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Nutrient Uptake Estimates for Woody Species as Described by the NST 3.0,
SSAND, and PCATS Mechanistic Nutrient Uptake Models
Wen Lin
Thesis submitted to the faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Master of Science
In
Forestry
J. Michael Kelly, Chair
Thomas R. Fox
John R. Seiler
July 17, 2009
Blacksburg, VA
Keywords: model comparison, multiple dimensional sensitivity analysis, soil moisture, I
max
Nutrient Uptake Estimates for Woody Species as Described by the NST 3.0, SSAND, and
PCATS Mechanistic Nutrient Uptake Models
Wen Lin
Abstract
With the advent of the personal computer, mechanistic nutrient uptake models have become
widely used as research and teaching tools in plant and soil science. Three models NST 3.0,
SSAND, and PCATS have evolved to represent the current state of the art. There are two major
categories of mechanistic models, transient state models with numerical solutions and steady
state models. NST 3.0 belongs to the former model type, while SSAND and PCATS belong to
the latter. NST 3.0 has been used extensively in crop research but has not been used with woody
species. Only a few studies using SSAND and PCATS are available. To better understand the
similarities and differences of these three models, it would be useful to compare model
predictions with experimental observations using multiple datasets from the literature to
represent various situations for woody species. Therefore, the objectives of this study are to: (i)
compare the predictions of uptake by the NST 3.0, SSAND, and PCATS models for a suite of
nutrients against experimentally measured values, (ii) compare the behavior of the three models
using a one dimensional sensitivity analysis; and (iii) compare and contrast the behavior of NST
3.0 and SSAND using a multiple dimensional sensitivity analysis approach. Predictions of
nutrient uptake by the three models when run with a common data set were diverse, indicating a
need for a reexamination of model structure. The failure of many of the predictions to match
observations indicates the need for further studies which produce representative datasets so that
the predictive accuracy of each model can be evaluated. Both types of sensitivity analyses
suggest that the effect of soil moisture on simulation can be influential when nutrient
concentration in the soil solution (C
Li
) is low. One dimensional sensitivity analysis also revealed
that I
max
negatively influenced the uptake estimates from the SSAND and PCATS models.
Further analysis indicates that this counter intuitive response of I
max
is probably related to low
soil nutrient supply. The predictions of SSAND under low-nutrient-supply scenarios are
generally lower than those of NST 3.0. We suspect that both of these results are artifacts of the
steady state models and further studies to improve them, such as incorporating important
rhizospheric effects, are needed if they are to be used successfully for the longer growth periods
and lower soil nutrient supply situations more typical of woody species.
iii
Acknowledgements
I would like to express thanks to committee members Dr. J. M. Kelly, Dr. T. R. Fox, and Dr.
J. R. Seiler for their valuable help into this work. I am honored to have them serve on my
committee and I learned a lot from them. I would also like to express thanks to Dr. C. A.
Copenheaver. She is a great teacher and always helpful. Many thanks to the professors of
Virginia Tech that showed me by their actions what an ideal scientific life looks like.
I own my appreciation to my family and friends, who supported and encouraged me over the
years. Finally I want to thank my best friend, Xiaowei Tang, without whom my life in a foreign
country would be much harder.
iv
Table of Contents
Abstract .............................................................................................................................. ii
Acknowledgements .......................................................................................................... iii
Table of Contents ............................................................................................................. iv
List of Tables .................................................................................................................... vi
List of Figures .................................................................................................................. vii
Chapter I. INTRODUCTION .......................................................................................... 1
Literature Cited ........................................................................................................... 4
Chapter II. LITERATURE REVIEW ............................................................................ 6
Introduction ................................................................................................................ 6
Soil Characteristics ..................................................................................................... 7
Nutrient availability in soils ............................................................................... 7
Transport processes ............................................................................................ 7
Mass flow ................................................................................................... 9
Diffusion .................................................................................................... 9
Simultaneous mass flow and diffusion .................................................... 10
Plant Properties ......................................................................................................... 10
Root morphology and growth .......................................................................... 10
Nutrient uptake kinetics ................................................................................... 11
Nutrient uptake mechanisms.................................................................... 11
Modeling nutrient uptake kinetics ........................................................... 12
Modeling Approaches .............................................................................................. 13
The basic principles of NUTRIENT UPTAKE and NST 3.0 .......................... 13
The basic principles of COMP8, SSAND, and PCATS................................... 14
Assumptions for the mechanistic models ......................................................... 16
Comparison of the two major categories of mechanistic models .................... 17
Current study on NST 3.0, SSAND, and PCATS ............................................ 18
Sensitivity Analysis of Mechanistic Nutrient Uptake Models ................................. 19
Nutrient Uptake Modeling of Woody Species ......................................................... 20
Literature Cited ......................................................................................................... 26
Chapter III. NUTRIENT UPTAKE ESTIMATES FOR WOODY SPECIES AS
DESCRIBED BY THE NST 3.0, SSAND, AND PCATS MECHANISTIC
NUTRIENT UPTAKE MODELS ................................................................................. 31
Abstract ..................................................................................................................... 31
Introduction .............................................................................................................. 32
v
Material and Methods ............................................................................................... 33
Basic principles and model assumptions ......................................................... 33
Data transformation .......................................................................................... 40
Methods of sensitivity analysis ........................................................................ 44
Results and Discussion ............................................................................................. 46
Calculation of uptake using data taken from the literature .............................. 46
Sensitivity analysis ........................................................................................... 52
One dimensional sensitivity analysis ....................................................... 52
Multiple dimensional sensitivity analysis ................................................ 60
Conclusions .............................................................................................................. 62
Literature Cited ......................................................................................................... 66
vi
List of Tables
Table 1: List of symbols and definitions used in mechanistic nutrient uptake model
equations. ............................................................................................................................ 8
Table 2: Nutrient uptake kinetic parameters for a variety of woody species taken from the
literature. ........................................................................................................................... 22
Table 3: Soil supply parameters from studies of a variety of woody species. .................. 23
Table 4: Root growth parameters from a variety of woody species and the mean water
flux at the root surface taken from the literature. .............................................................. 25
Table 5: List of symbols and definitions used in the NST 3.0, SSAND, and PCATS
mechanistic nutrient uptake model equations. .................................................................. 34
Table 6: Parameter values for loblolly pine and red maple based on observations reported
by Kelly et al. (1992) and Kelly et al. (2001) used for NST 3.0, SSAND and PCATS
simulations. ....................................................................................................................... 42
Table 7: Parameter values for hybrid poplar based on observations reported by Kelly and
Ericsson (2003) used for NST 3.0, SSAND and PCATS simulations. ............................. 43
Table 8: Literature values for soil moisture (θ) taken from field, nursery, and greenhouse
studies of loblolly and slash pine. ..................................................................................... 47
Table 9: Literature values for potassium concentration in soil solution (C
Li
) taken from
field, nursery, and greenhouse studies of loblolly and slash pine. .................................... 47
Table 10: Literature values of root growth rate (k) taken from field, nursery, and
greenhouse studies of loblolly and slash pine. .................................................................. 47
Table 11: Observed uptake of NO
3
-N, P, and K compared to simulated uptake as
predicted by NST 3.0, SSAND, and PCATS using data from Kelly et al. (1992), Kelly et
al. (2001), and Kelly and Ericsson (2003). The percentages represent the degree of
correspondence between the simulated and observed uptake value. ................................ 49
vii
List of Figures
Figure 1: Comparison of predicted potassium uptake by the NST 3.0, SSAND, and
PCATS models and observed potassium uptake using data from the studies by Kelly et al.
(1992), Kelly et al. (2001), and Kelly and Ericsson (2003). ............................................. 50
Figure 2: Comparison of predicted nitrate uptake by the NST 3.0, SSAND, and PCATS
models and observed nitrate uptake using data from the studies by Kelly et al. (1992),
Kelly et al. (2001), and Kelly and Ericsson (2003). ......................................................... 50
Figure 3: Comparison of predicted phosphorus uptake by the NST 3.0, SSAND, and
PCATS models and observed phosphorus uptake using data from the studies by Kelly et
al. (1992), Kelly et al. (2001), and Kelly and Ericsson (2003). ........................................ 51
Figure 4: One dimensional sensitivity analysis using SSAND with potassium uptake data
for loblolly pine seedlings from Kelly et al. (1992) and the diffusion coefficient of
potassium in water at 25 ºC taken from the Parsons (1959). ............................................ 54
Figure 5: One dimensional sensitivity analysis using PCATS with potassium uptake data
for loblolly pine seedlings from Kelly et al. (1992) and the diffusion coefficient of
potassium in water at 25 ºC taken from the Parsons (1959). ............................................ 55
Figure 6: One dimensional sensitivity analysis using NST 3.0 with potassium uptake data
for loblolly pine seedlings from Kelly et al. (1992) and the diffusion coefficient of
potassium in water at 25 ºC taken from the Parsons (1959). ............................................ 56
Figure 7: Simulated uptake by SSAND and NST 3.0 at five levels of I
max
and three levels
of C
Li
. For these simulations root growth rate and volumetric soil moisture have been set
to 39.3 cm day
-1
and 0.4 respectively. Other parameter values remained as listed in Table
6 for loblolly pine (Kelly et al. 1992). .............................................................................. 58
Figure 8: Response surface from a multiple dimensional sensitivity analysis of SSAND
and NST 3.0 using five levels of potassium concentration in the soil solution (C
Li
), five
levels of volumetric soil moisture (θ), and five levels of root growth rate (k). Other
parameter values remained as listed in Table 6 for loblolly pine (Kelly et al. 1992). The
unit of simulated uptake is µmol. ...................................................................................... 61
Figure 9: Response surface from a multiple dimensional sensitivity analysis of SSAND
and NST 3.0 using two levels of potassium concentration in the soil solution (C
Li
), five
levels of volumetric soil moisture (θ), and five levels of root growth rate (k) with cm
day
-1
as the units. Other parameter values remained as listed in Table 6 for loblolly pine
(Kelly et al. 1992). ............................................................................................................ 63
Figure 10: Simulated potassium uptake by SSAND and NST 3.0 with data taken from
Kelly et al. (1992). The simulation was conducted with nutrient concentration in the soil
solution (C
Li
) changing from 0.028 to 4.284 µmol ml
-1
while keeping other parameters
unchanged. ........................................................................................................................ 64
1
Chapter I. INTRODUCTION
Influenced by many chemical, physical, and physiological factors, plant nutrient
uptake is a very complex process (Barber 1995). A mechanistic nutrient uptake model
simulates this process using mechanistic or empirical equations based on basic
mechanisms. Based on the seminal efforts by Nye and Marriot (1969) and Baldwin et al.
(1973), and advanced by the subsequent work of Claassen and Barber (1976), Nye and
Tinker (1977), Barber and Cushman (1981), Claassen et al. (1986), Smethurst and
Comerford (1993), Yanai (1994), Smethurst et al. (2004), and Comerford et al. (2006),
mechanistic models were developed for the personal computer that allows nutrient uptake
by plant roots to be calculated. These models take into consideration the competition
between roots within a specified time, as well as the size and growth of the roots, the
kinetics of nutrient uptake, and the supply of nutrients from the soil to the root surface
(Barber 1995). Research on several crop species has demonstrated that the predictions of
these mechanistic models generally match the observed uptake under situations in which
the soil and plant conditions are relatively constant (Barber 1995; Tinker and Nye 2000).
Results with woody species have been more variable (Gillespie and Pope 1990; Van Rees
et al. 1990; Smethurst and Comerford 1993).
Validated mechanistic models allow data interpolation and extrapolation (Claassen
and Steingrobe 1999), and are able to provide predictions under various situations, which
may avoid the need for costly field trials (Barber 1995; Tinker and Nye 2000). They can
also be used to calculate values that are difficult to determine experimentally (Claassen
and Steingrobe 1999) in addition to revealing the factors that have the greatest influence
on the nutrient uptake processes (Barber 1995).
NST 1.0, SSAND, and PCATS are three mechanistic nutrient uptake models which
were presented in 1986, 2000, and 2004, respectively (Li and Comerford 2000; Smethurst
et al. 2004; Claassen et al. 1986). NST 3.0 is the improved version of NST 1.0, which
was developed in 1990s and not published in a journal (Claassen, N. Personal
communication. 2009, 31 July). NST models are based largely on the Barber-Cushman
model (Claassen et al. 1986), and belong to transient state models using a numerical
2
solution. SSAND and PCATS are steady state models based on the key equations
developed by Baldwin et al. (1973) and Nye and Tinker (1977), with the assumption that
the “concentration profile around the root can be considered to be in a steady state”
(Yanai 1994). Compared to SSAND, PCATS further simplifies calculation by running on
a fixed daily time-step (Smethurst et al. 2004). Transient state models with a numerical
solution are generally considered to be more accurate than steady state models (Smethurst
and Comerford 1993; Yanai 1994). However, steady state models can be constructed to
allow parameters to change during simulation and thus are able to respond to real-time
changes in parameters as well as allowing for the feedback between soil and plant
processes during simulation (Yanai 1994).
One way to evaluate the models is by comparing model predictions with measured
values. A second way is to use sensitivity analysis. The traditional way to conduct a
sensitivity analysis in the modeling context is to vary a single input parameter in a model
while keeping the others constant (Silberbush and Barber 1983). By plotting the change
ratio of the parameters compared to their original values on the horizontal axis, and that
of the predicted uptake to the original prediction on the vertical axis, it is possible to
evaluate the importance of each parameter by the slopes of the lines on the graph
(Silberbush and Barber 1983). As Williams and Yanai (1996) have suggested, this
method can be called a one dimensional sensitivity analysis. However, because the
relative importance of parameters defined this way can depend strongly on the values of
the other parameters, Williams and Yanai (1996) developed a multiple dimensional
sensitivity analysis in order to study model behavior across a broad range of possible
parameter values.
In their approach to multiple dimensional sensitivity analysis, Williams and Yanai
(1996) utilized data taken from the literature for both crop and tree species. They
identified four factors as the most influential parameters controlling uptake per unit
length of root: (i) the average dissolved nutrient concentration (C
av
), (ii) the maximal rate
of nutrient uptake (I
max
), (iii) the effective diffusion coefficient (D
e
), and (iv) the root
radius (r
0
). However, as noted by Fitter (2002), differences between the roots of woody
and herbaceous species can be important, especially in terms of root radius and
3
suberization. Both of these differences could influence estimated nutrient uptake. Also,
Williams and Yanai (1996) used a simplified model whose behavior may be different
from the NST 3.0, SSAND, and PCats models. For example, the study by Van Rees et al.
