fluctuations to vegetation
dimensionless groups define the interaction of the most response.

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628 The American Naturalist

Budyko’s Hydroclimatological Description having an increase in evapotranspiration. However, this

would not necessarily mean an improvement in plant con-
A well-known hydroclimatological relationship developed
ditions because it would also imply a change in the tem-
by Budyko (1974) describes the average terrestrial water
poral structure (frequency, duration, and intensity) of wa-
balance by means of a semiempirical curve (fig. 1, dots),
ter stress (Porporato et al. 2001).
which represents the fraction of the rainfall that is
Figure 1 also helps explain the effects of possible climate
evapotranspired as a nonlinear function of the dryness
changes on Budyko’s curve. As an example, depending on
index. Such a curve was tested on several river basins with
the degree to which evapotranspiration, rainfall regime,
different characteristics and synthesizes the average par-
and plant characteristics are affected by climate change,
titioning of the rainfall input into evapotranspiration and
alterations in the mean depth of rainfall per event will
runoff plus drainage. The theoretical solutions of the water
imply a vertical shift in the diagram while a shift along
balance (eq. [5]) are also shown in figure 1 for different
the X-axis will entail changes in potential transpiration
values of the governing parameter g that contains the av-
and mean rainfall rate.
erage rainfall depth and the soil water holding capacity
(through the plant rooting depth and soil texture). Re-
markably, for g near 5.5, the model reproduces Budyko’s Classification of Soil Water Balances
curve very well. This means that using typical values of
the parameters (e.g., average rainfall depth per event The response of the soil water balance to the forcing by
a p 1.5 cm, relative soil moisture at the wilting point the climate-soil-vegetation system is synthesized by the
s w p 0.2, relative soil moisture threshold for deep infil- PDF of the effective relative soil moisture. In particular,
tration and runoff s 1 p 0.85, and porosity n p 0.4), Bu- the behavior of the PDF as a function of the governing
dyko’s curve corresponds to a soil depth of approximately parameters (eq. [2]) provides a general classification of
30–35 cm. Such a value represents the average soil depth soil moisture regimes. Figure 2 shows that the PDF has
that is active from a hydrologic point of view. Interestingly, qualitatively different shapes according to the values of the
it also provides a good estimate of a typical effective root- governing parameters. In the terminology of statistical me-
ing depth, which is in agreement with recent root surveys chanics, these changes may be interpreted as noise-induced
in water-limited ecosystems (Jackson et al. 2000; Schenk transitions of the physical system (Horsthemke and Le-
and Jackson 2002). Figure 1 also shows that, all the other fever 1984). The boundaries indicated in figure 2 may thus
parameters remaining the same, a tendency to have deeper be used to define different hydroclimatic regimes: an “arid”
roots implies moving upward in the diagram and therefore regime (PDFs with 0 mode, i.e., a the wilting point), an

Figure 1: Fraction of total rainfall lost by evapotranspiration as a function of Budyko’s dryness index for different values of the parameter g. The
dots represent the semiempirical curve of Budyko, AETS/ARS p {DI[1 ⫺ exp (⫺DI)] tan h(1/DI)}0.5 . The continuous line underlying the dots corresponds
to g p 5.5. As explained in the text, this refers to an average effective rooting depth of approximately 35 cm. From the lowest to the highest, the
continuous curves refer to g p 0.5, 1, 2, 5.5, 20, and 1,000, respectively.

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 Water Balance and Ecosystem Response 629

Figure 2: Classification of soil water balance, based on the shape of the soil moisture probability density function (PDF), as a function of the two
governing parameters, l/h and g, that synthesize the role of climate, soil, and vegetation (see “Methods” for details). The dashed line, g p
(1/x∗)[(l/h) ⫺ 1], is the locus of points where the mode of the soil moisture PDF is equal to the threshold x∗ , which marks the onset of plant water
stress.

“intermediate” regime (corresponding to soil moisture duration of plant water stress and, in turn, on ecosystem
PDFs with a central maximum), and a “wet” regime (with productivity. When water stress appears, the daily leaf car-
the mode at x p 1, i.e., well-watered conditions). A fur- bon assimilation rate (A) is reduced from its maximum
ther distinction within the intermediate regime can be value (Amax) typical of well-watered conditions (Larcher
made on the basis of plant response to soil moisture dy- 1995; Bonan 2002). Assuming that A is equal to Amax for
namics. Defining x ∗ as a threshold (typically on the order x 1 x ∗ and that it linearly decreases to 0 as soil moisture
of 0.3–0.4) marking the onset of plant water stress (Larcher approaches the wilting point, the mean carbon assimilation
1995; Nilsen and Orcutt 1998; Porporato et al. 2001; Sperry rate during a growing season may be derived analytically
et al. 2002), the dashed line of slope 1/x ∗ in figure 2 be- as a function of climate, soil, and vegetation characteristics
comes the place where the mode of the effective relative using a derived distribution approach from the probabi-
soil moisture PDF is equal to x ∗ and thus where plants listic solution of soil moisture dynamics. While the ana-
are more likely to be at the boundary between stressed lytical details are reported elsewhere (Daly et al. 2004b),
and unstressed conditions. Accordingly, such a line may here the results are used to analyze the recent findings of
be used to divide water-stressed (or semiarid) types of a 4-year manipulative experiment (Knapp et al. 2002; Fay
water balance on the left side from unstressed ones on the et al. 2003) in which the ecosystem response of a native
right side. grassland to increased rainfall variability was investigated
by artificially reducing storm frequency and increasing
rainfall quantity per storm while keeping the total annual
Vegetation Response to Changes in Frequency
rainfall unchanged.
and Amount of Rainfall
Figure 3 shows a comparison of the experimental results
Climate change is presumed to impact the rainfall regime, (Knapp et al. 2002) with the theoretical mean carbon as-
especially in terms of frequency and intensity of rainfall similation as a function of the frequency of rainfall events
events. It is therefore of extreme interest to be able to for fixed total rainfall during a growing season. The ∼20%
predict the effects of such changes on the frequency and decrease in measured net assimilation for the altered rain-

