Abies pinsapo Boiss belongs to the Mediterranean group of firs that are the most ancient and less evolutionarily developed European species of this genus. Together with the Algerian species Abies numidica and the Turkish Abies cilicica, Abies pinsapo seems to be one of the first species to differentiate from the hypothesised common tertiary ancestor of all current Mediterranean species of this genus, including the widely spread Abies alba (Linares, 2011; Xiang et al., 2009). Some authors such as Alba-Sánchez et al., (2010) and Linares (2011) suggest an ancient segregation of the two Iberian Abies species: A. alba and A. pinsapo that could have remained isolated throughout the whole Quaternary. Abies pinsapo has occurred since then in two reduced areas on both sides of the Strait of Gibraltar, i.e. on European and African shores, between 1000 and 1800 m a.s.l. on the northern side, and between 1500 and 2150 m on the southern one. The genetic isolation and the limited area to which these populations have spread, aggravated by the intense habitat fragmentation as a result of ancient human intervention in the Mediterranean basin (Linares, 2011), have caused its current extremely low genetic diversity (Scaltsoyiannes et al., 1999) and its current status as endangered species. The small size of Abies pinsapo populations presents a considerable extinction risk. There are only three small areas of pinsapo forest in Spain, with a total extension of 3750 ha, from which less than 50% (1758 ha) are actually forests with closed canopies (MAGRAMA, 2015). Grazalema is one of these areas with nearly 500 ha of pure dense pinsapo stands that are the object of this study. The consequences of climate change involving an increase of temperature and a decrease of rainfall are likely to further reduce the chance of survival of these unique stands (Linares et al., 2009). Conservation management of these ecosystems requires a thorough understanding of its current dynamics that were heavily influenced by past human disturbances and subsequent conservation attempts (Linares & Carreira, 2009). It is assumed that the regeneration dynamics of pure Abies stands mostly rely on disturbance events opening up potential growing space (Morin, 1994). Abies dynamics involve long periods of biomass accumulation and increasing stocking (Matic, 2001) followed by eventual disturbance events (mainly pests and diseases) that provide sporadic opportunities for new cohorts of trees (Antos & Parish, 2002). Gap processes and patch dynamics have also been shown to play an important role in shade-tolerant stands that have been undisturbed for a long time, leading to a slower and more continuous turnover of the canopy (McCarthy, 2001; Antos & Parish, 2002). The dynamics of stands in intermediate overstocked periods are hardly understood. Competitive interaction and resulting differentiation and mortality processes are crucial issues for the dynamics and the stand structure at these developmental stages (Gray & He, 2009). Competition and also growth in dense forest stands are highly dependent on resource partitioning among trees. Trees acquire growth resources gathering them from their surrounding space and therefore resource partitioning has much to do with space partitioning.

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 Okabe et al. (2000) provides a good overview of the subject. Our model considers tree locations and the corresponding Voronoi cells are weighted by the size of each tree. We use different distance functions for computing distances between different trees. This way, we can simulate the different ability of neighboring trees to explore the surrounding space and can ensure an adequate partitioning of space among them.

For forest stands with closed canopies, the more size-symmetric the competition the higher the risk of stagnation (Oliver & Larson, 1996), especially if drought episodes become more frequent as it has been hypothesized for pinsapo stands in Spain (Linares, 2011). Over the last decades Spanish pinsapo stands have undergone a dramatic change in their dynamics due to a significant shift in management, i.e. from intensive timber management to strict conservation. Currently, after more than fifty years of almost total suppression of traditional ways of forest use including firewood, charcoal and livestock grazing, they have reached quite a high level of biomass accumulation (Coca et al., 2001; Linares et al., 2009) and there is no recruitment of small trees. On the other hand there is abundant advance regeneration in the understory waiting for opportunities (Arista, 1995; Cuadros et al., 2003).

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 (Weiner, 1990; Hara, 1992; Lundqvist, 1994; Bauer et al., 2004).

Size-asymmetric competition is often associated with even-aged stands and size-symmetric competition with uneven-aged ones (Lundqvist, 1994). In even-aged stands, canopy closure results in the extinction of light below canopy level and therefore in the suppression and death of slightly smaller individuals that remain below this light threshold (Weiner & Damgaard, 2006; Lundqvist, 1994; Oliver & Larson, 1996). However, in uneven-aged forests, the canopy is not totally closed at any height because otherwise there would not be enough light for smaller tree cohorts to develop. As a consequence self-thinning is not limited to small trees, rather it is distributed over the entire size range (Lundqvist, 1994). The nature of competition has much to do with stand structure and mostly with light extinction due to canopy closure. Nevertheless, size-asymmetric competition requires that some previous size differentiation among trees has established before canopy closure (Oliver & Larson, 1996). Otherwise, the competition among equal sized competitor trees would lead to an equal partitioning of resources (Uchmansky, 2000). In a forest stand with a closed canopy, this can lead to a high risk of stagnation, because many trees will approach light starvation as a result of insufficient growing space due to a lack of self-thinning (Oliver & Larson, 1996). This is the risk that threatens most artificially established dense even-aged pure stands. Silvicultural thinning is a way to avoid stagnation in such cases, just doing what nature is not able to do because of extreme homogeneity.

