A Consideration of the Constancy of Biomass Density in Plant Populations Undergoing Self-Thinning

The constancy of biomass density was considered in an entire plant population p by combining two adjacent populations p1 and p2 for which the self-thinning rule is assumed to be satisfied independently and each biomass density is also assumed to be the same constant value. Under these assumptions, the biomass density d in a population p was formulated as d = c((km−α + 1)(km+ 1)/(km1−α+1)(k+1)), where c is biomass density of p1 and p2, and k and m are stand area and density ratio of p1 to p2, respectively, and α is the self-thinning slope. In the case of m = 1, the value of d in the above equation is always larger than unity. This fact indicates that the biomass density in a combined population p is not equal to the biomass density c in each population p1 or p2 because of systematic error.


INTRODUCTION
Biomass density is generally defined as aboveground biomass per unit volume of space occupied by a forest stand.Since the volume of a forest can be approximated as the product of forest height and stand area, the biomass density of a forest has a dimension [weight volume −1 ] and is an index of the average organic-matter concentration packed in a unit volume of a forest [1,2].Since light penetration within a forest stand is affected by the crown or canopy structure [3], the biomass density per unit volume is a key factor in an effective gas exchange and photosynthetic production as well as in a forest stand structure.
Kira and Shidei [2] first demonstrated that the biomass density is approximately constant in fully closed forest stands.Kikuzawa [1] also reported that the biomass density reaches an upper limit of constant value in fully stocked forest stands.On the contrary, Osawa and Allen [4] and Xue et al. [5] pointed out for several forest stands that the constancy of biomass density may not always hold.Thus the constancy assumption of biomass density is still controversial.
In the present paper, we consider, mathematically, the constancy of biomass density from the following viewpoints: in two adjacent populations p 1 and p 2 satisfying the self-thinning rule [6][7][8][9] independently, each biomass density is assumed to be the same constant value.If the two populations p 1 and p 2 are combined as a single population under these assumptions, how does the biomass density change?This question is realistic because observed data often corresponds to the case of a combination of two plant populations [10].

FORMULATION
Formulation of basic data on two plant populations is given in Table 1.Here, S is the sum of a stand area occupied by two adjacent populations p 1 and p 2 , N is the total plant number of populations p 1 and p 2 , and X is the total plant mass of p 1 and p 2 , respectively, as follows: where ρ 1 and ρ 2 , and y 1 and y 2 are the stand density and biomass of populations p 1 and p 2 , respectively.
For convenience sake, the following relationships are taken into consideration: where k and m are the area and the density ratio between two populations p 1 and p 2 , respectively.The first assumption that two populations p 1 and p 2 satisfy the self-thinning rule independently leads to where K and α are a constant and a self-thinning exponent, respectively.Combining ( 3) and ( 4) gives the following relationship: The second assumption is that biomass density d 1 in population p 1 is equal to d 2 in population p 2 as follows: where c is a constant.The assumption of (6) indicates that Considering an entire population of p 1 and p 2 , mean plant height H and biomass y are defined as From ( 5), (7), and (8), biomass density (d) in a combined population of p 1 and p 2 , which is defined as y/H, is expressed as Here, let the (dimensionless) coefficient of c in (9) be as equal to, or smaller than unity, the quantitative relationship between d and c can be assessed.
Case 2 (m =1).If the denominator and numerator of f (k, m) in ( 10) are symbolized as f 1 and f 2 , the deference Δ between f 1 and f 2 is written as The value of self-thinning exponent (α) is theoretically considered to be 1/2 from a geometric basis [7,9] and, recently, to be 1/3 from a resource-allocation basis [6,8].Therefore, the present analysis considers the sign of (10) in the cases of α = 1/2 and α = 1/3 as follows.

CALCULATION AND PROPERTIES OF f = (k, m)
To examine the degree of deviation from the biomass density c in two populations, the values of f (k, m) in various values of density ratio (m) and area ratio (k) were calculated in the cases of self-thinning slope α = 1/2 (Table 2) and α = 1/3 (Table 3), respectively.The values of m and k for which the value of f (k, m) is less than 1.05 range from 0.6 to 1.8 and 0.5 to 2.0 in the cases of α = 1/2 and α = 1/3, respectively.However, controlling the f (k, m) value such that it is less than 1.02, the m values become smaller, ranging from 0.8 to 1.4 and 0.6 to 1.6 in the cases of α = 1/2 and α = 1/3, respectively.
Since f (k, m) is a coefficient of c in ( 9), the present numerical analysis may be useful for explaining slight differences in biomass density between populations as observed in, for example, the Chinese pine and larch stands reported by Xue et al. [5].

Table 1 :
Formulation data on two plant populations.