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9Competition dynamics and spatial uniformity of
plant community
Galitskii V.V. (galvv@ibbp.psn.ru)
Institute of Physical-Chemical and Biological Problems of Soil Science Russian Academy of Sciences
introduction
Competition among plants in a community - indirect interaction of plants in the course of partition and repartition of resources insufficient for all of the plants in the community (Clements and Weaver, 1929) - is the leading process in the development of any community of plants. Thinning - the decrease of community density by hundreds and thousands of times in the process of growth of community members - is the most demonstrative manifestation of this process.
It should be noted that phytocenologists (for example, Rabotnov, 1978), in the analysis of natural data, prefer to speak about competition for a concrete resource (light, water, elements of mineral nutrition). However, with the assumption (natural, on initial consideration) that the territory of a community is homogeneous, the area occupied by the community may be taken as a united resource. Resource is an additive quantity, and upon partition of the territory, the material resources located on it are shared proportionally.
Competition in a community of plants is manifested on a local level - interaction of neighboring plants that are close enough each to other results in change in their individual parameters (mass, shape, and so on). It is also manifested on an integral level, at the level of the community - characteristics of the entire community (average values and features of distributions of plant parameters, coefficients of variation, density of the community and intensity of thinning, correlation relationships, and so on) vary as a result of interindividual interactions.
Investigations of problems connected with competition in plant communities and, in particular, of measures of the "force" of local interaction of plants in a community have been carried out for at least a century (for example, Clements and Weaver, 1929; Sukachev, 1973, 1975; Pielou, 1960; Mack and Harper, 1977; Weiner, 1984; Duncan, 1995). Competition on a local level is not discussed here, the behavior of competition measures integral over a community is considered.
On selection of the integral measure
Hereinafter, the measure is understood as some method or algorithm of estimation of the value of some quality (here, the "force" or intensity of competition) of an object (here, a community of plants). It is obvious that there cannot be just one such method, but measures that give values from a fixed range, for example [0,1], for any situation can be considered preferable. Below, two groups of methods are discussed with reference to the topic under consideration.
The first group consists of theoretical measures directly following from the definition of competition and not necessarily best for use in practice. According to the definition, measures should be constructed from values describing the resource requirements of an individual plant of the community and values measuring the available resource that each plant has. We shall use the corresponding terms introduced in development of a dynamic model of a community of plants (Galitskii, 2003). Supposing the territory occupied by the community to be homogeneous, as a measure of the resources accessible to an i-th plant, it is natural to take the area of the appropriate
Voronoi polygon (Voronoi, 1908) Spn and, using the notion of a freely growing plant (Galitskii, 2003), to take the area of free growthAf,,(T) - one of the characteristics of a freely growing plant - as a measure of the resource necessary for a tree at age T.
-[comment [LM1]: plant?
According to Clements and Weaver (1929, p. 317), as an integral measure of competition in a community, it is necessary to take the ratio of the total deficiency of a resource over the community to the total requirement of members of the community for this resource:
C cl = i (Af- SpU)lNAF, (1)
AF,, > Spt,, '=!
where the first summation is made for all plants of the community experiencing competition and N is the total number of trees. It should be noted that this formulation "divides" individual shortages of resources into the entire community.
Using the same model concepts, it is possible to propose another formulation, not having a "socialist" shade, of an integrated measure of the competition experienced by individual plants of a community on the part of their neighbors - the average over the community ratio of individual shortage of a resource to individual requirement for it:
Cav = VN i (AF,, - Spl,,)lAF,.
AF ,>S„i,,
(2)
Both expressions are normalized on the interval [0.1]: if each plant has Sp,,, « Af,,(T), then both values are close to 1, and if for all plants Spi,, > AF/T), both values are equal to 0.
Since Af,,, Sp,,i, and N vary in time, Cci and Cav, generally speaking, depend on T also. It can be noted that, according to these expressions, the integral measure of competition is defined by the geometrical design of the community and one of the characteristics of the virtual object - a freely growing plant, dependent on the plant species and ecological conditions in the community.
The other group comprises practical measures for an estimation of the competition force, in which data obtained upon natural observations are used. Such competition measures should be constructed with the use of values showing the state of plants in the community (phytomass, biomass (Galitskii, 2003), height, stem diameter, etc.) and the values representing resource allocation in the community. The parameters showing the state of plants can in turn be divided with reference to the case in point into two groups.
