Annals of Botany 79 : 657–665, 1997
Modelling Asymmetrical Growth Curves that Rise and then Fall : Applications to
Foliage Dynamics of Sugar Beet (Beta vulgaris L.)
A. R. W E R K E R and K. W. J A G G A R D
IACR Broom’s Barn, Higham, Bury St Edmunds, Suffolk IP28 6NP, UK
Received : 11 October 1996
Accepted : 2 January 1997
This paper discusses the derivation and fitting of three empirical models with turning points for describing the growth
of plant components, such as shoot weight, leaf area and root length, that typically rise and then fall during the course
of the growing season. The models (Models I, II and III) have analytical solutions and may be viewed as extensions
of the Gompertz, Richards and Chanter growth equations. They differ by having an additional parameter which,
following a sigmoidal rise of the dependent variable, determines subsequent net rate of decline. The models were fitted
to sequential measures of foliage cover of sugar beet crops grown in the UK during 1980–1991. It was important that
this could be done with relative ease using standard statistical procedures. Partial linear transformations of two of
the models, with one non-linear parameter remaining, are described ; these were useful for estimating initial values for
the parameters. All three models described the data well, although the fitting of Model II invariably failed to converge.
For Models I and III common and separate parameters, amongst years, were estimated relating to date of emergence,
initial relative growth rate, maximum cover attained and rate of late season decline of foliage cover. The reduction
in the residual mean square on fitting separate, rather than common, parameters was usually significant. The models
accommodate several biological processes that yield similar shapes. This is demonstrated for Model I, in relation to
its formulation and to effects of small perturbations in the values of the parameters on the shape of the curves. Model
I, the simplest of the three models tested, has good fitting properties, and in practice was best suited to describing
foliage cover dynamics of sugar beet.
# 1997 Annals of Botany Company
Key words : Beta Šulgaris L., sugar beet, foliage cover, senescence, models, parameter estimation, growth functions.
INTRODUCTION
The classical growth models, such as the logistic, Gompertz,
monomolecular and Michaelis–Menten equations, have
been extensively used to describe a multitude of biological
processes. They can be fitted to data with relative ease, using
standard statistical software, and their parameters display
considerable biological interpretability. More recently, such
models have become particularly powerful in analyses of
treatment effects (e.g. Gilligan, 1989), and as integral
components in analyses of variance (Butler and Brain, 1990,
1993). The linear alternatives for analysing shape or trend
(e.g. Werker, Gilligan and Hornby, 1991), whilst more
flexible, are awkward to interpret.
A common feature amongst the classical growth models is
that growth only increases and tends to an asymptote.
Typically, as time (or other independent variable) increases,
the value of the dependent variable approaches a constant
which is not zero. In many cases this is appropriate. For
example, in agriculture, availability of data ceases at harvest,
when a yield component, or the severity of a disease, has
often reached a peak. However, this feature presents a
serious limitation to the description and analysis of certain
types of data which are not uncommon.
Many plant parameters, such as shoot weight, leaf area
and root length, typically rise, reach a maximum, and then
fall during the course of a growing season. A typical
0305-7364}97}060657­09 $25.00}0
example is the green leaf area of many cereal crops which
declines to virtually zero before the crop is harvested.
Temporal profiles of variables such as leaf area arise as a
consequence of the net effect of growth and senescence or
death. Moreover, growth and senescence rates change
during the course of the season in response to changes in the
life cycle of the plant, and to changes in the environment,
such as availability of assimilates and effects of pests and
diseases. Even in constant environments, the size of many
plant variables, other than the seed or storage components,
increases and decreases in response to a change in their
function. For example, in annual crops shoot and root
systems become redundant once the crop has seeded. For
these components, sustained periods of zero net growth
rates, depicted by asymptotic growth functions such as the
logistic or Gompertz, are seldom appropriate.
Despite their common occurrence, trajectories of this
nature, with a turning point, have received scant attention
in the literature. However, Gilligan (1990), for example,
analysed antagonistic interactions involving plant pathogen
populations that displayed a rise and fall during the course
of time. Richter, Spickermann and Lenz (1991) demonstrated a new, five parameter model to describe leaf biomass
of white cabbage ; they subsequently used the model to
analyse effects of fertilizer regimes, and explored a number
of extensions to accommodate other plant structures. Also,
Brain and Cousens (1989) have presented an equation to
bo970387
# 1997 Annals of Botany Company
658
Werker and Jaggard—Modelling Asymmetrical Growth CurŠes
describe plant response to herbicide damage where there is
a stimulation in response at low doses.
