Annals of Botany 78 : 203–214, 1996
Dynamics of Competition in Populations of Carrot (Daucus carota)
B O L I*†, A N D R E W R. W A T K I N S ON*§ and T O S H I H I K O H A R A‡
* Schools of Biological and EnŠironmental Sciences, UniŠersity of East Anglia, Norwich NR4 7TJ, UK and
‡ Department of Systems Science (Biology Section), The UniŠersity of Tokyo, Komaba 3-8-1, Meguro, Tokyo 153,
Japan
Received : 28 September 1995
Accepted : 15 February 1996
Populations of carrot (Daucus carota) were raised over a wide range of densities (79–5763 plants m−#) to examine the
dynamics of competition in terms of yield–density relationships and size variability, and to investigate the effects of
nutrient supply on competition. While the relationship between shoot yield and density was asymptotic, the
relationship between root and total yield and density tended to be parabolic. For a given time and density series the
relationship between yield per unit area and density could best be described by the model :
y¯
wm D
(1­aD)b
where y is the yield per unit area, D is density, wm, a and b are fitted parameters. The parameters wm and a increased
over time but nutrient availability affected only wm. An extension of the basic yield-density model is proposed to
describe the dynamics of the yield–density relationship over time :
y¯
kD
[1­c exp (®rt)] ²1­α[k}(1­c exp (®rt))]βD´b
in which t is time, k, c, r, α and β are fitted parameters, and the other parameters are as previously defined.
Size variability of individuals, measured by the coefficient of variation (CV), was influenced by both density and
time after sowing. The general relationship between CV and the logarithm of mean weight per plant, after first harvest,
could be described by a linear regression model, with a slope of approximately ®0±2. A theoretical justification for
a monotonically declining relationship between CV and mean plant weight is proposed. The magnitude of size
variability was ranked in the order : root weight " total weight " shoot weight. The differences in the magnitude of
size variability in yield components were due in part to allometric growth of plant parts. Nutrient availability,
however, had no influence on size variability.
# 1996 Annals of Botany Company
Key words : Allometry, carrot, coefficient of variation, competition, Daucus carota L., monoculture, nutrient
availability, size variability, yield}density relationship.
INTRODUCTION
Three effects of intraspecific competition as a result of
density-dependence have been recognized within plant
monocultures : (1) a reduction in the mean size of surviving
plants with increasing density (the competition-density
effect) ; (2) a decrease in the probability of survival (densitydependent mortality or self-thinning) ; and (3) an alteration
in the size structure of the population (Watkinson, 1980 ;
Weiner, 1988). A range of mathematical models has been
proposed to describe the consequences of competition in
plant monocultures sown over a range of densities (Willey
and Heath, 1969). None is universally accepted, although
the reciprocal equation proposed by Shinozaki and Kira
(1956) is widely used :
§ For correspondence.
† Present address : College of Life Sciences, Wuhan University,
Wuhan, Hubei (430072), The People’s Republic of China.
‡ Present address : The Institute of Low Temperature Science,
Hokkaido University, Sapporo 060, Japan.
0305-7364}96}080203­12 $18.00}0
y¯
D
a­bD
(1)
where y is yield per unit area, D is density and a and b are
fitted parameters. This equation defines an asymptotic
yield–density relationship often referred to as the law of
‘ constant final yield ’. Many studies (see review by Willey
and Heath, 1969), however, have shown that there are two
forms of yield-density relationship : asymptotic and parabolic. Obviously, the reciprocal equation fails to describe
parabolic yield–density relationship, but a simple modification allows it to do so (Bleasdale and Nelder, 1960).
Watkinson (1980) rescaled the model proposed by Bleasdale
and Nelder (1960) so that the relationship between yield ( y)
and density (D) can be described by the equation
y¯
wm D
(1­aD)b
(2)
where wm, a and b are fitted parameters. The value of wm is
# 1996 Annals of Botany Company
204
Li et al.—Dynamics of Competition in Monocultures
the predicted mean weight of a plant when D ' 1}a, while
the value of b determines the shape of the yield–density
curve ; when b ¯ 1 eqns (1) and (2) are identical.
Equations (1) and (2) provide a static description of the
effects of competition, which is of course a time-dependent
process. This can be illustrated both by an increase in
mortality in high density populations with the passage of
time (Watkinson, 1980) and by changes in the parameters in
the yield equation in non-thinning populations (Watkinson,
1984). Much attention has been paid to the time course of
density-dependent mortality (self-thinning) (see reviews by
Westoby, 1984 and Lonsdale, 1990). However, the dynamics
of the yield–density relationship in non-thinning populations
has been less extensively studied. Watkinson (1984) examined the dynamics of yield–density relationships in Vulpia
fasciculata and found that all three parameters in eqn (2)
increased over time. However, he did not translate the
parameters into functions of time. Several investigators (e.g.
Baeumer and de Wit, 1968 ; Scaife and Jones, 1976 ; Barnes,
1977 ; Hardwick and Andrews, 1983 ; Benjamin 1988 ;
Mutsaers, 1989 ; Aikman and Benjamin, 1994 ; Benjamin
and Aikman, 1995) have included time in competition
models, but only some have explicitly addressed the effects
of time on yield–density relationships.
Scaife and Jones (1976) assumed that under conditions of
ample water and nutrients, the ceiling yield might be
regarded as constant for a given level of radiation and
extended the reciprocal equation [eqn (1)] to :
" ­bD
(3)
w−" ¯ w−"e−kt­w−max
!
where w is the shoot dry weight at time t, w is the same at
!
time 0, and wmax is the asymptotic value of w, which would
be reached by a plant growing in isolation ; k is the early
relative growth rate of isolated plants. If we are only
concerned with the early vegetative growth of plants, when
the growth of plants may be exponential, the above equation
can be simplified to (Scaife, Cox and Morris, 1987) :
(4)
w−" ¯ w e−kt­bD
!
Similarly, a simple and effective model was proposed by
Hardwick and Andrews (1983), which was based on the
reciprocal model. Obviously their model and eqn (1) share
the same problems in power of description.
