Annals of Botany 83 : 11–18, 1999
Article No. anbo.1998.0782, available on line at http:\\www.idealibrary.com
Theoretical Relationships Between Mean Plant Size, Size Distribution and Self
Thinning under One-sided Competition
KIHACHIRO KIKUZAWA
Center for Ecological Research, Kyoto UniŠersity, Sakyoku, Kyoto, 606-8502, Japan
Received : 28 October 1997
Returned for revision : 23 February 1998
Accepted : 7 September 1998
As yet there is no comprehensive theory in plant population ecology to explain relationships between mean plant size,
size distribution and self-thinning. In this paper, a new synthesis of plant monocultures is proposed. If the reciprocal
relationship between plant biomass and plant population density among various stands of even-aged plant
populations holds, the same reciprocal relationship must exist between cumulative mass and cumulative number of
plants from the largest individual within a population, assuming strict one-sided competition (which is an extreme
condition for competition for light among plants). The two parameters of the relationship between cumulative mass
and cumulative number within a stand both correlate with maximum plant height in the stand. One parameter equals
the reciprocal of the potential maximum plant mass per area, which is expressed by the product of maximum plant
height and dry-matter density. The other parameter correlates with the potential maximum individual plant mass,
which is allometrically related to maximum plant height. As a stand develops, the growth rate of the smallest
individuals will become zero due to suppression from larger individuals, and they will die ; i.e. self-thinning will occur.
The slope of the self-thinning line is expressed through the coefficients of allometry between height and mass and
between dry matter density and height. When the former coefficient is 3 and the latter is 0, the gradient exactly
corresponds to the value expected from the 3\2 power rule, but it can take various values depending on the values
of the two coefficients. Competition among individuals determines size-density relationships among stands, which in
turn determine the size structure of the stand. The size structure constrains the growth of individuals and results in
self-thinning within the stand.
# 1999 Annals of Botany Company
Key words : Monoculture, plant population, self-thinning, competition, hierarchy, size-structure.
INTRODUCTION
There are several general quantitative relationships in plant
population ecology : populations of the same species growing
in similar geographical regions converge onto a constant
leaf mass per unit land area (Tadaki, 1977) and a constant
dry-matter density per unit volume of forest (Kira and
Shidei, 1967) after canopy closure. That leaf biomass of a
particular plant species per unit land area shows a more or
less constant value suggesting that the amount of light
energy which is shed on the land constrains the amount of
leaves. Dry-matter density is the amount of above-ground
organic matter per unit volume of forest ; it too is considered
more or less constant (Kira and Shidei, 1967) irrespective of
stand age. This trend implies the existence of an upper limit
where organic matter is fully packed in the unit volume of
a forest.
Three types of relationships have been studied in evenaged plant monocultures (Li, Watkinson and Hara, 1996) :
(1) the competition-density effect ; (2) self-thinning ; and (3)
size structure. Mean plant size is related to the reciprocal of
population density (Kira, Ogawa and Sakazaki, 1953 ;
Shinozaki and Kira, 1956). This formulation is supported
by many authors (Bleasdale and Nelder, 1960 ; Watkinson,
1984), with some modifications (Vandermeer, 1984). In a
crowded but actively growing stand, natural thinning usually
occurs while mean plant size increases (mean plant size is
usually expressed by a power function of plant density on
0305-7364\99\010011j08 $30.00\0
unit land area). Yoda et al. (1936) reported that the
exponent (the slope of the log-log plot) between population
density and mean plant size takes the value of k3\2, and
this thinning line was designated as the ‘ k3\2 power law ’
of natural thinning. Many empirical data sets of mean plant
size and plant density have supported this rule (White and
Harper, 1970 ; Westoby, 1984). The power value of 3\2 was
first derived by the comparison of the dimensions of mean
plant size and the reciprocal of allotted area to a mean
individual (Yoda et al., 1963). This basic interpretation was
extended to allometric relationships between mean plant
mass and mean leaf mass (White, 1981 ; Osawa and
Allen, 1993). However, Weller (1987) re-evaluated the law
and found many cases in which the gradient of the line
deviated from k3\2, casting doubt on the existence of a
constant slope. Research on self-thinning should address
the factors that determine whether the slope is k3\2 or
some other value, rather than whether or not there is a ‘ law ’
(Lonsdale, 1990 ; Kikuzawa, 1993 ; Osawa and Allen, 1993 ;
Hamilton, Matthew and Lemaire, 1995 ; Armstrong, 1997).
