Volume 50, Issue 3 pp. 280-290

Full Access

Relationship between the Virtual Dynamic Thinning Line and the Self-Thinning Boundary Line in Simulated Plant Populations

Kang Chen,

Kang Chen

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Hong-Mei Kang,

Hong-Mei Kang

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Juan Bai,

Juan Bai

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Xiang-Wen Fang,

Xiang-Wen Fang

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Gang Wang,

Corresponding Author

Gang Wang

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

*Author for correspondence.
Tel: +86 (0)931 891 2849;
Fax: +86 (0)931 891 2823;
E-mail: <[email protected]>.Search for more papers by this author

Kang Chen,

Kang Chen

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Hong-Mei Kang,

Hong-Mei Kang

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Juan Bai,

Juan Bai

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Xiang-Wen Fang,

Xiang-Wen Fang

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

Search for more papers by this author

Gang Wang,

Corresponding Author

Gang Wang

Key Laboratory of Arid and Grassland Ecology, Lanzhou University, Ministry of Education , Lanzhou 730000, China

*Author for correspondence.
Tel: +86 (0)931 891 2849;
Fax: +86 (0)931 891 2823;
E-mail: <[email protected]>.Search for more papers by this author

First published: 03 March 2008

https://doi.org/10.1111/j.1744-7909.2007.00618.x

Citations: 4

Supported by the National Natural Science Foundation of China (30270243).

Share a link

Email
Wechat
Bluesky

Abstract

The self-thinning rule defines a straight upper boundary line on log-log scales for all possible combinations of mean individual biomass and density in plant populations. Recently, the traditional slope of the upper boundary line, −3/2, has been challenged by −4/3 which is deduced from some new mechanical theories, like the metabolic theory. More experimental or field studies should be carried out to identify the more accurate self-thinning exponent. But it's hard to obtain the accurate self-thinning exponent by fitting to data points directly because of the intrinsic problem of subjectivity in data selection. The virtual dynamic thinning line is derived from the competition-density (C-D) effect as the initial density tends to be positive infinity, avoiding the data selection process. The purpose of this study was to study the relationship between the virtual dynamic thinning line and the upper boundary line in simulated plant stands. Our research showed that the upper boundary line and the virtual dynamic thinning line were both straight lines on log-log scales. The slopes were almost the same value with only a very little difference of 0.059, and the intercept of the upper boundary line was a little larger than that of the virtual dynamic thinning line. As initial size and spatial distribution patterns became more uniform, the virtual dynamic thinning line was more similar to the upper boundary line. This implies that, given appropriate parameters, the virtual dynamic thinning line may be used as the upper boundary line in simulated plant stands.

The self-thinning rule (also called −3/2 power rule, or Yoda's law) is a demographic relationship that has been widely studied in even-aged monocultures and it was first proposed by Yoda et al. (1963) to take the form:

(1)

where w is the mean shoot biomass per plant, ρ is the plant density, k and −β are the self-thinning coefficient and exponent, respectively. The exponent −β has been claimed to take the value −3/2 approximately regardless of species, age or site conditions, and k varies with species and growth conditions. The relationship of Equation 1 plotted on logarithmic coordinates shows a straight line of slope of −3/2, and this line is called the self-thinning line. Although the −3/2 power rule has been corroborated empirically and theoretically (White and Harper 1970; White 1981; Osawa and Sugita 1989; Sackville Hamilton et al. 1995; Bi et al. 2000; Li et al. 2000), its suitability for describing the self-thinning process has been questioned by some researchers (Weller 1987, 1990, 1991; Zeide 1987; Lonsdale 1990; Berger and Hildenbrandt 2000; Wang et al. 2004). Zeide (1987) argued that the time-trajectory of combinations of mean plant biomass and density in a self-thinning population is a curve concave down, and some researchers proposed models to describe the curve (Shibuya 1995; Hagihara 1998; Ogawa 2005). Researchers have found that the self-thinning rule of Equation 1 describes an upper boundary that cannot be exceeded by combinations of mean individual biomass and density in any self-thinning population of a species. In this paper, we call the self-thinning rule of Equation 1 the upper boundary line, and call the time-trajectory of combinations of mean plant biomass and density in a self-thinning population the dynamic thinning line. Recently, it has been argued that the −3/2 power rule should be the −4/3 law (Enquist et al. 1998; Niklas et al. 2003) based on the metabolic theory (West et al. 1997). Although some researchers (Adler 1996; Li et al. 2000) also provided a range of slope to describe the self-thinning rule, an accurate value is always preferred. More well-controlled experiments should be carried out in order to identify which is more accurate between −3/2 and −4/3. But traditional experimental and field studies on the self-thinning rule have two major problems: one is the subjectivity in data selection and the other is how to choose a suitable statistical method to fit the data (Weller 1987; LaBarbera 1989; Osawa and Sugita 1989); although several researchers proposed alternative methods (LaBarbera 1989; Lonsdale 1990; Sackville Hamilton et al. 1995; Bi et al. 2000) to solve these problems, these methods are all restricted by the subjectivity in data selection more or less.