The objective of this investigation was to develop simple
empirical growth functions to describe the temporal dynamics of plant components that rise, and then fall, during
the growing season. It was important that the parameters
had some biological realism and could be estimated easily,
using standard statistical software. Three growth models are
presented, all with turning points and zero asymptotes. The
models are fitted to sequential observations of foliage cover
of sugar beet crops.
Foliage cover is defined as the fraction of ground area
covered by leaves. It is a measure of crop canopy size, and
is used to estimate light interception by the crop (Steven et
al., 1986 ; Jaggard and Clark, 1990 ; Werker, Jaggard and
Webb, 1995). The alternative, and more conventional
measure of canopy size, leaf area index (LAI), can be readily
estimated from foliage cover by fitting calibration data to a
Mitscherlich curve (Steven, Biscoe and Jaggard, 1983 ;
Steven et al., 1986 ; Werker et al., 1995). Measurement of
foliage cover is quick and non-destructive, by infra-red
photography or measurements of crop reflectance made by
using hand-held or helicopter-mounted spectrophotometers
or from satellite images.
Model I
Consider the linked differential equations,
dy
¯ µy
dt
where µ ¯ β®δ in eqn (1) is the net relative growth rate,
which decays exponentially at a rate that is proportional to
its distance from the final net relative growth rate, µmin, as
t approaches infinity. Integrating eqn (2) gives,
µ ¯ µmin­( µ ®µmin) e−k(t−t!)
!
µ ®µmin
(1®e−k(t−t!)) ,
y ¯ y exp µmin(t®t )­ !
!
!
k
0
where β and δ are the relative rates of growth and senescence,
respectively. However, the segregation of processes of
growth and death can be misleading when the models are
simple and empirical, by inviting spurious interpretations of
parameter estimates based on goodness of fit. It is more
appropriate, in this context, to consider the dynamics of net
growth.
Three models are presented to describe the growth of a
plant component, y, that satisfy these conditions, notably
that the relative growth rate of y changes during the course
of the growing season in response to a changing environment, and to a change in the function of y as an
integral part of the life cycle of the crop. Model I is the
simplest : it has four independent parameters, and is related
to the Gompertz function. Models II and III have five
independent parameters : they are extensions of Model I and
they are related to the Richards function (Richter et al.,
1991) and the Chanter equation (discussed in France and
Thornley 1984), respectively. Model II equates to the model
of Richter et al. (1991), but differs here in its derivation
and in the interpretation of the parameters. Model III
includes a carrying capacity, a maximum value for y,
and has particular applications in describing crop foliage
cover.
1
(3)
where µ is the initial relative growth rate, and y the initial
!
!
size of y at time t . The rate constant, k, determines the
!
speed with which the initial net growth rate, µ , approaches
!
the final net growth rate, µmin, as t U ¢. When µmin ! 0, y
has a turning point where µ ¯ 0 at (tmax, ymax) given by,
0
MATERIALS AND METHODS
From a modelling perspective, it is common practice to
separate the processes of new growth and senescence.
Consider, for example, the growth rate of a plant component
y such that
dy
¯ βy®δy,
(1)
dt
(2)
dµ
¯®k( µ®µmin),
dt
1
0
1
®µmin
tmax ¯ t ® log
! k
µ ®µmin
!
µ µ
®µmin
ymax ¯ y exp !® min log
!
k
k
µ ®µmin
!
0
(4)
11 ,
and µ may be eliminated from eqn (3) to give,
!
µ ¯ µmin(1®e−k(t−tmax))
0
1
µ
y ¯ ymax exp µmin(t®tmax)® min (1®e−k(t−tmax)) .
k
(5)
In this case, the initial relative growth rate, µ ¯
!
µmin²1®exp [®k(t ®tmax)]´. Note that when µmin ¯ 0 then
!
eqn (3) reduces to the Gompertz function.
Model II
In this second model, additional flexibility may be found
by considering the change in the relative growth rate of y,
dµ}dt, as a quadratic function of µ, rather than as a linear
function as in eqn (2). Thus :
dy
¯ µy
dt
dµ
µ ®µ
¯®k( µ®µmin) max
,
dt
µmax®µmin
which on integration gives,
(6)
659
Werker and Jaggard—Modelling Asymmetrical Growth CurŠes
where
(µmax®µmin) (µ ®µmin) e−k(t−t!)