(1990) on slash pine (Pinus elliottii var. elliottii) and that of Kelly et al. (1992) on
loblolly pine (Pinus taeda) indicate that the influence of root growth rate (k) is prominent
in the sensitivity analysis by the NUTRIENT UPTAKE model by Oates and Barber
(1987).
Most studies to date have focused on tests of individual models. The number of
studies using SSAND and PCATS are limited. NST 3.0 has been widely used but only
with crop species. Therefore, using a common data set to compare the predictions of
nutrient uptake by NST 3.0, SSAND, and PCATS against observed values combined with
the use of sensitivity analysis would help us better understand model behaviors under
various situations and provide useful information for future model development.
Therefore, the objectives of this study are to: (i) compare the predictions of uptake by
the NST 3.0, SSAND, and PCATS models for a suite of nutrients against experimentally
measured values, (ii) compare the behavior of the three models using a one dimensional
sensitivity analysis; and (iii) compare and contrast the behavior of NST 3.0 and SSAND
using the multiple dimensional sensitivity analysis approach of Williams and Yanai
(1996). Common data sets, based on literature values only taken from studies of woody
species, will be used for these comparisons.
4
Literature Cited
Baldwin, J. P., P. H. Nye and P. B. Tinker. 1973. Uptake of solutes by multiple root
systems from soil III - a model for calculating the solute uptake by a randomly
dispersed root system developing in a finite volume of soil. Plant and Soil 38: 621-
635.
Barber, S. A. 1995. Soil nutrient bioavailability: a mechanistic approach. 2nd edition.
John Wiley & Sons, Inc., New York.
Barber, S. A. and J. H. Cushman 1981. Nitrogen uptake model for agronomic crops. p.
382-409 In I. K. Iskandar (ed.). Modeling waste water renovation-land treatment.
Wiley-Interscience, New York.
Claassen, N. and S. A. Barber. 1976. Simulation model for nutrient uptake from soil by a
growing plant root system. Agronomy Journal 68: 961-964.
Claassen, N. and B. Steingrobe 1999. Mechanistic simulation models for a better
understanding of nutrient uptake from soil. p. 327-367 In Z. Rengel (ed.). Mineral
nutrition of crops: fundamental mechanisms and implications. Food Products Press,
New York.
Claassen, N., K. M. Syring and A. Jungk. 1986. Verification of a mathematical-model by
simulating potassium uptake from soil. Plant and Soil 95: 209-220.
Comerford, N. B., W. P. Cropper, Jr., L. Hua, P. J. Smethurst, K. C. J. Van Rees, E. J.
Jokela and F. Adams. 2006. Soil Supply and Nutrient Demand (SSAND): a general
nutrient uptake model and an example of its application to forest management.
Canadian Journal of Soil Science 86: 655-673.
Fitter, A. 2002. Characteristics and functions of root systems. p. 15-32 In Y. Waisel, A.
Eshel and U. Kafkafi (ed.). Plant roots: the hidden half. Marcel Dekker, Inc., New
York.
Gillespie, A. R. and P. E. Pope. 1990. Rhizosphere acidification increases phosphorus
recovery of black locust: II. Model predictions and measured recovery. Soil Science
Society of America Journal 54: 538-541.
Li, H. and N. B. Comerford. 2000. SSAND Version 1.0, Release 1.06 (10-8-2001) User's
Guide. University of Florida. Gainesville, FL.
Nye, P. H. and F. H. C. Marriott. 1969. A theoretical study of the distribution of
substances around roots resulting from simultaneous diffusion and mass flow. Plant
and Soil 30: 459-472.
Nye, P. H. and P. B. Tinker. 1977. Solute movement in the soil-root system. University
of California Press, Berkeley.
5
Oates, K. and S. A. Barber. 1987. NUTRIENT UPTAKE: a microcomputer program to
predict nutrient absorption from soil by roots. Journal of Agronomic Education 16:
65-68.
Samal, D. 2007. Potassium Uptake Efficiency Mechanisms and Root Exudates of
Different Crop Species. Ph.D. Dissertation Georg-August-University Göttingen,
Germany.
Silberbush, M. and S. A. Barber. 1983. Sensitivity of simulated phosphorus uptake to
parameters used by a mechanistic-mathematical model. Plant and Soil 74: 93-100.
Smethurst, P. J. and N. B. Comerford. 1993. Simulating nutrient uptake by single or
competing and contrasting root systems. Soil Science Society of America Journal 57:
1361-1367.
Smethurst, P. J., D. S. Mendham, M. Battaglia and R. Misra. 2004. Simultaneous
prediction of nitrogen and phosphorus dynamics in a Eucalyptus nitens plantation
using linked CABALA and PCATS models. p. 565-569 In N. M. G. Borralho (ed.)
Eucalyptus in a changing world. Proceedings of an International conference of the
WP2.08.03 on silviculture and improvement of eucalypts, Aveiro, Portugal, 11-15
October 2004. IUFRO, Aveiro, Portugal.
Tinker, P. B. and P. H. Nye. 2000. Solute movement in the rhizosphere. Oxford
University Press, New York.
Van Rees, K. C. J., N. B. Comerford and W. W. Mcfee. 1990. Modelling potassium
uptake by slash pine seedlings from low-potassium-supplying soils. Soil Science
Society of America Journal 54: 1413-1421.
Williams, M. and R. D. Yanai. 1996. Multi-dimensional sensitivity analysis and
ecological implications of a nutrient uptake model. Plant and Soil 180: 311-324.
Yanai, R. D. 1994. A steady-state model of nutrient uptake accounting for newly grown
roots. Soil Science Society of America Journal 58: 1562-1571.
6
Chapter II. LITERATURE REVIEW
Introduction
Two general categories of nutrient uptake models, empirical and mechanistic models have
been developed to study nutrient uptake by plant roots (Rengel 1993). Empirical models are
based mainly on regressions as well as statistical means, often for practical use (Claassen and
Steingrobe 1999). Mechanistic models, on the other hand, require an understanding of the
mechanisms and a quantitative description of the phenomena (Rengel 1993). Mechanistic models
are therefore useful to test the correctness of our knowledge of the phenomena (Claassen and
Steingrobe 1999). Extrapolation of a verified mechanistic model is thus more reliable than that of
an empirical model (Claassen and Steingrobe 1999). Another scientific application is the
calculation of parameters that are difficult to obtain for either technical or economic reasons
(Claassen and Steingrobe 1999). Model runs can also be used to identify topics that warrant
further field or laboratory study. Finally, we can improve our understanding of the phenomena
by exploring model behavior without conducting field trials (Barber 1995; Claassen and
Steingrobe 1999).
Nutrient uptake by plant roots involves interdisciplinary studies: soil chemistry, soil physics,
and plant physiology (Barber 1995). The typical mechanistic nutrient uptake model describes the
supply of nutrients from bulk soil to root surfaces, root growth and morphology, and root uptake
kinetics (Barber 1995).
The modeling of nutrient uptake started in the early 1960s. Bouldin (1961) and Olsen et al.
(1962) proposed mathematical models to simulate diffusion of solutes through soils, which were
used to explain phosphate movement and uptake. Nye and Spiers (1964) subsequently developed
the partial differential equations used to describe simultaneous mass flow and diffusion for
nutrient uptake by a unit length of root. Nye and Marriot (1969) defined boundary conditions for
the equations and solved them numerically, while Baldwin et al. (1973), on the other hand,
solved the equations analytically with steady state approximations. Their work became the
foundation for mechanistic nutrient uptake models. Building on this base Claassen and Barber
(1976), Nye and Tinker (1977), Barber and Cushman (1981), Claassen et al. (1986), Smethurst
7
and Comerford (1993b), Yanai (1994), Smethurst et al. (2004), and Comerford et al. (2006)
proposed model revisions to cover the major sub-processes of nutrient uptake and to
accommodate a variety of additional conditions. Current models have been successfully used in
many areas of plant nutrition research.
In this review the soil and plant properties will be discussed first, followed by a mathematical
description of the models and the use of sensitivity analysis to identify key parameters. Finally, a
review of studies that have applied mechanistic nutrient uptake models to study woody species is
presented. Because many parameters are involved, the symbols and their definitions are listed in
Table 1.
Soil Characteristics
Nutrient availability in soils
Soil is a highly complex and heterogeneous system with many different components that
provide plants with water and nutrients. Nutrients exist in the soil in gaseous, liquid, and solid
forms. In this review only nutrients in liquid and solid forms will be discussed. Barber (1995)
defined an available nutrient as “the nutrient present in a pool of ions in the soil and can move to
the plant root during plant growth if the root is close enough”. In the early development of
mechanistic models, available nutrients were restricted to inorganic forms to simplify calculation
(Barber 1995), but latter models such as SSAND are able to include nutrient from organic forms
such as the mineralization of nitrogen (Comerford et al. 2006).
Transport processes
Interception, mass flow, and diffusion are the three components of nutrient movement to the
root surface (Marschner 1995). Interception is used to describe the uptake of soil nutrients at the
root interface when soil volume is displaced by root volume (Barber 1995). However, Tinker and
Nye (2000) consider the concept of interception to be somewhat arbitrary and argued that it can
be included in the diffusion component. Although conditions in the rhizosphere are sometimes
different from those in the bulk soil (Marschner 1995), the contribution of interception to
nutrient uptake is negligible for most nutrients (Barber 1995). Therefore, only mass flow and
diffusion are considered to be responsible for movement of nutrients to the root surface in
mechanistic modeling.
8
Table 1. List of symbols and definitions used in mechanistic nutrient uptake model equations.
Symbol Definition
b buffer power of nutrient
C nutrient concentration in soil
C
av
average nutrient concentration in soil solution
C
L
nutrient concentration in soil solution
C
L0
nutrient concentration in soil solution at the root surface
C
Li
initial concentration of the nutrient in the soil solution
C
min
concentration in solution below which net influx ceases
C
s
nutrient concentration in soil solid phase
D diffusion coefficient of solute
D
e
effective diffusion coefficient for the nutrient in the soil
D
L
diffusion coefficient of solute in water
E nutrient efflux of plant roots
F flux of solute
F
D
flux of solute by diffusion
F
M
flux of solute by mass flow
f
L
impedance factor of soil liquid-phase
f
s
impedance factor of soil solid-phase
I net influx of solute per unit area of root surface
I
l
net influx of solute per unit length of root
I
max
maximum net influx at high nutrient concentrations
k rate of root growth
K
m
Michaelis-Menten constant
L
0
initial root length
L
v
root length density
r radial distance in soil from the root surface
r
0
mean root radius
r
1
half-distance between root axes
t time
∆U the amount of nutrient uptake by root of unit length within a time
period
U the amount of nutrient uptake by root
U
p
the amount of nutrient uptake by root system within a time period
predicted by PCATS
v mean water influx
v
0
mean water influx at root surface
v
1
water influx at the distance r
1
x distance
∆x the extension of nutrient depletion zone in soil
α root absorbing power
θ volumetric soil moisture
ρ soil bulk density
9
Mass flow
Mass flow is the convective transport of nutrients through the soil to the root surface by
water flow as a result of transpiration (Barber 1995). The relative contribution of mass flow to
nutrient uptake depends on the nutrient, plant species, plant age, and time of day (Marschner
1995). For example calcium and magnesium supplied to plants by mass flow is significant, but
its contribution to potassium supply is negligible (Marschner 1995). The influx by mass flow
(F
M
) can be calculated by
[1]
where v is the mean water flux in soil driven by transpiration, and C
L
is the nutrient
concentration in the soil solution (Barber 1995).
Diffusion
Diffusion is the movement of nutrients from areas of high concentration to those of low
concentration (Barber 1995). It is the main mechanism for at least phosphorus and potassium
movement in the soil to plant roots (Marschner 1995). A depletion zone is produced when the
concentration of nutrient is lowered near the root surface due to root absorption (Jungk and
Claassen 1997). Diffusive flux F
D
can be described by Fick’s first law,
[2]
where D is the diffusion coefficient of the nutrient in soil, C is the nutrient concentration in soil
solution, and x is the distance. Diffusion in soils includes solute diffusion in soil solution and the
surface diffusion on the soil solid phase (Tinker and Nye 2000). Thus the diffusion coefficient is
calculated by
[3]
where D
L
is the diffusion coefficient of the solute in free solution, θ is the soil moisture, ρ is soil
bulk density, f
L
and f
s
are liquid- and solid-phase impedance factors, respectively, and S
s
is the
amount of solute adsorbed on a unit weight of solid (Tinker and Nye 2000). The first part of
equation [3] represents solute diffusion in solution, while the second part represents the surface
diffusion on the soil solid phase. Usually only diffusion in soil solution is considered in the
mechanistic nutrient uptake models, and the equation for the diffusion coefficient is reduced to
[4]
10
With the addition of the equation for buffer power b
[5]
equation [4] is rewritten as:
[6]
where D
e
is called the effective diffusion coefficient. It is also assumed that the liquid-impedance
factor f
L
is responsible for all the retarding effects from the solution and the solid phase during
the process of diffusion (Tinker and Nye 2000) and mainly reflects the tortuosity, water density,
and surface changes in the soil (Barber 1995).
Buffer power (b) reflects the relationship between nutrient concentration in the soil (C) and
in the soil solution (C
L
), and can be derived from sorption isotherms. Different sorption
equations have been proposed. For example,
, if the Freundlich equation is
adopted, where a, m, and n are regression constants. Therefore, b can be obtained from such
isotherm equations. Because the isotherm is usually non-linear, b is not constant when C
L
changes.
The extension of the depletion zone can be calculated by
[7]
where ∆x is the distance at which the decrease of concentration is 20% of the maximum decrease
at the root surface, and t is time (Syring and Claassen 1995).
Simultaneous mass flow and diffusion
Mass flow and diffusion occur simultaneously to supply nutrients to plant roots and cannot
be treated as separate processes. Nye and Spiers (1964) presented a partial differential equation
(equation 8) to describe simultaneous mass flow and diffusion, and this equation became the
foundation of the most mechanistic nutrient uptake models.
[8]
Plant Properties
Root morphology and growth
The root system that provides the plant with water and nutrients is very complex and
dynamic. The basic mechanistic nutrient uptake models simplify the system by describing it with
a few parameters. Root radius (r
0
) is used to describe the root morphology. Root hairs play an
11
important role in some plants and the radius of the root hair is included if uptake by root hairs is
considered. Initial root length (L
0
) and root growth rate (k) are used to describe root growth. Two
destructive harvests are usually required to obtain these parameters. Two mathematical methods
are available to describe root growth rate. The linear way is to calculate k by
[9]
where t and L represent the time and root length at individual harvests, 1 and 2 representing the
first and second harvest. The exponential way is to calculate k by
[10]
Half distance between root axes (r
1
) is used to describe the influence of inter-root competition on
nutrient uptake simulation. It is calculated by
[11]
where L
v
is the root length density (Barber 1995).