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630 The American Naturalist

Figure 3: Mean daily carbon assimilation rate as a function of the frequency of rainfall events for constant total amount of precipitation during a
growing season. The lines are the theoretical curves derived from the soil moisture probability density function, while the two points are field data
published by Knapp et al. (2002), who compared the response of a mesic grassland to ambient rainfall pattern versus an artificially increased rainfall
variability. The point on the right corresponds to the ambient conditions, and the point on the left corresponds to artificially modified conditions
while keeping the total rainfall the same. The continuous line is for mean total rainfall during a growing season of 507 mm, the dashed line for
600 mm, and the dotted line for 400 mm. The two insets show observed and theoretical soil moisture probability density functions for ambient
and altered conditions. The parameters used are lambient p 0.19 day⫺1, laltered p 0.08 day⫺1, n p 0.55 , Zr p 30 cm, Emax p 0.63 cm day⫺1, Amax p
39 mmol m⫺2 s⫺1, sw p 0.12, s∗ p 0.30, and s1 p 0.8.

fall pattern is well reproduced, going from a mean net Conclusions

assimilation of 23 mmol m⫺2 s⫺1 in natural condition to
∼18.4 mmol m⫺2 s⫺1 when total rainfall was the same but We have shown how the essential traits of the terrestrial
concentrated in fewer events. As shown by the effective water balance can be described by a simple stochastic
model that explicitly accounts for the rainfall unpredict-
relative soil moisture PDFs, the dramatic shift in the rain-
ability both in terms of frequency and amount of rainfall
fall frequency changes the grassland water balance from
events. A simple threshold to separate stressed and un-
an intermediate to a dry one (cf. fig. 2 and fig. 3). The
stressed conditions gives a first-order representation of the
analysis also shows that in such a grassland ecosystem plant nonlinear response to soil moisture dynamics. In
(Knapp et al. 2002), the impact on carbon assimilation of this manner, it is possible to explain the main features of
a decrease in total rainfall is more pronounced when such the soil water balance and the resulting ecosystem con-
a decrease is accompanied by a reduction in the frequency ditions under present-day and projected climatic scenarios.
of rainfall events. If the mean total rainfall is kept constant, These results may be useful to generalize and better un-
then the sensitivity of mean assimilation to the frequency derstand the results of field experiments as well as of more
of rainfall events becomes much more pronounced for dry elaborated numerical models.
periods. The present framework is also expected to be useful to

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 Water Balance and Ecosystem Response 631

investigate the impact of different plant physiological char- troduction to water-balance dynamics. Water Resources
acteristics (e.g., rooting depth, transpiration, and assimi- Research 14:705–712.
lation sensitivity to water stress) on the soil water balance Easterling, D. R., G. A. Meehl, C. Parmesan, S. A. Chang-
at a site or to link plants’ adaptation strategies to soil and non, T. R. Karl, and L. O. Mearns. 2000. Climate ex-
hydroclimatic conditions. Possible extensions of the model tremes: observations, modeling, and impacts. Science
to include seasonal components in rainfall and transpi- 289:2068–2074.
ration as well as to account for transient soil moisture Fay, P. A., J. D. Carlisle, A. K. Knapp, J. M. Blair, and S.
dynamics at the start of the growing season due to the L. Collins. 2003. Productivity responses to altered rain-
winter soil water recharge or spring snow melt could be fall patters in a C4-dominated grassland. Oecologia (Ber-
devised to analyze their interaction with the stochastic lin) 137:245–251.
hydrologic forcing during the growing season. Preliminary Federer, C. A., C. Vorosmarty, and B. Fekete. 2003. Sen-
examples along these lines can be found elsewhere (Rod- sitivity of annual evaporation to soil and root properties
riguez-Iturbe et al. 2001; Laio et al. 2002). in two models of contrasting complexity. Journal of
Hydrometeorology 4:1276–1290.
Acknowledgments Guswa, A. J., M. A. Celia, and I. Rodriguez-Iturbe. 2002.
Models of soil moisture dynamics in ecohydrology: a
We are grateful to P. Fay for providing the data for the
comparative study. Water Resources Research 38. http://
Kansas prairie and for useful discussions. I.R.-I. acknowl-
www.agu.org/journals/wr/wr0209/2001WR000826/.
edges the support of the National Science Foundation
Horsthemke, W., and R. Lefever. 1984. Noise-induced
through grants in Biocomplexity and the National Center
transitions. Springer, Berlin.
for Earth Surface Dynamics. A.P. acknowledges the Office
Jackson, R. B., J. S. Sperry, and T. E. Dawson. 2000. Root
of Science, Biological and Environmental Research Pro-
water uptake and transport: using physiological pro-
gram, U.S. Department of Energy, through the Great Plains
cesses in global predictions. Trends in Plant Science 5:
Regional Center of the National Institute for Global En-
482–488.
vironmental Change under cooperative agreement DE-
Knapp, A. K., P. A. Fay, J. M. Blair, S. L. Collins, M. D.
FC02-03ER63613.
Smith, J. D. Carlisle, C. W. Harper, et al. 2002. Rainfall
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632 The American Naturalist

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