!
(µmax®µ )­(µ ®µmin) e−k(t−t!)
!
!
µ ¯ µmin­
0
9
µ ®µmin
U ¯ µmin (t®t )­ !
(1®e−k(t−t!)).
!
k
:
µmax®µ
®µmin k(t®t )
! exp
!
! µ ®µ
µmax®µmin
max
min
y¯y
(7)
9
µ
:1
µ ®µmin
®µmax k(t®t )
!
exp
­ !
µmax®µmin
µmax®µmin
µ
max− min
−k
0
1
0
11
1
®µmin
µmax
®log
tmax ¯ t ® log
! k
µ ®µmin
µmax®µ
!
!
ymax ¯ y exp
!
­
0®µk
min log
0
0µ®µ
®µ 1
!
µmax
µmax
log
k
µmax®µ
min
κ
,
κ®y
! e−Umax
1­
y
!
where
0 0
(12)
1
1
µmin
®µmin
µ
log
® ! .
k
µ ®µmin µmin
!
Umax ¯®
Eliminating µ then gives,
!
(8)
11 .
y¯
κ
,
κ®ymax «
−U
1­
e
ymax
where
!
Eliminating µ from eqn (7) gives,
!
µ
U« ¯ µmin(t®tmax)® min (1®ek(t−tmax)).
k
µmin
(A®1)®Aek(t−tmax)
µ ¯ µmin­
µ
min
y ¯ ymax (Ae−(A−")k(t−tmax)®(A®1) e−Ak(t−tmax))− k(A−"),
A¯
ymax ¯
min
where
1
1
®µmin
tmax ¯ t ® log
! k
µ ®µmin
!
,
where the additional parameter, µmax, allows for a maximum
relative growth rate, where µ U µmax as t U®¢ (whereas in
Model I µ U ¢ as t U®¢). The change in the relative
growth rate is thus sigmoidal rather than exponential, and
its rate of change depends on k and the timing on µ and
!
t . Here, also, when µmin ! 0, y has a turning point where
!
µ ¯ 0 at (tmax, ymax) given by
0 0
When µmin ! 0, y has a turning point where µ ¯ 0 at
(tmax, ymax), given by,
(9)
µmax
.
µmax®µmin
When µmin ¯ 0, eqn (7) reduces to the Richards function.
(13)
This model is particularly appropriate for the description of
seasonal dynamics of foliage cover, because foliage cover
not only depends on leaf area, which itself rises then falls
during the course of the growing season, but it also depends
on the space available in which foliage cover can increase.
Here, κ has a theoretical value of one, which reduces the
number of parameters for estimation to four. When µmin ¯
0, Model III reduces to the Chanter growth equation
described in France and Thornley (1984).
Model III
Fitting models to data
The third model introduces a carrying capacity for y, such
that y U κ, while µ is positive ; if µ is constant, then dy}dt
reduces to the logistic function. Here, the relative growth
rate, µ, decays exponentially to a minimum value, µmin as in
Model I.
dy
(κ®y)
¯ µy
dt
κ
(10)
Models I, II and III were fitted to measurements of foliage
cover of sugar beet crops from field experiments at
IACR—Broom’s Barn. Foliage cover is a two-dimensional
measure of crop canopy and equates to the fraction of
ground area covered by leaves when viewed from above
(dimensions, m#m−#). The data span 12 growing seasons,
1980–1991, and comprise between five and 18 observations
within seasons, each constituting a mean of four or more
replicate observations. Foliage cover measurements were
obtained using infra-red photography (Biscoe and Jaggard,
1985) or from a hand-held spectrophotometer (Steven et al.,
1983).
The independent variable, T, is defined as the accumulated
daily mean air temperature above 3 °C from sowing. This is
appropriate because both seedling emergence and leaf area
expansion, on which the dynamics of crop foliage cover
depend, are temperature dependent ( Milford, Pockock and
dµ
¯®k( µ®µmin).
dt
Integrating eqn (10) gives,
y¯
κ
,
κ®y
! e−U
1­
y
!
(11)
660
Werker and Jaggard—Modelling Asymmetrical Growth CurŠes
Riley, 1985 ; Day, 1986 ; Durrant et al., 1988). Thus, dT}dt
is the average daily air temperature above 3 °C, on which
the rate of canopy expansion depends, such that dy}dt ¯
(dT}dt) µy, which simplifies to dy}dT ¯ µy.