Nutrient uptake kinetics
Nutrient uptake mechanisms
The transport of nutrients across the cell membrane is the rate-limiting step when the nutrient
supply is abundant (Williams and Yanai 1996). When nutrients arrive at the root surface, they
are available for root absorption. The process can be divided into two types, passive and active
uptake (Lodish et al. 2001). Passive uptake of nutrients refers to the transmembrane movement
of nutrients without the consumption of energy (Lodish et al. 2001). It includes diffusion and
facilitated diffusion along the concentration gradient between the inside and outside of the cell
membrane (Lodish et al. 2001). Facilitated diffusion is the diffusion of ions with the help of ion
channels or carrier proteins on the cell membrane (Lodish et al. 2001). For those nutrients whose
concentration inside the cell is higher than outside, they will be transported across the membrane
at the cost of energy, usually with the help of various membrane transporters (Lodish et al.
2001).
Marschner (1995) summarized the uptake isotherms and divided essential nutrients into three
categories: (i) the uptake of potassium, phosphorus, nitrate, and sulfur usually depends on the
12
external nutrient concentration before it becomes saturated; (ii) the uptake of sodium, calcium,
and magnesium also depends on the external concentration, but to a less extent, and there is no
obvious pattern for leveling off; and (iii) boron uptake is by diffusion, in direct proportion to the
external concentration. Recent studies of boron uptake by crop plants suggest that the mechanism
of boron uptake is more complex. It involves both active and passive mechanisms, depending on
the boron concentration in the soil solution at the root surface (Dannel et al. 2000; Pfeffer et al.
2001).
Modeling nutrient uptake kinetics
As early as the 1960s, root absorbing power (α) was used to connect the nutrient
concentration in soil solution and the influx of nutrient into cells using the equation of Nye and
Spiers (1964):
[12]
where C
L0
is the nutrient concentration in soil solution at the root surface. α was assumed to be
constant before the influx reaches its maximum (Nye and Spiers 1964; Nye and Marriott 1969;
Baldwin et al. 1973; Nye and Tinker 1977). That is, influx increases linearly as the nutrient
concentration in soil solution increases until the concentration reaches a critical point, above
which the influx will be constant. If net influx per unit length of root (I
l
) is used, the equation
(Baldwin et al. 1973; Nye and Tinker 1977) is transformed to
[13]
where r
0
is the root radius.
Reflecting advances in the understanding of cell biology, the mechanism of active uptake
was incorporated into the models by Claassen and Barber (1976). Active uptake is described by
the Michaelis-Menten equation:
[14]
where I is the nutrient influx per unit area of root surface, I
max
is the maximal influx at high C
L
,
K
m
is the Michaelis-Menten constant, C
min
is the solution concentration at which influx equals to
efflux, and E is the efflux of ions from roots into solution.
The relationship between C
L0
and I described by Michaelis-Menten equation indicates that
the root absorbing power constantly changes as C
L0
changes. To incorporate active uptake
13
kinetics in their model, Smethurst and Comerford (1993b) used a variable root absorbing power
(equation 15).
[15]
Modeling Approaches
Over the past four decades different mechanistic nutrient uptake models have been developed
to simulate nutrient uptake. Usually, these models consist of three basic components (Rengel
1993): (i) solute movement in the soil toward plant roots described by the continuity equation
(equation 8); (ii) nutrient uptake kinetics described by the Michaelis-Menten equation (equation
14); (iii) nutrient uptake as a result of root growth and inter-root competition by introducing root
growth and morphology parameters. Two categories of models have evolved, steady state and
transient models (Tinker and Nye 2000). NST 3.0 is an example of a transient model with a
numerical solution, while SSAND and PCATS are steady state models.
The basic principles of NUTRIENT UPTAKE and NST 3.0
Transient models utilizing numerical solutions are a well established approach to mechanistic
nutrient uptake models (Tinker and Nye 2000). The Barber-Cushman model is a well-known and
widely-used model in this category. NUTRIENT UPTAKE model and NST 1.0 are the personal
computer version of the Barber-Cushman model (Oates and Barber 1987; Claassen et al. 1968).
NST 3.0 is an improved version of NST 1.0. In this section, the principles of NUTRIENT
UPTAKE model are presented, followed by a brief introduction to NST 3.0.
The Barber-Cushman model is largely based on the work by Nye and Marriot (1969). Nye
and Marriot (1969) revised the continuity equation proposed by Nye and Spiers (1964) (see
equation 8) to describe the flux of nutrient in the soil to the root surface with the nutrient
concentration in soil solution (C
L
):
[16]
where v
0
is the water flux at the root surface, r is the radial distance from the root, and t is time.
Nye and Marriot (1969) defined boundary conditions and solved this equation numerically.
14
Summarizing the work by Claassen and Barber (1976) and Cushman (1979a; 1979b), Barber
and Cushman (1981) suggested new boundary conditions for the equation [16] to include inter-
root competition for nutrients:
(1) Inner boundary condition
[17]
where
is a transformation of the equation [14].
(2) Outer boundary condition:
If there is no inter-root competition,
[18]
If there is inter-root competition,
[19]
where v
1
is the water influx at a distance of r
1
.
The new boundary conditions incorporated inter-root competition as well as Michaelis-
Menten kinetics. When solved numerically, the enhanced mechanistic model evolved into
Barber-Cushman model. In 1983 Itoh and Barber developed a submodel to the Barber-Cushman
model to include nutrient uptake by root hairs.
In 1986 Claassen et al. published NST 1.0 model. In 1987 Oates and Barber published
NUTRIENT UPTAKE model. Both were based on the Barber-Cushman model. Later Claassen
and his colleagues developed NST 2.0 and NST 3.0, which were not published in a journal
(Claassen, N. Personal communication. 2009, 31 July). NST 3.0 incorporates the Freundlich
isotherm into the model so that the buffer power (b) changes as the nutrient concentration in soil
solution changes (Steingrobe et al. 2000).
The basic principles of COMP8, SSAND, and PCATS
Steady state models are the other standard method in mechanistic nutrient uptake modeling
(Tinker and Nye 2000). Baldwin et al. (1973) and Nye and Tinker (1977) proposed the key
equations in 1970s. Based on their work, Smethurst and Comerford (1993b) developed a
computer model, COMP8 (Competition model version 8), which was able to calculate nutrient
uptake between two competing and contrasting root systems. SSAND was a revision and
15
expansion of COMP8 by Comerford et al. (2006). Its main improvements lie in the functions of
predicting nutrient uptake as influenced by mycorrhizae and simulation of fertilization effects
(Comerford et al. 2006). Based on COMP8 and an earlier version of SSAND, another steady
state model, PCATS was developed to simulate nutrient uptake by a single species by Smethurst
et al. (2004). In this section, the principle of steady state models is briefly described; and the
features of COMP8, SSAND and PCATS are introduced.
Based on the continuity equation by Nye and Spiers (1964) (see equation 8), Baldwin et al.
(1973) and Nye and Tinker (1977) proposed the key equations of the concentration profile
around the root as well as the average concentration for use in a steady state model.
[20]
where C
Lr
is the nutrient concentration in soil solution at the distance r from the root.
By representing the average concentration across the depletion zone with C
av
, the relationship
between C
av
and C
L0
can be obtained by
[21]
Because the amount of nutrient uptake during the time period is given by
[22]
can be calculate by equation [23] and [24]. That is
[23]
At time interval , C
av1
= C
Li
. Therefore the new C
av2
can be calculated from the uptake at
time interval , and C
La2
can be calculated from C
av2
. This approach allows any time period to
be represented (Tinker and Nye 2000). Finally, the total amount of nutrient uptake can be
obtained by summing at each time interval.
16
Smethurst and Comerford (1993b) developed COMP8 based on the above equation and
another equation dealing with the competition between two root systems, with the improvements
that allow for (i)” a depletion zone that increases with time until it reaches the no-transfer
boundary, (ii) an adjustment in concentration to reflect newly encountered solutes in the
depletion zone, (iii) a variable root absorbing power α (see equation 15) to describe Michaelis-
Menten uptake kinetics, and (iv) a routine to account for the competition between two root
systems”. Two verification studies with slash pine and weeds were also conducted in the same
year (Smethurst and Comerford 1993a; Smethurst et al. 1993). However, this model has had little
use since 1993 because it was inadequate under some conditions possibly due to the inability of
the model to describe some components of the soil-root system such as root length development,
changing moisture contents, and the nutrient input from mineralization (Smethurst and
Comerford 1993b).
Based on COMP8, SSAND included several new functions allowing simulation of nutrient
uptake by roots under a variety of conditions such as mycorrhizal roots, fertilization, changing
soil water content, nutrients from different soil horizons, and dynamic mineralization rates
(Comerford et al. 2006). It includes sub-routines to calculate the nutrient demand for a target
plant growth, so that it can provide a recommendation on fertilization by comparing the
predicted uptake and demand.
Similar to SSAND, PCATS is able to simulate nutrient uptake by mycorrhizae as well as
uptake influenced by fertilization (Smethurst et al. 2004). However, it can only predict nutrient
uptake by one species. It also uses an analytical solution similar to COMP8 and SSAND, but
further simplifies the calculation by running on a fixed daily time-step (Smethurst et al. 2004).
Assumptions for the mechanistic models
A number of specific assumptions underlie the models.
The soil is homogeneous and isotropic (Rengel 1993).
Nutrients move to the root by a combination of mass flow and diffusion (Barber
1995).
Roots are smooth cylinders and the nutrient absorbing power is the same over all the
cylinders (Barber 1995).
17
Mycorrhizae, root hairs, root exudates, or microbial activity on the root surface do not
influence nutrient flux (Barber 1995).
Nutrient uptake can be described by Michaelis-Menten kinetics and the kinetics
parameters do not change over time (Barber 1995).
Influx is independent of the rate of water absorption (Barber 1995)
For NST 3.0, the soil moisture is essentially constant (Barber 1995). The roots are
distributed evenly in the whole soil volume, and no allowance is given for a changing
distance among roots as roots grow (Claassen and Steingrobe 1999); A root segment
can exploit only a limited volume of soil, and the root is at the center of this cylinder
(Claassen and Steingrobe 1999).
For SSAND and PCATS, Roots are assumed to be parallel and distributed regularly
throughout the soil volume (Baldwin 1973).
Comparison of the two major categories of mechanistic models
Usually transient state models with a numerical solution are considered to be more accurate
than steady state models (Smethurst and Comerford 1993b; Yanai 1994). Numerical methods are
those that “iteratively solve a system of simultaneous equations developed from approximations
of the differential equation for solute transport” (Smethurst and Comerford 1993b). “The steady
state approach assumes that the concentration profile around the root can be considered to be in a
steady state” (Yanai 1994), which is usually attained after long periods of time (Nye and Spiers
1964). Since natural phenomena are transient, the results obtained using this approach may not
be as accurate as the transient state model.
The major advantage of a steady state model over a transient state model lies in “the
independence of the mathematical solution to previous condition” during calculation (Yanai
1994) so that steady state models are able to respond to real-time changes in parameters.
Transient state models using a numerical solution are also called “deterministic” models
(Claassen and Steingrobe 1999) because the simulation does not accept time-varying input
(Yanai 1994). For example, soil moisture is not allowed to change during the calculation, though
it is unrealistic to think that soil moisture would remain constant in the field.
18
Current study on NST 3.0, SSAND, and PCATS
As a transient model utilizing a numerical solution, NST 3.0 has been used widely to predict
uptake of nutrients by various crop species (Sadana and Claassen 2000; Satnam and Sadana
2002; Samal et al. 2003; Sadana et al. 2005; Pypers et al. 2006). However, no studies on woody
species utilizing this model have been conducted.
Smethurst and Comerford (1993b) verified the ability of COMP8 to predict a one-species
scenario by comparing its responses with those of the NUTRIENT UTPAKE model. They also
published the verification of COMP8 by predicting potassium and phosphorus uptake by slash
pine in competition with weeds using both pot and field studies (Smethurst and Comerford
1993a; Smethurst et al. 1993). In the pot study, the model provided a good representation of
potassium and phosphorus uptake by pines and weeds under high nutrient concentration
treatments, but performed poorly under low nutrient concentration treatments (Smethurst and
Comerford 1993a). In the field study, the model predicted the phosphorus and potassium uptake
by pines adequately, but significantly over predicted uptake of both nutrients by weeds
(Smethurst et al. 1993). Ibrikci et al. (1994) studied phosphorus uptake by Bahiagrass with
COMP8. The prediction matched the observation for 18-day-old plants growing in Ap, E, and Bh
horizons, but a low level of agreement was found at 90 days for plants growing in Ap and E
horizon soils, both of which were known to be of low phosphorus supply (Ibrikci et al. 1994).
Four years later, Ibrikci and his colleagues (1998) found that the nitrogen uptake predicted for
COMP8 was 38% to 44% lower than that observed by field-grown corn (Ibrikci et al. 1998). The
potential error of adopting an I
max
value from the literature and COMP8’s inability to include
contributions from root hairs and nitrogen mineralization were suggested by Ibrikci et al. (1994)
as the reasons for their underestimates.
SSAND was published in 2000 (Li and Comerford 2000); the authors verified it by
predicting phosphorus uptake by 1-yr loblolly pine growing in southeastern Georgia and Florida
(Comerford et al. 2006). Borges-Gómez et al. (2008) used SSAND to predict potassium
requirement by habanero pepper in Yucatán, Mexico. But they used SSAND as a tool for
management and did not compare the model predictions to any observations of uptake. Singh
(2008) investigated the uptake of nitrogen, phosphorus, and potassium by hybrid poplar and
weeds in both the lab and field using SSAND. The disagreement between the simulated and
19
observed uptake was significant except for the simulation of nitrogen uptake by hybrid poplar in
the control field site (Singh 2008). Only one paper is available for the model PCATS. The
authors verified the model by comparing the PCATS estimates of uptake to these provided by
NUTRIENT UPTAKE model (Smethurst et al. 2004).
To summarize, the differences between NST 3.0, SSAND, and PCATS are not well
understood in terms of predictive accuracy and model behavior. The different solution methods
used in the models may lead to different simulations. Reported research to date related to the use
of SSAND and PCATS is limited. Agreement between the predictions of their predecessor
model, COMP8, and observations was not good under some situations such as low nutrient
supply scenarios (Smethurst and Comerford 1993a). Although NST 3.0 performed well with crop
species, no tests have been done with woody species. Because no study has been conducted to
compare the efficacy of the three models, it would be beneficial to evaluate the three models by
comparing the predictions with observations using multiple datasets representing various
situations taken from the literature.
Sensitivity Analysis of Mechanistic Nutrient Uptake Models
The traditional sensitivity analysis can also be called a one dimensional sensitivity analysis.