Although mean population behaviour does not consider
differences among individuals, size often varies considerably
between individuals within a stand, even if individuals are
the same age. These size differences among individuals may
be the result of genetic differences in growth rate or be due
to chance differences e.g. in the timing of seed-emergence
(Weiner and Thomas, 1986). Initial small differences can be
accelerated by competition. Competition among individuals
# 1999 Annals of Botany Company
12
Kikuzawa—Hierarchy and Self-thinning
for light is considered to be one-sided (Cannell, Rothery and
Ford, 1984), because larger individuals shade smaller ones
but the reverse seldom occurs. Hence size hierarchy among
individuals becomes pronounced (Weiner, 1985) resulting in
a positively skewed frequency-distribution of individual
sizes. Hozumi, Shinozaki and Tadaki (1968) developed a
method to describe the size frequency distribution by using
cumulative number (N ) and cumulative biomass (Y ) from
the largest individual in a stand and designated this as the
MNY method. This method was further developed by
Yamakura (1984) and applied for forest management
(Kikuzawa, 1982).
Several attempts have been made to determine the
relationships between the above laws. Hozumi et al. (1968)
and Yamakura (1984) estimated the ranges of parameter
values of their size distributions under conditions of selfthinning or density effect. Westoby (1981) examined the
relationship between size frequency distribution and selfthinning and argued that the mortality process depends on
the suppression of the smallest individuals in the frequency
distribution by larger individuals. Weiner and Thomas
(1986) reviewed self-thinning data and found size-inequality
among plants in a population increases with competition
before mortality is extensive, but then decreases as mortality
proceeds. They concluded that mortality of the smallest
individuals reduces size variability. As Weiner and Thomas
(1986) stated, ‘ A comprehensive theory which explains the
relationships between the growth of individual plants,
density-yield relationships, size distributions and selfthinning is waiting to be uncovered ’. In this paper, I will
derive the size distribution from competition-density effects,
or the relationship between mean plant-size (or total
biomass) and plant population density, assuming one-sided
competition among individuals. I will then derive the selfthinning rule under the assumptions of individual plantheight and plant weight allometry and constant dry-matter
density and will show that self-thinning is the consequence
of shading of small individuals by larger ones under the
constraint of size hierarchy. I will test my theory using
several data sets from pine and birch forests.
ASSUMPTIONS
In this paper, I will assume the following : (1) there is a
reciprocal relation between stand biomass and plant density.
Kira et al. (1953) first found a simple relationship between
mean plant size and population density in an experimental
population of soy bean. This relationship was later
formulated by a reciprocal equation which derived from a
logistic growth equation (Shinozaki and Kira, 1956), 1\w l
ApjB where w is mean plant size and p is plant density per
unit land area and A and B are parameters. By multiplying
both sides of the equation by 1\p, we obtain the relationship
between the biomass of a population per unit land area ( y)
and population density,
1\y l B\pjA
(1)
This law governs total biomass among plant populations
grown at different densities for the same period of time and
appears quite general within plant populations, but it does
not consider variation in individual plant sizes within a
population. (2) Strictly one sided-competition. Individual
plants compete for resources such as light, water and
nutrients. When competing for light, larger individuals have
an advantage since they can shade smaller ones, while
smaller plants cannot shade larger ones (Cannell et al.,
1984 ; Weiner, 1985). Recently, some authors have considered competition for light not to be one-sided in the strict
sense, but rather asymmetrical, where smaller individuals
affect larger ones to some extent (Hara, 1988), but larger
ones are definitely at an advantage. Here I assume that
competition is completely one-sided. (3) Constant drymatter density. Kira and Shidei (1967) divided stand
biomass ( y) by the height of the stand (H ) and designated
this value dry-matter density. They found that dry matter
density of various stands across a range of stand heights
takes a more or less constant value, around 1 kg m−$, except
for extremely dwarf stands with individuals of small but
bent architecture. Here I will use the height of the highest
individuals in the stand (Hmax) instead of mean height (H ).