The first group are parameters having cumulative character, i.e., those whose momentary value is defined by the entire previous history (of competition) of the plant and monotonously changes (only increases) with age. From among the parameters listed above, these include phytomass, height, and stem diameter. If we address the model of a plant (Galitskii, 2003) on which the following analysis is based, the model of plant phytomass dynamics is the integral over time of the value named the biomass. It does not include directly the parameters related to the resources that the plant has. Therefore, use of phytomass and analogous "cumulative" characteristics in construction of a momentary measure of competition intensity, which strictly speaking is what is discussed here, obviously, would be irrational. At the same time, it is possible to suppose that their use will be, on the contrary, natural on consideration of the integral role of competition in the community for its foregoing history, i.e., on consideration of the "historical" role of the spatial structure in its development.
The second group of parameters of the plant are those that directly depend on the resource provision of the plant. Among the parameters mentioned above, these include the plant biomass, which is understood as the physiologically active part of its full mass. According to the model of a plant (Galitskii, 2003) used here, the rate of biomass change is directly connected with the resources accessible to a plant, and it would be most appropriate in construction of a measure of competition intensity. However, experimental data for such a characteristic of plants for concrete
Comment [LM2]: Differs from the Russian--??
^Comment [LM3]: OK?
communities, apparently, are absent, and hereinafter we shall limit ourselves to use of the biomass proper.
From among the integral characteristics of a community mentioned above (average value and features of plant biomass distribution in the community, coefficient of variation, density of the community and its intensity of thinning, and correlation between biomass and area of territory occupied by plants), in construction of practical measures of competition, preference can be given to the last - the correlation relationship. The point is that, as was stated above, the measure should be dimensionless and normalized. Concerning other characteristics, this would be possible to achieve knowing the biomass of a freely growing plant and/or some initial characteristics of the community, which in the case of natural observations is practically difficult. Moreover, they react to competition with some delay, as follows from the plant biomass model (Galitskii, 2003).
Competition in dynamics
Although as early as F. Clements it was pointed out that competition is a "dynamic process," this aspect of competition in essence has still not been considered because of an absence of methods of definition of the resource requirements of an individual plant and the total needs of plants in course of development of a community, as well as the difficulty (or impossibility) of observing a community over sufficient intervals of time. Moreover, study of natural tree stands usually begins at that tree stand age when the process monotonously fades (the model of this situation was developed by Khilmy, 1957), and for cultures it usually comes to an end when the process has just begun to develop. The theoretical analysis of this problem, most significant for rather spatially uniform communities, demands the use of spatial dynamic models and certain computer possibilities. For example, in the well-known JABOWA model of a forest ecosystem (Botkin et al., 1972) the total death rate of trees is set (!) to obtain the given density of a population at the given age. Earlier, with the 2D model of a plant community, for a sufficiently spatially uniform community, the possibility of nonmonotone dynamics of the production characteristics related to self-thinning, which accompanies competition in such a model community, was demonstrated (Galitskii, 1979, 1982).
In (Abaturov, Galitskii, 1995), upon computer processing of maps of sites in even-aged spruce cultures located in Moscow suburbs, results were obtained that were interpreted as a manifestation of a nonmonotonic competition process in such initially uniform tree stands (Galitskii, Abaturov, 1996). In the present paper, this phenomenon is analyzed with the 2D model developed in (Galitskii, 2003a). Below, the natural data are used (Abaturov, Galitskii, 1995; Galitskii, Abaturov,1996; Galitskii, 2003b) and the model analysis is given.
Model analysis of the competition dynamics in the plant community
In (Abaturov, Galitskii, 1995), maps of placement for tree stands of one age are fixed only for one moment of time (Abaturov, Galitskii, 1995) and the functionAf(T), generally speaking, is unknown. Thus, the direct and dynamic estimation of CW with the use of the available natural data of mapping and, hence, validation of nonmonotonicity are impossible.
Therefore, processing of results of modeling similar to that used for natural data (Abaturov, Galitskii, 1995) was one of the primary goals of the analysis of uniform community dynamics with 2D models of the plant and the community (Galitskii, 2003).