A theoretical line derived from the competition-density (C-D) effect may help researchers avoid the data selection process. The C-D effect describes the relationship between the mean individual biomass and density in different even-aged monoculture stands at a point in time. Hagihara (1999) established a reciprocal model of the C-D effect in self-thinning populations in a manner similar to the logistic theory of the C-D effect (Shinozaki and Kira 1956). Based on Hagihara's equation, Xue and Hagihara (1998, 2002) derived mathematical forms of mean stem volume and asymptotic density at certain times, as initial density tends to be infinity when they studied the C-D effect in self-thinning stands of Pinus densiflora and P. massoniana. The asymptotic density ρ and mean stem volume w at a given time (t) are formulated as:

(2)

and

(3)

where ɛ(t), A_t and B are coefficients from Hagihara's equation. Xue and Hagihara (1998, 2002) found that, in P. densiflora stands the time-trajectory of the mean stem volume and asymptotic density was less steep than −3/2 and then along a slope of ca. −3/2, and in P. massoniana stands the slope of the trajectory was ca. −1.2 and then reached a value of ca. −1.8. In the present study, we define w as the mean plant biomass and use the term virtual dynamic thinning line instead of time-trajectory of mean plant biomass and asymptotic density. It seemed that the slope of the virtual dynamic thinning line approximately described the self-thinning exponent. However, there are very limited studies on the relationship between the virtual dynamic thinning line and the upper boundary line. If the virtual dynamic thinning line can be regarded as the upper boundary line, researchers can avoid the subjectivity in data selection in self-thinning studies by deriving the data points from the C-D effect.

In this paper we investigate the relationship between the virtual dynamic thinning line and the self-thinning upper boundary line in simulated even-aged plant populations. First, a spatially explicit, individual-based plant competition model based on empirical equations was developed; then the model was validated according to simulation results; finally the possibility of use of the virtual dynamic thinning line as the boundary line was discussed.

In the present study, we developed a new spatially explicit, individual-based plant competition model based on the widely accepted empirical equations. There are three main elements in this model: a general plant growth equation, the allometric relationships between the scales of a plant and competition influence. In our model, plants grow in three dimensions, and the combination of position, size (or biomass), height and stem diameter is used to identify a specific plant. In the absence of competition, the growth equation should produce sigmoidal growth in biomass similar to that observed in plants (Weiner et al. 2001). Among the functions that conform to this assumption, we chose:

(4)

where m is the plant mass at time t, M is the maximum (asymptotic) plant mass, and r is the initial (maximum) relative growth rate in mass per unit mass. The growth trajectories for plant populations tend to show simple allometric relationships among height h, stem diameter D and aboveground biomass m when plants are uncrowded (West et al. 1991; Weiner and Thomas 1992; Weiner and Fishman 1994), i.e.

(5)

and

(6)

where a, b, p, and q are constants.

In crowded populations, plants compete for resources with neighbor individuals and their performances are suppressed because of local competition. The indices of neighborhood interference from suppressors were first formulated by Weiner (1982, 1984), used and improved by other researchers (Zhang et al. 1989; Zhang 1993). In our model, we consider only the competition for sunlight, which is “asymmetric” or “one-sided” (Cannell et al. 1984; Weiner and Thomas 1986; Kikuzawa 1999). For every individual plant, there is a circular area with the initial coordinates of the plant as the center. If other plants grow in this circle, they will be suppressed by the central plant. The central plant is called the suppressor, and the circular area is the “area of suppression” (AOS) of the suppressor. The AOS concept is illustrated in Figure 1. As a plant grows, its AOS becomes larger accordingly. The radius (R) of the AOS of a plant is assumed to be:

(7)

for simplicity, where D is the stem diameter of the plant, and c is a constant. The suppressor has no effect on the plants growing outside the AOS, conforming to the locality nature of plant competition. Within the AOS, the suppression effect placed on a plant by the suppressor is determined by the distance to the suppressor and their relative sizes. This relationship can be formulated as:

(8)

where I is the suppression effect from a suppressor plant, f is the relative size factor, and g is the distance factor. The size asymmetry of competition is reflected in the definition of f:

(9)

where h, D and S are, respectively, the height, the stem diameter and the area of stem cross-section of the suppressed plant and h_s, D_s and S_s are, respectively, the height, the stem diameter and the area of stem cross-section of the suppressor. Plant height and stem diameter are effective aspects of relative size in our model. The exponent 2 in S²_s has two meanings: first, even if the suppressor and the subject plant have the same size, f still increases with growth; second, this assumption can make I dimensionless. g decreases reciprocally with respect to the distance d between the suppressed plant and the suppressor, and has the form:

(10)

At the edge or outside of the AOS, g is zero. Substituting Equations 7, 9 and 10 into Equation 8, we get:

(11)

A plant may be suppressed by more than one suppressor, so the total suppression effect placed on a plant is:

(12)

where NI is the index of neighborhood interference, and i represents the ith suppressor of the subject plant.

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

Diagram of the area of suppression (AOS) concept.
White circles represent AOSs of plants, the central solid points in white circles are the original positions of corresponding plants, and the capital letters denote plants or their AOSs. In the AOS model, if the original position of plant X is in the AOS of plant Y, plant Y will suppress the growth of plant X. We say plant X is in the AOS of plant Y, and X and Y are called the suppressed plant and the suppressor plant, respectively. The intensity of suppression influence placed on plant X by plant Y is determined by their relative sizes and distance. In this diagram, plant A is a suppressor of plants B and C, but is only suppressed by plant B; plants B and A are mutual suppressors; plant C is suppressed by plant A, but is not a suppressor of plant A; plant D is isolated, and its existence and growth don't influence and aren't influenced by other plants. This diagram illustrates that neighboring individuals are not always mutually suppressing in the AOS model.

In crowded populations, the growth equation of Equation 4 is rewritten as:

(13)

where k is a constant. If a plant has no suppressors, i.e. NI = 0, Equation 13 is equivalent to Equation 4. Mortality occurs when dm/dt < 0, i.e. NI > 1/k, and the biomass of a plant is assumed to be zero after the plant dies.