Stand biomass ( y) will be expressed as,
y l dHmax
(2)
where d is dry matter density and is independent of height,
d l kHmax!. Here I will relax this formulation to obtain the
more general form :
(3)
d l K Hmaxa
"
where K is a constant and a is a parameter. When a is zero,
"
eqn (3) coincides with the constant dry matter density. (4)
Allometric relationship between mass and height. It is well
known that there is a relationship, expressed by a power
function, between individual plant mass (W ) and diameter
or height of the plant, which is designated allometry
(Niklas, 1994). When we consider plant mass and height,
this is expressed as :
W l K Hb
(4)
#
where K is a constant and b takes a value around 3, since
#
plant mass has a dimension of [L$], while height is [L"]. The
allometric relationship among individuals of the same age is
shown to change when plants are competing (Weiner and
Thomas, 1992 ; Weiner and Fishman, 1994 ; Nagashima and
Terashima, 1995). Here I will use eqn (4) as a trajectory of
development of the largest individual.
THEORETICAL CONSIDERATIONS
DeriŠation of Y–N relation
Consider two populations of the same age but different
planting densities and suppose one-sided competition (Fig.
1 A and B). The number of plants per unit land area in a
dense stand (A) is n and in a sparse stand (B) is m. The plant
biomass per unit land area and population density of the
two plots must be related by eqn (1) (Fig. 1 D). Smallest n–m
plants in the crowded stand do not affect the growth of
largest m plants in the same stand (Fig. 1 A), since
competition is one-sided and thus smaller plants have no
effect on larger plants. We can say, therefore, that the larger
m individuals in the crowded stand (Fig. 1 A) are identical in
size to m individuals in the sparse stand (Fig. 1 B) where m
13
Kikuzawa—Hierarchy and Self-thinning
A
B
C
D
Y
y
a
b
1/Y = B/N + A
1/y = B/p + A
N
p
Y–N Relation
within a stand
y–p Relation
among stands
F. 1. Schematic representation of comparison of two stands with different initial planting densities but the same age. A, Dense stand where
smaller individuals () do not affect the size of larger individuals () due to one-sided competition. Therefore the size of larger individuals is
identical to that of individuals in a sparse stand where the number of plants planted equals the number of larger plants in the dense population.
B, Sparse stand where the number of plants planted equals the number of large individuals in the dense stand. They are assumed to be identical
to the largest plants in the dense stand. C, Largest plant number (N ) and largest plant size (Y ) relationship in the dense stand, which is identical
to plant number per plot ( p) and total plant size per plot ( y) relationship. D, Population density ( p) and population biomass ( y) relationship which
was obtained by comparison of dense and sparse stands.
plants were initially planted. Within the dense stand the
largest m plants and the total n plants must also be related
by eqn (1) since the largest m plants in the dense stand are
identical to the total m plants of the sparse stand. This
argument is possible upon any number (m) of largest
plant in the dense stand, thus we can conclude that same
relationship shown in eqn (1) must hold between cumulative
plant size (Y ) and cumulative plant number (N ) from the
largest individual (Fig. 1 C).
1\Y l B\NjA
(5)
To describe skewed size structure within a forest, Hozumi
et al. (1968) independently obtained eqn (5) as an empirical
function in which N and Y are expressed as follows :
Nl
&
Wmax
W
f(W ) dW
(6)
Yl
&
Wmax
Wf(W )dW
(7)
W
where f(W ) is a distribution density function of individual
weight W (Hozumi et al. 1968).
DeriŠation of the relationship between parameters A and B
I will examine the meanings of parameters A and B in eqn
(5). Parameter A is the reciprocal of stand biomass (Ymax )
when N reaches infinity
A l 1\Ymax ,
(8)
while B is the reciprocal of mean plant size when N reaches
zero, or the reciprocal of the maximum individual mass
(Wmax) of the stand
B l 1\Wmax
(9)
Both parameters A and B, together with Wmax and Ymax
will change with the development of the stand and are thus
14
Kikuzawa—Hierarchy and Self-thinning
considered to be functions of time t. They can be expressed
as functions of maximum plant height [H(t)max] at each time
period (t), since plant height increases with time.
By applying eqn (2) to the Ymax–Hmax relationship, we
obtain Y(t)max l dH(t)max. By substituting eqns (3) and (8),
the following is obtained :
")
A(t) l K−" H(t)−(a+
(10)
max
"
By applying eqn (4) to HmaxkWmax allometry, we obtain :
W(t)max l K H(t)bmax
#
Substitution of eqns (9) and (10) into eqn (11) gives
(11)
B(t) l K A(t)b/(a+")
$
where K l Kb/(a+") K−"
$
"
#
(12)
DeriŠation of self-thinning line
Here I will consider changes in stand biomass and
population density under conditions where stand structure
is expressed by eqn (5) and its parameters change with time
following eqn (12). We can assume that the order of
individual sizes within a stand will not change with time
since we presume from assumption (2) (one-sided competition) that larger individuals have greater growth than
smaller individuals.