To obtain initial estimates for the parameters, Models I
and III were transformed to reduce the number of nonlinear parameters to just one, k. The remaining parameters
were derived from the regression coefficients A, B and C
after fitting the equation,
Y ¯ A­Bt­C[exp (®kt),
(14)
where, for Model I, eqn (5),
0 1
1
kC
;
Y ¯ log ( y) ; µmin ¯ B ; tmax ¯ log
k
B
log ymax ¯ A­µmin
0
(15)
1
1
­t
,
k max
The remaining estimated parameter, µmin, represents the
relative rate of late season decline of crop foliage cover after
reaching a maximum value, ymax.
All three models were fitted to the data using the general
statistical package Genstat (Payne et al., 1993). Subsequently, amongst years, separate values for the parameters :
(1) t , (2) t and µmin, and finally (3) t , µmin and ymax, were
!
!
!
estimated. The parameter κ (Model III) was held constant at
κ ¯ 1 for all fits. Five out of the 12 years had insufficient
data to permit estimation of µmin. The fitting process was
modified such that for these years µmin acquired a value
equal to the mean µmin of the remaining years. Significance
testing between successive fits, that is, whether separate
values amongst years for individual parameters are justified,
was done by examining the magnitude of the change in
residual mean square between fits in which common and
separate values for the parameters were estimated (see
Brown and Rothery, 1993, for a description of this
technique).
and for Model III, eqn (13),
0
1
A similar transformation of Model II, eqns (7) and (9), was
not possible and estimation of all parameters, with the
exception of ymax, required non-linear fitting methods.
These transformations, however, do not permit estimation
of t , which is an important parameter in the present
!
context. If y is defined as the fraction of foliage cover at
!
emergence, then t equates to the time of crop emergence ; a
!
variable that is frequently measured in field experiments.
Thus, t , µ , ymax and µmin were estimated, all as non-linear
! !
parameters, by combining eqns (3), (7) and (11) with (4), (8)
and (12), respectively. The parameter ymax was estimated
instead of k, by rendering k as a function of ymax, y , µ and
! !
µmin (Model I), and also µmax (Model II) or κ (Model III), by
re-arranging eqns (4), (8) and (12), respectively. For
example, for Model I,
0
1
µ ®µmin
(1®e−k(t−t!))
y ¯ y exp µmin(t®t )­ !
!
!
k
where
0
1
®µmin
µ ®log
µ
!
µ ®µmin min
!
.
k¯
ymax
log
y
!
Model I
Model II
1000 2000
1000 2000
Model III
0.0000
dl/dT
1
(16)
ymax
1
¯ A­µmin ­tmax .
k
κ®ymax
Graphical displays of y for Models I, II and III are given in
Fig. 1, together with dy}dT (the apparent growth rate), µ
(the relative growth rate) and dµ}dT (the rate of change of
the relative growth rate). The essential shape of all three
–0.0001
–0.0002
0.050
l
0
0 1
1
kC
µmin ¯ B ; tmax ¯ log
;
k
B
0.025
0.000
0.0050
dy/dT
log
y
0κ®y
1;
0.0025
0.0000
0.8
(17)
The parameter y was fixed at y ¯ 0±000015, defined as the
!
!
mean area of sugar beet cotyledons at 50 % crop emergence.
This was necessary because the parameters t , y and µ ,
! !
!
relating to the initial condition of eqns (3), (7) and (9), are
interdependent. Fixing y permits t and µ to be estimated ;
!
!
!
they are defined as the time of emergence and the relative
rate of expansion of foliage cover at emergence, respectively.
y
Y ¯ log
RESULTS
0.4
0
1000 2000
T, accumulated temperature
F. 1. Graphical displays of the dependent variable, y, the apparent
growth rate dy}dT, the relative growth rate, µ, and the rate of change
of the relative growth rate, dµ}dT, against accumulated temperature,
T. The values of the parameters are, y ¯ 0±000015, t ¯ 100, µ ¯
!
!
!
0±0050, µmax ¯ 0±0051, ymax ¯ 0±90, κ ¯ 1±0, µmin ¯®0±0050 and k ¯
0±00346, 0±0171 and 0±00286 for Models I, II and III, respectively.