This approach was first used to by Silberbush and Barber in 1983 to show the influence of
different parameters on model simulations. It was accomplished by changing a single model
input parameter while keeping the others constant (Silberbush and Barber 1983). By plotting the
change ratio of the parameters compared to their original values on the horizontal axis, and the
predicted uptake to the original prediction on the vertical axis, it is possible to evaluate the
importance of each parameter by the slope of the line on the graph. This procedure has been
largely followed in all subsequent analysis. Although Claassen and Steingrobe (1999) proposed
to treat initial soil solution concentration (C
Li
) and buffer power (b) together in the analysis
because the two parameters are related to each other, the principle is the same.
Yanai proposed the idea of multiple dimensional sensitivity analysis in 1994 and did several
two dimensional sensitivity analyses. Two years later Williams and Yanai conducted a multiple
dimensional sensitivity analysis using a simplified steady state model to simulate nutrient uptake
by a unit length of root (Williams and Yanai 1996). Because the model they used did not take
root growth and inter-root competition into account, it contained 7 parameters, while a typical
20
mechanistic nutrient uptake model has 11 parameters. By changing each parameter at 4 levels,
16,384 parameter datasets were created. Using ANOVA, they found up to four-way interactions
among the parameters with C
av
, I
max
, D
e
, and r
0
exerting the most significant influences on the
simulations (Williams and Yanai 1996). They then represented each of 5 parameters (these 4
parameters plus water influx into roots) at 5 levels and plotted the response surface using graphs
(Williams and Yanai 1996). The response surfaces developed using this method indicated clear
relationships between the selected parameters. For example, the most influential parameters
depend on the concentration of nutrients supplied by soil processes (Williams and Yanai 1996).
When the concentration is low, the soil parameters dominate the uptake process. When the
concentration is high, the parameters representing the ability of the plant root to take up
nutrients, such as I
max
and r
0
, will determine how much nutrient is taken up (Williams and Yanai
1996).
Therefore, the multiple dimensional sensitivity analysis is a powerful tool in helping us
understand model behavior under various situations. However, since the simplified mechanistic
model used by Williams and Yanai (1996) did not include inter-root competition and root
growth, using this method to analyze NST 3.0, SSAND, and PCATS with the key parameters
C
av
, I
max
, D
e
, r
0
, plus the parameters of root growth and inter-root competition would provide a
more complete evaluation of the model behaviors and explore the potential differences between
the three models more thoroughly.
Nutrient Uptake Modeling of Woody Species
Gillespie and Pope (1990) used the Barber-Cushman model combined with a model on
rhizosphere acidification to study phosphorus uptake by black locust. It is the first study to
employ a minimalistic mechanistic nutrient uptake model to study a woody species. Gillespie
and Pope (1990) found that the predictions matched the observations well if the influence of
rhizosphere acidification was considered. In the same year, Van Rees et al. (1990) reported a
study on potassium uptake by slash pine with both the Barber-Cushman and the Baldwin-Nye-
Tinker models. The Baldwin-Nye-Tinker model (BNTM) was developed by Baldwin et al. in
1973 and became the foundation for COMP8 in 1993. Van Rees et al. (1990) found that the
prediction by BNTM was 5% higher than that by the Barber-Cushman model and that both
models only worked well under situations treated with fertilizers.
21
In collaboration with S. A. Barber, Kelly and Barber (1991) reported the magnesium uptake
kinetics parameters of loblolly pine as well as the influence of seedling age on these parameters.
In the following year, Kelly et al. (1992) verified the use of the Barber-Cushman model on
phosphorus and potassium uptake by loblolly pine seedlings, but the simulated uptake of
magnesium was substantially underestimated. From the mid 1990s to the early 2000s, Kelly and
his coworkers used NUTRIENT UPTAKE model to study the uptake of several macro-nutrients
by various tree species under different growing treatments (Kelly et al. 1995; Kelly et al. 2000;
Kelly and Kelly 2001; Kelly et al. 2001; Adam et al. 2003; Kelly and Ericsson 2003). The
research largely focused on the influence of environmental factors and plant growth on
parameter values, such as the seasonal dynamics of soil supply capacities (Kelly et al. 1995) and
the uneven growth of roots during a growing season (Kelly et al. 2001). Some limitations of
NUTRIENT UPTAKE model were also pointed out, such as the fixed root growth rate (Kelly et
al. 2001) and the inability to include the contributions of decomposition and mineralization
during the simulation (Kelly and Ericsson 2003). Adam et al. (2003) also used these parameters
and the established methodology as tools to study the influence of temperature on uptake kinetics
(Adam et al. 2003).
Unlike the Barber-Cushman model, the steady state model has been constantly revised since
1990s, and most of the studies were conducted with woody species. Because COMP8, Yanai’s
model in 1994, SSAND, and PCATS are important achievements in the development of the
steady state model, and they are discussed in the section discussing the current studies of
SSAND and PCATS, these studies are not repeated here. Yanai (2003) used the steady state
mechanistic model she proposed in 1994 to calculate the nutrient concentration differences
between rhizosphere and bulk soil in a Norway spruce stand. She found that in contrast to the
model calculation, the observed nutrient concentrations in the rhizosphere were generally higher
than those in bulk soil (Yanai et al. 2003).
Representative ranges for parameters needed conduct a multiple dimensional sensitivity
analysis are presented in Tables 2, 3, and 4 and list values from the literature for nutrient uptake
kinetics, soil supply, and root growth parameter.
22
Table 2. Nutrient uptake kinetics parameters for a variety of woody species taken from the literature.
Nutrient Species Common name I
max
µmol cm
2
s
-1
K
m
µmol cm
-3
C
min
µmol cm
-3
Source
NH
4
-N Populus sp. hybrid poplar 0.000004 0.049 0.001 (Singh 2008)
Picea glauca white spruce 0.0000207 0.20568 (Hangs et al. 2003)
Populus tremuloides aspen 0.00001254 0.21712 (Hangs et al. 2003)
NO
3
-N Populus sp. hybrid poplar 0.0000034-0.0000285 0.093-0.712 0.001 (Kelly and Ericsson 2003)
Acer rubrum red maple 0.0000157-0.00005908 0.204-0.523 0.001 (Kelly et al. 2000)
Acer rubrum red maple 0.0000195-0.0000318 0.000088-0.00019 0.000018-0.000066 (Adam et al. 2003)
Acer rubum red maple 0.0000309 0.32 0.001 (Kelly et al. 2001)
Picea glauca white spruce 0.0000045 0.34451 (Hangs et al. 2003)
Populus tremuloides Aspen 0.00000581 0.3365 (Hangs et al. 2003)
K Populus sp. hybrid poplar 0.0000176 0.0269 0.003 (Kelly and Ericsson 2003)
Populus sp. hybrid poplar 0.00000266 0.034 0.001 (Singh 2008)
Acer rubrum red maple 0.0000038 10.46 0.003 (Kelly and Kelly 2001)
Pinus elliottii var. elliottii slash pine 0.00000361 0.029 0.001 (Van Rees et al. 1990)
Pinus elliottii var. elliottii slash pine 0.00000361 0.029 0.001 (Van Rees and Comerford 1990)
Pinus taeda loblolly pine 0.0000014 0.03 0.001 (Kelly et al. 1992)
Pinus taeda loblolly pine 0.00000365 0.0237 0.0002 (Kelly et al. 1995)
P Populus sp. hybrid poplar 0.00000151 0.00087 0.001 (Kelly and Ericsson 2003)
Populus sp. hybrid poplar 0.00000113 0.038 0.001 (Singh 2008)
Acer rubrum red maple 0.00000549 15.02 0.001 (Kelly and Kelly 2001)
Pinus elliottii var. elliottii slash pine 0.000000643 0.00545 0.0002 (Smethurst and Comerford 1993a)
Pinus taeda loblolly pine 0.000000268 0.016 0.0006 (Kelly et al. 1992)
Pinus taeda loblolly pine 0.00000064 0.00545 0 (Comerford et al. 2006)
Robinia pseudoacacia black locust 0.0000017 0.0018 0.0007 (Gillespie and Pope 1990)
Mg Pinus taeda loblolly pine 0.000000112 0.00858 0.001 (Kelly and Barber 1991)
Pinus taeda loblolly pine 0.000000079 0.00869 0.001 (Kelly and Barber 1991)
Pinus taeda loblolly pine 0.000000129 0.00983 0.001 (Kelly and Barber 1991)
Pinus taeda loblolly pine 0.000000129 0.00983 0.001 (Kelly et al. 1992)
23
Table 3. Soil supply parameters from studies of a variety of woody species.
Nutrient C
Li
b D
e
Plant name Source
µmol ml
-1
cm
2
s
-1
NH
4
-N 0.0037-0.714
hybrid poplar (Singh 2008)
0.001-0.075 4.9-209.3 0.000000015-0.00000131 multiple species (Kelly and Mays 1999)
0.052-0.215
0.0000064-0.000015 Norway spruce (Yanai et al. 2003)
NO
3
-N 0.0761-2.31 1.14-1.98 0.000000684-0.000113 hybrid poplar (Kelly and Ericsson 2003)
0.047-10.714 hybrid poplar (Singh 2008)
0.002-2.33 0.04-69.09 0.000000191-0.0002 multiple species (Kelly and Mays 1999)
0.51 1.35 0.0000632 red maple (Kelly et al. 2001)
K 0.124-0.132
cottonwood (Wang et al. 2004)
0.0963-0.519 1.4-5.095 0.000000263-0.000000994 hybrid poplar (Kelly and Ericsson 2003)
1.432-3.529 hybrid poplar (Singh 2008)
0.27 10.55 0.00000329 loblolly pine (Kelly et al. 1992)
0.99-8.54 0.16-1.19 0.000000664-0.00000635 loblolly pine (Kelly et al. 1995)
0.018-0.561 multiple species (Kelly and Mays 1999)
0.028-0.142 0.0000048-0.000017 Norway spruce (Yanai et al. 2003)
0.112 20 0.000000358 red maple (Kelly and Kelly 2001)
0.13 7.02 0.00000171 red maple (Kelly et al. 2001)
0.035-0.06 3.00-3.44 0.000000291-0.000000628 slash pine (Van Rees et al. 1990)
0.093-0.203 3.64-6.26 0.00000002-0.000000104 slash pine (Van Rees and Comerford 1990)
0.0382-0.338 0.55-2.13 0.00000019-0.00000128 slash pine (Smethurst and Comerford 1993a)
0-0.36 0.7-4.3
slash pine (Smethurst et al. 1993)
P 0.00161-0.00484 black locust (Gillespie and Pope 1990)
0.0287-0.0339 cottonwood (Wang et al. 2004)
0.0021-0.0388 15.41-180.5 0.000000003-0.000000041 hybrid poplar (Kelly and Ericsson 2003)
0.016 hybrid poplar (Singh 2008)
0.19 5.84 0.000000817 loblolly pine (Kelly et al. 1992)
0.0032
loblolly pine (Comerford et al. 2006)
24
Table 3. (Continued)
Nutrient C
Li
b D
e
Plant name Source
µmol ml
-1
cm
2
s
-1
P
0.001-0.01 84.5-1451.5 0.0000000091-0.000000029 multiple species (Kelly and Mays 1999)
0.001 199 0.0000000203 red maple (Kelly and Kelly 2001)
0.001 166.8 0.0000000143 red maple (Kelly et al. 2001)
0.0226-1.21 0.23-0.85 0.00000019-0.0000013 slash pine (Smethurst and Comerford 1993a)
0-0.29 0.7-1520 slash pine (Smethurst et al. 1993)
Ca 0.644-0.706
cottonwood (Wang et al. 2004)
0.019-0.599 multiple species (Kelly and Mays 1999)
0.004-0.009
0.000019-0.000067 Norway spruce (Yanai et al. 2003)
Mg 0.560-0.609 cottonwood (Wang et al. 2004)
1.35 1.32 0.000000145 loblolly pine (Kelly et al. 1992)
0.012-0.37 multiple species (Kelly and Mays 1999)
0.018-0.047 0.0000017-0.000006 Norway spruce (Yanai et al. 2003)
S 0.028-0.051 0.0000026-0.000009 Norway spruce (Yanai et al. 2003)
25
Table 4. Root growth parameters from a variety of woody species and the mean water flux at the root surface taken from the literature.
Plant name L
0
L
v
r
0
r
1
k v
0
Source
cm cm cm
-3
cm cm cm s
-1
cm s
-1
hybrid poplar 1094
0.02 0.573-0.664 0.000347-
0.00113 0.000000746 (Kelly and Ericsson 2003)
hybrid poplar
0.001-11.88 0.01-0.03
0.00000195 (Singh 2008)
box elder
0.0675
(Comas et al. 2002)
red maple
0.035-0.0397
(Adam et al. 2003)
red maple 3842-16179
0.028-0.042
(Kelly et al. 2000)
red maple 1696
0.044 0.99 0.000475 0.00000517 (Kelly et al. 2001)
sugar maple
0.0685
(Comas et al. 2002)
white spruce
0.041
(Hangs et al. 2003)
Norway spruce
0.05 0.36-0.55
0.0000016-0.0000021 (Yanai et al. 2003)
jack pine
0.039
(Hangs et al. 2003)
Eastern hemlock
0.0783
(Comas et al. 2002)
slash pine 16.6-31.6
0.036-0.06 1.39-3.15 0.000012-
0.000184 0.00000379 (Van Rees et al. 1990)
slash pine
0.1-0.3 0.0234-0.0932
0.0000032 (Smethurst and Comerford
1993a)
slash pine 11.9-41.3
0.027-0.046 0.62-2.59 0.000064-
0.000455
0.00000223-
0.00001035
(Van Rees and Comerford
1990)
loblolly pine 10-517.3
0.043-0.052 2.47-6.02 0.00004-0.00013 0.000000566 (Kelly et al. 1995)
loblolly pine 285
0.035 2 0.000162 0.000000566 (Kelly et al. 1992)
loblolly pine
0.4 0.04
0.000002 (Comerford et al. 2006)
Virginia pine
0.0741
(Comas et al. 2002)
cottonwood 301.1
(Wang et al. 2004)
aspen
0.043
(Hangs et al. 2003)
white oak
0.0504
(Comas et al. 2002)
red oak
0.0605
(Comas et al. 2002)
black locust 0.941 0.000000001 (Gillespie and Pope 1990)
26
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31
Chapter III. NUTRIENT UPTAKE ESTIMATES FOR WOODY SPECIES AS
DESCRIBED BY THE NST 3.0, SSAND, AND PCATS MECHANISTIC NUTRIENT
UPTAKE MODELS
Abstract
Mechanistic nutrient uptake models have been developed based on the principles of nutrient
movement in the soil, nutrient uptake kinetics, and root growth and morphology. There are two
major categories of mechanistic models, transient state models with a numerical solution and
steady state models. NST 3.0 belongs to the former set of models, while SSAND and PCATS
belong to the latter. Because the different approaches to solution between transient and steady
state models may lead to different simulation results, and studies using the three models to assess
woody species are limited, it would be useful to evaluate the three models by comparing model
predictions based on common data sets taken from the literature against the observed
measurements. Therefore, the objectives of this study are to: (i) compare the predictions of
uptake by the NST 3.0, SSAND, and PCATS models for a suite of nutrients against
experimentally measured values, (ii) compare the behavior of the three models using a one
dimensional sensitivity analysis; and (iii) compare and contrast the behavior of NST 3.0 and
SSAND using a multiple dimensional sensitivity analysis. The predictions of nutrient uptake by
the three models using the same data set were diverse, indicating a need to reexamine model
structure. Further studies using representative datasets are also needed so that the predictive
accuracy of each model can be evaluated. Both types of sensitivity analyses suggested that the
effect of soil moisture on simulation can be influential when nutrient concentration in the soil
solution (C
Li
) is low. One dimensional sensitivity analysis also revealed that I
max
negatively
influenced estimates of nutrient uptake in the SSAND and PCATS models. Further analysis
indicated that this phenomenon was related to the low nutrient supply ability of the soils. The
predictions of SSAND under low-nutrient-supply scenarios are generally lower than these of
NST 3.0 (the greatest underestimate was 100%). We suspect that both of these results are
artifacts of the steady state models and further studies to improve them are required. Modeling
32
the influence of rhizospheric effects warrants more emphasis because of the longer growth
periods and low nutrient supply situations more typical related to woody species.