Another consequence of plant competition in monocultures is size hierarchy development. For ecological,
evolutionary and even economic reasons, studies of size
hierarchies are of considerable importance (Weiner and
Thomas, 1986). The question of size hierarchy development
in monocultures sown over a range of densities is of
particular importance for commercial producers of crops
such as vegetables, as the consumer often prefers uniform
produce. In the case of carrots, studies have been made of
the growth response of field-grown carrot plants to the
spatial distribution of neighbours (Sutherland and Benjamin, 1993), the development of size hierarchies (Currah,
1975), the control of root sizes in the field (Benjamin and
Sutherland, 1992) and the allometric relationship between
root and shoot weight of field populations (Stanhill, 1977 a,
b ; Benjamin and Wren, 1978 ; Currah and Barnes, 1979 ;
Hole et al., 1983). Less is known about the dynamics of
competition over time and the effects of nutrient levels on
plant performance.
The aim of this study is to examine the dynamics of
competition in terms of the yield–density relationship and
the development of size hierarchy in order to determine, (1)
the changes in the relationship between yield and density
over time ; (2) the effects of resource availability on yield–
density relationships ; (3) the changes in dry matter
distribution between the shoot and storage root over time
and at different nutrient levels ; and (4) the effects of
competition on the size variability of individuals in
greenhouse-raised monocultures of carrot.
MATERIALS AND METHODS
Experimental procedure
Seeds of carrot [Daucus carota L. variety Early Scarlet Horn
(Unwins Seeds Ltd.)] were sown in Levington Multipurpose Compost (Fisons plc) in 127 mm diameter pots on
23 Feb. 1992 at densities of 1, 3, 7, 19, 37, and 73 seeds per
pot. These densities are equivalent to 79, 237, 553, 1500,
2921, and 5763 seeds m−#. Seeds were sown in a hexagonal
pattern, so that each individual was surrounded by six
equidistant neighbours. This spatial pattern was achieved
with the aid of a paper template, the seeds being sown
through holes in the template. Thirty-six pots of each
density were sown to allow for three nutrient treatments,
four harvests and three replicates. To facilitate statistical
analysis, density 1 had nine replicates. The nutrient
treatments included three nutrient levels—low (L), medium
(M) and high (H), in which 100 ml of one-ninth strength (L),
one-third strength (M) and full strength (H) of the Standard
Stock Solution made from Solinure No. 7 (Fisons plc. 1986)
were given to the appropriate pots weekly after the first
harvest. The pots were arranged in a randomised block
design in a heated greenhouse with supplementary lighting
(Sodium light) to provide a 16 h day. Five days after
sowing, seedlings started to emerge, and seedling emergence
reached its peak a week after sowing. Pots were watered
regularly to keep the compost moist.
Four consecutive harvests were made on 23 Mar., 25
Apr., 9 May and 23 May. At each harvest, dry weight of
shoots and roots and number of leaves of each individual
were determined for each pot. The dry weight determinations
were made after the shoots and roots had been dried to
constant weight at 70 °C.
Data analysis
On the basis of a comparison of the most commonly used
yield-density models, eqn (2) was fitted to the data (Li,
1995). The quasi-Newton estimation method (SYSTAT,
1992) was used to fit this model, as the base-line yield–density
model, to the data on total, shoot and root dry weight over
four harvests. The parameters in this model were further
analysed to examine the changes over time so that an
extension of this model could be derived to include time.
In order to assess plant size distributions quantitatively,
many measures have been proposed (Hutchings, 1986). In a
Li et al.—Dynamics of Competition in Monocultures
comparison of the skewness coefficient, coefficient of
variation and Gini coefficient as measures of inequality
within populations, Bendel et al. (1989) proposed that the
coefficient of variation (CV) is a more appropriate predictor
of size variability, if preference is given to measures of
relative precision. The reason is that the CV is more
sensitive to observations in the right-hand tail of the
distribution. Consequently, in this paper, the CV is used as
the measure of size variability. The sample CV is defined as
CV ¯ s}xa
where
s¯
'
a ¯ αwβm
n
RESULTS
Yield–density relationship
The data on total, shoot and root dry weight in populations
of carrot sown over a range of densities over four harvests
were fitted to eqn (2). Estimated parameters are presented in
Table 1. Table 1 shows that only five of the 30 estimates of
the parameter b were significantly different from unity,
although the range of estimates was from 0±88 to 8±21.
Nevertheless, it should be noted that there was a tendency
for the estimates of b to be greater than unity ; only seven of
the 30 estimates were less than one. Where there were
deviations from unity, it was for root weight at intermediate
stages of growth. These results indicate that the relationships
between shoot yield per pot and density were asymptotic,
whilst these for total and root yields tended to be parabolic.
There was a linear relationship between the parameter wm
in eqn (2) and the observed mean yield of carrot at density
D ¯ 1 plants per pot, w , indicating that wm reflects the
!
growth of isolated plants (Fig. 1). However, as plant size
increased, w , increasingly underestimated the value of wm.
!
The addition of nutrients resulted in a greater value of wm at
any given harvest (Table 1), i.e. larger plants and a higher
asymptotic yield per pot. In the case of parabolic yield–
density relationships, it led to higher maximum yields
(Table 1).
The relationship between wm and time could be described
by the logistic equation (Fig. 2). This implies that wm in eqn
(2) may be described as a function of time, wm,t, by the
logistic function
k
[1­c exp (®rt)]
(5)
where t is time and c, k and r are fitted parameters.
Both wm and a increased with time (Table 1) and varied
concomitantly with each other (Fig. 3). The parameter a in
eqn (2) can be interpreted as the area an isolated plant
requires to achieve the yield of wm (Watkinson, 1980, 1984).
In consequence, the parameter a may be made a function of
(6)
where α and β are constants with estimated values (³s.e.) of
0±056³0±010 and 0±57³0±08, respectively. The estimated
value of β approximates to # and is thus consistent with a #
$
$
allometric relationship between a and wm according to the
dimensional relationship between area and volume :
a ¯ αw#m/$
and
xa ¯ 3 xi}n.
i="
Analyses of variance were carried out on the CV to test
for the effects of plant density, time (weeks after sowing)
and nutrient availability on size variability.
wm,t ¯
wm. The relationship between a and wm can be described by
the allometric relationship (Fig. 3)
n
3 (xi®xa )#}(n®1),
i−"
205
(6 a)
Similarly, Aikman and Benjamin (1994) have assumed that
the projected zone area of the crown of an isolated plant is
proportional to plant mass to the 2}3 power. Combining
eqn (2) with eqns (5) and (6) allows a dynamic relationship
between total yield per pot at time t ( yT,t) and density to be
produced :
yT,t ¯
kD
(7)
[1­c exp (®rt)] ²1­α[k}(1­c exp (®rt))]βD´b
in which all of the parameters remain as previously defined,
of which only k, c, r and b need to be estimated, α and β are
assumed to have values of 0±056 and 0±57, respectively. The
model fitted to the carrot data on total yield per pot
explained over 98 % of the variation in the data. The
estimated parameters are given in Table 2 and the
corresponding fitted curves are presented in Fig. 4. For each
nutrient treatment, the estimated yield curves are parabolic.