In crowded populations, the m-D allometric trajectory of growth of Equation 6 holds (Weiner and Thomas 1992; Weiner and Fishman 1994; Yang and Titus 2002; Niklas et al. 2003; Brown et al. 2004), while the h-D allometric relationship of growth is influenced by competition (Weiner and Thomas 1992; Weiner and Fishman 1994). Equation 6 remains the same form in crowded stands, whereas Equation 5 is reconstructed to reflect competition interference from suppressors. Weiner and Thomas (1992) proposed two models of the h-D relationship between dynamic allometric trajectories and static among-individual allometric relationships in competing and non-competing populations. Based on Weiner and Thomas's model 1, we rewrite Equation 5. First, we differentiate both sides of the logarithmic form of Equation 5 with respect to physical time t and get:

(14)

According to Weiner and Thomas's model 1, h-D allometric trajectories have the same asymptotic slope on log-log coordinates. Incorporating NI, Equation 14 is changed to:

(15)

where e and j are constants. Then Equation 15 is transformed to:

(16)

Equation 16 describes how the differential growth of h and D of a plant changes with competition in crowded stands. Equations 5, 6, 12, 13 and 16 are basic assumptions for our model plant species.

The AOS model is an improved neighborhood model. In traditional neighborhood models (Pacala and Silander 1985; Pacala 1986, 1987; Zhang et al. 1989; Mabvurira and Miina 2002), the radius of neighborhood is fixed throughout the ontogeny of a plant. Whether two plants are neighbors is only determined by their proximity, and plants are called neighbors mutually. In the AOS model, however, AOS of a plant increases with growth of the plant, reflecting the increasing demand for more resources as plants grow larger, and plants are not necessarily neighbors mutually. Another difference between the AOS model and traditional neighborhood models is that the “neighborhood” (AOS) is defined from the point of the suppressor in the AOS model but from the point of the suppressed individual in traditional neighborhood models. When plants compete for sunlight, larger plants shade smaller neighbors and intercept more solar radiation, whereas smaller plants cannot determine which plants will shade them. So, competition for light can be studied in a more natural way by defining the AOS of a suppressor other than the neighborhood of a suppressed plant.

Results

The C-D effect

Hagihara (1999) developed the reciprocal equation of the C-D effect in self-thinning stands, and the equation has the form:

(17)

where A_t and B are coefficients and they change with time. Coefficient B is defined as

(18)

where τ is the biological time and is first defined by Shinozaki (1961). The reciprocal relationships between mean individual biomass w and plant density ρ in different populations at the ages of 40, 50, 60, 70 and 80 years were evident, and were well described with Equation 17 (Figure 2). Mean individual biomasses in lower density populations were higher than those in higher density stands. With the progress of competition, mean individual biomasses in all seven populations increased, but those in lower density populations increased more than in more crowded populations. At the same time, plant densities of all seven populations decreased, but more plants died in crowded populations than in lower density populations, indicating the existence of an environmental capacity.

Reciprocals of A_t and B in Equation 17 mean, respectively, the asymptote of yield (=wρ) and the asymptote of mean individual biomass at a given point in time. B should decrease monotonically during the development of populations. In the present runs of our model, B increased after 80 years, so only data until 80 years were used in analyzing the C-D effect. A_t gradually decreased from 0.321 m²/kg to 0.055 m²/kg with physical time t, while B decreased more abruptly from 5.216 kg⁻¹ to 0.009 kg⁻¹, close to zero (Figure 3).

Assumption 5 of Hagihara's equation assumes that, the relationship between realized density ρ and initial density ρ_i at a given time conforms to the following equation:

(19)

where ɛ is a coefficient independent of ρ and ρ_i and it changes with time. The ρ-ρ_i relationships in simulated populations were well described with Equation 19 (Figure 4). The ρ-ρ_i curves in Figure 4 concave down with increasing initial density, implicating the existence of an asymptote of density at different given times. ɛ in Equation 19 is proposed to meet the following ɛ-τ relationship (Shinozaki 1961):

(20)

where p and μ are constants, and the biological time τ is derived from Equation 18. Inserting Equation 20, Equation 19 can be rewritten as (Shinozaki 1961):

(21)

The relationships between realized density ρ and biological time τ in all seven populations with different initial densities were well fitted using Equation 21 (Figure 5). Densities of different populations tended to converge to a small range of density, conforming to results of previous field studies (Yoda et al. 1963; Xue and Hagihara 2002).

Dynamic thinning lines

Ogawa (2005) developed a quantitative model to describe the time-trajectory of mean plant biomass w and plant density ρ in a self-thinning population when he studied a self-thinning sugi (Cryptomeria japanica D. Don) plantation. Several earlier models of the same kind (Shibuya 1995; Hagihara 1998) can be regarded as special cases of Ogawa's model, so Ogawa's equation was used in our study to validate the AOS model. The equation has the form:

(22)

where a, c, l, m, and n are coefficients. Let β=m/l and then inline image

, Equation 22 is rewritten as:

(23)

Equation 23 indicates that, during the course of self-thinning, the w-ρ trajectory asymptotically approaches a straight line of slope −β on log-log scales.

