1 Introduction

As plants in a population develop, plant growth becomes limited by the rate of availability of resources, and individuals are forced sharing of limiting resources with a compensating plastic reduction in individual plant development (Aikman et al., 1980). For populations with different densities that are grown under the same habitat conditions, the limit increases with increasing density. This results in decreases of mean plant weight with increasing density, i.e. the competition-density (C-D) effect. The reciprocal equations for describing the C-D effect were developed by some researchers (Bleasdale et al., 1960; Nelder, 1962; Bleasdale, 1967; Farazdaghi et al., 1968; Watkinson, 1980; 1984; Vandermeer, 1984; Silvertown et al., 1993), these equations originated in the logistic theory of the C-D effect established by Shinozaki et al. (1956). This theory can be used to model the mean stem volume-density relationship of forests, so that forest managers predict the response of mean stem volume to density and modify the density-management prescriptions of forests.

Populus × euramericana is a fast-growing species. This hybrid was created in Italy and was introduced to China in 1971, and recently has become one of the major silviculture tree species in China. However, the growth analysis on the C-udy C-D effect in P.xeuramericana plantations has not been reported.In this study, the reciprocal equation of the C-D effect was applied for the growth analysis of P. × euramericana plantations examined by Zheng et al. (1990). First, the relationship between biological time, which was defined as the integral of the coefficient of growth in the general logistic equation with respect to physical time, and physical time was examined. Then, the trends of the coefficients in the reciprocal equation with biological time were pursued. Finally, the coefficient of growth and the final yield in the general logistic equation were analyzed with respect to growth stage.

2 Materials and methods 2.1 Experimental sites

Experimental area was located in the terrace of Yi River, which lies 35°19′~35°46′N, 118°07′~118°43′E, in Linyi County, Shandong Province, China. Mean annual temperature is 12.8℃ and mean annual frostfree period is 208 days. Mean annual sunshine time, rainfall and relative humidity are 2 478.4 h, 836.8 mm and 68%, respectively. Soil is Cohrepts consisting of a 30 cm topsoil of sandy loam and a 4 m subsoil of clay layer. Chemical characteristics of the soil were as follows: total N and total P were respectively 0.33~0.36 and 0.75~1.16 g·kg^-1, and alkalizate hydrolytic N, extractable P and exchangeable K were 20.03~21.39, 3.18~3.20 and 43.32~48.09 mg·kg^-1, respectively. Organic matter was 5.7~6.2 g·kg^-1 and pH was 7~7.5 (Zheng et al., 1990). The previous vegetation in the experimental area was a locust (Robinia pseudoacacia L.) plantation, which was cleared out before the plantations of Populus × euramericana were established. The experimental area was overall dug up to 1 m depth and soil per hectare was mixed with 300 m³ fine sand before seedlings were planted. 1-year-old seedlings were planted in the soil and planting depth was 80 cm. Each seedling was fertilized using 15 kg dry human wastes and 1 kg calcium superphosphate, and then was watered. The plantations were watered 2~3 times and topdressed with urea each year. The amount of urea applied was increased year-by-year, rising to 0.45 kg·tree^-1 at 3-year-old and 0.75 kg·tree^-1 from 4-year-old.

2.2 Experimental design

In 1983, Populus×euramericana(Dode) Guinier cv. 'San Martion' plantations were established with three levels of initial density: 285, 495 and 1 560 trees·hm^-2. The investigated plot area was 0.1 hm² for 285 and 495 trees·hm^-2 plantations and 0.07 hm² for 1 560 trees·hm^-2 plantation. Tree height and stem diameter at breast height were measured from 1 to 7 years. Plantation age was counted from planting seedling time without including seedling age. Self-thinning did not occur in those plantations over the experimental period (Zheng et al., 1990).

