— Vorest has proved to be a good tool for simulating natural stands with closed canopies.

— The Grazalema

Voronoi diagrams divide space into cells according to the closest distance to a tree location. For every tree location, its Voronoi region contains the points of the stand area that are closer to it than to any other tree location. The original Voronoi diagrams, developed by Georgy Voronoi in the early 1900s, consider Euclidean space. Since then, many generalizations of the Voronoi diagrams have been developed considering different spaces, different points or different metrics. The book by

For forest stands with closed canopies, the more size-symmetric the competition the higher the risk of stagnation (

Size-symmetric competition in forest stands has been defined as a situation where each individual competes and obtains growth resources in proportion to its size whilst size-asymmetric competition is defined as a situation where larger individuals have a disproportionate competitive effect on smaller individuals leading to the mortality of smaller ones (

Size-asymmetric competition is often associated with even-aged stands and size-symmetric competition with uneven-aged ones (

The rapid restocking that Grazalema

The objective of this paper is both to present an improved version of the growth model Vorest (

The study site is a 460 ha natural stand of

Grazalema is, as a whole, an irregular forest with wide size ranges (in terms of diameter at breast height, or

The main monitoring layout includes two 2100m^{2}-sized rectangular plots, representative of the Grazalema

The increase in mean and maximum stand

The model description follows a reduced version of the ODD (overview, design concepts and details) standard protocol for describing individual- and agent-based models proposed by

The purpose of the model is to simulate the current space-time development of natural forest stands with closed canopies and no significant, active recruitment. The main driving forces are growth, competition, and mortality.

The model handles two hierarchical levels: The lower level consists of individual trees and the upper level of forest stands (plots). The basic entities are individual trees that are characterized by the attributes and state variables specified in

The spatial resolution is one decimeter. The model landscape corresponds to forest plots or stands between 1,000 m^{2} and a few hectares. Tree locations are measured in meters (to the nearest decimeter) and

Vorest is a simulation model implemented as a computer program in C

Tree growth is considered in the model in terms of diameter increment (

When the main forest canopy is completely closed, the growing space is shared between the neighboring trees and typically most trees occupy a smaller growing space than they would do if they grew in the open. This behaviour leads to a corresponding reduction in tree growth compared to an open-grown tree of the same size (

The area that an open grown tree of the same

We define the

Vorest estimates the

The Voronoi diagram subdivides space acording to the proximity to a given set of points (

We call

In each simulation step, the following processes are run in the following order:

Synchronous size (

Identification of natural neighbors of each tree and updating the weights of each tree.

Computing the range of the potential growing space (PGS) of each tree according to its current size.

Synchronous simulation of natural mortality.

Calculation of the occupied growing space (OGS) for each tree.

a) Basic principles

The basic principles underpinning the model can be summarized as follows: The growth of each tree within the stand is calculated by applying a reduction coefficient to its potential growth to incorporate the effect of competition (

Natural mortality occurs after a period of reduced or zero growth, which is related to the size of the tree.

b) Emergence

The model simulates the growth of trees and their resulting size, survival or mortality and the space they occupy. A number of emergent properties, some highly relevant to forest management, are derived from the simulation, such as the spatial structure of the stand, both in terms of tree locations and dimensional aspects, as well as of population characteristics such as size distribution or stocking and the stand self-thinning dynamics.

c) Interaction

The main interaction between trees is competition. The partitioning of growing space between trees is simulated by linking resource availability with physical space. The area actually assigned to a tree can increase or decrease during the simulation, consistently varying its growing rate.

d) Stochasticity

Currently, there is no stochasticity in the model. All simulated processes are deterministic.

The input data for the initialization are provided in a text file (ASCII). The file includes the coordinates, species and initial diameter at breast height (

The growth model is based on the potential modifier method (

Where _{i,t} is the diameter at breast height of tree

Actual increment is simulated as a fraction of this potential increment depending on the competitive status of the tree.

Competition is related to the magnitude of the reduction in growing space that a tree undergoes due to the presence of adjacent trees that act as competitors. Vorest considers the “natural neighbors” as competitors of a subject tree as defined by the weighted Voronoi diagrams (see

As a surrogate for potential growing space (PGS), Vorest incorporates functions published in the literature (

where _{i,t} of a tree _{i,t}) in relation to the mean

The individual distance function assigned to each tree is continuously adapted at each time step to generate the weighted Voronoi diagrams. This individual distance function is derived from the Euclidean distance divided by the weight assigned to the tree (_{i,t}) according to

At each time step the weighted Voronoi diagrams define two issues: Which are the nearest neighbors of each tree (

_{i,t} is the occupied growing space of tree _{i,t} is the potential growing space of tree

If all neighbors of a tree are sufficiently far from a target tree (i.e. when the zones of influence (PGS) of every neighbor do not overlap with that of the target tree), the occupied growing space (OGS) of the tree equals the potential one (PGS) and thus relative area (

_{i,t} is the diameter increment of tree

Natural mortality can be either the consequence of sustained suppression or of ageing. In any case it is the ultimate result of poor growing conditions (

_{i,t} is the _{i,t}_{-5} denotes the

α and β are model parameters.