By differentiating eqn (5) with W, we obtain :
dY\dW l (dY\dN ) (dN\dW ) l [B\(ANjB)#](dN\dW ).
From definitions of Y and N in eqns (6) and (7), dY\dW l
Wf and dN\dW l f, we obtain the following (Hozumi et al.
1968) :
W(t) l B(t)\[A(t)NjB(t)]#.
Substituting eqn (12), we get :
W(t) l oK A(t)b/(a+")q\oA(t)NjK A(t)b/(a+")q#
$
$
Differentiation of this equation with respect to t gives :
dW\dt l (dA\dt)[K N(b\(aj1)k2)Ab/(a+")
$
k(bK #\(aj1))A(#b/(a+")−")]\(ANjK Ab/(a+"))$. (13)
$
$
Now let consider the condition where self-thinning occurs in
eqn (13). Assume individuals will die when the growth rate
or dW\dt equals zero. In an actively growing stand, dA\dt
is not zero, nor is (ANjK Ab(a+"))$. The condition which
$
satisfies dW\dt l 0 in eqn (13), therefore, is obtained as :
N(t) l K A(t)(b/(a+")−")
!
%
where N(t) is number of individuals which satisfy dW\dt l
!
0 and K l bK (aj1)\[b\(aj1)k2]. Substituting this
%
$
equation and eqn (12) into eqn (5), and letting Y which
satisfies dW\dt l 0 be Y(t) , we obtain :
!
(14)
Y(t) l K N(t)[−(a+")/(b−(a+"))]
!
!
&
where K l oK−(a+")/(b−(a+")jK K−b/(b−(a+"))q−". When a l 0
&
%
$ %
[assumption (2)] and b l 3 [assumption (4)], the power in
eqn (14) becomes k1\2, which corresponds to the k3\2
power law.
MATERIALS AND METHODS
Birch stands in Hokkaido, northern Japan (Kikuzawa,
1988) and a pine stand in Tochigi, central Japan (Tadaki et
al. 1979) were used for analysis. The questions addressed
here are as follows : does the cumulative plant size
cumulative plant number relationship follow equation (5) ? ;
is parameter a in eqn (3) nearly zero in stands where selfthinning is occurring ? ; is there an allometric relationship
expressed by (4) between observed maximum plant mass
and observed maximum plant height ? if so, does the
parameter b in eqn (4) take a value near 3 ? ; does the
relationship expressed by eqn (12) exist between parameters
A(t) and B(t) in eqn (5) ? ; is the gradient of the A(t)–B(t)
relationship in naturally thinned stands on a double
logarithmic scale approximated by b\(aj1) ? ; and can the
self-thinning line be approximated by eqn (14) ?
Data from four plots were analysed. The surveyed area
was 100 m# in plots 1, 2 and 4, and 25 m# in plot 3. Plots 1–3
are birch stands in which Japanese mountain birch (Betula
ermanii Cham.) and a few other tree species (B. maximowicziana Regel and Phellodendron amurense Rupr.)
regenerated naturally after a scarification. Plots 1 and 2 are
in Tobetsu (western Hokkaido) and plot 3 is in Ashibetsu
(west-central Hokkaido). Ages of stands at establishment
were 12 years in plots 1 and 2 and 6 years in plot 3. In plot
2, trees were artificially thinned from 200 trees per plot to 19
trees per plot at the time of establishment. Plots 1 and 2
were established in 1983 and plot 3 in 1985. The number of
trees in each plot, stem diameter at breast height (DBH) or
at the ground surface (D ) (when trees were smaller than 1n3
!
m in plot 3) and height of each tree were monitored every
year for 10, 9 and 8 years for plots 1, 2 and 3, respectively
(Kikuzawa, 1988, 1993).
Plot 4 is a pine (Pinus dinsiflora Sieb. et Zucc.) stand
described by Tadaki et al. (1979) in which pines were
naturally regenerated after a clear logging. The plot was
established in 1953 and data sets from 1957 to 1976 were
used in this study. The number of trees in the plot, tree
height and DBH were monitored every 2 years for 20 years
beginning when the stand was 14 years old.