Werker and Jaggard—Modelling Asymmetrical Growth CurŠes
of little consequence in most years, but Model I produced a
better fit to the data in those years when there was a
particularly sharp decline in foliage cover soon after
attaining a maximum value (e.g. 1990, Fig. 3). Model III,
however, predicted emergence dates (t ) closer to values
!
obtained from field measurements. For example, Durrant et
al. (1988) found that 50 % emergence occurred at between
104 and 150 d degrees above 3±5 °C from sowing. Estimating
separate values amongst years for time of emergence (t ),
!
and for the maximum foliage cover attained ( ymax) typically
reduced the residual deviance by half (Table 1). The rates of
decline (µmin) also differed amongst years despite the fact
that in five out of the 12 years there were insufficient data in
this region for reliable estimation of µmin.
1.0
y, foliage cover
0.8
0.6
0.4
0.2
0
661
500
1000
1500
2000
2500
T, accumulated temperature > 3°C from sowing
F. 2. Fitted profiles for Models I (——) and III (- - - -), with constant
parameters amongst years, to measures of foliage cover of sugar beet
crops grown at IACR Broom’s Barn during 1980–1991.
models is that of an asymmetrical bell, depicting a sigmoidal
rise and fall, given that µmin ! 0. Distinctions between
Models I and II are depicted in their respective relative
growth rates ( µ), which are sigmoidal for Model II, and
exponential for Model I. The additional parameter µmax
adds considerable control over both the speed and the
timing in which the relative growth rate changes from its
initial value to its final value. The dµ}dT plot shows this as
a pulse which may be positioned anywhere along the x-axis,
initiating a change in the relative growth rate. Models I and
III on the other hand, display infinite relative growth rates
when y is infinitesimally small. Model III differs from
Models I and II where y approaches κ, when the apparent
growth and decay rates slow down, allowing for a more
sustained period during which y is near ymax.
The four parameter models, Model I and Model III with
fixed κ ¯ 1, described the data well, and estimation of the
parameters, whether common (Fig. 2, Table 1), or separate
(Fig. 3, Table 1) amongst years, was achieved with relative
ease. By contrast, successful fitting of Model II, which has
the additional parameter µmax, was not possible. Despite
failure to converge, some of the fitting attempts yielded
parameter estimates which, when substituted into Model II,
produced trajectories for predicted foliage cover very close
to those predicted by Model I. In such cases the convergence
process ceased on a high value for µmax, typically in excess of
twice the value of µ . When µmax is large by comparison to
!
µ , then Model II approximates to Model I.
!
Differences in the goodness of fit between Models I and
III were minimal, with Model I having marginally smaller
residual mean squares (Table 1). In practice, Model I was
simpler to fit, typically requiring half the number of
evaluations before convergence was achieved. The distinct
difference in shape between Models I and III, in the region
where foliage cover approaches its potential maximum, was
DISCUSSION
Three related models are presented for describing and
analysing growth of plant components that typically rise,
and then fall, during the course of the growing season. The
models have analytical solutions and may be viewed as
extensions of classical growth models, containing parameters
that have a strong biological basis. Alternative formulations
of the models are given, similar to those that exist for
classical models. Partial linear transformations of two of the
models are also described (log and logit transformations),
and these have proved useful for estimating initial values for
some of the parameters. Two of the models can describe
foliage cover data, using standard statistical procedures,
including estimation of common and separate parameter
values between data-sets. Difficulties were encountered in
fitting the model of Richter et al. (1991) (Model II). The
additional parameter, µmax, could not be estimated reliably,
and during the fitting process the tendency was for this
parameter to acquire values in the direction where Model II
approximates to Model I.
The parameters of the models may be interpreted similarly
to those of the classical growth models. They have the
additional parameter, µmin, that determines how fast y
decays after reaching its maximum ; this gives the functions
their characteristic turning point. Thus, as with the
Gompertz function, y has a variable relative growth rate
that starts at µ , at time t , when y ¯ y , and that tends to
!
!
!
µmin as t increases. The rate constant, k, determines how fast
this change occurs. When µmin ¯ 0, this equates to the
Gompertz function, and y levels out to an asymptote, ymax.