There was a tendency for the addition of nutrients to result
in a decrease in the value of b, while k, c and r increased.
Shoot and root yield can be derived from total yield using
the allometric relationship between the weight of a plant
part and the whole plant :
or
wR ¯ µwγT
(8)
ln wR ¯ ln µ­γ ln wT
(8 a)
where wR is root weight and wT total weight per plant (g),
and µ and γ are constants. It should be noted that the
application of the above allometric relationship to both
shoot and root data, while statistically convenient over a
limited range, is somewhat problematic, as there is a
mathematical contradiction in assuming that ws ¯
µs(ws­wR)γ and wR ¯ µr(ws­wR)γ. Nevertheless, eqns (8 and
8 a) provided a very good fit to the data for all three nutrient
levels at each harvest time ; the estimates of the parameters
are given in Table 3. The addition of nutrients had no effect
on either the slope or the intercept of the allometric
equation (Table 3) but ANOVA reveals that the slopes
differed significantly amongst harvests. At harvests 1, 3 and
4 the slopes of the allometric relationship were not
significantly different from unity (P " 0±10), whereas at
harvest 2 they were significantly less than unity (P ! 0±01).
The difference in the intercepts amongst harvests was even
more significant, with the intercept increasing through time.
In order to model allometric dynamics in root vegetables,
Barnes (1979) derived the following model to relate shoot
weight (ws) to root weight (wR) per plant over time (t) :
ln ws ¯ ζ®ηt­ρ ln wR
(9)
where ζ, η and ρ are fitted parameters. This model implies
206
Li et al.—Dynamics of Competition in Monocultures
T     1. Parameter estimates for the model y ¯ wm D(1­aD)−b, where wm (g), a, and b are the estimated parameters
(³s.e.). T, weeks after sowing ; N, nutrient regime (L, low ; M, medium ; H, high) ; YC, components of yield ; w , obserŠed dry
!
mean yield per pot at density 1 (g) ; yom, obserŠed maximum yield per pot (g) and F, form of yield curŠe* (I, increasing ; A,
asymptotic ; P, parabolic)
Parameter estimates
T
N
YC
4
4
4
8
8
8
8
8
8
8
8
8
10
10
10
10
10
10
10
10
10
12
12
12
12
12
12
12
12
12
—
—
—
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
Total
Shoot
Root
w
!
0±18
0±17
0±02
4±95
3±30
1±65
5±16
3±71
1±45
5±55
4±03
1±51
9±53
5±03
4±51
10±63
5±80
4±82
11±09
5±97
5±12
14±03
5±31
8±72
14±31
5±96
8±37
16±50
6±30
10±20
yom
wm
a
b
R#
F
4±5
4±2
0±4
25±5
16±1
9±4
26±7
16±3
10±8
29±9
19±2
11±0
37±1
16±4
20±8
39±5
20±6
21±0
44±0
21±5
23±4
40±8
17±3
24±8
48±0
20±2
27±9
53±0
22±5
32±9
0±20³0±02
0±18³0±01
0±02³0±00
3±94³0±55
3±00³0±52
1±08³0±09
4±07³0±46
4±30³0±86
1±10³0±06
6±70³0±51
4±75³1±17
1±92³0±48
10±35³1±08
6±34³1±83
4±65³0±41
13±38³0±95
7±89³1±69
6±07³0±21
16±74³3±79
9±69³1±47
6±55³2±80
22±31³7±28
10±46³5±17
12±46³3±38
20±16³0±93
9±11³1±37
11±60³1±09
24±70³3±23
12±50³3±90
13±10³2±30
0±03³0±01
0±03³0±01
0±04³0±03
0±05³0±02
0±08³0±04
0±03³0±01
0±05³0±02
0±23³0±10
0±01³0±00
0±20³0±04
0±24³0±13
0±13³0±08
0±14³0±03
0±32³0±18
0±08³0±02
0±23³0±03
0±42³0±17
0±15³0±15
0±30³0±14
0±41³0±12
0±16³0±16
0±61³0±36
1±10³0±93
0±45³0±22
0±36³0±03
0±58³0±16
0±29³0±05
0±53³0±13
1±07³0±58
0±35³0±12
1±06³0±21
1±09³0±19
0±99³0±36
1±63³0±28
1±48³0±25
2±13³0±29
1±62³0±23
1±03³0±09
8±21³0±44
1±01³0±04
1±00³0±11
1±10³0±19
1±30³0±09
1±06³0±11
1±63³0±15
1±11³0±04
0±97³0±07
1±29³0±09
1±06³0±09
1±02³0±05
1±19³0±30
0±96³0±07
0±86³0±07
1±03³0±08
1±04³0±02
0±93³0±03
1±11³0±04
0±95³0±03
0±85³0±04
1±04³0±06
0±997
0±982
0±989
0±998
0±991
0±997
0±975
0±966
0±999
0±994
0±998
0±976
0±987
0±978
0±958
0±984
0±986
0±966
0±994
0±986
0±987
0±990
0±990
0±980
0±991
0±999
0±986
0±965
0±994
0±989
A
A
A
A
A
P
A
A
P
A
A
A
P
A
P
A
A
P
A
A
A
A
A
A
A
A
A
A
I
A
ln (wm of total weight)
* The arbitrary criteria for distinguishing the form of the yield-density curve are as follows : (1) when LCL (the lower confidence limit at 95 %)
of the estimated parameter b in eqn (2) ! 1 and UCL (the upper confidence limit at 95 %) " 1, the curve is asymptotic ; (2) when UCL ! 1, the
curve is increasing ; and (3) when LCL " 1, the curve is parabolic.