2.3 Logistic theory and its related model

Shinozaki et al. (1956) developed the logistic theory of the C-D effect. The theory is constructed from the following two basic assumptions: one is that mean plant weight w grows following the general logistic equation

(1)

where λ(t) is the coefficient of growth, which is independent of initial density ρ_i, and W(t) is the asymptote of w. Another is that final yield Y(t) (= W(t)·ρ_i) is constant irrespective of ρ_i(the law of constant final yield). As a result, the following reciprocal equation of the C-D effect is derived

(2)

Here A and B are coefficients at a given growth stage, and the coefficients A and B are respectively defined as,

(3)

and

(4)

where w₀ is initial mean plant weight, which is independent of density ρ, and τ is called biological time (Shinozaki, 1961) defined as the integral of λ(t) with respect to physical time t:

(5)

Considering Eq. 4, the biological time τ can be calculated from the following equation (Shinozaki et al., 1956)

(6)

The τ-t relationship is approximated by the following hyperbolic equation (Hozumi, 1977),

(7)

where the reciprocal of g denotes the intrinsic growth rate at the initial growth stage, the reciprocal of h denotes the ceiling value of τ as t tends to infinity and L stands for a lag time.

In the logistic growth curve of Eq. 1, λ(t) represents the intrinsic growth rate. The observed average value of λ(t) for the period between two successive sampling years can be calculated by the following equation (Hozumi, 1973)

(8)

which is derived on the basis of Eqs 5 and 6. Meanwhile, the λ(t) can be derived by differentiating both sides of Eq. 7 with respect to physical time t (Hozumi, 1977)

(9)

On the other hand, the average of the final yield Y(t) for the period between two successive sampling years can be calculated by the following equation (Shinozaki et al., 1962)

(10)

which is derived on the basis of Eq. 3 and 4.

The above equations were employed to calculate various growth characteristics of P. × euramericana plantations. Since mean stem volume relates isometrically to mean plant weight (e.g. White, 1981), mean stem volume substitutes for mean plant weight in the present study.

3 Results 3.1 The C-D effect

The relationships of mean stem volume w to density initial ρ_i in the P. × euramericana plantations was shown in Fig. 1. Mean stem volume decreased with increasing plantation density at each growth stage. Difference in mean stem volume among plantations with different densities increased with increasing growing stage. The relationship of w and ρ_i was well fitted in the C-D curve given by Eq. 2 on logarithmic coordinates. The C-D curve shifted upward with the progress of time and tended to become linear over the range of data eventually.

3.2 Biological time and physical time

The initial mean stem volume w₀ for the P. × euramericana was estimated to be 0.004 7 m³. The relationship between biological time τ and physical time t was shown in Fig. 2. With increasing physical time t, the biological time τ increased rapidly during early growth stages and an increase in τ became slow gradually during later growth stages. The τ-t relationship was well approximated by the curve given by Eq. 7. The constants g, h and L in Eq. 7 were respectively calculated to be 0.561 a, 0.120 4 and 0.026 0 a.

3.3 Changes in the coefficients A and B with biological time

Fig. 3 presented the tendency of the coefficient A, the reciprocal of which means the asymptote of yield (=w·ρ_i) at a given growth stage. With increasing τ, initially the coefficient A abruptly increased from zero to a maximum value, and then decreased gradually. Fig. 4 depicted the change of the coefficient B, the reciprocal of which means the asymptote of mean stem volume at a given growth stage. The coefficient B decreased exponentially with increasing τ and approached zero. The B had its maximum when τ was zero. The B-τ relationship was well approximated by Eq. 4.

3.4 Coefficient of growth λ(t) and final yield Y(t)

As shown in Fig. 5, there was a clear decrease in λ(t) with increasing plantation age. The λ(t)-t relationship was well fitted by the curve given by Eq. 9. A decrease of the λ(t)-value shown by the curve corresponded to that of the observed λ(t)-value.The change in final yield Y(t) calculated by Eq. 10. The Y(t) increased gradually during the earlier and medium stages, and then increased rapidly and reached a peak at the later growth stages(Fig. 6).