In the seven populations with normal initial size and random spatial pattern, the time-trajectories of combinations of mean plant biomass and plant density were seven curves concave down (Figure 6). Mean plant biomass increased as density-dependent mortality occurred, and the combinations of mean plant biomass and density eventually approached a common asymptotic line. This implicitly indicates the existence of an upper boundary line, and also illustrates the difference between dynamic thinning lines and the upper boundary line. The fitting of Equation 23 to data from the population with 29 785 initial individuals is illustrated in Figure 7, which indicates that the dynamic thinning line is well described with Ogawa's model. The values of k and −β in Equation 23 resulting from the fitting in Figure 7 are listed in Table 2.

Table 2. Values (value ±SE) of exponent −β and coefficient k in different types of self-thinning equation

Type of the self-thinning	−β	k
Upper boundary line^a	−1.349 ± 0.020	4.537 ± 0.140
Dynamic thinning line in the population with 29 785 initial plants^b	−0.935 ± 0.605	13.238 ± 63.754
Virtual dynamic thinning line^a	−1.290 ± 0.007	3.120 ± 0.053

^a−β and k are from Equation 1; ^b−β and k are from Equation 23.

Virtual dynamic thinning line and the upper boundary line

Because plants in populations with the same initial size and hexagonal spatial pattern reached the largest sizes at the end of the simulation period, such populations were used to obtain the upper self-thinning boundary line. Density-dependent mortality occurred in four populations with the highest initial densities among the seven such populations. Combinations of density ρ and the corresponding largest mean plant biomass w are fitted well using Equation 1. The values of exponent −β and coefficient k are shown in Table 2 in detail.

The seven random populations were used to study the virtual dynamic thinning line defined by Equations 2 and 3. The w-ρ combinations on the virtual dynamic thinning line are well described with Equation 1 (Figure 8), and the exponent −β and coefficient k in the regression are shown in Table 2. The upper boundary line and the virtual dynamic thinning line are also plotted in Figure 6 on log-log scales. The upper boundary line lies above and is slightly steeper than the virtual dynamic thinning line, and dynamic thinning lines approach asymptotically the virtual dynamic thinning line as initial density gets higher and as competition goes on (Table 2, Figure 6). We also found in our study that the slopes and intercepts of the upper boundary line and the virtual dynamic thinning line were closer as initial spatial and size patterns were more uniform.

Discussion

Model selection

There are two main groups of individual-based models used in studies of the dynamics of plant populations: fixed-radius neighborhood models and zone-of-influence-like models, and the latter also include ordinary zone of influence (ZOI) models, ecological field models and field-of-neighborhood (FON) models. Fixed-radius neighborhood models (Pacala and Silander 1985; Pacala 1986, 1987; Zhang et al. 1989; Mabvurira and Miina 2002) are pioneer individual-based models. In these models, each plant is the center of a fixed-radius circle, and other plants within this circle are neighbors that compete with and influence the performance of the central plant. Fixed-radius neighborhood models are so simplified that they are seldom used now. Ordinary ZOI models (Wyszomirski 1983; Miller and Weiner 1989; Wyszomirski et al. 1999; Weiner et al. 2001) also assume a circular area around each plant, but the area increases as the plant grows and the meaning of the area is explicit: a plant obtains resources over the zone. If the ZOIs of two plants overlapped, they compete for the resources within the overlapping area. One shortcoming of ZOI models is that the influence intensity of a plant in the ZOI does not change. If the overlapping area of two plants remains unchanged, the distance between them does not matter, imagining ZOI of a smaller plant completely within that of a larger plant. Ecological field models (Wu et al. 1985; Mou et al. 1993; Li et al. 2000) and FON models (Berger and Hildenbrandt 2000) can be regarded as the recent developments of ZOI models. These models are similar, but the concept of FON provides more mechanistic resolution and less complexity than the ecological field theory. In these models, not only geometry of interaction but also intensity of interaction over space is defined. FON models have been used to study self-thinning (Berger et al. 2002; Berger and Hildenbrandt 2003), productivity (Berger et al. 2004), and size structures in monocultures (Bauer et al. 2004).

The objective of this study was to evaluate the relationship between the upper boundary line and the virtual dynamic thinning line. The main resource competed for among plants is assumed to be sunlight, and competition for light is asymmetric (Cannell et al. 1984; Weiner and Thomas 1986; Kikuzawa 1999). Taller plants preempt the sunlight and suppress the performances of underlying individuals, but shorter plants cannot shade taller plants. If a neighbor of a plant is defined as the suppressor of the plant, then a very tall plant is a neighbor of a very little plant nearby, but the little plant has little influence on the tall plant and cannot be called a neighbor of the tall plant. We wanted a model with such neighborhood definition as an intrinsic feature when we started this study. In ordinary ZOI models and FON models, plants are always mutual neighbors, so they were not used directly. Adler (1996) also developed an individual-based model to study the self-thinning rule, and we also evaluated his model. Adler's model is elegant when used to study the upper boundary line, but the “effective number” used in the model is less than or equal to the census number, with equality only when all plants have the same size. Adler's model cannot be used to get the virtual dynamic thinning line which is derived from the C-D effect, so it could not be used in this study. For our purposes, we developed a spatially explicit, individual-based model based on the “area of suppression” (AOS) concept. In the AOS model, intensity of competition between two plants depends on their relative sizes and the distance as in the FON approach, but plants influence each other directly, not through the resources in the overlapping area of ZOIs, as in traditional neighborhood models. We use the suppression index other than the resource substitution of ZOI for ease of definition of one-sided AOS and for a less intensive computing workload. Because the AOS of a plant is used just to identify the plants whose performances are suppressed by the plant, it holds the simplicity and computing efficiency of traditional neighborhood models. The allometric relationships of h-D and m-D are incorporated into the AOS model to make a more realistic model plant species. In our study, the C-D effect in simulated plant populations conformed well to Hagihara's model and findings in self-thinning Pinus densiflora and P. massoniana stands (Xue and Hagihara 1998, 2002), and dynamic thinning lines conformed to previous studies (e.g. Zeide 1987; Adler 1996) and were well described with Ogawa's model. It seems that the AOS model can conform well to empirical and theoretical equations (Hagihara's equation of the C-D effect and Ogawa's model of dynamic thinning line) given appropriate values of the parameters.