Estimated wm
30
20
10
0
4
8
12
16
wo
F. 1. The relationship across shoot, root and total dry matter (see
Table 1) between wm estimated from eqn (2) and the observed mean
yield per pot (g) of carrot at density 1, w , wm ¯ 1±50w ®0±89, R# ¯
!
!
0±94, n ¯ 30, P ! 0±0001. (——) Regression line, (± ± ± ± ±) the line of
equality (slope ¯ 1).
that the slope of the simple allometric equation is constant
over time, but that the intercept is a linear function of time.
Following Barnes’ basic assumption that the slope is
4
A
B
C
2
0
–2
–4
0
4
8
12
0 4 8 12
0
t (weeks after sowing)
4
8
12
F. 2. The time-dependence of the parameter wm in eqn (2) for total
carrot weight (g dry matter). The equations of the fitted curves for low
(A), medium (B) and high (C) nutrient levels are wm,t ¯ 28±8(1­
2847 e−!±(&t)−", R# ¯ 1, n ¯ 4, wm,t ¯ 24±7(1­3167 e−!±)"t)−", R# ¯ 1,
n ¯ 4 and wm,t ¯ 27±7(1­5536 e−!±*$t)−", R# ¯ 1, n ¯ 4, respectively.
constant over harvests, eqn (8 a) can be written in the
general form :
(10)
ln wR ¯ f(t)­γ ln wT
where f(t) is the intercept of eqn (8 a) and is a function of
time. As γ is around unity, f(t) must be less than or equal to
zero to ensure that less than or at most 100 % of the total
Li et al.—Dynamics of Competition in Monocultures
0
A
8
ln a
–1
4
–2
0
B
60
L
–3
M
H
C
L
M
H
D
L
M
H
30
–1
0
1
ln wm
2
3
Dry weight (g pot–1)
–4
–2
207
4
F. 3. The allometric relationship between the parameters a and wm
calculated from the relationship between total yield and density (Table
1). (± ± ± ± ±) the forced # allometric relationship, described by the equation
$
a ¯ 0±045w#m/$, R# ¯ 0±80, n ¯ 10, P ! 0±001 ; (——) the best-fit relationship, described by the equation a ¯ 0±056w!m±&(, R# ¯ 0±84, n ¯ 10,
P ! 0±001.
where
60
30
biomass is allocated to roots at any growth stage. Therefore,
f(t) can be defined as a boundary function with an upper
limit ¯ 0 when t U¢ :
f(t) ¯ ln [1®exp (φ®ψt)],
0
0
60
φ®ψt ! 0 (11)
30
Combining eqns (10) and (11) gives the allometric model
relating root weight to total weight (wT) and time :
ln wR ¯ ln [1®exp (φ®ψt)]­γ ln wT
0
1
or equivalently
wR ¯ [1®exp (φ®ψt)] wγT.
(12)
4
16 64
1 4 16 64
1 4
Carrot density (plants pot–1)
16 64
F. 4. Dynamics of the relationship between the yield per unit area (g)
of carrots and plant density (plants per pot) at four harvests (A, harvest
1 ; B, harvest 2 ; C, harvest 3 and D, harvest 4) at three nutrient levels
(L, low ; M, medium and H, high). Curves show the predicted yields
from the yield-density-time model [eqns (7) and (12)]. (- -D - -) total
yield ; (—E—) shoot yield and (± ± ± * ± ± ±) root yield.
The data for all three nutrient levels at each harvest time
are presented in Fig. 5, from which it is clear that this model
also provides an adequate fit to the experimental data of
carrots. The estimates of the parameters in eqn (12) are
given in Table 4. Substituting eqn (12) into eqn (7) gives
root yield and hence shoot yield per unit area ; the fitted
models, using the parameter estimates given in Tables 2 and
4, are presented in Fig. 4. The model provides a generally
good fit to the data, although it fails to mimic the parabolic
root yield–density relationships observed at intermediate
stages of growth.
total (P ! 0±001, P ! 0±001), shoot (P ! 0±001, P ! 0±001),
root weight (P ! 0±001, P ! 0±001) and the total number of
leaves (P ! 0±001, P ! 0±05). In contrast, nutrient levels had
no influence on size variability of any of these yield
components.
Figure 6 summarizes the relationship between CV and
mean weight per plant. CV increases with density, at the first
harvest, despite the fact that interference had little impact
on mean plant weight. Subsequently the CV then increased
with time before declining slightly with increasing plant
Size Šariability
Three-way ANOVA revealed that density and time
respectively had a very significant effect on size variability of
T     2. Estimates (³s.e.) of the parameters in eqn (7 ) for the relationship between total yield per unit area, density and
time. R#, coefficient of determination, F, the shape of yield–density curŠe (see Table 1 )
Estimates of the parameters
Nutrient
regime
k
c
r
b
95 % confidence
intervals of b
R#
F
Low
Medium
High
16±9³1±4
20±9³1±0
21±2³1±1
3162³2134
3757³1330
5182³2633
0±89³0±09
0±88³0±05
0±95³0±07
1±12³0±01
1±09³0±01
1±06³0±01
1±12³0±03
1±09³0±01
1±06³0±02
0±984
0±996
0±993
P
P
P
208
Li et al.—Dynamics of Competition in Monocultures
T     3. The parameter estimates for µ and γ in the
allometric equation (wR ¯ µwγT) relating root dry weight
(wR, g per plant) to total dry weight (wT, g per plant) at four
successiŠe harŠests in greenhouse-raised populations of carrot.
The data were fitted using the log-transformed Šersion of the
model [i.e. eqn (8 a)] ; the coefficient of determination (R#) and
the probability (P) of the linear regression are giŠen
Time
(weeks)
Nutrient
regime
Slope
γ³s.e.*
Intercept
ln µ
µ†
R#
P
4
8
8
8
10
10
10
12
12
12
—
Low
Medium
High
Low
Medium
High
Low
Medium
High
1±09³0±05
0±92³0±03
0±91³0±09
0±92³0±06
0±99³0±03
1±00³0±03
0±97³0±04
1±02³0±01
1±03³0±01
0±98³0±03
®2±25
®1±05
®1±08
®1±01
®0±62
®0±68
®0±51
®0±52
®0±53
®0±50
0±11
0±35
0±34
0±36
0±54
0±51
0±53
0±60
0±60
0±62
0±99
0±99
0±96
0±98
0±99
0±99
0±99
1±00
1±00
1±00
! 0±0001
! 0±0001
! 0±0001
! 0±0001
! 0±0001
! 0±0001
! 0±0001
! 0±0001
! 0±0001
! 0±0001
* The slopes across nutrient levels at a given harvest were not
significantly different while there were significant differences across
harvests (F , ¯ 27±8, P ! 0±001).