Calculations of dynamics for communities in which trees are initially placed near to knots of a rectangular grid - uniform placement - and also for communities with random (Poisson) initial placement at identical values of other parameters were carried out with the 2D model of a community. Uniform placement was realized on a site of 1 x 1 size of a square grid, with one tree randomly placed per location of 0.1 x 0.1 of the square 100 x 100 grid's step. The initial number was taken to be 10 000 trees. The initial density varied with change in the size of a square site. Hereafter, the following designations (in parentheses is the relative density of
Comment [LM4]: Date? Also, previous reference appears to be missing---?
Comment [LM7]: Sentence OK?
Comment [LM6]: Sentence OK?
Comment [LM5]: Sentence OK?
placement of trees) of model sites are used: uniform g89 (0.83), g93 (0.94), g90 (1.00), g94 (1.06), and g95 (1.66) and Poissonp90 (1.00). In calculations, the following values of biological, growth, and physiological parameters of the plant model (Galitskii, 2003) were used: A0=0.004, A:=100, |=2.5, a=10.0, B type of growth, k=0.8, eM=0.2, tm=0.1, and cG=0.1; the shape of the polygon was not taken into account.
Here it is pertinent to notice that the plant model used gives at the output values of biomass and phytomass, and it is not related directly with any dimensions of a plant. The data of natural observations available to us, on the contrary, were results of linear measurements, from which, in particular, values of the areas of crown projections were obtained. To interpret the results of the model analysis, the following formulae for recalculation, based on notions about allometric relationships between characteristics of a tree, were used. On processing of
T
T
Fig. 1. a) Dynamics of measures of the competition intensity on (Clements and Weaver, 1929) - CCi (1) and average oppression because of shortage of the area for growth - Cav (2) for uniform communities gXX of various density and poisson p90, b) comparison CCl, Cav, amounts of plants Nopr oppressed by competition and model coefficient of correlation (Scr-Sp) - rBS at lag=8 for community g90; an explanation of a designation lag look in item Correlation of the individual characteristics .
model results, the value scr=B2/1 (| is an allometric parameter; its isometric value in the case of the B type of growth is |=2.0 (Galitskii, 2003)) was taken as the area of a crown projection of a tree with biomass B. Analogously, the value d i.3 =P 1/a+^>, where P is the physiologically passive part of the tree mass, or the phytomass (Galitskii, 2003), was taken as the "stem diameter."
Measures of competition
In Fig. 1a, the model dependences CW(T) for several uniform communities of various initial density and for one Poisson community are exhibited. A uniform community, unlike the Poisson community, really demonstrates nonmonotone dynamics of the integral measure of competition Cav. After the first, largest peak, corresponding to the mass entry into competition of the community's members, peaks of decreasing amplitudes occur with some periodicity, and with time the curve of the uniform community approaches the corresponding dependence of the Poisson community. With a decrease in planting density, curves Cav are shifted on the axis T, essentially not changing.
T
T
Fig. 2. a) Model dynamics of number of alive plants n/N0, average phytomass AVP and biomassAVB, coefficients of variation of phytomass CVP and biomass CVB for communities g90 p90; b) dynamics of coefficients of variation of phytomass for communities gXX of various density, on insertion - in the increased scale near to section A-A.
Dependences Ca(T) and Cav are shown in Fig. 1b for comparison. It is possible to see that the competition measure Cci "exaggerates" competition intensity in comparison with Cav for the most part, except for those cases when the main part of the community consists of oppressed members almost equally experiencing a shortage of resources; then both measures are rather close. In the figure, the time dependence of the number of such members Nopr(T) is also presented. The behavior of this dependence (on the appropriate scale) is rather close to the dependence CW(T), while Cca(T) lags a little.
Statistics
The dynamics of statistical characteristics of model communities was calculated. It is necessary to separate out the coefficient of variation and the criteria of distribution normality of stem diameter D i,3 and crown projection area Scr used for ranging of sites observed in nature.