Apart from various estimations, Vorest also provides a graphical simulation output, which shows the spatial distribution of living and dead trees and the growing space occupied and available in the stand. We addressed the spatial edge effect by periodic edge correction methods (

a) Growth model

We used Grazalema plot F to fit the potential growth model. From measured

b) Competition model

To fit the competition model we needed to estimate initial values of

To fit the final growth model

c) Mortality model

As natural mortality is a sporadic event in forest stands and our time series only spanned nine years, we added mortality data of plots M1 to M6 to those of plot F in order to increase the availability of mortality data.

A threshold value of relative ^{5}) typical of live trees of the species ^{5} for dead and live trees.

We can see that ^{5} does not allow to sharply split dead and live trees, even if the average differences are statistically significant (mean ^{5} = 0.017 for dead trees and 0.033 for live trees; ^{5} of these suppressed live trees of 0.022.

We analysed the relationship between ^{5} and ^{5} value of dead trees. We realized that a fixed value of ^{5} as mortality threshold would not fit well to this condition and consequently defined a decreasing threshold survival function for ^{5} as in ^{5 }mortality values for trees with larger

We have fitted α and β parameters of this function using again the statistical package quantreg of R based on a quantile of 0.05 of the data of live trees (

The model performance was analyzed comparing simulated versus actual data both for growth and mortality.

To evaluate and validate growth simulation, goodness of fit has been assessed with the statistics root mean squared error

The predictive ability of the growth model has been evaluated by statistical estimates of bias (mean error, ME)

_{i} and

The growth model is also assessed graphically by means of plotting expected

To evaluate and validate the mortality model two similar statistics were used: Mean deviation (MD)

_{i} is a dummy variable that takes the value 0 if tree _{i} is another dummy variable that accounts for the simulated death or survival of trees. Comparing the number of dead trees in the simulation with the actual mortality in the same period is another global assessment of the mortality model.

In order to answer the initial question about the current stagnation risk of Grazalema

From the simulated stands we have derived

_{i }is the proportion of trees belonging to the

The values of the evaluation statistics for the

The data of another plot (plot V), not used to fit any of the submodels, was included in the validation of the model. The plot consisted of 127 trees at the beginning of the simulation in 1998.

For the validation of the mortality model we obtained values of

The observed trend of the diameter distributions is similar in two ways: There is a marked decrease in the initial peak located around 15 cm, and there is a progressive expansion of the frequencies along an increasing interval towards the upper classes. The initial distributions reflect the presence of two distinct cohorts with a mean

Vorest has proved a good tool to simulate current Grazalema

Nevertheless, the version of Vorest presented in this paper deals with the specific dynamic stage that the Grazalema

The model presented here has considered a stand development stage involving a closed canopy with a high level of competition that is currently blocking new recruitment. The aim of the simulated 60 years of stand development was to elucidate the level of stability we can expect in the stand in the near future without the intervention of silvicultural treatments or other kind of disturbances.

The long term dynamics of fir forests are currently much debated in many countries. It is generally assumed that the shade tolerance and the branching pattern provide the genus with a high competitive advantage.

Seedling banks play a major role in the regeneration process (Morin & Laprise 1997;

We have shown that, in our case, high levels of stocking together with high frequencies in lower diameter classes do not cause stagnation and the dynamics lead to an increasing size diversity. In an old growth spruce-fir stand,

The presence of one older cohort in the Grazalema

Using the present version of Vorest we have found that weighted Voronoi diagrams are a good method of modelling resource partitioning among trees and provide correct feedback loops steering the growth-competition processes in conditions of high stand density.

The simulation with Vorest has shown that there is enough asymmetry in competition relationships among

The slow suppression process is typical of shade tolerant species and in our case a comparatively low mortality-growth threshold has been confirmed.

Our simulation suggests a likely decline in competition pressure that probably can allow for new episodes of recruitment to occur, even if no catastrophic regeneration episodes take place.

Our study also revealed that there is currently a low risk of stagnation in the

As other authors (e.g.

It has frequently been demonstrated that tree growth in stands with closed canopies is largely a function of local neighborhood competition and that the nature of this competition is mainly size-asymmetric (