The rapid restocking that Grazalema pinsapo stands have undergone in the last decades and the resulting canopy closure achieved poses the question about the nature of the dynamics that they are currently experiencing and their needs for silvicultural treatments. Size differentiation and self-thinning processes may be most instrumental in maintaining the stand in a healthy good condition. An absence of these processes would require silvicultural interventions in order to avoid stagnation and decline.

The objective of this paper is both to present an improved version of the growth model Vorest (Abellanas et al., 2007; Abellanas et al., 2012) that allows to deterministically simulate the natural dynamics of forests stands with closed canopies and to apply it for diagnosing the nature of the dynamics that the studied Grazalema pinsapo stand is currently undergoing. Our aim is to identify the nature of the prevailing competition regime and the level of stagnation risk this Grazalema pinsapo stand it is potentially confronted with. Our findings inform what silvicultural management is required for conserving these unique stands.

Vorest has proved a good tool to simulate current Grazalema pinsapo dynamics through an appropriate simulation of self-thinning of dominated trees and by continuously re-adjusting the growing capacity of both, previously dominant and newly promoted trees. The growing space allocation among trees using weighted Voronoi diagrams has proved to be a good approach to simulate the mainly size-asymmetric competition that dominates differentiation and self-thinning in the Grazalema pinsapo stands.

Nevertheless, the version of Vorest presented in this paper deals with the specific dynamic stage that the Grazalema pinsapo stand is currently undergoing based on its history and is not supposed to cover extensive simulation periods. As stated by Antos & Parish (2002), in this kind of forests, variation in dynamics and in the processes controlling changes can be large in the long term.

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. Abies dynamics involve long periods of biomass accumulation and increasing stocking (Matic, 2001). Both dominated stems and seedlings, grow comparatively slowly and can persist for very long periods until their conditions improve (Antos et al., 2000; Antos & Parish, 2002). Consistent with this statement, we have found very low mortality related growth-rate threshold values for pinsapo trees.

Seedling banks play a major role in the regeneration process (Morin & Laprise 1997; Stewart et al., 1991; Woods, 1984; Antos & Parish, 2002). The recruitment of advance regeneration can occur as a result of major disturbances affecting canopy cover as well as of less catastrophic events leading to patch dynamics and gap processes or to even almost subtle endogenous events affecting only one or a few trees (Antos & Parish, 2002).

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, Antos & Parish (2002) have also found that there is no need for major disturbances to maintain a long-term stability in this kind of forest. In their case study, it probably was due to the low density of the subalpine forest that there was no sign of stagnation features after a long time period without exogenous disturbing events. In our case and for a limited time scale, the observed trend is probably the result of the presence of a few larger trees acting as seed trees.

The presence of one older cohort in the Grazalema pinsapo stand, has probably made the increasing size evenness possible, blurring the known effect towards bimodality that is the typical consequence of prevailing size-asymmetric competition (Weiner, 1986; Bauer et al., 2004; Eichhorn, 2010). On the other hand the simulation has suggested a likely decline in mortality rates for the next decades (not shown) that together with the reduction in basal area and number of trees per hectare could indicate a global decrease in competition load.

Pinsapo stands of Grazalema are currently undergoing a process of mortality by suppression caused by a relatively sudden restocking based on the radical change of management imposed a few decades ago by the establishment of strict protection rules. However, our simulation has also indicated a sufficient promotion of some of the youngest cohort trees that progressively colonise the upper diameter classes, thus changing stand biomass distribution towards a more uniform partitioning among the size classes and a simultaneous increase of the weight of intermediate and larger trees. This trend allows expecting an increasing stability of the stand in the near future even in the absence of exogenous disturbances.

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 pinsapo trees to promote an active self-thinning. But due to the high shade tolerance of firs and likely due to the presence of only few interspersed big trees of previous cohorts, these dynamics have also entered an increasing size differentiation process.

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 pinsapo stands of Grazalema, due to an increase in structural diversity that tends to prosper because of self-thinning and the promotion of younger cohort trees by old seed trees.

As other authors (e.g. Antos & Parish, 2002) have shown, the absence of disturbances not necessarily leads to a stagnation of stands with closed canopies, if size differentiation is encouraged.

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 (Oheimb et al., 2011; Grabarnik & Särkkä, 2011). The nature of competition and not only its intensity has proved an important factor driving forest dynamics and the resulting structure (Toda et al., 2010). As Bauer et al. (2004) have shown, it seems that size-asymmetric competition is indeed a general outcome of tree neighborhood interactions.