$'
† The anti-intercepts ( µ) across nutrient levels at a given harvest
were not significant while there were significant differences across
harvests (F , ¯ 508±6, P ! 0±0001).
ln (root dry weight per plant) (g)
$'
3
A
C
B
0
–3
–6
–3
densities after the second harvest (Fig. 7). The general
relationship between the CV of individual plant weight and
the logarithm of mean weight per plant, after the first
harvest, could be described by a linear regression model,
with a slope not being different from ®0±20 (P ¯ 0±59,
n ¯ 9, Fig. 6).
For plant parts, the magnitude of size variability was on
average (across densities and nutrient levels) in the order :
root weight (CV ¯ 0±64) " total weight (CV ¯ 0±54) "
shoot weight (CV ¯ 0±50). The total number of leaves had
much lower variability, and across all treatments the mean
CV was 0±18. Furthermore, the difference in the size
variability of yield components could in part be explained
by the allometric relationship between the weight of a plant
part and the whole plant. The relationship between the
weight of a plant part (wP) and the whole plant (wT) of
individuals at each density and at each nutrient level could
be described by eqn (8 a), or equivalently
wP ¯ µwγT
(13)
ln wP ¯ ln µ­γ ln wT
(13 a)
The mean slope (γ³s.e.) of the relationship between shoot
and total weight was 0±82³0±02, which was significantly
less than unity (P ! 0±0001, n ¯ 60), whereas that between
root and total weight was 1±25³0±02 (n ¯ 60), which was
significantly greater than unity (P ! 0±0001). The slope for
the former was therefore significantly less than that for the
latter (P ! 0±0001, n ¯ 60). When plotting the slope of the
allometric relationships against the ratio of CV of a plant
part yield to that of the total yield, the former explained up
to 28 % of the variation in the ratio (Fig. 8).
DISCUSSION
–1
1
3 –3 –1
1
3 –3 –1
ln (total dry weight per plant) (g)
1
3
F. 5. The relationship between root dry weight per plant and total dry
weight per plant through four successive harvests [harvest 1(E),
harvest 2 (*), harvest 3 (+) and harvest 4 (D)] at three nutrient levels
(A, low ; B, medium and C, high nutrient level). The estimated
parameters for the lines derived from eqn (12) are presented in Table 4.
T     4. Estimates (³s.e.) of the parameters in eqn (12) for
the time-dependent allometric relationship between root and
total yield. R# coefficient of determination
Three phases in the response of plant populations to density
can be distinguished (Antonovics and Levin, 1980 ; Watkinson, 1984). At very low densities, plant competition, if any,
may be very weak, and has little or no effect on the growth
of plants. As density increases, competition becomes more
intense and produces a plastic reduction in plant size.
Density-dependent mortality of the whole plants, known as
self-thinning, occurs as the density further increases. In this
study, mortality did occur, but was very low ; even at the
highest density (73 plants per pot), mortality was only about
1 % (see Li, 1995). The effects of competition observed in
this experiment were therefore restricted to reduced plant
performance and the alteration in the size structure of the
population.
Estimates of the parameters
Nutrient
regime
φ
ψ
γ
R#
Low
Medium
High
0±30³0±03
0±27³0±03
0±30³0±03
0±10³0±01
0±09³0±01
0±10³0±01
1±01³0±03
1±01³0±03
1±00³0±02
0±998
0±996
0±998
weight. The slight decline in size variability might be the
result of plant senescence, which was indicated by a
consistent decrease in the total number of living leaves at all
Dynamics of yield–density relationship
Both wm and a increased with time (Table 1) and varied
concomitantly with each other. The relationship between a
and wm could roughly be described by a simple area}volume
allometry (Fig. 3), which complies with the area-weight
interpretation of a and wm (Watkinson, 1980, 1984). The
addition of nutrients increased the value of wm in eqn (2),
the maximum dry weight of an isolated plant at harvest, and
consequently resulted in higher yields. In populations of V.
Li et al.—Dynamics of Competition in Monocultures
1.0
209
A
L
M
H
0.8
0.6
0.4
0.2
0.1
Coefficient of variation
1.0
1
10
0.1
1
10
0.1
1
10
B
L
M
H
0.8
0.6
0.4
0.2
0.1
1.4
1
10
0.1
1
10
0.1
1
10
C
L
M
H
1.2
1.0
0.8
0.6
0.4
0.2
0.01
0.1
1
10
0.01
0.1
1
10
Plant weight (g) [log scale]
0.01
0.1
1
10
F. 6. Changes in CV with mean total (A), shoot (B) and root (C) weight per plant (g) for sequential harvests (4, 8, 10 and 12 weeks) of
greenhouse-raised populations of carrot at three nutrient levels (L, low ; M, medium and H, high) at six densities [(D) density 1 ; (+) density 3 ;
(^) density 7 ; (E) density 19 ; (*) density 37 and (_) density 73]. Excluding the data at the first harvest, the relationships between CV and ln
mean weight can be described by a linear model. For clarity, the fitted lines are arranged in the top right corner. The fitted equations of the lines
(for all the regression n ¯ 18, P ! 0±0001) for the relationship between CV and mean plant weigh (wk ) are :
A L : CV ¯ 0±72®0±17 ln wk , R# ¯ 0±92, M : CV ¯ 0±78®0±21 ln wk , R# ¯ 0±87 and H : CV ¯ 0±77®0±20 ln wk , R# ¯ 0±84 ;
B L : CV ¯ 0±55®0±16 ln wk , R# ¯ 0±81, M : CV ¯ 0±59®0±19 ln wk , R# ¯ 0±92 and H : CV ¯ 0±57®0±17 ln wk , R# ¯ 0±88 ;
C L : CV ¯ 0±73®0±21 ln wk , R# ¯ 0±90, M : CV ¯ 0±73®0±23 ln wk , R# ¯ 0±87 and H : CV ¯ 0±74®0±21 ln wk , R# ¯ 0±83.
fasciculata, Watkinson (1984) observed that, at a given
point in time, while wm increased, a decreased with increasing
nutrient availability, i.e. a larger weight could be supported
by a given area due to the addition of nutrients. In contrast,
the addition of nutrients had no effect on a in the current
study and consequently a common allometry held across all
nutrient treatments (Fig. 3). This may reflect the relatively
minor impact that nutrient addition had on total yield.