Introduction
Mechanistic modeling of nutrient uptake started in the early 1960s when Bouldin (1961) and
Olsen et al. (1962) proposed mathematical models to simulate the diffusion of solutes through
soils. Models of plant uptake based on the seminal efforts by Nye and Spiers (1964), Nye and
Marriot (1969), and Baldwin et al. (1973), and advanced by the subsequent work of Claassen and
Barber (1976), Nye and Tinker (1977), Barber and Cushman (1981), Claassen and Steingrobe
(1999), and Steingrobe et al. (2000), are becoming important research tools. Mechanistic
modeling took a significant step forward with advent of the personal computer as reflected in the
work of Oates and Barber (1987), Smethurst and Comerford (1993b), Yanai (1994), Smethurst et
al. (2004), and Comerford et al. (2006). Usually, these models consist of three basic components
represented by a set of equations that describe (Rengel 1993): (i) solute movement in the soil
toward plant roots using the continuity equation that expresses “the change in mass of a
substance in a small volume over a small time” (Tinker and Nye 2000); (ii) nutrient uptake
kinetics using the Michaelis-Menten equation; and (iii) nutrient uptake as a result of root growth
and inter-root competition through the use of root growth and morphology parameters.
Depending on the way in which the continuity equation is solved, two major categories of
models have evolved, steady state or transient state models (Tinker and Nye 2000). NST 3.0 is
an example of a transient state model with a numerical solution, while SSAND and PCATS are
steady state models.
Research on several crop species has demonstrated that the predictions of these mechanistic
models generally match the observed uptake under situations in which the soil and plant
conditions are relatively constant (Barber 1995). However, results with woody species have been
more variable (Gillespie and Pope 1990; Van Rees et al. 1990; Smethurst and Comerford 1993b).
One way to evaluate the models is by comparing model predictions with measured values. A
second approach is the use of sensitivity analysis. The traditional way to conduct a sensitivity
analysis in the modeling context is by changing a single input parameter in the model while
keeping the others constant (Silberbush and Barber 1983). Williams and Yanai (1996) suggested
this method be called a one dimensional sensitivity analysis. However, because “the relative
33
importance of parameters defined this way can depend strongly on the values of the other
parameters”, Williams and Yanai (1996) developed a multiple dimensional sensitivity analysis in
order to study model behavior across a broad range of possible parameter values.
Differences between NST 3.0, SSAND, and PCATS are not well understood in terms of
predictive accuracy and model behavior. The different development methods used in the models
may lead to different simulations. Research to date related to the use of SSAND and PCATS is
limited. Agreement between the uptake predictions of their predecessor model, COMP8, and
observed values were not good in low nutrient supply scenarios (Smethurst and Comerford
1993a). Although NST 3.0 has performed well with crop species, no tests have been done with
woody species. Because no study has been conducted to compare the efficacy of the three
models, it would be beneficial to evaluate the three models by comparing model predictions with
nutrient uptake observations using multiple datasets representing various situations taken from
the literature and by conducting sensitivity analyses.
Therefore, the objectives of this study are to: (i) compare the predictions of uptake by the
NST 3.0, SSAND, and PCATS models for a suite of nutrients against experimentally measured
values, (ii) compare the behavior of the three models using a one dimensional sensitivity
analysis; and (iii) compare and contrast the behavior of NST 3.0 and SSAND using the multiple
dimensional sensitivity analysis approach of Williams and Yanai (1996).
Material and Methods
Basic principles and model assumptions
Mechanistic nutrient uptake models are very complex nonlinear models. They were
developed based on understanding from multiple disciplines including soil chemistry, soil
physics, and plant physiology and improved constantly by the advances in these fields. NST 3.0,
SSAND, and PCATS belong to two major categories of mechanistic models, and represent the
current state of the art. In order to make a successful comparison, it is necessary to review the
basic principles and assumptions of these mechanistic models. Table 5 lists the primary symbols
and parameters that are used in the mechanistic models described in this paper.
A number of both general and specific assumptions underlie the models.
The soil is homogeneous and isotropic (Rengel 1993).
34
Table 5. List of symbols and definitions used in the NST 3.0, SSAND, and PCATS mechanistic
nutrient uptake model equations.
Symbol Definition
b buffer power of nutrient
C nutrient concentration in soil
C
av
average nutrient concentration in depletion zone
C
L
nutrient concentration in soil solution
C
L
0
nutrient concentration in soil solution at the root surface
C
Li
initial concentration of the nutrient in the soil solution
C
min
concentration in solution below which net influx ceases
D diffusion coefficient of solute
D
e
effective diffusion coefficient for the nutrient in the soil
D
L
diffusion coefficient of solute in water
E nutrient efflux of plant roots
F flux of solute
F
D
flux of solute by diffusion
F
M
flux of solute by mass flow
f impedance factor or tortuosity factor
I net influx of solute
I
max
maximum net influx at high nutrient concentrations
k rate of root growth
K
m
Michaelis-Menten constant
L
0
initial root length
L
v
root length density
r radial distance in soil from the root surface
r
0
mean root radius
r
1
half-distance between root axes
t time
U the amount of nutrient uptake by a root or root systems
v mean water influx
v
0
mean water influx at root surface
v
1
water influx at the distance r
1
α
root absorbing power
θ
volumetric soil moisture
35
Nutrients move to the root by a combination of mass flow and diffusion (Barber
1995).
Roots are smooth cylinders and the nutrient absorbing power is the same over all the
cylinders (Barber 1995).
Mycorrhizae, root hairs, root exudates, or microbial activity on the root surface do not
influence nutrient flux (Barber 1995).
Nutrient uptake can be described by Michaelis-Menten kinetics and the kinetics
parameters do not change over time (Barber 1995).
Influx is independent of the rate of water absorption (Barber 1995)
For NST 3.0, the soil moisture is essentially constant (Barber 1995). The roots are
distributed evenly in the whole soil volume, and no allowance is given for a changing
distance among roots as roots grow (Claassen and Steingrobe 1999); A root segment
can exploit only a limited volume of soil, and the root is at the center of this cylinder
(Claassen and Steingrobe 1999).
For SSAND and PCATS, Roots are assumed to be parallel and distributed regularly
throughout the soil volume (Baldwin 1973).
As a transient state model utilizing a numerical solution, NST 3.0 has been widely used to
predict nutrient uptake by various crop species (Sadana and Claassen 2000; Satnam and Sadana
2002; Samal et al. 2003; Sadana et al. 2005; Pypers et al. 2006). However, we are not aware of
any studies utilizing this model on a woody species. Transient state models utilizing a numerical
solution are a well established approach to mechanistic nutrient uptake models (Tinker and Nye
2000). The Barber-Cushman model is a well-known and widely-used model in this category.
NUTRIENT UPTAKE model and NST 1.0 are the personal computer version of the Barber-
Cushman model (Claassen et al. 1986; Oates and Barber 1987). Since NST 3.0 is an improved
version of NST 1.0, and all the data sets obtained from the literature for testing model predictive
accuracy were developed for use with NUTRIENT UPTAKE model, the principles of
NUTRIENT UPTAKE model and NST series are presented.
Nye and Spiers (1964) proposed the continuity equation that described the flux of nutrient in
the soil to the root surface:
where r is the radial distance from the root,
concentration in the soil solution, v
radius, b is the buffer power of the nutrient, and t is time. Nye and Marriot (1969)
boundary conditions and
solved this equation numerically.
Building on the work of Nye and Marriot (1969), Claassen and Barber (1976), and Cus
(1979a; 1979b)
, Barber and Cushman (1981) revised the
include inter-
root competition for nutrients
(1) Inner boundary condition:
Where F is the flux of the nutrient,
high nutrient concentrations, E is the nutrient efflux of plant roots, and
tran
sformation of the Michaelis
(2) Outer boundary condition:
If there is no inter-
root competition,
If there is inter-
root competition,
where v
1
is the water influx at the distance r
The boundary conditions incorporated inter
When solved numerically, the enhanced mechanistic model evolved into the Barber
model.
In 1983, Itoh and Barber made
simulate nutrient uptake by root hairs. In 1987, Oates and Barber published
UPTAKE model
, which is a personal computer version of
revision by Itoh and Ba
rber (1983).
based largely on the Barber-
Cushman model
2.0 and NST 3.0, which were not published in a journal (Claassen, N. Personal communication.
36
where r is the radial distance from the root,
D is the diffusion coefficient,
C
concentration in the soil solution, v
0
is the water flux at the root surface, r
0
radius, b is the buffer power of the nutrient, and t is time. Nye and Marriot (1969)
solved this equation numerically.
Building on the work of Nye and Marriot (1969), Claassen and Barber (1976), and Cus
, Barber and Cushman (1981) revised the
boundary conditions of
root competition for nutrients
under both inner and outer boundary conditions
(1) Inner boundary condition:
Where F is the flux of the nutrient, α is the root absorbing power, I
max
is maximum net influx at
high nutrient concentrations, E is the nutrient efflux of plant roots, and
sformation of the Michaelis
-Menten equation.
(2) Outer boundary condition:
root competition,
root competition,
is the water influx at the distance r
1
.
The boundary conditions incorporated inter
-root competition and
Michaelis
When solved numerically, the enhanced mechanistic model evolved into the Barber
In 1983, Itoh and Barber made
a change to the Barber-
Cushman model so that it is able to
simulate nutrient uptake by root hairs. In 1987, Oates and Barber published
, which is a personal computer version of
the Barber-
Cushman model with the
rber (1983).
In 1986 Claassen et al. published
NST 1
Cushman model
.
Later Claassen and his colleagues developed NST
2.0 and NST 3.0, which were not published in a journal (Claassen, N. Personal communication.
[1]
C
L
is the nutrient
is the average root
radius, b is the buffer power of the nutrient, and t is time. Nye and Marriot (1969)
defined
Building on the work of Nye and Marriot (1969), Claassen and Barber (1976), and Cus
hman
boundary conditions of
equation 1 to
under both inner and outer boundary conditions
:
[2]
is maximum net influx at
is a
[3]
[4]
Michaelis
-Menten kinetics.
When solved numerically, the enhanced mechanistic model evolved into the Barber
-Cushman
Cushman model so that it is able to
simulate nutrient uptake by root hairs. In 1987, Oates and Barber published
NUTRIENT
Cushman model with the
NST 1
.0 model, which was
Later Claassen and his colleagues developed NST
2.0 and NST 3.0, which were not published in a journal (Claassen, N. Personal communication.
37
2009, 31 July). NST 3.0 incorporates the Freundlich isotherm into the model so that the buffer
power (b) changes as the nutrient concentration in soil solution changes (Steingrobe et al. 2000).
Steady state models are another major standard category in mechanistic nutrient uptake
modeling (Tinker and Nye 2000). Baldwin et al. (1973) and Nye and Tinker (1977) proposed the
key equations. Based on their work, Smethurst and Comerford (1993b) developed a model,
COMP8 (Competition model version 8), which was able to calculate nutrient uptake between
two competing and contrasting root systems. SSAND is a revision and expansion of COMP8 by
Comerford et al. (2006). Its main improvements lie in the functions of predicting nutrient uptake
as influenced by mycorrhizae and simulation of fertilization effects (Comerford et al. 2006).
Based on COMP8 and an earlier version of SSAND, another steady state model, PCATS was
developed by Smethurst et al. (2004) to simulate nutrient uptake by a single species. In the
following section, the principles of steady state models are briefly described, and the features of
COMP8, SSAND and PCATS are introduced.
Based on the continuity equation of Nye and Spiers (1964), Baldwin et al. (1973) and Nye
and Tinker (1977) proposed the key equations for the concentration profile around the root, as
well as the average concentration for use in a steady state model.
[5]
Where C
Lr
is the nutrient concentration in soil solution at the distance r from the root, and C
L0
is
the nutrient concentration in soil solution at the root surface.
By representing the average concentration across the depletion zone with C
av
, the relationship
between C
av
and C
L0
can be obtained by
[6]
Because the amount of nutrient uptake during the time period is given by the equation:
[7]
can be calculated using equations 6 and 7:
38
[8]
At time interval
, C
av1
= C
Li
. Therefore the new C
av2
can be calculated from the uptake at
time interval
, and C
L02
can be calculated from C
av2
. This approach allows any time period to
be represented. Finally, the total amount of nutrient uptake can be obtained by summing at
each time interval.
Another key equation which Smethurst and Comerford (1993b) based the development of
COMP8 on is not discussed here because it is related to nutrient uptake by two competing roots.
Smethurst and Comerford (1993b) also made the following improvements: (i) “a depletion zone
that increased with time until it reached the no-transfer boundary, (ii) an adjustment in
concentration to reflect newly encountered solutes in the depletion zone, (iii) a variable root
absorbing power
α
(equation 9) to describe Michaelis-Menten uptake kinetics, and (iv) a routine
to account for the competition between two root systems”.