Possibility of using the virtual dynamic thinning line as the upper boundary line

Xue and Hagihara (1998) found that, in self-thinning P. densiflora stands, data points under the asymptotic density condition moved along a line with a slope slightly less steep than −3/2 and then along a line with a slope of ca. −3/2. It seems there may be a relationship between the virtual dynamic thinning line and the self-thinning upper boundary line. But they did not provide the exact slope of the virtual dynamic thinning line, and they did not discuss further the relationship between the two lines. When Xue and Hagihara (2002) studied the C-D effect in self-thinning P. massoniana stands, they also found that data points on the virtual dynamic thinning line first moved along a slope of ca. −1.2, and then reached the position of the self-thinning line with a slope of ca. −1.8. They explained that the difference in the results was because of different ages, indicating the complexity in field studies. We tried to find the possible relationship between the virtual dynamic thinning line and the upper boundary line from the perspective of a general model species for simplicity. In our study, the virtual dynamic thinning line and the upper boundary line are both straight lines on log-log scales and the slopes are almost the same values with only very little difference (0.059). We found in this study that the difference would be even less if there were more data points using to fit the upper boundary line and initial spatial and size patterns in the populations used to derive the virtual dynamic thinning line were more uniform. Although Table 2 shows that the exponent of the upper boundary line, −1.349, was not significantly different from −4/3 and different significantly from −3/2, this does not mean −4/3 is better than −3/2. We did not study the accurate self-thinning exponent in this study, and the exponent −1.349 may be the result of the parameter settings. Figure 6 shows that the intercept of the virtual dynamic thinning line is less than that of the upper boundary line. This difference in intercept may be because of different initial plant size and spatial patterns. The upper boundary line was obtained as plants grew with unique initial size and uniform (hexagonal) initial spatial distribution pattern. Under this condition, plant individuals were influenced at the same level, and the maximum mean plant biomass at a given density was the largest. While plants in the seven populations used to derive the virtual dynamic thinning line grew with normalized initial sizes and random spatial patterns. Under such conditions, the mean plant biomass in a population cannot reach the ideal largest value, so does the mean plant biomass in the virtual population used to fit the virtual dynamic thinning line. If populations are established with the plants, which are of more similar size and distributed more uniformly at the beginning of population development, the virtual dynamic thinning line will lift up closer to the upper boundary line.

The results of our study also show the difference among the upper boundary line, the dynamic thinning line and the virtual dynamic thinning line. The upper boundary line can only be derived from ideal populations with uniform initial size and spatial distribution, so the upper boundary line may be regarded as a theoretical upper boundary of combinations of mean plant biomass and density in self-thinning populations. But in experimental and field studies, it is hard to get appropriate data to fit the right upper boundary line, and this may partly be the reason why there are different values of the self-thinning exponent in the published reports. The dynamic thinning line is a population-specific line, which describes the time-trajectory of combinations of mean plant biomass and density in a given population as density-dependent mortality occurs. Dynamic thinning lines gradually approach but cannot reach the upper boundary line with the self-thinning process. Because plants are unmovable and there may be gaps in populations, the mean plant biomass in a population at a density does not always reflect the capacity of environments, and different populations have different dynamic thinning lines. The virtual dynamic thinning line is derived from the C-D effect as initial plant density tends to be positive infinity, so it is also a theoretical line like the upper boundary line. If there were no canopy gaps and density-dependent mortality occurred at any time in the population with infinite initial density, the mean plant biomass at any density would be regarded as an environmental characteristic, and then the virtual dynamic thinning line would be the same as the upper boundary line. In our study, it seemed that the virtual dynamic thinning line tended to be used as the upper boundary line in simulated plant populations given appropriate parameter values and as initial spatial and size patterns tended to be uniform.

Because the virtual dynamic thinning line bridges the gaps between the C-D effect and self-thinning, and between the dynamic thinning line and upper boundary line, it combines static and dynamic aspects of intraspecific competition into one whole body and it should be treated more seriously than ever. However, although Xue and Hagihara mentioned the relationship between the virtual dynamic thinning line and the upper boundary line when they studied the C-D effect in two self-thinning pine species, researchers have seldom studied the relationship. There are two dominating values, −3/2 and −4/3, of the self-thinning exponent in plant populations, and identifying which is more accurate and eventually identifying the underling mechanism is still one of the concerns of plant ecologists. But because of the ideal conditions exposed in this study, it is hard to derive the accurate upper boundary line by fitting directly to experimental or field data. If the relationship between the virtual dynamic thinning line and the upper boundary line is corroborated, then we can obtain the upper boundary line through the virtual dynamic thinning line without worrying about whether a data point is from a population with full density, avoiding the subjectivity in data selection as in traditional self-thinning studies. In our study, it seemed that the virtual dynamic thinning line could act as the upper boundary line in simulated plant stands given appropriate parameter values. The results of our study should be examined with further well-controlled experiments investigating the virtual dynamic thinning line. These experiments should pay attention to initial spatial and size patterns, which influence slightly the similarity of the virtual dynamic thinning line and the upper boundary line as revealed in our study.

