^{1}

^{2}

^{*}

This paper presents a successful application of techniques from the adjustment theory for modeling interaction in fine root biomass dispersion. Using special distance and species dependent weightings the influence function for fine root biomass dispersion of two species is estimated. Using the estimated influence functions the fine root biomass is predicted at the locations where the real data was sampled. Goodness of fit of our model is evaluated by comparing sample values and predicted values. However, the results show successful coincidence between sampled and predicted values. Finally, we present an example for the root dispersion in a mixed stand of beeches and spruces in Saxony/Germany.

Forest scientists as well as wood processing industry are interested in understanding and modeling interactions among trees in stands. The spatial distribution of the resource pool trees rely on affects the availability of these resources like water, nutrients and radiation for a tree. Now, such a characteristic as the fine root biomass depends on locations of the corresponding trees, their interactions and competition effects among them. Investigations of the spatial dispersion of the fine root biomass can contribute to an improved knowledge about these effects. In this paper, the multi-tree case with trees from two different species is discussed. At fixed locations at ground level we want to model the total mean of the fine root biomass in a fixed soil depth belonging to the trees from both species. Obviously, root dispersal is caused by the trees. The total biomass of fine roots consists of the contributions of the individual trees. Here, interactions can affect the root biomass. Therefore, we model the fine root biomass by a weighted sum of individual biomass contributions. Now, the fine root biomass measurements should additionally be weighted related to the relative number of neighbouring trees of the same species. This two-step weighting is necessary for modeling as well intra-specific interactions among trees of the same species as inter-specific interactions between different species. Obviously, our model can also be generalised for more than two species.

There are different mathematical methods for modeling interaction effects, see (

The study was carried out in a mixed spruce and beech stand consisting of 11 beech (species 1) and 17 spruce trees (species 2) in Germany (Saxony) near to Dresden. The study site is part of a greater nearly homogeneous spruce stand. In 2003, soil cores were taken at 226 given sampling points with collections of the fine root biomass from the forest floor organic and mineral horizon, see

Sampling procedure started with the collection of roots from the forest floor organic and mineral horizons using a steel frame (diameter 50 mm). The samples per core hole (organic horizon, mineral soil up to a depth of 80 cm) were filled in separated plastic bags. According to (

In

A deterministic alternative to the mark correlation function discussed in

Let us explain the main idea of the influence function method based on an example from forestry. We assume that each tree characteristic, for example the fine roots biomass, depends not only on the corresponding tree, but also on other trees, namely on their locations and characteristics. It seems to be realistic to assume that this influence only depends on distances between measurement points of fine roots biomass and the tree. We denote the maximal tree distance with still existing influence

Further, we assume that there are two tree species with two different fine root biomass influence functions. The measurements of fine root biomass are taken around each tree following radial transect lines. The whole plot is divided into two groups: fine root biomass from trees of the first and of the second species, see

The next model assumption refers to the linearity of the average of individual influences of neighbouring trees for a fine root biomass value. Let ^{j} _{0} _{i}, i = 1, …, N

The weights ^{j} ^{j}^{j}

We should note that another number of unknown variables in

leads to m values of _{k} ^{j} _{min} > 0is needed. Using seven or nine variables does not change the form of influence functions significantly, see

At second, each measurement ^{j} ^{j}

The solution of the following linear system of equations ^{j}

where

The elements of the matrix _{k}

The exact solution of

We applied the method described above for our data. From

It should be noted again that using more than 5 variables does not change the form of the influence functions significantly. The corresponding results can be seen in

In our paper we show that intra- and inter-specific interaction and competition effects among trees and species can be modeled by a special two-step weighting approach. From an ecological point of view this is necessary because forest dynamics is shaped by suppression, support and survival of the strongest species.

In general, a mathematical model is a problem-related simplification and requires some restrictions and assumptions. Goodness-of-fit should be proved for each model. In our approach two influence functions are estimated. Based on them, the values of fine roots biomass are predicted at the locations where real measurements are given. The comparison of real (observed) and predicted (model) values is carried out calculating their means, variances and the correlation between real and model values. The obtained results underline that our model is well suited for describing the fine root dispersion in a mixed stand of two species.

Applying the influence function approach is very helpful, especially for such cases if additional restricting demands of other mathematical approaches could not be satisfied. For example, there are too small sampling sizes or vague measurements, etc. An influence function is free from restrictions: It can be as well positive as negative, it is not normalized. It is a simple and meaningful tool for a preliminary analysis of spatial multidimensional data.

Our approach provides a more general and flexible approach in comparison with the approaches by

Further, other tree-dependent characteristics, which are of interest for forest scientists, can be analyzed in a similar way. Therefore, we want to contribute to the broader dissemination of our approach in forestry and ecological research.

The authors are grateful to S. Wagner and A. Zeibig, who collected the root data we used.

Locations of trees and fine root biomass measurements (transects) at the Rabenau site. The trees of the first species (beech) are coloured with red stars. The trees of the second species (spruce) are coloured with blue stars. Yellow diamonds show locations of measurements. Both axes X and Y are in [m].

Soil core from Rabenau.

Fitted fine root biomass influence functions (Y-axis). The red lines correspond to the first tree species (beech). The blue lines describe the estimated influence function for the second tree species (spruce) for

Fitted fine root biomass influence functions (Y-axis). The red lines correspond to the first tree species (beech). The blue lines describe the estimated influence function for the second tree species (spruce) for

Real measurements of fine root biomass for the first species (a) and for the second species (b) at Rabenau. Estimated fine root biomass for the first species (c) and for the second species (d) for ^{3}].

Some statistical characteristics providing goodness-of-fit of our model.

R | Species | Observed measurements | Values estimated with |
|||
---|---|---|---|---|---|---|

10 [m] | Species 1 | Mean | 60.9248 | 60.9561 | 60.8873 | 61.5880 |

Variance | 4.7628 x 10^{3} |
1.7635 x 10^{3} |
1.7387 x 10^{3} |
1.8707 x 10^{3} |
||

Correlation | 1.0 | 0.6152 | 0.6107 | 0.6395 | ||

Species 2 | Mean | 97.2478 | 99.9726 | 98.3769 | 98.4068 | |

Variance | 1.3658 x 10^{4} |
0.7236 x 10^{4} |
0.7660 x 10^{4} |
0.8045 x 10^{4} |
||

Correlation | 1.0 | 0.7513 | 0.7567 | 0.7712 | ||

15 [m] | Species 1 | Mean | 60.9248 | 61.6426 | 61.9068 | 61.809 |

Variance | 4.7628 x 10^{3} |
1.7532 x 10^{3} |
2.0310 x 10^{3} |
2.1523 x 10^{3} |
||

Correlation | 1.0 | 0.6031 | 0.6575 | 0.6768 | ||

Species 2 | Mean | 97.2478 | 100.317 | 98.61 | 98.794 | |

Variance | 1.3658 x 10^{4} |
0.7072x 10^{4} |
0.7729 x 10^{4} |
0.7558 x 10^{4} |
||

Correlation | 1.0 | 0.7297 | 0.7583 | 0.7522 |