[9]
Two verification studies with slash pine and weeds were conducted in the same year
(Smethurst and Comerford 1993a; Smethurst et al. 1993). However, this model has had little use
since 1993 because it was judged to be inadequate under some conditions possibly due to its
inability to describe some components of the soil-root system such as root length development,
changing soil moisture content, and nutrient input from mineralization (Smethurst and
Comerford 1993b; Singh 2008).
Based on COMP8, SSAND included several new functions allowing simulation of nutrient
uptake by roots under a variety of conditions such as mycorrhizal roots, fertilization, changing
soil water content, nutrients from different soil horizons, and dynamic mineralization rates
(Comerford et al. 2006). SSAND includes a sub-routine to calculate the nutrient demand for
target plant growth, so that it can provide a recommendation on fertilization by comparing the
predicted uptake and demand.
Similar to SSAND, PCATS is able to simulate nutrient uptake by mycorrhizae and uptake
influenced by fertilization (Smethurst et al. 2004). However, it can only predict nutrient uptake
39
by one species. PCATS shares similar principles with COMP8 and SSAND, but further
simplifies the calculation by running on a fixed daily time-step (Smethurst et al. 2004).
Usually transient state models with a numerical solution are considered to be more accurate
than steady state models (Smethurst and Comerford 1993b; Yanai 1994). Numerical methods are
those that “iteratively solve a system of simultaneous equations developed from approximations
of the differential equation for solute transport” (Smethurst and Comerford 1993b). “The steady
state approach assumes that the concentration profile around the root can be considered to be in a
steady state” (Yanai 1994), which is usually attained after long times (Nye and Spiers 1964).
Since natural phenomena are transient, the results obtained using this approach may not be as
accurate as the transient state model.
The major advantage of a steady state model over a transient state model lies in “the
independence of the mathematical solution to previous condition” during calculation (Yanai
1994) so that steady state models are able to respond to real-time changes in parameters.
Transient state models using a numerical solution are also called “deterministic” models
(Claassen and Steingrobe 1999) because the simulation does not accept time-varying input
(Yanai 1994). For example, soil moisture is not allowed to change during the calculation, though
it is unrealistic to think that soil moisture would remain constant in the field.
Data Used for Comparison of NST 3.0, SSAND, and PCATS
Data were taken from three studies conducted by Kelly et al. (1992), Kelly et al. (2001), and
Kelly and Ericsson (2003). The study by Kelly et al. (1992) verified the use of the NUTRIENT
UPTAKE model on the simulation of phosphorus, potassium, and magnesium uptake by 1-0
loblolly pine seedlings growing in a modified A horizon soil (Lily series) for 180 days. Two sets
of data were taken from this study.
The study by Kelly et al. (2001) largely focused on the influence of environmental factors
and plant growth on parameter values used by NUTRIENT UPTAKE model. They measured the
model parameters of 1-year-old black cherry, northern red oak, and red maple seedlings in pots
with A horizon soils from two forest sites in the Great Smoky Mountains National Park in
Tennessee. Two sets of data on red maple grown in the Cove Mountain soil were taken from this
study. The study of phosphorus uptake was not included because of the authors did not detect
any phosphorus in the soil solution obtained.
40
Uptake of nitrate, phosphorus, and potassium by hybrid poplar cuttings was simulated for
either 30 or 105 days by Kelly and Ericsson (2003) using the NUTRIENT UPTAKE model. A
steady state technique to assure maximal relative growth rate at three fertilizer additions
(different amounts of 17-6-12 fertilizer addition equivalent to 0, 75, and 150 kg ha
-1
of nitrogen)
were included in the experiment. Nine data sets were taken from this study.
Data transformation
The inputs for NST 3.0, SSAND, and PCATS differ in some ways from those used for
NUTRIENT UPTAKE model. SSAND provides the user great flexibility to define various
scenarios. For example, the user of SSAND and PCATS can define more than one soil horizon
that the plants explore and can specify different parameters in each horizon. The number of
parameters also depends on the way the user collects data. Because all data sets obtained from
the studies of Kelly et al. (1992), Kelly et al. (2001), and Kelly and Ericsson (2003) were
developed for use with NUTRIENT UPTAKE model, data transformations were required.
Unlike NUTRIENT UPTAKE model which uses a fixed value of buffer power (b), all three
models allow users to input parameters to define a sorption equation so that b can be calculated
as the nutrient concentration in the soil solution changes as plant uptake occurs. This function is
slightly different among the models. NST 3.0 and PCATS allow the user to define only a
Freundlich equation for the absorption isotherm. SSAND uses different adsorption and
desorption isotherms, and allows the user to choose either a Freundlich or Langmuir equation to
fit these sorption isotherms. Because the studies of Kelly et al. (1992), Kelly et al. (2001), and
Kelly and Ericsson (2003) followed the methodology of Barber (1995) and the buffer power of
the soil is the nutrient concentration per unit soil volume (C) divided by the concentration in soil
solution (C
L
):
[10]
The isotherm equations for the three models were set as
[11]
Like NUTRIENT UPTAKE model, NST 3.0 requires the user to input a fixed value for the
tortuosity factor (f). SSAND provides several functions that the user can choose to define a
formula for calculating f with volumetric soil moisture
θ
. PCATS works similarly to SSAND in
this aspect, except that it provides only one slightly different function. The formulas used in our
41
simulation were first proposed by Kovar and Barber (1990). That is, if the soil is > 75% sand and
θ
< 0.15 m
3
m
-3
at field capacity,
[12]
Otherwise
[13]
When an absorption isotherm is introduced into the model, buffer power (b) is no longer
fixed and changes as C
Li
changes. The effective diffusion coefficient (D
e
) can also vary during
simulation because it is related to b. Since NST 3.0, SSAND, and PCATS require the user to
define sorption isotherms, D
e
is not an input parameter in these models and is replaced by
volumetric soil moisture (
θ
) and the diffusion coefficient in water of the nutrient (D
L
).
NST 3.0 uses the half distance between root axes (r
1
) at the end of the experiment rather than
the average r
1
over the experimental period. SSAND and PCATS do not require the user to input
r
1
. Instead, PCATS requires the input of root length density on a daily basis. SSAND can
calculate nutrient uptake by assuming no root growth occurred with a specified root length
density, or it can use an Excel file to input daily root length density using the same format as
PCATS.
Both SSAND and PCATS require user input on soil bulk density and soil volume. PCATS
further requires the user to define the percentage of soil volume that is occupied by roots.
Because of the lack of information in the literature, the soil volume was assigned as the pot size
in the SSAND simulation, and the percentage of the soil volume that was occupied by roots was
assumed to be 1. The same assumptions were used in the PCATS simulation in order to keep the
input the same as SSAND.
Since PCATS is designed specifically for predicting phosphorus uptake by plants, its
simulated results (U
P
) are calculated as µmols of phosphorus uptake during the time period.
Therefore, the actual simulated results (U) are transformed with the following formula:
[14]
The values for the parameters used in each model are listed in Tables 6 and 7, values that are
not listed are discussed below. For NST 3.0, three parameters (a, b, and c) are used to define the
sorption isotherm using the simplified isotherm equation 11, b equals to the buffer power, while
42
Table 6. Parameter values for loblolly pine and red maple based on observations reported by Kelly et al. (1992) and Kelly et al. (2001)
used for NST 3.0, SSAND and PCATS simulations.
Parameters Units
Loblolly pine (Kelly et al.
(1992)) Red maple (Kelly et al. (2001))
K P NO
3
-N K
D
L
†‡§
cm
2
s
-1
1.98E-05 8.90E-06 1.90E-05 1.98E-05
θ
†‡
unitless 0.062 0.44
v
0
( or LI rate)
†‡
cm s
-1
5.66E-07 5.17E-06
C
li
†‡
µmol ml
-1
0.27 0.19 0.51 0.13
b
†‡
unitless 10.55 5.84 1.35 7.02
I
max
†‡
µ mol cm
-2
s
-1
1.40E-06 2.68E-07 3.09E-05 3.80E-06
K
m
†‡
µmol ml
-1
0.03 0.016 0.32 10.46
C
min
†‡
µmol ml
-1
0.001 0.0006 0.001 0.003
r
0
†‡
cm 0.035 0.044
r
1
†
cm 1.65 0.903 1.186
L
0
†
cm 285 1696
k
†
cm day
-1
14 41.04
simulated time†‡ day 180 30 176
Soil volume
‡
cm
3
24000 7500
Bulk density
‡
g cm
-3
0.75 0.74
† parameters used by NST 3.0
‡
parameters used by SSAND and PCATS
§ The values of D
L
for nitrate and potassium were taken from Parsons (1959), and that of phosphorus from Edwards and Huffman
(1959).
43
Table 7. Parameter values for hybrid poplar based on observations reported by Kelly and Ericsson (2003) used for NST 3.0, SSAND
and PCATS simulations.
Parameters Units No fertilizer addition 75kg ha
-1
of fertilizer addition 150kg ha
-1
of fertilizer addition
NO
3
-N P K
NO
3
-N P K
NO
3
-N P K
D
L
†‡§
cm
2
s
-1
1.90E-05 8.90E-06 1.98E-05 1.90E-05 8.90E-06 1.98E-05 1.90E-05 8.90E-06 1.98E-05
θ
†‡
unitless
0.27
0.28
0.28
v
0
( or LI rate)
†‡
cm s
-1
7.46E-07
7.46E-07
7.46E-07
C
L
i
†‡
µmol ml
-1
0.0761 0.0021 0.0963 1.52 0.0086 0.217 2.31 0.0388 0.519
b
†‡
unitless 1.14 180.5 5.095 1.98 52.23 2.62 1.42 15.41 1.4
I
max
†‡
µmol cm
-
2
s
-1
2.85E-05 1.51E-06 1.76E-05 2.85E-05 1.51E-06 1.76E-05 2.85E-05 1.51E-06 1.76E-05
K
m
†‡
µmol ml
-1
0.712 0.00087 0.0269 0.712 0.00087 0.0269 0.712 0.00087 0.0269
C
min
†‡
µ mol ml
-1
0.001 0.001 0.003 0.001 0.001 0.003 0.001 0.001 0.003
r
0
†‡
cm 0.02
0.02
0.02
r
1
†
cm 0.42
0.26
0.26
L
0
†
cm 1094
779
976
k
†
cm day
-1
29.98
97.63
94.18
Simulation
time
†‡
day 105
105
105
Soil volume
‡
cm
3
2357
2357
2357
Bulk density
‡
g cm
-3
1.4 1.41 1.4
† parameters used by NST 3.0
‡
parameters used by SSAND and PCATS
§ The values of D
L
for nitrate and potassium were taken from Parsons (1959), and that of phosphorus from Edwards and Huffman
(1959).
44
a and c were set as 1 and 0 for every run. The number of compartments used for numerical
calculations was set to 40 for each run. For SSAND, the adsorption and desorption isotherm
were assumed to be the same. The Freundlich equation was chosen as the adsorption and
desorption type, and the same simplified isotherm equation was used. Equation 12 or 13 was
used to define the impedance factor (f), depending on volumetric soil moisture and soil texture.
For PCATS, the parameters for the sorption isotherm were set the same as in the other two
models. Because the unique function provided by PCATS for calculating impedance factor is
different, the value of the parameter of the impedance factor equation was adjusted so that the
calculated impedance factors of PCATS were the same as the others. For both SSAND and
PCATS, which require the user to define root length density day by day, the root length density
for the first day was calculated by dividing initial root length by soil volume. For subsequent
days the root length density increases linearly with the step value equal to the root growth rate
(cm day
-1
) divided by soil volume.
Methods of sensitivity analysis
The traditional or one dimensional sensitivity analysis was first used to by Silberbush and Barber
(1983) to show the influence of different parameters on model simulations. It was accomplished
by changing a single model input parameter while keeping the others constant (Silberbush and
Barber 1983). By plotting the change ratio of the parameters compared to their original values on
the horizontal axis, and the predicted uptake to the original prediction on the vertical axis, it is
possible to evaluate the importance of each parameter by the slope of the lines on the graph. This
procedure has been largely followed in subsequent analyses.
Following the methodology developed by Silberbush and Barber (1983), a one dimensional
sensitivity analysis was conducted using the data of Kelly et al. (1992) for potassium uptake by
loblolly pine seedlings. Each of the parameters was changed by a factor of 0.5, 0.75, 1.25, 1.5, or
2 times the original level while the remaining parameters were held constant. Those parameters
that are dependent on other parameters were not included in the sensitivity analysis. Claassen
and Steingrobe (1999) suggested doing sensitivity analysis with the C
Li
and b combined. Since
an isotherm equation for the Kelly et al. (1992) study was not available, and the simplified
relationship between C
L
and b (equation 11) is not theoretically accurate, the sensitivity analysis
45
was done using individual C
Li
and b. Therefore, ten parameters were evaluated in the one
dimensional sensitivity analysis of each model.
Yanai (1994) proposed the concept of multiple dimensional sensitivity analysis and
conducted several two dimensional sensitivity analyses using the steady state model developed
by Baldwin et al. (1973) and Nye and Tinker (1977). Williams and Yanai (1996) conducted a
multiple dimensional sensitivity analysis using a simplified steady state model across a range of
values described in the literature. By changing each parameter at 4 levels, 16,384 parameter
datasets were created. Using ANOVA, they identified the average nutrient concentration in the
soil solution (C
av
), maximal rate of nutrient uptake (I
max
), root radius (r
0
), and the effective
diffusion coefficient (D
e
), to be the most important parameters that control uptake per unit length
of root (Williams and Yanai 1996). Root radius r
0
becomes less important when uptake rates are
expressed on a surface basis (Williams and Yanai 1996). They then represented selected
parameters at 5 levels and plotted the response surface using a series of graphs. The response
surfaces developed using this method indicated clear relationships between the selected
parameters.
Because SSAND and PCATS share similar principles in modeling and response patterns in
the one dimensional sensitivity analysis, only SSAND and NST 3.0 were compared using the
multiple dimensional sensitivity analysis. Results obtained from the one dimensional sensitivity
analysis indicated that C
Li
,
θ
, k, r
0
, and v
0
were the most influential parameters. Williams and
Yanai (1996) found that C
Li
(or C
av
), D
e
, and I
max
were the most influential parameters in their
multiple dimensional sensitivity analysis with a simplified steady state model. Therefore, C
Li
and
θ
were chosen for further evaluation in the current study because of their prominent effects in the
one dimensional sensitivity analysis. Williams and Yanai (1996) did not take root growth (k) or
other root morphological parameters into account. But studies on modeling of nutrient uptake by
woody species (Van Rees et al. 1990; Smethurst and Comerford 1993b; Comerford et al. 2006;
Singh 2008), including our own one dimensional sensitivity analysis, showed that k was an
important parameter for nutrient uptake by woody species. Because all three models simulate
nutrient uptake on a surface area basis, plus the fact that the range of average root radius (r
0
) for
a species is narrow, r
0
was not included in the current multiple dimensional sensitivity analysis.