Materials and Methods

In our model, we assign conventional units to measures of mass, area and time for ease of interpretation; however, these units are essentially arbitrary and should not be taken literally. The virtual experimental field was a 300 m × 300 m plain and homogeneous square. In order to avoid edge effects, only the plants in the central 100 m × 100 m region were “measured”. The embedded approach was used in this model, because it was found to be more realistic than the torus, linear expansion or reflection methods (Haefner et al. 1991). Simulated plant populations developed for 100 years at seven initial densities, with 7 360, 9 078, 12 932, 16 560, 20 064, 25 950 and 29 785 plants in the whole field, and in two initial spatial patterns, random or uniform (hexagonal). Plant growth equations were discretized using the fourth-order Runge-Kutta algorithm with a step interval of 0.2 year, and data were collected once a year (every five steps). Initial biomasses (m₀) were 0.1 kg while deriving the upper boundary line, and distributed normally with a mean of 0.1 kg and a standard deviation of 0.015 while studying the dynamic thinning line and the virtual dynamic thinning line. For the parameters in the AOS model, we investigated the parameter space, but we just provided one set of parameter values in the present study for simplicity and brevity. Asymptotic biomass M was set to 100.0 kg for simplicity, and the value of r assured that biomass m reached approximately M at the end of the simulation period. Exponent q in the m-D allometric relationship was set to 2.67 according to Enquist et al. (1998), close to findings of previous studies (Osawa and Allen 1993; Sackville Hamilton et al. 1995; Yang and Titus 2002). Exponent b in the h-D allometric relationship was set to a value based on the findings of Weiner and Fishman (1994). Coefficients a in h-D and p in m-D relationships were set to values so that the maximum height and stem diameter of a plant reach about 8 m and 30 cm, respectively, when the plant grows in isolation. These scales seem to be appropriate in consideration of the size of the experimental field. Other parameters, i.e. c, k, e, j, were set to values so that plants have AOSs of proper size and die at certain density threshold. This specific point in parameter space gives us a general model species, considering biological reality and computing workload at the same time. Another reason we do not discuss the parameter space in this paper is that there exist parameter values just because of mathematical reasonability. For example, a condition that the asymptotic height is 100 m is reasonable in the model itself, but does not seem appropriate in a 300 m × 300 m experimental field. Table 1 lists parameter values for Equations 5, 6, 12, 13, and 16 in detail.

Table 1. Values of the parameters in Equations 5, 6, 12, 13 and 16

Parameter	Description	Value
a	Allometric coefficient in Equation 5	30.08
b	Allometric exponent in Equation 5	1.10
p	Allometric coefficient in Equation 6	2 489.35
q	Allometric exponent in Equation 6	2.67
c	Coefficient in Equation 12	20.00
r	Maximum relative growth rate in Equation 13	0.13
M	Asymptotic plant mass in Equation 13	100.00
k	Coefficient in Equation 13	30.075
e	Coefficient in Equation 16	2.00
j	Coefficient in Equation 16	0.01

Hagihara's model (1999) of the C-D effect in self-thinning populations and Ogawa's equation (2005) of a dynamic thinning line were used to validate the AOS model. These equations were chosen because: (i) Hagihara's model is suitable in self-thinning populations; (ii) Ogawa's equation incorporates some earlier models (Shibuya 1995; Hagihara 1998); and (iii) these equations describe the dynamics of self-thinning populations in static and dynamic aspects, respectively, and combination of these equations can effectively validate a plant competition model.

The output of our model consisted of the biomasses, heights and locations of all the plants in the central 100 m × 100 m region over time. The density and mean plant biomass of the plants alive in each population were calculated, and then the upper boundary line, dynamic thinning lines, the C-D effect and the virtual dynamic thinning line were analyzed. Because our simulations run in the deterministic mode, least square regression (LSR) methods other than principal components analysis (PCA) or reduced major axis (RMA) regression were used to analyze data. Nonlinear regression method based on the Levenberg-Marquardt algorithm of Origin 6.1 was used to fit simulation data.

To test the effects of random initial spatial pattern and normalized sized distribution, selected runs were repeated five times with different random number seeds. Variation between these replicates was so small that replication of other runs was cancelled.

The complete source code in C of our model will be provided on request, and researchers are invited to use it for further studies.

(Handling editor: Jian-Xin Sun)