Alternatively, the data from carrots can be viewed as
supporting the ‘ altered-speed ’ hypothesis for the effects of
resource availability on yield–density relationships, while
that for V. fasciculata suggests the ‘ altered-form ’ hypothesis
(see Morris and Myerscough, 1984). Evidence in the
literature is available to support both hypotheses, but it is
not clear under which circumstances and for which species
the alternative responses apply.
The parameter b had different values for different yield
components and over time averaged 1±03 for shoot, 1±17 for
total and 1±97 for root yield. There was, however, no
systematic variation in b, except in that b was higher at
intermediate stages of growth for root yield at low and
medium nutrient levels ; there was a similar tendency for
210
Li et al.—Dynamics of Competition in Monocultures
Living leaves plant–1
10
A
B
C
7
4
1
4
6
8
10
12 4
6
8
10
Weeks after sowing
12
4
6
8
10
12
F. 7. The effects of plant density [densities 1(D), 3 (+), 7 (^), 19 (E), 37 (*) and 73 (_)] and time on number of living leaves per plant at
low (A), medium (B) and high (C) nutrient levels.
Ratio of CVP/CVT
A
B
1.6
1.0
0.4
0.5
0.7
0.9
1.1
0.8
1.0
Slope of the allometric relationship
1.2
1.4
1.6
F. 8. The relationship between the ratio of CV of a plant part (CVP ; A, shoots and B, roots) to that of the whole plant (CVT) and the slope
(γ) of the allometric relationship between the weights of a plant part (wP) and the whole plant (wT) [eqn (13 a)]. The fitted equations of the lines
in A : y ¯ 0±27­0±79x, n ¯ 60, R# ¯ 0±25, P ! 0±0001 and B : y ¯ 0±48³0±60x, n ¯ 60, R# ¯ 0±28, P ! 0±0001.
total yield. There was no indication of b increasing from 0
to 1, as observed in V. fasciculata during the early stages of
growth before canopy closure (Watkinson, 1984). Experiments would have to be carried out during the first 4 weeks
of growth to see if a similar increase in the value of b
occurred during the canopy development of carrot.
The allometric relationship between a and wm offered the
possibility of including time in the total yield–density model,
eqn (2), since the growth of isolated plants, estimated as wm,
could be described by a logistic function. On the basis of the
available data, it is assumed here that b is invariant with
time. The dynamic version of the yield–density model [eqn
(7)] provided a good fit to the experimental data on total
yield for each of the three nutrient treatments. The shoot
and root yields were then predicted on the basis of the
allometric relationship between root and shoot and total
yield (Fig. 4).
A number of studies (e.g. Stanhill, 1977 a, b ; Benjamin
and Wren, 1978 ; Barnes, 1979 ; Currah and Barnes, 1979 ;
Hole et al., 1983) have been carried out to examine allometric
relationships in carrots. Barnes (1979) proposed a quantitative model to describe changes in the allometric
relationships between root and shoot yield over time, in
which the slope of the simple allometric equation is assumed
to be constant over time [eqn (9)]. Other studies (Currah and
Barnes, 1979 ; Hole et al., 1983) including this one (Table 3),
have shown changes in the slope of the allometric
relationship over time. This is, in part, why the dynamic
total yield model [eqn (7)], combined with the dynamic
allometric model [(eqn (12)], cannot accurately describe the
Li et al.—Dynamics of Competition in Monocultures
yield–density relationships for parts at some stages of
growth.
As the dynamic relationship between total yield and
density appears to be parabolic (b " 1), the yield–density
relationships of shoots and roots may be either parabolic or
asymptotic, depending on the relationship between the
weight of the plant part and the whole plant, or the root :
shoot ratio. The experimental data indicate that the root
yield–density relationship (Table 1) is more parabolic than
the relationship for shoots although the slope of the
allometric relationship, γ, is not significantly greater than
unity. This would seem to indicate that the yield–density
relationship for the shoots and roots should be similar,
which is not the case. Clearly the shape of the root and
shoot yield–density relationship depends critically on the
slope of the allometric relationship and how the root : shoot
ratio varies in relation to density. Although little effort has
been directed towards the effects of density on the root : shoot
ratio, a study conducted with carrot by Hole et al. (1983)
shows that plant density influenced the root : shoot ratio of
carrot. Of the four commercial varieties of carrot tested, two
(Super Sprite and Kingston) showed a significant exponential decrease in the root : shoot ratio with increasing
density. There was also a marginally significant negative
exponential decrease in the root : shoot ratio with increasing
density in the rest of the varieties (although they did not test
for this effect). The consequence of this effect is that a simple
allometric equation [eqn (8 a)] cannot account for the
allometric relationship across densities ; a non-linear
function (e.g. quadratic) is necessary. In this study, the
allometric relationships across densities were somewhat
curvilinear in some cases (Fig. 5). Even if only slight
curvilinearity exists, a forced (log-transformed) linear
allometric relationship will create a bias in the shape of the
plant part yield–density relationship, as observed in this
study. Thus, the scenario in which the log-transformed
allometric relationship is used to derive part yield from the
total yield–density relationship requires a ‘ perfect ’ linear
allometric relationship between a plant part and the whole
plant, because a ‘ little ’ relative variation (on a log scale) in
the allometric relationship may translate into a considerable
absolute variation (on an arithmetic scale) in yield. This is
the reason why, although the allometric relationship
provides a good fit to the experimental data here (Fig. 5), a
deviation in expected yield is still apparent from the observed
data (Fig. 4).
Size Šariability
The data on the size variability in dry weight provide
strong support for the general observation (Hara, 1984 a, b ;
Weiner and Thomas, 1986 ; Weiner, 1988) that size
variability increases with population density. These observations are consistent with the dominance and suppression
hypothesis of plant size hierarchy development. Nutrient
availability, however, had no influence on the development
of the size hierarchy of yield components. Similarly Turner
and Rabinowitz (1983) observed that fertilization had no
effect on the skewness of size distributions of individuals in
Festuca populations. The reason for the lack of response in
211
the experiment reported here is unclear, but it should be
noted that the fertilizer treatments in this experiment had a
relatively minor impact on plant yield (Fig. 4), a major
determinant of variability.