Individual tree sizes (stem volume or above-ground
phytomass) were estimated using allometric relationships
between sizes and DBH or D and height obtained from ten
!
to 160 sample trees which were felled near the plots
(allometries for birch stands were given in Kikuzawa, 1988).
For plots 1–3, stem volume was used and for plot 4 above
ground phytomass was used. Trees were arranged from
largest to smallest in each plot, and cumulative number (N )
and cumulative size (Y ; volume or mass) were calculated
from the largest.
Dry matter density (or stand volume\mass density, d )
was calculated by dividing total tree mass or volume in a
plot by the maximum tree height in the plot for each plot in
each age.
Simple least square linear regressions were applied on
log transformed x, y data sets. Non linear regressions on
"!
eqn (5) were carried out using quasi Newton’s method.
Significance levels adopted in this paper were P 0n001 and
P 0n0001.
15
Kikuzawa—Hierarchy and Self-thinning
RESULTS
Regressions of eqn (5) to actual data sets were fairly good,
as exemplified in Fig. 2. There was no significant correlation
between stand volume density (d ) and maximum tree height
(Hmax) in plot 1 in any year (Fig. 3 A) ; i.e. parameter a in eqn
(3) is nearly zero, which supports assumption (3).
W(t)maxkH(t)max allometry in plot 1 (Fig. 3 B) also fits the
expectation from assumption (4) i.e. the allometric
coefficient is nearly 3. As a result, parameter b in the
B(t)–A(t) relationship shown in eqn (12) also takes a value
of approx. 3 (Fig. 3 C), which fits fairly well with
expectations. As expected from eqns (9) and (8), the stem
volume of the largest tree in the plot, and the total stem
volume within the plot were almost proportional to
reciprocals of parameters B and A in eqn (5), respectively
(data not shown). Total stand volume (Y ) was regressed
!
against total stand number (N ) in each year, resulting in the
!
self-thinning line. The slope of the self-thinning line was
k0n424 (Fig. 3 D). However, the self-thinning line predicted
by eqn (14) deviated somewhat from the data.
Similar results were obtained in plots 3 and 4, where
parameter a in the d–Hmax regression is very small and
natural thinning actively occurs, although the observed
thinning lines deviated slightly from the expected ones. The
natural thinning line in plot 4 (Fig. 4 I) is shallower than
1\2.
In plot 2, parameter a in the d–Hmax relationship is greater
than zero (Fig. 4 A). In this plot, self-thinning hardly occurs
(Fig. 4 C), so no thinning line could be obtained.
DISCUSSION
Nearly half a century has passed since the first monumental
work on the effects of density on plant size by Kira et al.
(1953). Since then, many studies (Watkinson, 1984 ; Begon,
Harper and Townsend, 1996) have been carried out to
support the density effect on plant size expressed by eqn (1).
Yoda et al. (1963) studied the mean plant size (or total
plant size per unit land area) and plant density trajectory
of actively growing plant populations. The trajectory on a
double logarithmic scale usually takes a value around
k3\2. They considered the basis for this trajectory to be the
ratio of dimensions of mean plant mass to mean area
occupied by a plant.
The relationship between the two laws has not yet been
clarified. The solution must be pursued not in the mean size
of a population but in the size structure of a plant
population, since interactions among individuals bring
about density effects and self-thinning. Two kind of
0.2
0.6
B
0.15
Y (m3 100 m–2)
Y (m3 100 m–2)
A
0.1
0.4
0.2
0.05
0
10
5
15
20
0
N (100m )
20
N (100m )
1200
C
D
1000
Y (kg 100 m–2)
1000
Y (kg 100 m–2)
15
2
1200
800
600
400
200
0
10
5
2
800
600
400
200
20
40
60
2
N (100 m )
80
100
0
20
40
60
80
100
2
N (100 m )
F. 2. Cumulative tree size (Y, stem volume or above ground mass) and cumulative tree number (N ) relationships in representative stands and
ages. A, Y–N relationship in plot 2 at age 12. 1\Y l 3n609\Nj41n22 (m$ 100 m−#, 100 m−#) ; r# l 0n994 ; P 0n0001. B, Y–N relationship in plot
2 at age 20. 1\Y l 1n002\Nj11n98 (m$ 100 m−#, 100 m−#) ; r# l 0n9906 ; P 0n0001. C, Y–N relationship in plot 4 at age 14.