Regularities similar to natural ones revealed themselves in model communities as well. Thus, the distribution normality for scr disappears, and earlier than that for d 3. In Fig. 2a, the dynamics of some statistical characteristics for uniform g90 and Poissonp90 communities are shown, in particular, coefficients of variation for d 1.3 - CVp - and for Scr - CVb. In general, the behavior of these model coefficients does not contradict natural observations (Table 2, Galitskii, 2003b;
Abaturov, Galitskii, 1995): with the emergence of competition in the community, they increase, though in the course of model competition their dynamics is more complicate (especially for Scr) than was possible to assume. The difference in behavior of these communities is most appreciably expressed in the behavior of the coefficient of variation for Scr. In time, the characteristics of both communities approach each other. In Fig. 2b, the dynamics of the coefficients of variation for d 1-3 for the gXX communities is shown (in the inset, the part of interest to us is shown on an enlarged scale). In time, the (small) differences between communities practically disappear. Obviously, the coefficient of variation cannot be used as a competition measure but rather as a directional indicator of the course of competition.
Correlation of the individual characteristics
In Figs. 3a-3c, model dynamics of correlation coefficients (scr-Spl) and (d i.3-Spl) for the areas of the polygons calculated for the given moment of time T (lag=0) and the previous moment T - lag (lag=8) are shown. In the latter case, the situation was imitated in which, in processing of data of natural observations (construction of a Voronoi mosaic), positions and diameters of tree stems dead at the given moment were also taken into account. It can be seen that the dependences of the coefficients with change in the initial density qualitatively do not vary too strongly and with reduction of the density are shifted toward greater T. This, together with the substantial nonmonotonicity of the dependences, makes possible such sharp distinctions between values of coefficients of
correlation in a ranged series of sites (Table 3, Galitskii, 2003b).
In the figures, the moment of time (section A-A) is marked when the situation with model correlation coefficients is rather close to the situation of natural observations whose results are described in (Table 3, Galitskii, 2003b). For this moment, in the inset in Fig. 3a, the corresponding values of model and natural coefficients of correlation are shown. The graphs in Fig. 3a show that, in section A-A, variant g89 (the analogue of site H56) is on the first wave of aggravation of competition and variant g95 (the analogue of site AL7), on the second.
In Table 1, values of other coefficients of the pair correlation of model communities for the moment of section A-A are shown. It can be noted that the correspondence is quite satisfactory not only for the row of coefficient (scr-Spl), lag=8, whose values were the basic ones in choice of the location of section A-A, but also for the majority of other rows. The location of the section is chosen practically uniquely to within a step of integration (0.1) of the differential equations of the model. The theoretical minima of the dependences
T
T
T
Fig. 3. a) and b) Model dynamics of the correlation coefficients (Scr-Sp) for communities gXX at lag=8 and lag=0 accordingly. On insertion at fig. 3a comparison of values (Scr-Spl) at lag=8 for the model communities in section A-A (circles) and coefficients of correlation (Scr-Spl) for sites in spruce cultures (black squares); c) model dynamics of correlation coefficients (d 13-Spl) for communities gXX at lag=8.
(scr-Spl) on time, appropriate for comparison with the natural data, are rather sharp, and the difference in coefficients of correlation (Scr-Spl) for sites D22, R22, and AL5 (Table 3) is very small. Therefore, it is possible to accept that these sites are rather similar, including in initial density of communities, which practically is not known.
Table 1.
Coefficients of correlation between characteristics of individual plants of the model communities: "diameter of a stem" - d 1.3 and "the area of a crowns projection" - scr for the moment of section A-A (compare to Tab. 3); an explanation of a designation lag look in section Correlation of the individual characteristics .
correlation coeff. g89 g93 g90 g94 g95
(scr-d 3), lag=0 0.730 0.605 0.716 0.714 0.698
(scr-Spl), lag=8 0.368 0.113 -0.024 -0.027 0.303
(scr-Spl), lag=0 0.093 0.044 -0.026 0.202 0.164
(d 1.3 -Spl), lag=8 0.608 0.309 0.135 0.123 0.441
(d 1.3 -Spl), lag=0 0.004 -0.037 -0.034 -0.028 0.031
The model parametric analysis of correlations is not presented here, but some realizations (distinguished by samples of the random-number generator) of the variants described here and some other variants show that the general character of the dependence (scr-Spl) on time is retained. In particular, a change in the parameters of the model of a plant (Galitskii, 2003) - the value a (describing the intensity of expenditures of a plant on maintenance of biomass) by an order of magnitude or the parameter em (the level of dying off of a plant) by twofold - results in a change in the position of section A-A (Fig. 3) by approximately 1. This allows us, on the basis of the natural and model data presented, to conclude that, in the functions of free growth Af(T) and BF(T) used in the model (Galitskii, 2003), it is possible to estimate the parameter of the time scale Aj as the value Aj~360 years and the age of the practical end of growth of the biomass of a freely growing tree (spruce) as -1000 years.