Time is another important factor responsible for size
variability (Weiner and Thomas, 1986) because the intensity
of competition varies with time, and because densitydependent mortality of plants (self-thinning) is timedependent (Firbank and Watkinson, 1985). It was observed
in this experiment that at a given density, size variability
increased until the second or third harvest and then
decreased (Fig. 6). In some studies (e.g. Mohler, Marks and
Sprugel, 1978 ; Weiner, 1985 ; Weiner and Thomas, 1986), it
has been shown that size variability increases until the onset
of self-thinning (density-dependent mortality), when it then
declines. In the populations reported here, density-dependent mortality was, however, negligible and restricted to
the two highest densities (Li, 1995). Yet declines in size
variability with time occurred at all densities. An alternative
explanation for the decline in inequality lies in plant
senescence. Figure 7 shows that the number of living leaves
per plant consistently declined after the second harvest
under all nutrient levels and at all the six densities. Although
Fig. 4 shows no consistent decline in shoot biomass after the
second harvest, living shoot biomass per plant was actually
decreasing ; estimates of shoot biomass at the final two
harvests included both dead and living shoots.
The combined effect of density and time on plant
variability can be examined by looking at the relationship
between mean plant weight and the CV of individual plant
weight. Before senescence, mean plant size increases with
time, whereas it decreases with increasing density. Here it
was observed that size variability initially increased and
then declined with increasing mean plant size through time
but consistently increased with plant size as a result of
decreases in density.
Interestingly, the exclusion of the data at the first harvest
led to a linear relationship between CV of individual plant
weight and the logarithm of mean plant weight per plant
over a wide range of densities. A similar negative relationship
was found in self-thinning populations of Abies baslsamea,
Pinus ponderosa and Tagetes patula (Weiner and Thomas,
1986). The slope of the decrease in this study was, on
average, ®0±196 (Fig. 6). This value is identical to the slope
for Abies baslsamea, ®0±195, and also very close to that for
Pinus ponderosa, ®0±208. Although Weiner and Thomas
used the Gini coefficient rather than CV as a measure of size
variability, the comparison is valid because the difference
between these two measures is predominantly in scale. It is
possible that the ®1}5 line describes a boundary, beneath
which any combination of mean plant mass and the CV can
occur, but above which there are no permissible combinations. It is possible that when the CV reaches the ®1}5
line, it will stay there or decline along the line. It is also a
trajectory which the various time-courses approach and
then follow. The mechanisms generating the decrease were
different : in the case reported by Weiner and Thomas (1986)
the decrease was caused by the density-dependent mortality
of the smallest plants whereas, in this study there was a
plastic response to density and plant senescence.
212
Li et al.—Dynamics of Competition in Monocultures
An important implication of this finding is that we can
predict size variability from either the mean plant weight or
density. Note also that the size variability of potato tubers
decreases monotonically with an increase in the density of
tubers (MacKerron, Marshall and Jefferies, 1988). On the
basis of the diffusion equation model proposed by Hara
(1984 a), it can be shown that CV can be expected to become
a monotonically decreasing function of mean plant weight
(see Appendix). Here, we have shown empirically that, from
the time of the second harvest, the CV can be predicted from
mean plant weight, wk , by the equation
CV ¯ k­α ln wk
(14)
where α E®0±20, or the density of a population, D, from
the equation
CV ¯ k­α ln wm(1­aD)−b
(15)
for non-thinning populations, and
CV ¯ k­α ln cD−K
extension of the results obtained here to field-grown carrots.
The carrots in the experiments would have become potbound. It should also be noted that the carrots were grown
over one period of time, at a time of year with increasing
irradiance, March to May, although there was supplementary lighting to give a constant day length. Further
experiments are clearly needed to determine how the time
functions reported here relate to other growth conditions,
radiation regimes and temperatures.
A C K N O W L E D G E M E N TS
We thank Dr A. J. Davy for help and advice during all
aspects of this work, Mr R. Freckleton for statistical advice
and Mr D. Alden for managing the experiments. One of
authors (B.L.) thanks the Chinese State Education Commission, the British Council and Sir Y. K. Pao for providing
financial support.
(16)
for self-thinning populations, because mean weight per
plant of the non-thinning populations is readily predicted
from density by eqn (2), or from the self-thinning relationship wk ¯ cD−K (Yoda et al., 1963) where c and K are
fitted parameters. Under what conditions the above relationships apply is, however, unclear. During the early stages of
growth it appears that the relationship may be non-linear
(Fig. 6).
Amongst plant parts, there was a consistent positive
skewness in the distributions of shoot, root, and total
weights from the first harvest to final harvest (Li, 1995).
This is the common pattern of size distribution amongst
plants (Koyama and Kira, 1956 ; Obeid, Machin and Harper,
1967 ; Hutchings, 1986 ; Hara, 1988). The magnitude of the
size variability was however different for these parts. On
average, the variability in size distributions could be ranked
in the order : root weight " total weight " shoot weight.
This is, in part, a consequence of allometric growth of
plants parts (Fig. 8). The relationship between the weight of
a plant part (wP) and the whole plant (wT) can be described
by eqn (13 a). It is clear that wP will have the same CV as wT
only when γ is equal to unity, i.e. the weight of a plant part
is directly proportional to that of the whole plant. In this
study, for individual plants γ was found to be greater than
unity for the allometric relationship between the weight of
roots and the whole plant of individuals and less than unity
for that between the weight of shoots and the whole plant.
This could, however, explain only a small fraction of the
variability in the ratio of the CV of a plant part to that of
total weight (Fig. 8).