1\Y l 0n00157\Nj0n277 (kg 100 m−#, 100 m−#) ; r# l 0n9984 ; P 0n0001. D, Y–N relationship in plot 4 at age 33. 1\Y l 0n000715\Nj0n0140
(kg 100 m−#, 100 m−#) ; r# l 0n9978 ; P 0n0001.
16
Kikuzawa—Hierarchy and Self-thinning
0.1
0.4
Wmax (m3)
Y (m3 100 m–2 m)
A
0.01
7
10
0.1
0.01
17
B
3
10
Hmax (m)
10
300
30
Hmax (m)
D
C
B (m–3)
Yo (m3 100 m–2)
100
1
10
3
0.3
1
2
3
–3
A (100 m m )
0.1
10
100
1000
–2
Ho (100 m )
F. 3. Relationships between parameters and variables of a young birch stand in plot 1. A, A relationship between maximum tree height (Hmax)
n!*)(
in a stand and stand volume density (d ) along stand development. d l 0n0535H!max
(m$ 100 m−$, m) ; r# l 0n102 ; P l 0n368. B, A relationship
n!)&
between Hmax and largest tree volume (Wmax) along stand development. Wmax l 0n0000215H$max
(m$, m) ; r# l 0n921 ; P 0n0001. C, A relationship
between parameters A and B along stand development B l 26n35A#n)$# (m−$, 100 m# m−$) ; r# l 0n9914 ; P 0n0001. D, A relationship between
total tree number (N ) and total tree volume (Y ) in the stand along stand development. Y l 6n524N−!n%#* (m$ 100 m−#, 100 m−#) ; r# l 0n9715 ;
!
!
!
!
P 0n0001. Broken line is an expected thinning line from eqn (14).
interactions, one- or two-sided, have been shown (Weiner
and Thomas, 1986). If plants compete for light, competition
is more one-sided, since larger plants suppress small ones
but smaller plants cannot suppress larger ones. However, if
larger plants are shaded by smaller neighbours in the lower
part of their crown, competition cannot be absolutely onesided, but asymmetrically two-sided (Yokozawa and Hara,
1992 ; Kikuzawa and Umeki, 1996). Moreover, if plants
compete for underground resources, competition is twosided (Hara, 1986 ; Weiner and Thomas, 1986). Therefore,
one sided-competition is an extreme case. However, in plant
populations, competition for light is most important and
thus one-sided competition is not rare. Westoby (1981)
suggested that crowding will increase the size hierarchy
among individuals in a population and suppression of
smaller individuals by larger ones will bring about natural
thinning. Weiner and Thomas (1986) revealed that smaller
individuals actually die in the process of self-thinning.
Kikuzawa (1993) found that smaller trees died by suppression from larger trees in a birch stand.
In this study, I derived the Y–N relationship in eqn (5),
first proposed by Hozumi et al. (1968), from the density
effect expressed in eqn (1), assuming of one-sided competition Under assumptions of independence of dry matter
density on maximum plant height (Hmax) in the stand and an
allometric relationship between maximum plant mass (Wmax)
and Hmax, parameters A and B will change with time
following the relationship expressed in eqn (12). When the
Y–N relation shifts on a trajectory constrained by eqn (12),
smaller plants will die because their growth rate becomes
zero as a consequence of suppression by larger plants. These
are the causal relationships between density effect, size
structure and self-thinning.
During stand development when competition has not yet
occurred, the trajectory of plant biomass density rises
vertically, since no mortality usually occurs. When interactions among individuals start to occur, the trajectory
deviates left of the vertical line. As competition becomes
more severe, the gradient of the trajectory line becomes
shallower, and at last reaches a steady state where the selfthinning line is expressed as an oblique line with a fixed
negative gradient. Yoda et al. (1963) first considered the
gradient of self-thinning to be k3\2 in mean plant size Šs.
plant population relations and k1\2 in total mass Šs.