It can also be noted that the positions of the extrema of the time dependences of (scr-Spl) and the measure of competition Cav described above are rather close, which can serve as one more argument in favor of this measure of competition. This is illustrated in Fig. 1b.
For the obtained moment of section A-A, it is possible to compare the values of the coefficients of variation for d 1.3 and Sc r of variants gXX (Table 2) with the corresponding coefficients of variation of the stem diameter and the area of a tree crown projection (Table 2, Galitskii, 2003b; Abaturov, Galitskii, 1995) that were used for ranging of sites in a natural study.
Table 2.
Model values of coefficients of variation of "the diameter of a stem" - d 13 and "the areas of a crowns projection" -scr for variants gXX at the moment of section A-A
variation coeff. g89 g93 g90 g94 g95
d 1.3 0.0448 0.0466 0.0468 0.0488 0.0526
sc r 0.210 0.272 0.243 0.245 0.208
The model values are much less than the natural ones - for diameters by 4-6 times and for crowns by 1.5-2 times - although, as for the natural values, the variation of diameters is less than the variation of the areas of projections of tree crowns, and for diameters the variation grows with the initial density of the community.
Correlation with neighbors
In Table 3, model values of correlation coefficients between some characteristics of a focal tree and characteristics of its neighbors for the moment of section A-A (see Fig. 3) are shown. As characteristics of neighbors, the average values of characteristics of the neighbors of the focal tree on a momentary mosaic, and in some cases with some lag as well, were taken.
Table 3.
Model coefficients of correlation of parameters of a focal tree: "the diameter of a stem" - d13 and "the area of a crowns projection" - scr with average (over lined) values of the same parameters of neighbors on a mosaic for homogeneous (gXX) various density and poisson (p90) communities at the moment of section A-A; an explanation of a designation lag look in item Correlation of the individual characteristics .
correlation coeff. g89 g93 g90 g94 g95 p90
(d1.3- d 1.3,n X lag=0 0.049 -0.0079 -0.015 -0.0084 -0.031 0.041
(scr- J cr,n ), lag=0 -0.0048 0.042 0.046 0.050 0.046 0.067
(scr- spl,n X lag=0 -0.098 -0.036 -0.024 0.075 0.0063 0.102
(scr- Ipl,n X lag=8 0.016 0.015 0.015 0.0013 -0.028 0.021
(<¿1.3- s pl,n ), lag=0 -0.056 0.0069 -0.0022 0.0035 -0.044 0.038
(d13- Jpl,n ), lag=8 0.016 0.044 0.027 0.029 0.045 -0.0071
Comparison of the first two rows of the table with the corresponding values of Table 4 (Galitskii, 2003b) shows that distinctions are rather significant, first of all, in values of coefficients: model values are much smaller. The last four rows of Table 3 contain correlation coefficients between characteristics of focal plants and average values of resources of neighbors. These coefficients are also rather small. The dynamics of model coefficients of correlation with neighbors looks rather casual, especially for biomass. As an example, in Figs. 4a and 4b, model
dependences of (scr-scrn) and (d 1-3-d 1.3,n) on time for the homogeneous g90 and Poissonp90
placements of plants are shown.
Comment [LM8]:
Conclusion
Thus, results of the model analysis indicate that the effect of nonmonotone change in the correlation coefficient of the areas of crown projections of trees in a ranged series of sites in spruce cultures of the Moscow suburbs of identical (90 years) age with a resource (the area of the Voronoi polygon) available to each tree, which was found in natural studies, is not an artifact. It can be considered validly as a consequence of nonmonotonicity of competition in uniform plant communities.
In this connection and as this model effect takes place with a rather large variation of model parameters, it is possible to consider sufficiently proven the obtained estimates of the parameter of the temporal scale of the model A1 - 360 years and, accordingly, the time of growth of the biomass of freely growing spruce trees in the Moscow suburbs -1000 years, which were made on
Comment [LM9]: (see above, comment 1)
Comment [LM10]: OK?
the basis of data of mapping of crown projections in spruce cultures of age 90 years (Abaturov, Galitskii, 1995). Apparently, this way of estimating a temporal parameter could also be applied to cultures of other species.