However, when µmin ! 0, y has a turning point at (tmax,
ymax), and then decays and tends to zero. Model II has the
additional parameter µmax, such that µmax and µmin are
theoretical limits of the relative growth rate of y. The
parameter k determines how fast the relative growth rate
changes from µmax to µmin. It can be fast, acting like a switch
and generating an acute turning point, or it may be more
gradual, when Model II approximates to Model I. When
µmin ¯ 0, Model II equates to the Richards function (Richter
et al., 1991), which in its general form has been found to be
much less applicable than its special cases, notably the
logistic, Gompertz and von Bertalanffy functions. In Model
III the growth rate of y is limited by some maximum value
662
Werker and Jaggard—Modelling Asymmetrical Growth CurŠes
T     1. Estimated parameter Šalues and residual sums of squares (with d.f. in parentheses) on fitting Models I [eqns (3) and
(4)], II [eqns (7) and (8)] and III [eqns (11) and (12)] to measurements of foliage coŠer of sugar beet crops grown at IACR
Broom’s Barn during 1980–1991
Model and estimated parameter values and range of values
amongst years (means of ranges in parentheses)
I
I
I
I
II
III
III
III
III
Common
Derived
Common
Separate
Derived
Common
Separate
Derived
Common
Separate
Derived
Common
Common
Derived
Common
Separate
Derived
Common
Separate
Derived
Common
Separate
Derived
t ¯ 140
µmin ¯®0±00017
!
k ¯ 0±00587
µmin ¯®0±00017
ymax ¯ 0±904
t ¯ 66–230 (140)
!
k ¯ 0±00617
ymax ¯ 0±908
µ ¯ 0±0689
!
t ¯ 79–232 (155)
!
µmin ¯®0±0041–®0±00007 (®0±00017)
k ¯ 0±00606–0±00621 (0±00616)
µ ¯ 0±0746
!
t ¯ 93–267 (185)
!
µmin ¯®0±0037–®0±00007 (®0±00016)
ymax ¯ 0±717–0±964 (0±903)
k ¯ 0±00660–0±00683 (0±00694)
µmin ¯®0±00017
tmax ¯ 1158
k ¯ 0±00583
t ¯ 68
µmin ¯®0±00168
!
k ¯ 0±00260
µmin ¯®0±00145
ymax ¯ 0±892
t ¯ 32–188 (115)
!
k ¯ 0±00306
ymax ¯ 0±894
µ ¯ 0±0467
!
t ¯ 58–203 (126)
!
µmin ¯®0±00231–®0±00070 (®0±00145)
k ¯ 0±00300–0±00331 (0±00315)
µ ¯ 0±0470
!
t ¯ 17–224 (133)
!
µmin ¯®0±00280–®0±00072 (®0±00145)
ymax ¯ 0±743–0±946 (0±892)
k ¯ 0±00287–0±00344 (0±00316)
Residual sums of squares
(d.f. in parentheses)
ymax ¯ 0±902
µ ¯ 0±0656
!
rss ¯ 1±115 (120)
µ ¯ 0±0689
!
rss ¯ 0±5736 (109)
rss ¯ 0±4153 (103)
ymax ¯ 0±903
ymax ¯ 0±894
rss ¯ 0±2472 (92)
µmax ¯ 76±8
rss ¯ 1±114 (119)
µ ¯ 0±0399
!
rss ¯ 1±130 (120)
µ ¯ 0±0456
!
rss ¯ 0±5734 (109)
rss ¯ 0±4451 (103)
rss ¯ 0±2617 (92)
In all cases, the parameter y was fixed at y ¯ 0±000015, defined as the mean leaf area of sugar beet cotyledons at 50 % emergence. For Models
!
!
I and III the parameter k is not estimated ; it is derived from the remaining parameters (see text). Where separate parameter values were estimated
amongst years, only the range and means (in parentheses) are given. Note : fitting of Model II failed to converge.
for y, or the carrying capacity κ. κ has the effect of slowing
down both the growth rate, and rate of decay, of y when y
is near κ. This is quite appropriate for variables such as
foliage cover which are limited by space and which cannot
exceed one. Foliage cover is a function of leaf area : the area
may continue to increase or to decay for a period without
affecting foliage cover itself.
Figure 4 summarizes the sensitivity of Model I to small
changes in the values of the parameters for three formulations
of the model given by eqns (3), (5) and (17). These show
some practical benefits afforded by eqn (17), notably with
respect to fitting the models to data. For example, changes
in the parameters y , t , µ , ymax and µmin are least correlated,
! ! !
and have very distinct and identifiable effects on the shape
of curves. Conversely, given eqn (3), k and µ are inversely
!
correlated and have large effects, typically on ymax (Fig. 4 A
and C). When ymax is held constant and either µ or k is
eliminated (eqns (5) or (17), respectively), then µ or k affect
!
the rate with which y increases from y to ymax by either
!
displacing t (eqn (5), Fig. 4 K) or tmax (eqn (17), Fig. 4 G).