17
Kikuzawa—Hierarchy and Self-thinning
1
A
B
0.1
0.01
5
10
0.1
0.01
20
Yo (m3 100m–2)
5
Wmax (m3)
1
10
Hmax (m)
D
0.1
10
15
3
–2
Yo (m 25m )
0.01
0.001
1
10
100
F
0.1
0.01
10
100
100
G
100
2
10
Hmax (m)
20
No (25 m )
H
10
1
1000
–2
10000
50
100
10
Hmax (m)
Wmax (kg)
d (kg 100 m–2 m)
1
1
E
Hmax (m)
200
0.1
No (100 m–2)
0.01
0.005
1
Hmax (m)
Wmax (m3)
d (m3 25 m–2 m)
0.1
100
C
Yo (kg 100m–2)
d (m3 100 m–2 m)
0.5
100
10
Hmax (m)
I
1000
100
10
100
1000
10000
No (100 m–2)
F. 4. Relationships between parameters and variables in young birch (plots 2 and 3) and pine (plot 4) stands. A, Relationship between maximum
n&%
tree height in a stand (Hmax) and stand volume density (d ) in plot 2. d l 0n0000739 H#max
(m$ 100 m−$, m) ; r# l 0n690 ; P l 0n006. B, Relationship
n!&
between Hmax in a stand and stem volume of the largest tree (Wmax) in plot 2. Wmax l 0n00003097 H$max
(m$, m) ; r# l 0n873 ; P 0n001. C,
Relationship between number of trees in a stand (N ) and total tree volume (Y ) in the stand in plot 2. D, d against Hmax in plot 3.
!
!
n#"&
n!"
d l 0n0155 H!max
(m$ 25 m−$, m) ; r# l 0n583 ; P l 0n027. E, Wmax against Hmax in plot 3. Wmax l 0n00002379 H$max
(m$, m) ; r# l 0n966 ; P 0n0001.
F, Y against N in plot 3. Y l 1n225 N−!n%!% (m$ 25 m−#, 25 m−#) ; r# l 0n947 ; P 0n0001. Broken line is an expected thinning line from eqn (14).
!
!
!
! !n!$'* (kg m−$, m) ; r# l 0n047 ; P l 0n54. H, W against H
n$&
G, d against Hmax in plot 4. d l 118n5 H−max
in plot 4. Wmax l 0n0294 H$max
(kg, m) ;
max
max
r# l 0n943 ; P 0n0001. I, Y against N in plot 4. Y l 3625 N−!n#'% (kg 100 m−#, 100 m−# ) ; r# l 0n953 ; P 0n0001. Broken line is an expected
!
!
!
!
thinning line from eqn (14).
population relations. In this study, the self-thinning line
is expressed by eqn (14) and the slope by k(aj1)\
[bk(aj1)]. If b l 3 and a l 0, as expected from assumptions (3) and (4), the gradient becomes k1\2. However, the
gradient can take various values depending on a and b. If a
equals zero but b is greater than 3, the self-thinning line
becomes shallower than k1\2 (Fig. 4 I). On the other hand,
when a has a positive value, the slope of the self-thinning
line becomes steeper. From eqn (14), no plant death is
expected to occur when b aj1. When b l 3, if a is greater
than 2, no self-thinning will be found as is seen in plot 2
(Fig. 4 C). Hence eqn (14) could simulate the deeper to
shallower change of the thinning gradient by the systematic
changes in parameters.
In the fully stocked forest stands, parameter a in eqn (3)
may be zero and dry matter density of stands reaches an
upper limit of constant value. However, dry matter density
could be smaller than the upper limit in stands which do not
reach full stock and parameter a in eqn (3) could take a
positive value. Parameter b implies the dependency of plant
mass (W ) on plant height (H ). This dependency has been
shown to change when plants are competing (Weiner and
Thomas, 1992 ; Weiner and Fishman, 1994 ; Nagashima and
Terashima, 1995). However, the allometry between mass
18
Kikuzawa—Hierarchy and Self-thinning
and height of the largest individual is considered to be more
or less constant under strict one-sided competition, since
neighbouring individuals do not affect growth of the largest
individual. In summary, parameter a is considered to
decrease and b to be constant with the development of
stand. Thus the slope of eqn (14) is expected to become
shallower with the development of a stand. This trend is
consistent with the trajectory of actual forest stands (Tadaki,
1963).
A C K N O W L E D G E M E N TS
Field work was supported by the late Norio Mizui, as well
as T. Asai, K. Seiwa, M. Ishida, M. Shibuya, N. Nitta, K.
Umeki, and H. Koyama. T. Takada, T. Hara and Y.
Harada provided helpful mathematical suggestions. I also
thank N. Yamamura and A. Osawa for reading the draft of
the manuscript and anonymous reviewers for helpful
suggestions. This work was supported by the fund
T09304073 of Japanese Ministry of Education, Science,
Sports and Culture.
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