References

Adler FR (1996). A model of self-thinning through local competition. Proc. Natl. Acad. Sci. USA 93, 9980–9984.
10.1073/pnas.93.18.9980
CAS PubMed Web of Science® Google Scholar
Bauer S, Wyszomirski T, Berger U, Hildenbrandt H, Grimm V (2004). Asymmetric competition as a natural outcome of neighbour interactions among plants: results from the field-of-neighbourhood modelling approach. Plant Ecol. 170, 135–145.
10.1023/B:VEGE.0000019041.42440.ea
Web of Science® Google Scholar
Berger U, Hildenbrandt H (2000). A new approach to spatially explicit modelling of forest dynamics: spacing, ageing and neighbourhood competition of mangrove trees. Ecol. Model 132, 287–302.
10.1016/S0304-3800(00)00298-2
Web of Science® Google Scholar
Berger U, Hildenbrandt H (2003). The strength of competition among individual trees and the biomass-density trajectories of the cohort. Plant Ecol. 167, 89–96.
10.1023/A:1023965512755
Web of Science® Google Scholar
Berger U, Hildenbrandt H, Grimm V (2002). Towards a standard for the individual-based modeling of plant populations: self-thinning and the field-of-neighborhood approach. Nat. Res. Modeling 15, 39–54.
10.1111/j.1939-7445.2002.tb00079.x
Google Scholar
Berger U, Hildenbrandt H, Grimm V (2004). Age-related decline in forest productivity: modelling the effects of growth limitation, neighbourhood competition and self-thinning. J. Ecol. 92, 846–853.
10.1111/j.0022-0477.2004.00911.x
Web of Science® Google Scholar
Bi H, Wan G, Turvey ND (2000). Estimating the self-thinning boundary line as a density-dependent stochastic biomass frontier. Ecology 81, 1477–1483.
10.1890/0012-9658(2000)081[1477:ETSTBL]2.0.CO;2
Web of Science® Google Scholar
Brown JH, Gillooly JF, Allen AP, Savage VM, West GB (2004). Toward a metabolic theory of ecology. Ecology 85, 1771–1789.
10.1890/03-9000
Web of Science® Google Scholar
Cannell MGR, Rothery P, Ford ED (1984). Competition within stands of Picea sitchensis and Pinus contorta. Ann. Bot. 53, 349–362.
10.1093/oxfordjournals.aob.a086699
Web of Science® Google Scholar
Enquist BJ, Brown JH, West GB (1998). Allometric scaling of plant energetics and population density. Nature 395, 163–165.
10.1038/25977
CAS Web of Science® Google Scholar
Haefner JW, Poole GC, Dunn PV, Decker RT (1991). Edge effects in computer models of spatial competition. Ecol. Model 56, 221–244.
10.1016/0304-3800(91)90201-B
Web of Science® Google Scholar
Hagihara A (1998). A practical model for the time-trajectory of mean phytomass and density in the development of even-aged pure stands. J. For. Plann. 4, 65–69.
Google Scholar
Hagihara A (1999). Theoretical considerations on the C-D effect in self-thinning plant populations. Res. Popul. Ecol. 41, 151–159.
10.1007/s101440050017
Web of Science® Google Scholar
Kikuzawa K (1999). Theoretical relationships between mean plant size, size distribution and self thinning unter one-sided competition. Ann. Bot. 83, 11–18.
10.1006/anbo.1998.0782
Web of Science® Google Scholar
LaBarbera M (1989). Analyzing body size as a factor in ecology and evolution. Annu. Rev. Ecol. Syst. 20, 97–117.
10.1146/annurev.es.20.110189.000525
Web of Science® Google Scholar
Li B-L, Wu H, Zou G (2000). Self-thinning rule: a causal interpretation from ecological field theory. Ecol. Model 132, 167–173.
10.1016/S0304-3800(00)00313-6
Web of Science® Google Scholar
Lonsdale WM (1990). The self-thinning rule: dead or alive? Ecology 71, 1373–1388.
10.2307/1938275
Web of Science® Google Scholar
Mabvurira D, Miina J (2002). Individual-tree growth and mortality models for Eucalyptus grandis (Hill) Maiden plantations in Zimbabwe. For. Ecol. Manage. 161, 231–245.
10.1016/S0378-1127(01)00494-7
Web of Science® Google Scholar
Miller TE, Weiner J (1989). Local density variation may mimic effects of asymmetric competition on plant size variability. Ecology 70, 1188–1191.
10.2307/1941388
Web of Science® Google Scholar
Mou P, Mitchell RJ, Jones RH (1993). Ecological field theory model: a mechanistic approach to simulate plant-plant interactions in southeastern forest ecosystems. Can. J. For. Res. 23, 2180–2193.
10.1139/x93-271
Web of Science® Google Scholar
Niklas KJ, Midgley JJ, Enquist BJ (2003). A general model for mass-growth-density relations across tree-dominated communities. Evol. Ecol. Res. 5, 459–468.
Web of Science® Google Scholar
Ogawa K (2005). Time-trajectory of mean phytomass and density during a cource of self-thinning in a sugi (Cryptomeria japonica D. Don) plantation. For. Ecol. Manage. 214, 104–110.
10.1016/j.foreco.2005.03.067
Web of Science® Google Scholar
Osawa A, Allen RB (1993). Allometric theory explains self-thinning relationships of mountain beech and red pine. Ecology 74, 1020–1032.
10.2307/1940472
Web of Science® Google Scholar
Osawa A, Sugita S (1989). The self-thinning rule: another interpretation of Weller's results. Ecology 70, 279–283.
10.2307/1938435
Web of Science® Google Scholar
Pacala SW (1986). Neighborhood models of plant population dynamics. IV. Single species and multispecies models of annuals with dormant seeds. Am. Nat. 128, 859–878.
10.1086/284610
Web of Science® Google Scholar
Pacala SW (1987). Neighborhood models of plant population dynamics. III. Models with spatial heterogeneity in the physical environment. Theor. Popul. Biol. 31, 359–392.