!
Similarly, changes to µmin not only affect the rate of decay of
y, but also ymax eqn (3), Fig. 4 D, t (eqn (5), Fig. 4 L) or tmax
!
(eqn (17), Fig. 4 H). Also, changes in y affect ymax (eqn (3),
!
Fig. 4 B), or tmax (eqn (17), Fig. 4 F). Thus, using eqn (17),
subsidiary changes in the shape of the curves when increasing
or decreasing y , t , µ or µmin are on tmax. The remaining
! ! !
parameters, t (eqns (1) and (17), Fig. 4 E) and tmax (eqn (5),
!
Fig. 4 I) identically displace the curve along the x-axis.
The models were formulated by considering net growth
rates, and no attempt was made to partition net growth into
new growth and senescence. Consider, for example, the
relative growth rate of y for Model I [eqn (3)]. There are a
number of ways in which the function of µ may be partitioned into new growth ( β) and senescence (δ ). For example,
if β ¯ µ exp [®k(t®t )] and δ ¯ µmin²1®exp [®k(t®t )]´
!
!
!
satisfies eqns (1) and (3), then the relative growth rate
decays exponentially from µ to zero, and senescence
!
starts at zero and increases with ever-decreasing gradient to
µmin ; this is reasonable. Equally feasible, however, is that
the relative rate of senescence is constant, δ ¯ µmin, with
the relative rate of new growth, β ¯ B exp [®k(t®t )],
!
decreasing to zero, as in the first example (thus, the
Werker and Jaggard—Modelling Asymmetrical Growth CurŠes
0.8
0.4
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
y, foliage cover
0.8
0.4
0.8
0.4
0.8
0.4
0
1000
2000
1000
2000
1000
2000
T, accumulated temperature > 3°C from sowing
F. 3. Fitted profiles for Models I (——) and III (- - - -), with variable
parameter values for t , µmin and ymax amongst years, to measures of
!
foliage cover of sugar beet crops grown at IACR Broom’s Barn during
1980–1991.
initial relative growth rate ¯ B ¯ µ ®µmin). A third
!
alternative is the reverse of the second, where the relative
growth rate, β ¯ µmin is constant (with µmin " 0) and
senescence, δ ¯ D exp [®k(t®t )] starts at zero and tends to
!
D, (where D ¯ ( µ ®µmin) and D ! 0). These models satisfy
!
a number of scenarios and without specific data (e.g.
Vandendriessche et al., 1990), attributing parameter values
to ‘ new growth ’ and ‘ senescence ’ is not justified. The
models accommodate more general processes, such as
growth limited by the concentration of a second substrate,
µ, for example, nitrogen. Alternatively, the growth rate of y
depends on a partitioning function µ, that decays with time
as other plant components, in particular storage organs,
increasingly obtain a larger proportion of the assimilates.
The models were satisfactory in describing the seasonal
dynamics of foliage cover of sugar beet crops grown under
standard farm practices. Yet, the analyses showed that
significant improvements were obtained in the fits when
separate values were estimated amongst years for up to
three of the four parameters. This seems oversensitive in
what is, in effect, random variation. However, variability in
two of the parameters, t and µmin, was expected. Crop
!
emergence (t ) is delayed when there is insufficient moisture
!
in the soil (Durrant et al., 1988). This is common because
sowing can only be carried out in dry conditions and
because, in eastern England where these crops were grown,
there is often very little rain for 1 or 2 weeks thereafter. At
the other end of the growing season, the impact of foliar
663
diseases (in particular virus yellows), and the scarcity of
mineral nitrogen affect the rate of late season decay of
foliage cover, µmin. The extent of virus yellows in any year
is very variable and depends much on the weather during
the preceding winter (Harrington, Dewar and George,
1989). In well-managed crops, the scarcity of mineral
nitrogen during autumn means that nitrogen is translocated
from old leaves to sustain crop growth (Burcky and Biscoe,
1983), and this can accelerate senescence. In addition to
senescence, the yellowing of the leaves further reduces
estimates of foliage cover when these are derived from
measures of canopy reflectance (Steven and Jaggard, 1995),
which are concerned with measuring ‘ green ’ leaf cover. This
may explain why Model I fitted the data better than Model
III : the rapid decline in ‘ green ’ foliage cover observed in
some years cannot be accommodated by Model III.
