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Dept. of Soil Science and Agricultural Chemistry,
It is “an assembly of concepts in the form of
a mathematical equation that portrays
understanding of a natural phenomenon.”
Emprical models – statistical means
and regression- black box models
Mechanistic models- biophysical,
biochemical and physiological
NUTRIENT UPTAKE MODELS
• Soil nutrient transformation models emphasize on soil
• Crop uptake models emphaize on plant characters
especially on root characters and also soil characters.
• Models are formed for (i) identifying the process which
describe nutrient uptake and (ii) evaluating quantitatively
the effect of different parameters of nutrient uptake.
• Extrapolation of a verified mechanistic model is more
reliable than that of an empirical model(Claassen and
• 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,
Steady state models
• Flow of nutrients through a soil- steady state
• Soil – volume element conducting nutrient flow
• Neither a sink nor a source of nutrients
• based on diffusion theory
• either a sink or source influencing / modifying its flux
• describes the dynamic process of nutrient uptake & the
concomitant changes in the ion concentration &
distribution in the rhizosphere.
Interception is used to describe the uptake of soil nutrients at the
root interface when soil volume is displaced by root volume
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
Therefore, only mass flow and diffusion are considered to be
responsible for movement of nutrients to the root surface in
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.
Mass flow is the convective transport of nutrients through the
soil to the root surface by water flow as a result of transpiration
The relative contribution of mass flow to nutrient uptake
depends on the nutrient, plant species, plant age, and time of
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 (FM) can be calculated by
where v is the mean water flux in soil driven by transpiration,
and CL is the nutrient concentration in the soil solution.
Diffusion is the movement of nutrients from areas of high
concentration to those of low concentration (Barber
It is the main mechanism for at least phosphorus and
potassium movement in the soil to plant roots
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 FD
can be described by Fick’s first law,
where D is the diffusion coefficient of the nutrient in soil,
C is the nutrient concentration in soil solution, and x is
SIMULTANEOUS MASS FLOW AND DIFFUSION
Mass flow and diffusion occur simultaneously to supply
nutrients to plant roots and cannot be treated as separate
Nye and Spiers (1964) presented a partial differential
equation to describe simultaneous mass flow and
diffusion, and this equation became the foundation of the
most mechanistic nutrient uptake models.
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) to describe the flux
of nutrient in the soil to the root surface with the nutrient
concentration in soil solution (CL)
Nye and Marriot (1969) defined boundary conditions and
solved this equation numerically.
NYE AND TINKER MODEL
Nye and Tinker, 1977
The uptake of nutrient per unit length (U) is given by,
r = radius of the root
α = root absorbing power
C1= ion concentration at the root surface
Summarizing the work by Claassen and Barber
(1976) and Cushman (1979), Barber and Cushman
(1981) suggested new boundary conditions to
include inter-root competition for nutrients.
The new boundary conditions incorporated inter-
root competition as well as Michaelis-Menten
When solved numerically, the enhanced
mechanistic model evolved into Barber-Cushman
Crop uptake is assumed to follow Michaelis-menten
relationship between concentration and flux.
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
Both were based on the Barber-Cushman model.
MICHAELIS- MENTEN KINETICS
In = Rate of ion uptake
Imax = maximum rate of ion uptake when concentration is
Cla = concentration of nutrient ion in soil solution at the
Km = Michaelis – menten constant which represents a
concentration of nutrient ion in soil solution when In=Imax/2.
Based on Nye and Tinker model, 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 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).
For transient models –NST 1.0, NST 3.0.
NYE AND TINKER, 2000
where b is the soil buffer power,
hence c = bc is the total amount of solute bound to the
θ is the volumetric soil water content,
thus is the overall amount of solute per unit volume of soil;
D is the solute diffusivity in water;
f is the soil impedance factor,
Dfθ is the effective diffusion coefficient for the solute in
the soil; and v is the water flux in the soil.
MODELLING CROP N UPTAKE
Benbi, Prihar and Cheema (1991) – used mass flow
approach & predicted N uptake for wheat matched with
measured N uptake (wet treatments).
In dry moisture regime other process of N uptake like
diffusion & root interception should be used.
BASED ON CROP DEMAND
Several models - driven by demand from the plant.
They calculate N demand for growth rates & concentration of N
Whitmore and Addiscott (1987) – computed demand using a
simple variant of the logistic function to compute the pattern of
N uptake (Y), with thermal time (x):
Y=(A-1/X + e-kx )-n
A - maximum of Y, n - shape factor, k - rate constant.
• Tillotson and Wagenet (1982) and Hutson and Wagenet
(1992) – modelled N uptake – based on Nye and Tinker
• Warncke and Barber (1973) – value of α to be estabilished
for different soils, crops and species under a range of
• De Willigen, 1991 – more complex models give better
simulations than simpler ones.
TRANSFORMATION INCLUDING UPTAKE
Satisfactorily describes P uptake flux (Fp, mol cm-1 root
h-1 ) as a function of P concentration in a well-stirred
solution (Cpb mol l-1 ).
Fp = Fp max * Cpa /(Km +Cpa )
Fp max and Km denote the maximum Fp and Cpa at
which Fp = 0.5*Fmax .
P uptake models
Plant roots have a major influence on P dynamics in
A= (b/a) w/2πDp
a,b = midway distance between root and root radius
Dp = diffusion co-efficient
B = Fpmax /W
W = solution velocity towards the root
Bar and Yosef (1999)
K concentration in
K in soil
K flux to
mass flow &
biomass Fixed K
Multi- Parametric model for K uptake
Mc Caskill and Blair (1988) have developed a pseudo-
mechanistic computer simulation model for predicting
perennial pasture growth and S uptake.
But not verified.
Later Heng (1991) and Phimsarn(1991), developed
simple models of pasture S uptake using relationship
between actual daily ET(AET), average root density in
the soil profile, soil solution S levels, sulphate buffering
Mathematical models of micro-nutrient uptake by
plants have been based on models designed for
uptake of major nutrients.
This poses a number of problems, as the behaviour
of micro-nutrients in the rhizosphere is different,
particularly with respect to interactions between
solid and solution compartments.
There is no standard procedure for the
measurement of soil parameters that affect the
supply of micro-nutrient from the soil.
Dynamic Plant Uptake Model
DGT (Diffusive Gradient in Thin-flims) tecnique is