Dynamics of Competition in Populations of Carrot (Daucus carota)

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

LITERATURE CITED Aikman DP, Benjamin LR. 1994. A model for plant and crop growth, allowing for competition for light by the use of potential and restricted projected crown zone areas. Annals of Botany 73 : 185–194. Antonovics J, Levin DA. 1980. The ecological and genetic consequences of density-dependent regulation in plants. Annual ReŠiew of Ecology and Systematics 11 : 411–452. Baeumer K, de Wit CT. 1968. Competitive interference of plant species in monoculture and mixed stands. Netherlands Journal of Agricultural Science 16 : 103–122. Barnes A. 1977. The influence of the length of the growth period and planting density on total crop yield. Annals of Botany 41 : 883–895. Barnes A. 1979. Vegetable plant part relationships. II. A quantitative hypothesis for shoot}storage root development. Annals of Botany 43 : 487–499. Bendel RB, Higgins SS, Teberg JE, Pike DA. 1989. Comparison of skewness coefficient, coefficient of variation, and Gini coefficient as inequality measures within populations. Oecologia 78 : 394–400. Benjamin LR. 1988. A single equation to quantify the hierarchy in plant size induced by competition within monocultures. Annals of Botany 62 : 199–214. Benjamin LR, Aikman DP. 1995. Predicting growth in stands of mixed species from that in individual species. Annals of Botany 76 : 31–42. Benjamin LR, Sutherland RA. 1992. Control of mean root weight in carrots (Daucus carota) by varying within-row and between-row spacing. Journal of Agricultural Science 119 : 59–70. Benjamin LR, Wren MJ. 1978. Root development and source–sink relations in carrot, Daucus carota L. Journal of Experimental Botany 29 : 425–433. Bleasdale JKA. 1967. The relationship between the weight of a plant part and total weight as affected by plant density. Journal of Horticulture Science 42 : 51–58. Bleasdale JKA, Nelder JA. 1960. Plant population and crop yield. Nature 188 : 342. Currah IE. 1975. Some factors affecting the size of plants in the carrot crop. PhD thesis, University of London. Currah IE, Barnes A. 1979. Vegetable plant part relationships. I. Effects of time and population density on the shoot and storage root weights of carrot (Daucus carota L.). Annals of Botany 43 : 475–486. Firbank LG, Watkinson AR. 1985. A model of interference within plant monocultures. Journal of Theoretical Biology 116 : 291–311. Hara T. 1984 a. A stochastic model and the moment dynamics of the growth and size distribution in plant populations. Journal of Theoretical Biology 109 : 173–190. Hara T. 1984 b. Dynamics of stand structure in plant monocultures. Journal of Theoretical Biology 110 : 223–239.

Li et al.—Dynamics of Competition in Monocultures Hara T. 1988. Dynamics of size structure in plant populations. Trends in Ecology and EŠolution 3 : 129–133. Hardwick RC, Andrews DJ. 1983. A simple model of the relationship between plant density, plant biomass, and time. Journal of Applied Ecology 20 : 905–914. Hole CC, Barnes A, Thomas TH, Scott PA, Rankin WEF. 1983. Dry matter distribution between the shoot and storage root of carrot (Daucus carota L.). I. Comparison of varieties. Annals of Botany 51 : 175–187. Hutchings MJ. 1986. The structure of plant populations. In : Crawley MJ, ed. Plant ecology. Oxford : Blackwell Scientific Publications, 97–136. Koyama H, Kira T. 1956. Intraspecific competition among higher plants. VIII. Frequency distribution of individual plant weight as affected by the interaction between plants. Journal of the Institute of Polytechnics, Osaka City UniŠersity Series D 7 : 73–94. Li B. 1995. Studies of weed-crop competition. PhD thesis, University of East Anglia, Norwich, UK. Lonsdale WM. 1990. Self-thinning rule, dead or alive ? Ecology 71 : 1373–1388. MacKerron DKL, Marshall B, Jefferies RA. 1988. The distributions of tuber sizes in droughted and irrigated crops of potato. II. Relation between size and weight of tubers and variability of tuber-size distributions. Potato Research 31 : 279–288. Mohler CL, Marks PL, Sprugel DG. 1978. Stand structure and allometry of trees during self-thinning of pure stands. Journal of Ecology 66 : 599–614. Morris EC, Myerscough PJ. 1984. The interaction of density and resource level in monospecific stands of plants : a review of hypotheses and evidence. Australian Journal of Ecology 9 : 51–62. Morris EC, Myerscough PJ. 1987. Allometric effects on plant interference. Annals of Botany 59 : 629–633. Mutsaers HJW. 1989. A dynamic equation for plant interaction and application to yield–density–time relations. Annals of Botany 64 : 521–531. Obeid M, Machin D, Harper JL. 1967. Influence of density on plant variations in fibre flax (Linum usitatissimum). Crop Science 7 : 471–473. Scaife MA, Cox EF, Morris GEL. 1987. The relationship between shoot weight, plant density and time during the propagation of four vegetable species. Annals of Botany 59 : 325–334. Scaife MA, Jones D. 1976. The relationship between crop yield (or mean plant weight) of lettuce and plant density, length of growing period, and initial plant weight. Journal of Agricultural Science 86 : 83–91. Shinozaki K, Kira T. 1956. Intraspecific competition among higher plants : VII. Logistic theory of the C-D effect. Journal of Institute and Polytechnics, Osaka City UniŠersity Series D 7 : 35–72. Stanhill G. 1977 a. Allometric growth studies of the carrot crop. I. Effects of plant development and cultivars. Annals of Botany 41 : 533–540. Stanhill G. 1977 b. Allometric growth studies of the carrot crop. II. Effects of cultural practices and climatic environment. Annals of Botany 41 : 541–552. Sutherland RA, Benjamin LR. 1993. The influence of the arrangement of neighbours on the spatial component of resource capture by an individual plant in even-aged monocultures. Annals of Botany 71 : 131–134. SYSTAT. 1992. SYSTAT, Version 5.2 Edn. Evanston, IL : SYSTAT, Inc. Turner MD, Rabinowitz D. 1983. Factors affecting frequency distributions of plant mass : the absence of dominance and suppression in competing monocultures of Festuca paradoxa. Ecology 64 : 469–475. Watkinson AR. 1980. Density-dependence in single-species populations of plants. Journal of Theoretical Biology 83 : 345–357. Watkinson AR. 1984. Yield-density relationships : the influence of resource availability on growth and self-thinning in populations of Vulpia fasciculata. Annals of Botany 53 : 469–482. Weiner J. 1985. Size hierarchies in experimental populations of annual plants. Ecology 66 : 743–752. Weiner J. 1988. Variation in the performance of individuals in plant populations. In : Davy AJ, Hutchings MJ, Watkinson AR, eds.

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Plant population ecology. Oxford : Blackwell Scientific Publications, 59–81. Weiner J, Thomas SC. 1986. Size variability and competition in plant monocultures. Oikos 47 : 211–222. Westoby M. 1984. The self-thinning rule. AdŠances in Ecological Research 14 : 167–225. Willey RW, Heath SB. 1969. The quantitative relationship between plant population and crop yield. AdŠances in Agronomy 21 : 281–321. Yoda K, Kira T, Ogawa H, Hozumi K. 1963. Self-thinning in overcrowded pure stands under cultivated and natural conditions (intraspecific competition among higher plants. XI). Journal of Biology, Osaka City UniŠersity 14 : 107–129.

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# #

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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.