Estimating the parameter κ (the upper limit that foliage
cover can reach) improved the goodness of fit of Model III,
but resulted in estimates of κ " 1 which is clearly not
possible. Indeed, in practice κ ! 1 because 100 % foliage
cover in sugar beet crops is rare. Model III is perhaps more
suited to describing the absolute foliage cover, irrespective
of its colour.
Foliage cover is a measure of the potential growth rate of
the crop, rather like LAI which is the more traditional
variable associated with determining light interception. The
relationship between foliage cover and LAI is well described
by a Mitscherlich (or monomolecular) function (Steven et
al., 1986 ; Werker et al., 1995). For sugar beet grown in the
UK, foliage cover provides a direct measure of radiation
interception (Steven et al., 1986 ; Jaggard and Clark, 1990).
Thus, extending eqn (2) to include dry matter production as
a function of radiation intercepted for sugar beet crops
grown in the UK, gives,
dW
¯ εSy
dt
dy
¯ µy
dt
(18)
dµ
¯®k( µ®µmin),
dt
where W is the total dry matter, S is the incident solar
radiation, and ε a conversion coefficient (Steven et al.,
1983). More recently, interest has developed in using
measures of foliage cover, in particular those obtained from
satellite images, to ‘ update ’ or ‘ tune ’ crop growth models
(Bouman, 1992 ; Steven and Jaggard, 1995). Usually, this
requires a sequence of measures of foliage cover to estimate
the growth curve. Baret et al. (1988) suggest three to five
measurements are required depending on the complexity of
the model. Model I has four parameters and therefore
requires a minimum of four measurements to calculate
values for all the parameters. A statistical fit requires five
measures. When there are few points, some of the parameters
could be fixed (for example, µ ). Alternatively, historical
!
data could be used to restrict the range of possible values the
parameters can acquire in the fitting process.
664
Werker and Jaggard—Modelling Asymmetrical Growth CurŠes
E
A
I
0.8
k × 0.9, 1.1
(eqn 3)
t0 × 0.5, 2
(eqn 3 & 17)
0.4
B
tmax × 0.9, 1.1
(eqn 5)
F
J
0.8
y, foliage cover
0.4
y0 × 0.9, 1.1
(eqn 3)
C
y0 × 0.5, 2
(eqn 17)
ymax × 0.9, 1.1
(eqn 5 & 17)
G
K
0.8
µ0 × 0.9, 1.1
(eqn 3)
0.4
k × 0.5, 2
(eqn 5)
µ0 × 0.5, 2
(eqn 17)
D
H
L
0.8
0.4
0
µmin × 0.5, 2
(eqn 3)
µmin × 0.5, 2
(eqn 17)
1000
1000
2000
µmin × 0.5, 2
(eqn 5)
2000
1000
2000
T, accumulated temperature
F. 4. Effects of increasing ([[[[[[) or decreasing (- - - -) y , t , µ , k, ymax or µmin with respect to Model I given eqn (3) (plots A–E), eqn (5) (plots
! ! !
I–L) or eqn (17) (plots E–H and J) on the shape of curves. The default curve (——) is identical throughout and its parameter values are given
in Table 1, Model I, with all parameters common amongst years.
CONCLUSIONS
This paper has demonstrated the derivation, interpretation
and fitting of three non-linear models to describe the growth
of plant components that typically rise, and then fall, during
the course of a growing season. The models are related to
the Gompertz, Richards and Chanter growth equations and
include the additional parameter that determines the rate of
decline of a crop or plant component after reaching a
maximum. Model I, the simplest of the three, has four
independent parameters and described, with relative ease,
data on the seasonal dynamics of foliage cover of sugar beet
crops spanning 12 years. Two five-parameter models were
also tested. One failed to optimize when fitting to data, the
other, Model III, described the data well, but the need for
the additional parameter could not be justified on the basis
of goodness of fit, despite this model being theoretically
suited to data constituting proportions, such as foliage
cover.
A C K N O W L E D G E M E N TS
This work is financed from the UK Home Grown Sugar
Beet Research and Education Fund. IACR receives grantaided support from the Biotechnology and Biological
Sciences Research Council of the United Kingdom.
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