Although v
0
is among the most influential parameters in our one dimensional sensitivity analysis,
46
it is not included in the multiple dimensional sensitivity analysis because the uptake of potassium
is mainly by diffusion (Barber 1995), the importance of v
0
, a major parameter for mass flow, is
not as important. While Williams and Yanai (1996) defined I
max
as one of the most influential
parameters, the range of I
max
for loblolly and slash pine was narrow, from 1.4E-6 µmol cm
-2
s
-1
(Kelly et al. 1992) to 3.65E-6 µmol cm
-2
s
-1
(Kelly et al. 1995). Although its actual range may be
much wider, it is hard to artificially define a range for this parameter. Therefore, I
max
was
excluded in the multiple dimensional sensitivity analysis. As a result, C
Li
,
θ
, and k were selected
as the three parameters for evaluating the multiple dimensional sensitivity analysis.
The ranges of values for these parameters were taken from the literature. Because the number
of studies on measuring model parameters of loblolly pine is limited, data on slash pine were
included to build reasonably representative ranges of the three parameters. The values and
sources of the data are listed in Tables 8, 9, and 10. Based on these values, the range for
volumetric soil moisture (
θ
) was 0.06 to 0.4, the range for nutrient concentration in the soil
solution (C
Li
) was 0.028 µmol ml
-1
to 8.54 µmol ml
-1
, and the range for root growth rate (k) was
1 cm day
-1
to 39.3 cm day
-1
.
Each of the three parameters was varied linearly at five levels across the range, giving 125
“observations” in the data set for each of the two models. Using the obtained results, two
graphical representations similar to those of Williams and Yanai (1996) were developed to show
the relationship of the parameters and their influences on the simulations.
Results and Discussion
Calculation of uptake using data taken from the literature
Most model simulations underpredict nutrient uptake by 2% to 100% (Table 11). Five
simulations of potassium uptake were run with NST 3.0, SSAND, and PCATS (Figure 1). For
four out of the five simulations NST 3.0 and PCATS predicted 8 to 54% of the observed uptake.
The prediction of uptake by hybrid poplar under the no fertilizer treatment was 125% and 83%
for NST 3.0 and PCATS, respectively. SSAND predicted 7 to 50% of the observed uptake in all
simulations.
47
Table 8. Literature values for soil moisture (
θ
) taken from field, nursery, and greenhouse studies
of loblolly and slash pine.
Species and study type
θ
Data source
Slash pine, field study 0.24-0.3 (Van Rees et al. 1990)
Slash pine, nursery study 0.12-0.17 (Van Rees et al. 1990)
Slash pine, field study, soil depth measured: 0-10 cm 0.1-0.3 (Smethurst et al. 1993)
Slash pine, field study, soil depth measured: 10-26 cm 0.1-0.4 (Smethurst et al. 1993)
Slash pine, field study, soil depth measured: 26-50 cm 0.2-0.4 (Smethurst et al. 1993)
Slash pine, field study, soil depth measured: 50-70 cm 0.3-0.4 (Smethurst et al. 1993)
Slash pine, field study 0.17-0.23 (Smethurst and
Comerford 1993a)
Loblolly pine, field study 0.15 (Comerford et al. 2006)
Loblolly pine, greenhouse study 0.062 (Kelly et al. 1992)
Table 9. Literature values for potassium concentration in soil solution (C
Li
) taken from field,
nursery, and greenhouse studies of loblolly and slash pine.
Species C
Li
(µmol ml
-
1
) Data source
Loblolly pine 0.27 (Kelly et al. 1992)
Loblolly pine 0.99-8.54 (Kelly et al. 1995)
Slash pine 0.035-0.203 (Van Rees et al. 1990)
Slash pine 0.0382-0.338 (Smethurst and Comerford 1993a)
Slash pine 0-0.46 (Smethurst et al. 1993)
Table 10. Literature values of root growth rate (k) taken from field, nursery, and greenhouse
studies of loblolly and slash pine.
Species k (cm s
-
1
) Data source
Slash pine 0.000012-0.000184 (Van Rees et al. 1990)
Slash pine 0.000064-0.000455 (Van Rees and Comerford 1990)
Loblolly pine 0.00004-0.00013 (Kelly et al. 1995)
Loblolly pine 0.000162 (Kelly et al. 1992)
48
Four simulations of nitrate uptake were run with the three models (Figure 2). The
percentages for the model predictions divided by the observed uptake for each model is relatively
constant. NST 3.0 and SSAND underpredicted nitrate uptake by about 50%. PCATS predicted
nitrate uptake by hybrid poplar in the three fertilizer treatments at 98%, 98%, and 85% of the
observed. Red maple uptake of nitrate as described by PCATS was 61% of the observed uptake.
Four simulations of phosphorus uptake were run with each model (Figure 3). SSAND and
PCATS responded very similar and predicted 20% of the observed uptake in the loblolly pine
study, and less than 1% of the observed uptake in the hybrid poplar study. On the other hand,
NST 3.0 predicted 61% of the observed phosphorus uptake for the loblolly pine study. The
prediction by NST 3.0 of phosphorus uptake by hybrid poplar improved as the fertilizer addition
increased and the best prediction (110% of the observed uptake) occurred with the highest
fertilizer addition.
NST 3.0, SSAND, and PCATS produced diverse results using the same data (Table 11). For
nitrate uptake simulation, PCATS had the closest estimates to the observed uptake, while the
performance of NST 3.0 and SSAND are similar. For phosphorus uptake simulation, the
performance of NST 3.0 was greatly improved when applied to situations with fertilization, and
this was in accordance with the observation of Van Rees et al. (1990) that the Barber-Cushman
model worked well in the tree nursery when fertilizers were added. Both SSAND and PCATS
predicted phosphorus uptake of hybrid poplar to be less than 1 µmol, while the observed uptake
ranges from 52 µmol to more than 1000 µmol. No obvious pattern was observed in the
simulations of potassium uptake by the three models.
In an earlier model comparison study, Van Rees et al. (1990) evaluated potassium uptake by
slash pine seedlings growing in a greenhouse, a tree nursery, and the field using the Barber-
Cushman model and the Baldwin-Nye-Tinker model. The authors found that the simulated
uptake of the latter model was 5% higher than the former in all of their studies. As successors of
the Barber-Cushman and Baldwin-Nye-Tinker models, NST 3.0, SSAND, and PCATS produced
simulated results with a much lower level of agreement.
Although NST 3.0 and PCATS performed relatively well with some runs, there was not a
general pattern in the performance of the models in predicting major nutrient uptake in the three
studies of coniferous and deciduous species taken from the literature. For example, potassium is
49
Table 11. Observed uptake of NO
3
-N, P, and K compared to simulated uptake as predicted by
NST 3.0, SSAND, and PCATS using data from Kelly et al. (1992), Kelly et al. (2001), and
Kelly and Ericsson (2003). The percentages represent the degree of correspondence between the
simulated and observed uptake value.
Experiments
Nutrient
Observed
uptake
Simulated uptake (µmol)
(µmol)
NST 3.0
SSAND
PCATS
Loblolly pine (simulation time: 180 days)
K
6663 t
1020 aa
15%
466 aa
7%
510 aaaaat
8%
P
1332 t
596 aa
45%
265 aa
20%
270 aaaaat
20%
Red maple (simulation time for N: 30 days; for K: 176 days)
NO
3
-N
8810 t
4160 aa
47%
4277 aa
49%
5370 aaaaat
61%
K
1890 t
983 aa
52%
950 aa
50%
1016 aaaaat
54%
Hybrid poplar (simulation time: 105 days)
0 kg ha
-
1
of
fertilizer
addition
NO
3
-N
346 t
167aa
48%
216 aa
62%
309 aaaaat
89%
P
170 t
52.3i
31%
0 aa
0%
0.0004t
0%
K
719 t
900 aa
125%
158 aa
22%
600 aaaaat
83%
75 kg ha
-
1
of
fertilizer
addition
NO
3
-N
11000 t
6000 aa
55%
5411 aa
49%
10814 aaaat
98%
P
581 t
489 aa
84%
0.05
0%
0.001t
0%
K
4302 t
1090 aa
25%
662 aa
15%
1562 aaaat
36%
150 kg ha
-
1
of fertilizer
addition
NO
3
-N
14429 t
6590 aa
46%
6701 aia
46%
12235aaaaat
85%
P
940 t
1030 aa
110%
1.01
0%
0.0006
0%
K
5624 t
1380 aa
25%
1364 aa
24%
2467 aaaat
44%
50
Figure 1. Comparison of predicted potassium uptake by the NST 3.0, SSAND, and PCATS
models and observed potassium uptake using data from the studies by Kelly et al. (1992), Kelly
et al. (2001), and Kelly and Ericsson (2003).
Figure 2. Comparison of predicted nitrate uptake by the NST 3.0, SSAND, and PCATS models
and observed nitrate uptake using data from the studies by Kelly et al. (1992), Kelly et al. (2001),
and Kelly and Ericsson (2003).
0
500
1000
1500
2000
2500
3000
0 1000 2000 3000 4000 5000 6000 7000
Simulated uptake (µmol)
Observed uptake (µmol)
NST 3.0
SSAND
PCATS
1:1 line
0
2000
4000
6000
8000
10000
12000
14000
16000
0 2000 4000 6000 8000 10000 12000 14000 16000
Simulated uptake (µmol)
Observed uptake (µmol)
NST 3.0
SSAND
PCATS
1:1 line
51
Figure 3. Comparison of predicted phosphorus uptake by the NST 3.0, SSAND, and PCATS
models and observed phosphorus uptake using data from the studies by Kelly et al. (1992), Kelly
et al. (2001), and Kelly and Ericsson (2003).
0
200
400
600
800
1000
1200
0 200 400 600 800 1000 1200 1400
Simulated uptake (µmol)
Observed uptake (µmol)
NST 3.0
SSAND
PCATS
1:1 line
52
considered to be easier to model than other major nutrients because the components of the
potassium cycle in the soil are less complex than nitrogen and its reactions with soil particles or
root surfaces are simpler than phosphorus (Tinker and Nye 2000). But the ranges of the
percentages of the predicted against observed values by the three models (Table 11) were wide.
The fact that the best simulations of NST 3.0 and SSAND occurred in the hybrid poplar study
under the no fertilizer treatment contradicts the opinion that mechanistic nutrient models work
better under the conditions when soil supply of nutrients is abundant (Van Rees et al. 1990).
Therefore, our study indicates that the prediction of nutrient uptake by NST 3.0 SSAND, and
PCATS using the same dataset can be diverse, but it is unclear that why such differences
occurred. The lack of estimates on mycorrhizal uptake of nutrients may be an important reason
that most simulations underestimated nutrient uptake. Further studies producing representative
datasets that both reflect soil and plant characteristics of woody species scenarios are required.
Sensitivity analysis
One dimensional sensitivity analysis
SSAND and PCATS produce similar patterns in the one dimensional sensitivity analysis
(Figures 4 and 5). Listed in order of their importance, the most influential parameters were:
θ
>
C
Li
> k > v
0
> r
0
. The analysis also shows that both I
max
and C
min
have similar negative influences
on uptake estimates from the two models. For example, increasing I
max
by a factor of 2 decreases
the predicted uptake by 35% and 26% for SSAND and PCATS, respectively. The change ratios
for
θ
, C
Li
, v
0
, K
m
, C
min
, and I
max
in SSAND indicate a greater influence on simulated results than
those of PCATS. The influence of k, L
0
, and b on the simulated results of the two models was
almost the same. Only the influence of r
0
is less in SSAND than in PCATS (Figures 4 and 5).
The one dimensional sensitivity analysis response of NST 3.0 is different from those of
SSAND and PCATS (Figure 6). If listed in the order of significance, the most influential
parameters in this sensitivity analysis of NST 3.0 were C
Li
> r
0
= k >
θ
> v
0
. Note that the
influence of
θ
is not as prominent in NST 3.0 as it is in SSAND and PCATS. The uptake kinetics
parameters (I
max
, K
m
, and C
min
) have no influence on NST 3.0 simulations, while for SSAND and
PCATS, I
max
and C
min
had equally significant negative influences and K
m
had a slight positive
influence. Finally, NST 3.0 is the least sensitive model of the three to changes in parameter
values. For example, if the value of a parameter is increased by a factor of 2, the subsequent
53
change ratio of results for NST 3.0 ranges from 1 to 2, while that of SSAND ranges from 0.65 to
2.55, and PCATS ranges from 0.74 to 2.42.
Williams and Yanai (1996) explained the interactions of soil and plant factors in controlling
nutrient uptake by “distinguishing whether the rate of nutrient uptake is more limited by the
potential rate of nutrient delivery to the root, which depends on soil properties, or by the
potential rate of nutrient uptake into the root, which depends on root physiology”. In this one
dimensional sensitivity analysis of NST 3.0, SSAND, and PCATS for potassium uptake by
loblolly pine seedlings, the uptake kinetics parameters either have little or negative influences,
and nutrient concentration in soil solution and volumetric soil moisture were among the most
influential parameters. Therefore, it appears that potassium uptake by loblolly pine seedlings
taken from Kelly et al. (1992) was limited mainly by soil nutrient supply.
The influence of volumetric soil moisture, as represented by
θ
, is a relatively new parameter
in the model sensitivity analysis. It is included in NST 3.0, SSAND, and PCATS, but was not
used in the NUTRIENT UTPAKE model. Most nutrient uptake studies using SSAND or NST
3.0 exclude
θ
in the sensitivity analysis (Sadana and Claassen 1999; Sadana and Claassen 2000;
Gill et al. 2005; Comerford et al. 2006). There are two studies where
θ
was included in the
sensitivity analysis. These authors found that
θ
was among the most influential parameters in
both studies (Smethurst and Comerford 1993b; Singh and Sadana 2002). Williams and Yanai
(1996) identified C
av
, I
max
, and D
e
as the most influential parameters in their multiple
dimensional sensitivity analysis on a root surface area basis with a simplified steady state model.
Because
, the value of D
L
is relatively fixed, and f is a function of
θ
, the parameters
replacing D
e
in NST 3.0, SSAND, and PCATS include
θ
and b. From the equation we also notice
that D
e
is quadratically related to
θ
. And this may explain why
θ
is more influential than b in the
sensitivity analysis.