From the considered formulations of measures of competition intensity - the measure Cci, fixing the total lack of resources over a community and following from the definition of competition in a community of plants given by Clements and Weaver, and the measure Cav proposed here, reflecting the average degree of oppression of plants in a community because of lack of resources, the second measure may be considered preferable. The dynamics of this measure fully correspond to the change in time both of the amount of oppressed plants Nopr and of the coefficient of correlation (scr-Spl) between the area of a crown projection (biomass) of a plant and the resources (the polygon area) available to it in the community.
The model analysis of results of natural studies shows also the insufficiency of the two-dimensional models of plant and community used. Thus, although model coefficients of variation of "diameters of tree stems" and "areas of crown projections," as well as their natural analogues, are increased on manifestation of competition in the community, their dynamics is more complex than was possible to assume; first of all, their values are much less. The initial (given at the moment of the beginning of competition) normality of distributions of biomass and phytomass disappears in the order appropriate to the natural study; however, this occurs excessively quickly - at the first steps of work of the model. Correlation coefficients of characteristics of plants with characteristics of their neighbors, in particular, of "average" neighbors, are smaller than their natural analogues.
The two-dimensional form of the models used and, as a consequence, their excessive "rigidity," with sufficient conspicuity can be considered as the reason of these discrepancies. Thus, the model plant that has fallen into an oppressed state and not had time to receive an increase in the area of growth from its dying neighbors dies off rather quickly - over several time units (recall that the parameter of the time scale used in the model is A:=100). In nature, the plant experiencing a lack of resources has an opportunity to try to solve this problem at the expense of the third dimension, i.e., growth in height, thus translating a "symmetric" competition into an "asymmetric" one (terms from (Weiner, Thomas, 1986)). It ekes the time interval of dying off, increasing the coefficients of variation since so radical a cutoff of the left tail of distributions does not occur, and the affinity of distributions to the normal type is retained longer. The discrepancy of natural and model correlations with neighbors is also a consequence of the simplicity of the models. The point is that, as was noted earlier (Galitskii, 2003), in the model of a community, plants actually interact only at the moments of repartition of polygons, when their neighbors die off, and between these moments each plant grows and dies off independently on its Voronoi polygon. This weakens correlation relationships between neighboring plants, which as it is are not too great since on average six neighbors cooperate with each focal plant and many plants, previously not cooperating, appear neighboring because of dying off of the neighbors. In the real world, the competition of trees also occurs during growth in height, along the third dimension, i.e., in much greater intervals of time.
These problems, as well as the similar problems arising in the model analysis of the -3/2 rule (Galitskii, 1998), point to the necessity of development of three-dimensional models of a plant and a community of plants. Development of a three-dimensional model of a plant and then of a community is a significantly more complicated problem, to which various approaches are possible.
Among these are merging of modeling of the material processes connected with functioning of a tree and algorithmic modeling of the dynamics of the form of the tree based on so-called growth grammars, describing the spatial and functional structures of a tree - an approach intensively developing in a number of European centers (Kurth, 1994; De Reffye et al., 1995; Perttunen et al., 1996). Such an approach, rather simply realized in three dimensions, allows
(similarly to the approach used in the given work) one rather pictorially to formulate, realize, and interpret models, both of an individual plant and of small communities. It is thought that this promising approach can turn out to be an effective (if not the main) source of characteristics and parameters of models of a freely growing plant and algorithms of interaction of plants used in models of large communities.
Models of large communities of plants consisting of tens of thousands of elements and more, necessary for studying the behavior of communities in connection with the properties of the plants comprising them and in taking into account the structure of the community and the properties of the environment, require the development of models of an individual plant describing adequately enough the interaction of a plant with its neighbors in the community and being simple in internal construction. The simplicity should allow one to determine correctly the destiny of an individual plant in conditions of the community. Work with the models of a plant and a community used here allows us to propose that they can be a base for development (according to the principle of the "minimal angle of view" (Galitskii, Tjurjukhanov, 2001)) of quasi-three-dimensional models in which layers of biomass and phytomass of a plant will be "visible" and that will describe as well the "movement" of the biomass of a plant along the third dimension but not include "surplus" information on finer details of the construction.
Acknowledgements
I am very thankful to Mrs. L. Martin for improving English. The Russian Foundation of Basic Researches (grant #03-04-48616) supported the study.
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