From the current study it is evident that research into
competition should embrace not only yield–density relationships and the effects of competition on dry matter
distribution between plant parts (e.g. Bleasdale, 1967 ;
Watkinson, 1980, 1984 ; Morris and Myerscough, 1987) but
size variability. This study provides an insight into the
competition–allometry–size variability relations. However,
further considerations are obviously needed to build the
theoretical relationship between competition and allometry
and size variability. Caution too needs to be applied in the
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APPENDIX
This appendix derives the CV-mean plant weight relationship expected on the basis of the diffusion equation model
developed by Hara (1984 a) :
¦
1 ¦#
f(t, w) ¯
[C(t, w) f(t, w)]
¦t
2 ¦w#
®
¦
[G(t, w) f(t, w)]®M(t, w) f(t, w) (A1)
¦w
where f(t, w) is the size distribution of plant weight w at time
t, G(t, w) is the mean growth rate of individuals of plant
weight w at time t, C(t, w) is the variance of growth rate of
individuals of plant weight w at time t, and M(t, w) is the
mortality rate of individuals of plant weight w at time t. Let
©g(t, w)ª denote the mean of any continuous function of t
and w, g(t, w), as follows :
©g(t, w)ª ¯
&
wmax
g(t, w) f(t, w) dw
wmin
&
wmax
(A2)
f(t, w) dw
wmin
where wmax and wmin represent maximal and minimal w,
respectively. Note that ©g(t, w)ª is a function of time t only.
Thus ©wª 3 wk in eqn (14). Assuming b ¯ 1 for simplicity
(see Table 2) and from eqns (2) and (6), we have
©wª ¯
y
wm
¯
D (1­αwβm D)
(A3)
where y is yield per unit area, D is density and wm is given
by eqn (5) as a function of time t. From eqn (A3),
d©wª d©wª dwm 1­αDwβw(1®β) dwm
¯
¯
¯ A(t) ©wª#
dt
dwm dt
wm #
dt
©wª
0 1
¯ A(t) ²©w#ª®µ (t)´ ¯ ©A(t) ²©wª#®µ (t)´ª
#
#
(A4)
where
A(t) ¯
1­αDwβm(1®β) dwm
w#
dt
(A5)
m
and µ (t) is the variance of size distribution at time t given
#
as
µ (t) ¯ ©(w®©wª)#ª ¯ ©w#ª®©wª#
#
214
Li et al.—Dynamics of Competition in Monocultures
On the other hand, from Hara (1984 a) and assuming
M(t, w) ¯ 0 (no mortality), we have
From eqn (5), wm(t) U k as t U­¢, and wm(0) ¯ k}(1­c).
Therefore, from eqn (A10), as t U­¢,
d©wª
¯ ©G(t, w)ª
dt
1
1­αkβD
.
oµ (¢) ¯ oµ (0)
#
# 1­c 1­αkβ(1­c)−βD
(A6)
(A11)
On the other hand, as t U­¢ in eqn (A3),
From eqns (A4) and (A6),
G(t, w) ¯ A(t)w#®A(t) µ (t)
(A7)
#
Note that A(t) " 0 and as t U­¢, A(t) U 0 and hence
G(t, w) U 0. This confirms to the general case of G(t, w)
empirically obtained in monocultures of annuals under
crowded conditions (Hara, 1984 a, b) and the simulated
G(t, w) based on the canopy photosynthesis model (Hara,
1986 ; Yokozawa, 1992).
The co-variance of X and Y(X, Y : functions of t and w),
cov ²X, Y ´, is given by cov ²X, Y ´ ¯ ©(X®©X ª) (Y®©Y ª)ª
¯ ©XY ª®©X ª ©Y ª. From Hara (1984 a) and by assuming
C(t, w) ¯ 0 (for simplicity) and M(t, w) ¯ 0 (no mortality),
the time rate of change in µ (t) is given as follows :
#
©wª (t U¢) ¯
1
oµ (¢) ¯ oµ (0)
#
# 1­c
k
²1®(1­c)−β´ ©wª (t U¢)­k(1­c)−β
dµ
# ¯ 2A(t) ©wª ²©wª#®©w#ª´ ¯ 2A(t) ©wª (®µ ) (A8)
#
dt
Therefore,
11
dµ ¯ A(t) ©wª dt
(A9)
®
#
2µ
#
Note that ©wª and A(t) are both functions of wm [eqns (A3)
and (A5), respectively] and hence functions of time t
through eqn (5). Integrating both sides of eqn (A9) from
time 0 to t [from µ (0) to µ (t) for µ and from wm(0) to wm(t)
#
#
#
for wm], gives
w (t)
1­αDwβm(t)
" ln µ (t) ¯ " ln µ (0)®ln m ­ln
#
#
#
#
w (0)
1­αDwβ (0)
m
m
(A10)
(A13)
Therefore, the CV as time t tends to infinity is given as
CV(t U¢) ¯
oµ (¢)
#
©wª (t U¢)
¯
Assuming ©w$ª ¯ ©wª$ as an approximation for the
moment dynamics (e.g. Goel and Richter-Dyn, 1974), we
have
(A12)
Eliminating D from eqns (A11) and (A12), gives
dµ
# ¯ 2 cov ²(w®©wª), G(t, w)´
dt
¯ 2²©wG(t, w)ª®©wª ©G(t, w)ª´
¯ 2²A(t) ©w$ª®A(t) ©wª ©w#ª´. [from eqn (A7)]
k
1­αkβD
1
(A14)
#
κ ²©wª (t U¢)´ ­κ ©wª (t U¢)
"
#
where
1­c®(1­c)"−β
(1­c)"−β
,κ ¯
.
(A15)
#
koµ (0)
oµ (0)
#
#
If we assume that µ (0) (variance of size distribution just
#
before seedling emergence) is constant irrespective of density
D, then κ and κ in eqn (A15) are constants and hence the
"
#
CV-mean plant weight relationship approaches a fixed
boundary given by eqn (A14) as the stand develops (i.e. as
t U­¢). Because κ " 0 and κ " 0 (note that c " 0 and 0
"
#
! β ! 1 ; Table 2 and Fig. 3), CV (t U¢) is a monotonically
decreasing function of mean plant weight ©wª (t U¢). This
agrees with the empirically obtained equation, eqn (14).
κ ¯
"
ADDITIONAL LITERATURE CITED
Goel NS, Richter-Dyn N. 1974. Stochastic methods in biology. New
York : Academic Press.
Hara T. 1986. Growth of individuals in plant populations. Annals of
Botany 57 : 55–68.
Yokozawa M, Hara T. 1992. A canopy photosynthesis model for the
dynamics of size structure and self-thinning in plant populations.
Annals of Botany 70 : 305–316.