The negative influence of I
max
on simulated results in the one dimensional sensitivity analysis
of SSAND and PCATS is surprising. No similar reports have been found. Three studies on
nutrient uptake by woody species with a steady state model are available. Smethurst and
Comerford (1993b) and Comerford et al. (2006) found no negative influence of I
max
in COMP8
and SSAND simulations. Singh (2008) reported on the use of SSAND to simulate ammonium,
nitrate, phosphorus, and potassium uptake by hybrid poplar growing on two soils. The study used
54
Figure 4. One dimensional sensitivity analysis using SSAND with potassium uptake data for
loblolly pine seedlings from Kelly et al. (1992) and the diffusion coefficient of potassium in
water at 25 ºC taken from the Parsons (1959).
0
0.5
1
1.5
2
2.5
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Simulated result change ratio
Parameter change ratio
θ
CLi
1:1 line
k
v0
r
L
K
b
C
I
0
0
0
m
min
max
Li
55
Figure 5. One dimensional sensitivity analysis using PCATS with potassium uptake data for
loblolly pine seedlings from Kelly et al. (1992) and the diffusion coefficient of potassium in
water at 25 ºC taken from the Parsons (1959).
0
0.5
1
1.5
2
2.5
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Simulated result change ratio
Parameter change ratio
θ
C
1:1 line
k
v
r
L
K
b
C
I
0
0
0
m
min
max
Li
56
Figure 6. One dimensional sensitivity analysis using NST 3.0 with potassium uptake data for
loblolly pine seedlings from Kelly et al. (1992) and the diffusion coefficient of potassium in
water at 25 ºC taken from the Parsons (1959).
0
0.5
1
1.5
2
2.5
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Simulated result change ratio
Parameter change ratio
C
1:1 line
r
k
θ
v
L
b
I
K
C
0
0
0
m
min
max
Li
57
2 harvests and 16 simulations. The slope of the line representing I
max
in the sensitivity analysis of
ammonium uptake at the second harvest on both soils, and of potassium uptake in the second
harvest on one of the soils, were slightly negative. Unfortunately the author did not discuss this
phenomenon and summarized parameter values regardless of soil type and harvests such that it is
difficult to explore further with their data. Because the negative influence of I
max
in the Singh
(2008) study was found at the second harvest when soil nutrients were depleted, and the most
influential parameters in the one dimensional sensitivity analysis were related to soil properties,
we suspect that the uptake process was limited by soil supply of nutrients and it seems that the
negative influence of I
max
is related to a limited supply of nutrients by the soil.
To explore this phenomenon, we examined the effects of I
max
on uptake simulation with
artificial datasets. In these data sets, root growth rate were set as 39.3 cm day
-1
, and volumetric
soil moisture was set as 0.4 (see multiple dimensional sensitivity analysis), so that they are not
limiting soil supply or plant uptake process. The value of C
Li
was varied linearly at 5 levels,
ranging from 0.028 to 8.54 µmol ml
-1
. In this case, we assume the nutrient supply of the soil
increases as C
Li
increases. I
max
values for potassium uptake by loblolly and slash pine taken from
the literature (Van Rees et al. 1990; Kelly et al. 1992; Kelly et al. 1995) range from 1.4E-6 to
3.65E-6 µmol cm
-2
s
-1
. Because the actual I
max
may be very different, five levels of I
max
, 1.4E-7,
1.4E-6, 3.65E-6, 1.4E-5, and 1.4E-4 µmol cm
-2
s
-1
, were used in order to get a comprehensive
understanding of model behavior.
The results of this analysis are shown in Figure 7. When C
Li
was equal to 0.028 µmol ml
-1
,
uptake simulated by SSAND was significantly lower than that of NST 3.0. The prediction by
SSAND increased as I
max
increased from 1.4E-7 to 1.4E-6 µmol cm
-2
s
-1
, and then dropped as
I
max
increased further, while the prediction of NST 3.0 increased as I
max
increased. Thus the
pattern of simulated uptake by SSAND as I
max
increased shows a bell shape, compared to an
exponential shaped curve by NST 3.0. When C
Li
was increased to 2.156 µmol ml
-1
, both models
had similar predictions and patterns before I
max
reached 1.4E-5 µmol cm
-2
s
-1
. The prediction of
NST 3.0 still increased as I
max
increased, while the prediction by SSAND dropped when I
max
increased from 1.4E-5 to 1.4E-4 µmol cm
-2
s
-1
. The patterns of model behavior under different
levels of C
Li
higher than 4.284 µmol ml
-1
were similar. The predictions of both models were
close and increased as I
max
increased when I
max
ranged from 1.4E-7 to 1.4E-5 µmol cm
-2
s
-1
. The
58
Figure 7. Simulated potassium uptake by SSAND and NST 3.0 at five levels of I
max
and three
levels of C
Li
. For these simulations root growth rate and volumetric soil moisture have been set
to 39.3 cm day
-1
and 0.4 respectively. Other parameter values remained as listed in Table 6 for
loblolly pine (Kelly et al. 1992).
0
2000
4000
6000
8000
10000
1.0E-07 1.0E-06 1.0E-05 1.0E-04
Simulated uptake (µmol)
Imax(µmol cm-2 s-1)
CLi=0.028 µmol ml-1
SSAND
NST
0
100000
200000
300000
400000
500000
1.0E-07 1.0E-06 1.0E-05 1.0E-04
Simulated uptake (µmol)
Imax(µmol cm-2 s-1)
CLi=2.156 µmol ml-1
SSAND
NST
0
200000
400000
600000
800000
1.0E-07 1.0E-06 1.0E-05 1.0E-04
Simulated uptake (µmol)
Imax(µmol cm-2 s-1)
CLi=4.284 µmol ml-1
SSAND
NST
59
prediction of SSAND was lower than that of NST 3.0 when I
max
was equal to 1.4E-4 µmol cm
-2
s
-1
. Therefore, the behavior of SSAND is influenced by levels of C
Li
, which indicates the ability
of the soil to supply nutrients in our assumption. Our suspicion that the negative influence of I
max
on SSAND simulation relates to the low nutrient supply by the soil is supported.
Previous studies with transient state models with a numerical solution have not reported a
negative influence of I
max
in the sensitivity analysis (Van Rees et al. 1990; Teo et al. 1992;
Sadana and Claassen 1999; Sadana and Claassen 2000; Sterckeman et al. 2004; Gill et al. 2005;
Comerford et al. 2006). In addition, the graphical representation in the multiple dimensional
sensitivity analysis of a simplified steady state model by Williams and Yanai (1996) did not
show a negative response in simulated uptake when I
max
increased. According to the Michaelis-
Menten uptake kinetics equation, nutrient influx into roots is positively related to I
max
. Because
nutrient uptake by plants is the accumulation of influx over time and root surface area, it is
illogical to think that increased I
max
would lead to a decrease in uptake. Therefore, we examined
the key equation of the steady state model.
If A is used to represent
, and B for
, then the key equation 8
of the steady state model can be rewritten as
[15]
If D is used to represent
, then equation 9 can be rewritten as
[16]
Then
[17]
The partial derivative of
∆
U with respect to I
max
is
[18]
60
When
,
because
, and
. Therefore
. When
,
and
.
Therefore B is positive as well. Because
,
,
,
, the partial derivative
(equation 18) is always positive, indicating that the
function
is monotonically increasing. That is,
increases as I
max
increases. Since the total
amount of nutrient uptake is obtained by summing
at each time interval, the total amount of
nutrient uptake calculated this way increases as I
max
increases. The structure of SSAND and
PCATS is more complex. But this simple calculation indicates that the negative influence by I
max
may be an artifact in the development of SSAND and PCATS.
Multiple dimensional sensitivity analysis
Figure 8 shows the relationship of simulated uptake to nutrient concentration in the soil
solution (C
Li
), which increases along the x-axis of each graph for both models. Each graph shows
the results at a fixed level of root growth rate (k). The y-axis shows the simulated uptake in
µmol, while the x-axis covers the range of C
Li
values, and on each graph there are five different
lines, one for each level of volumetric soil moisture (
θ
). Thus each line represents the change in
uptake at given
θ
and k, with increasing C
Li
. Each group of 5 graphs shows the results for a
particular mechanistic model. The graphs are arranged such that k increases from the top row to
the bottom. In general, uptake increases with C
Li
until it reaches its maximum at high C
Li
.
Comparison of the five lines on each graph reveals that with those with a higher
θ
show the
strongest response to increasing C
Li
. But at high levels of C
Li
, an increased
θ
does not produce
any differences in uptake. At low C
Li
, the uptake is strongly reduced. Comparison of graphs in
the same row shows the effect of changing k. At low k, the maximal simulated uptake is strongly
reduced. The maximal simulated uptake increases proportionally as k increases. If the simulated
uptake by both models is fitted to a linear regression, the regression equations are similar. For
example, if U
M
(µmol) represents maximal simulated uptake, the equation for NST 3.0 is
.
61
SSAND NST 3.0
Figure 8. Response surface from a multiple dimensional sensitivity analysis of SSAND and NST
3.0 using five levels of potassium concentration in the soil solution (C
Li
), five levels of
volumetric soil moisture (θ), and five levels of root growth rate (k). Other parameter values
remained as listed in Table 6 for loblolly pine (Kelly et al. 1992). The unit of simulated uptake is
µmol.
0
5000
10000
15000
20000
0 2 4 6 8 10
Simulated uptake
k=1 cm day
-1
θ=0.06 θ=0.145
θ=0.23 θ=0.315
θ=0.4
0
5000
10000
15000
20000
0 2 4 6 8 10
k=1 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
Simulated uptake
k=10.6 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
k=10.6 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
Simulated uptake
k=20.2 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
k=20.2 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
Simulated uptake
k=29.7 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
k=29.7 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
Simulated uptake
k=39.3 cm day
-1
0
5000
10000
15000
20000
0 2 4 6 8 10
k=39.3 cm day
-1
C
Li
(µmol ml
-
1
) C
Li
(µmol ml
-
1
)
62
Figure 9 shows the relationship of the same parameters used in the multiple dimensional
sensitivity analysis depicted in Figure 8 in another way. In this case the x-axis covers the range
in θ values and each graph represents a change in the C
Li
value. The graphs of simulated uptake
at higher C
Li
values were not included in Figure 9 because for each k, the simulated uptake at the
higher C
Li
were the same as the uptake at C
Li
=2.156 µmol ml
-1
with a high θ. When C
Li
is low
(0.028 µmol ml
-1
), SSAND produces lower estimates than NST 3.0. For example, when C
Li
is
0.028 µmol ml
-1
and θ is 0.06, the predicted potassium uptake by SSAND is less than 0.1 µmol,
while uptake simulated by NST 3.0 ranges from 24 to 258 µmol. When C
Li
is increased to 2.156
µmol ml
-1
, the increase in θ did not lead to a significant change in simulated uptake by NST 3.0,
but the uptake simulated by SSAND is significantly reduced at low θ (Figures 8 and 9). It seems
that SSAND is more sensitive to θ, as was indicated in the one dimensional sensitivity analysis
as well. However, the multiple dimensional sensitivity analysis also indicates the influence of θ
on the SSAND simulation is only valid when C
Li
is low (Figure 9).
Because the transient state model with a numerical solution is assumed to be more accurate
than the steady state model (Smethurst and Comerford 1993b; Smethurst et al. 2004), the
performance of SSAND can be evaluated by comparing its simulated results with those of NST
3.0 in our multiple dimensional sensitivity analysis. As shown in Figure 9, the predicted uptake
by SSAND is lower than that of NST 3.0 before the simulations reach their maximums (an
example is given in Figure 10). Given this underestimate, and the inability of SSAND to predict
phosphorus uptake in the hybrid poplar study, it is suspected that SSAND is less able to predict
uptake accurately when the supply of nutrients by the soil cannot meet the plant’s needs as
defined by the uptake kinetics parameters.
Conclusions
In summary, NST 3.0, SSAND, and PCATS differ both in predictive accuracy and model
behavior. PCATS successfully predicted nitrate uptake, NST 3.0 predicted phosphorus uptake
well, while SSAND underpredicted all nutrient uptake severely. Results were also diverse,
indicating the need of reconsideration of the assumptions and solutions of the two mechanistic
model categories. Although NST 3.0 and PCATS performed relatively well with some runs, the
model predictions of the others failed to match the observation. The underestimates of many of
63
SSAND NST 3.0
Figure 9. Response surface from a multiple dimensional sensitivity analysis of SSAND and NST
3.0 using two levels of potassium concentration in the soil solution (C
Li
), five levels of
volumetric soil moisture (θ), and five levels of root growth rate (k) with cm day
-1
as the units.
Other parameter values remained as listed in Table 6 for loblolly pine (Kelly et al. 1992).
0
4000
8000
12000
16000
20000
0 0.1 0.2 0.3 0.4
Smulated uptake (µmol)
CLi=0.028 µmol ml-1
k=1 k=10.6
k=20.2 k=29.7
k=39.3
0
4000
8000
12000
16000
20000
0 0.1 0.2 0.3 0.4
CLi=0.028 µmol ml-1
0
4000
8000
12000
16000
20000
0 0.1 0.2 0.3 0.4
Simulated uptake (µmol)
θ
CLi=2.156 µmol ml-1
0
4000
8000
12000
16000
20000
0 0.1 0.2 0.3 0.4
θ
CLi=2.156 µmol ml-1
64
Figure 10. Simulated potassium uptake by SSAND and NST 3.0 with data taken from Kelly et al.
(1992). The simulation was conducted with nutrient concentration in the soil solution (C
Li
)
changing from 0.028 to 4.284 µmol ml
-1
while keeping other parameters unchanged.
0
2000
4000
6000
8000
012345
Simulated uptake (µmol)
CLi (µmol ml-1)
NST 3.0
SSAND
65
the runs may be the results of running simulations without including the nutrient uptake
by mycorrhizaes. But this alone cannot explain some successful simulations. Therefore,
further studies that can produce representative datasets to evaluate the predictive
accuracy against observed values for each model are needed for future model evaluation.
Both types of sensitivity analyses indicate that soil moisture (θ) plays an important
role in uptake simulation when the nutrient concentration in the soil solution (C
Li
) is low.
This has not been noted in previous studies. Under low-nutrient-supply scenarios, I
max
can influence the predictions of SSAND and PCATS negatively, and the uptake
predictions of SSAND are generally lower than those of NST 3.0. We suspect that these
are artifacts of the steady state models and further studies are needed to improve their
ability to represent nutrient uptake under low-nutrient-supply scenarios.
In the process of soil exploration and nutrient uptake by plant roots, the influence of
rhizospheric effects such as differences in rhizosphere pH and redox potential are
important (Gillespie and Pope 1990; Marschner 1995). The three mechanistic nutrient
uptake models do not include routines to describe such subprocesses except for the
effects of mycorrhizaes. Because these effects are probably not negligible when the
models are applied to longer growth periods and lower soil nutrient supply situations that
are more typical of woody species, further studies to incorporate important rhizospheric
effects other than mycorrhizae are suggested.
66
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