10.1016/0040-5809(87)90012-8
Web of Science® Google Scholar
Pacala SW, Silander J (1985). Neighborhood models of plant population dynamics. I. Single species models of annuals. Am. Nat. 125, 385–411.
10.1086/284349
Web of Science® Google Scholar
Sackville Hamilton NR, Matthew C, Lemaire G (1995). In defense of the −3/2 boundary rule: a re-evaluation of self-thinning concepts and status. Ann. Bot. 76, 569–577.
10.1006/anbo.1995.1134
Web of Science® Google Scholar
Shibuya M (1995). A simple and practical model for mean size-density trajectories of tree stand. J. Jpn. For. Soc. 77, 247–253.
Google Scholar
Shinozaki K (1961). Logistic Theory of Plant Growth. Ph.D. thesis, Kyoto University.
Google Scholar
Shinozaki K, Kira T (1956). Intraspecific competition among higher plants. VII. Logistic theory of the C-D effect. J. Inst. Polytech. Osaka City Univ. D. 7, 35–72.
Google Scholar
Wang G, Yuan J, Wang X, Xiao S, Huang W (2004). Competitive regulation of plant allometry and a generalized model for the plant self-thinning process. Bull. Math. Biol. 66, 1875–1885.
10.1016/j.bulm.2004.04.005
PubMed Web of Science® Google Scholar
Weiner J (1982). A neighborhood model of annual plant interference. Ecology 63, 1237–1241.
10.2307/1938849
Web of Science® Google Scholar
Weiner J (1984). Neighborhood interference amongst Pinus rigida individual. J. Ecol. 72, 183–195.
10.2307/2260012
Web of Science® Google Scholar
Weiner J, Fishman L (1994). Competition and allometry in Kochia scoparia. Ann. Bot. 73, 263–271.
10.1006/anbo.1994.1031
Web of Science® Google Scholar
Weiner J, Thomas SC (1986). Size variability and competition in plant monocultures. Oikos 47, 211–222.
10.2307/3566048
Web of Science® Google Scholar
Weiner J, Thomas SC (1992). Competition and allometry in three species of annual plants. Ecology 73, 648–656.
10.2307/1940771
Web of Science® Google Scholar
Weiner J, Stoll P, Muller-Landau H, Jasentuliyana A (2001). The effects of density, spatial pattern, and competitive symmetry on size variation in simulated plant populations. Am. Nat. 158, 438–450.
10.1086/321988
CAS PubMed Web of Science® Google Scholar
Weller DE (1987). A reevaluation of the −3/2 power rule of plant self-thinning. Ecol. Monogr. 57, 23–43.
10.2307/1942637
Web of Science® Google Scholar
Weller DE (1990). Will the real self-thinning rule please stand up? A reply to Osawa and Sugita. Ecology 71, 1204–1207.
10.2307/1937389
Web of Science® Google Scholar
Weller DE (1991). The self-thinning rule: dead or unsupported? A reply to Lonsdale. Ecology 72, 747–750.
10.2307/2937216
Web of Science® Google Scholar
West PM, Wells KF, Cameron DM, Rance SJ, Turnbull CRA, Beadle CL (1991). Predicting tree diameter and height from above-ground biomass for four eucalypt species. Trees 6, 30–35.
Google Scholar
West GB, Brown JH, Enquist BJ (1997). A general model for the origin of allometric scaling laws in biology. Science 276, 122–126.
10.1890/03-0757
CAS PubMed Web of Science® Google Scholar
White J (1981). The allometric interpretation of the self-thinning rule. J. Theor. Biol. 89, 475–500.
10.1016/0022-5193(81)90363-5
Web of Science® Google Scholar
White J, Harper JL (1970). Correlated changes in plant size and number in plant population. J. Ecol. 58, 467–485.
10.2307/2258284
Web of Science® Google Scholar
Wu H, Sharpe PJH, Walker J, Penridge LK (1985). Ecological field theory: a spatial analysis of resource interference among plants. Ecol. Model 29, 215–243.
10.1016/0304-3800(85)90054-7
Web of Science® Google Scholar
Wyszomirski T (1983). A simulation model of the growth of competing individuals of a plant population. Ekologia Polska 37, 73–92.
Google Scholar
Wyszomirski T, Wyszomirski I, Jarzyna I (1999). Simple mechanisms of size distribution dynamics in crowded and uncrowded virtual monocultures. Ecol. Model 115, 253–273.
10.1016/S0304-3800(98)00182-3
Web of Science® Google Scholar
Xue L, Hagihara A (1998). Growth analysis of the self-thinning stands of Pinus densiflora Sieb. Et Zucc. Ecol. Res. 13, 183–191.
10.1046/j.1440-1703.1998.00256.x
Web of Science® Google Scholar
Xue L, Hagihara A (2002). Growth analysis on the C-D effect in self-thinning Masson pine (Pinus massoniana) stands. For. Ecol. Manage. 165, 249–256.
10.1016/S0378-1127(01)00622-3
Web of Science® Google Scholar
Yang Y, Titus SJ (2002). Maximum size-density relationship for constraining individual tree mortality functions. For. Ecol. Manage. 168, 259–273.
10.1016/S0378-1127(01)00741-1
Web of Science® Google Scholar
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). J. Biol. Osaka City Univ. 14, 107–129.
Google Scholar
Zeide B (1987). Analysis of the 3/2 power law of self-thinning. For. Sci. 33, 517–537.
Web of Science® Google Scholar
Zhang YX (1993). Application and improvement of the neighborhood interference model. Acta Phytoecol. Geobot. Sin. 17, 352–357.
Google Scholar
Zhang D, Zhao S, Zhang P, Chen Q (1989). On neighborhood effects in a successional community and an improved index of neighborhood interference. Acta Ecol. Sin. 9, 53–58.
Google Scholar

Citing Literature

Volume50, Issue3

March 2008

Pages 280-290

Relationship between the Virtual Dynamic Thinning Line and the Self-Thinning Boundary Line in Simulated Plant Populations

Abstract