4 Discussion

The C-D effects of P. × euramericana plantations are well described by Eq. 2, which have been successfully used in soybean (Shinozaki et al., 1956), Pinus densiflora Sieb. et Zucc., Cunninghamia lanceolata (Lamb.) Hook. and Pinus massoniana Lamb. stands (Xue et al., 1998; 1999; 2001a; 2001b; 2002). Eqs. 2, 4, 7 and 9 respectively fit observed values of w, B, τ and ε well, indicating that this analysis methodology can be used to other species stands. Using the methodology, the intraspecific competition and growth characteristics of these species or the same species grown at different sites can be revealed, this will enrich the knowledge of plant demographic processes.

Considering Eq. 2, yield y (=wρ_i) is given by the following equation

(11)

As density ρ_i tends to infinity, Eq. 11 becomes

(12)

so that the reciprocal of A stands for the asymptote of y at a given growth stage. The value of A rapidly increased in earlier growth stage and thereafter decreased gradually (Fig. 3).On the other hand, it can be known from Eq. 4 that the value of B decreases with increasing τ. As density ρ_i tends to zero, Eq. 2 becomes

(14)

so that the reciprocal of B stands for the asymptote of mean stem volume w at a given growth stage. Therefore, that B decreased with increasing τ (Fig. 4) means the asymptote of w increased with increasing τ.

The g-value for the Populus × euramericana was smaller than that of Pinus densiflora reported by Xue et al.(1998). This indicates that the P. × euramericana is high in intrinsic growth rate at the initial growth stage. The P. × euramericana is a fast-growing hybrid species compared to the P. densiflora, and the increase of mean stem volume is larger in P. × euramericana than in the P. densiflora at the initial growth stage. The h-value for the P. × euramericana was larger than that of the P. densiflora. This indicates that the ceiling value of τ is smaller in the P. × euramericana than in P. densiflora. This is clear from Eq. 5 that growth stage progresses, coefficient of growth λ(t) of the P. densiflora becomes higher than that of the P. euramericana. The lag time L of the P. × euramericana almost equals that of the P. densiflora. This lag time is considered to be the period of time required by saplings for adapting to new site conditions after being transplanted, after which they start to grow.

Initially the mean stem volume of high-density plantation was almost the same as those of medium- and low-density plantations (Fig. 7), but the growth of high- density plantation continued to decline, so that the difference of mean stem volume among different density plantations became large with time. The mean stem volume of high-density plantation was about 57% and 45% of that of medium- and low-density plantations when the plantations were 4 years old, and the ratios respectively declined to about 47% and 37% when the plantations were 7 years old. However, tree numbers per hectare of low- and medium-density plantations are only 18% and 32% of that of high-density plantation, respectively, resulting in their volume yields are only 54% and 64% of that of high-density plantation.

High-density led to a relatively long period of intense competition among trees. This is not favorable for the development of vigorous, high-quality P. × euramericana trees. Self-thinning will likely occur in the high-density plantation in the near future. To alleviate the problems of overcrowding, light thinning is recommended. This thinning method can produce the most desirable combination of individual-tree volume growth and stand-level volume growth and obtain some thinning volume. The sum of thinning volume and crop volume is slightly greater than the volume of non-thinning plantation, the former together with individual-tree size could result in a little greater income than the latter (Zheng et al., 1990).

This paper analyzes growth characteristics of P. × euramericana plantation under competition using some theoretical growth models, this is useful for understanding growth characteristics of this pieces under intraspecific competition. The aim of planting trees is to get the most income, so that in production practice the determining of plantation density should be based on individual-tree volume, volume yield per area and timber prices of various tree sizes which maybe changes with time. It is possible that divorce between the result of theory analysis and practice could be caused. Therefore, further considerations are obviously needed to build the growth models combining theory with practice.