Prognosis of the diameter distribution... PROGNOSIS OF THE DIAMETER DISTRIBUTION AND CARBON STOCK IN A SECONDARY ATLANTIC FOREST BY MARKOV CHAIN

– Growth and yield modeling at the diameter distribution level is an important tool to understand forest dynamics and to predict whether a forest will act as a CO 2 emissions source or sink. Therefore, the objective of the present study was to determine the diameter distribution and carbon stock of a forest fragment using the Markov chain to evaluate the impact of growth dynamics on forest carbon capture. Twenty plots of 10x50m were inventoried, between 2010 and 2015, counting the stems with dbh > 5 cm. Diameter distribution was projected for 2015 and 2020, considering the whole fragment and the ecological groups (pioneer and non-pioneer tree species). The volume was determined using allometric equation while biomass and carbon was determined by multiplying the volume by the basic wood density and the carbon content, respectively. The diameter distribution estimated for the fragment and ecological groups did not differ statistically from the values observed using the Kolmogorov-Smirnov test (p-value < 0.05). The number of stems ha -1 estimated in the fragment was from 1,692 in 2015 to 1,841 in 2020. For the pioneer and non-pioneer species, the number of stems ha -1 was from 476 and 1,203 in 2015 to 472 and 1,362 in 2020, respectively. The carbon stock increased 5.69 MgC ha -1 , with the greater contribution coming from the non-pioneer species. Therefore, it was concluded that the diameter distribution would remain “J-inverted” and the forest fragment would continue to act as a carbon sink over the coming years.


1.INTRODUCTION
Tropical forests play a fundamental role, providing ecosystem services and maintaining plant biodiversity . However, these environmental functions have been severely affected in recent decades due to intense fragmentation of these areas (Scarano and Ceotto, 2015). One of the tropical forests most affected by this fragmentation was the Atlantic Forest. Currently, there is only 12.5% of its original coverage spread across forest fragments smaller than 100 ha (Fundação SOS Mata Atlântica and INPE, 2015).
Given this scenario, studies focusing on growth and yield modeling are extremely important for the biodiversity and ecosystem service maintenance of these forest fragments (Lima and Leão, 2013). Independent of its structural complexity, the aim of modelling is to allow the projection of production into the future based on current conditions (Chassot et al., 2011), with diameter distribution models, in particular the transition matrix, being the most common for tropical forests (Teixeira et al., 2007;Vasconcelos et al., 2009;Batista et al., 2016).
Markov chain is a stochastic process, which is based on the concept of state (Souza and Soares, 2013). Therefore, it is presumed that a tree located in a determined diameter class at time t has a certain probability of remaining in this class or of migrating to another at time t+ as a function of its current state (Dalla Lana et al., 2015).
Despite the transition matrix being easily applicable, it presents some limitations such as the realization of projections only in years multiple to the monitoring period and the exclusion of possible changes to the forest structure over time (Schneider and Finger, 2000). However, this tool presents good accuracy in the diameter distribution projection in tropical forests over the short term (Stepka et al., 2010;Ebling et al., 2012).
Using the projection of diameter distribution associated with the probability of transition to other states, it is possible to identify the organizational processes that occur in forest ecosystems such as mortality, recruitment, growth and succession (Shen et al., 2013). Additionally, the projections made with the Markov chain can be useful as a tool to indicate if the forest will present as a sink or as a source of carbon dioxide for the environment in the future (Teixeira et al., 2007;Souza et al., 2014a).
Therefore, the objective of the present study is to determine the diameter distribution and carbon stock of a forest fragment using the Markov chain to evaluate the impacts of the growth dynamics on the carbon capture of the forest.

Description of the study area
The forest fragment has 44.11 hectares and is located in the Technology Park, Viçosa -MG with geographic coordinates 42° 51' W and 20° 42' S and an average altitude of 721 m (Souza et al., 2014b). The vegetation is located in a region classified as Semideciduous Seasonal Mountainous Forest (IBGE, 2012). According to the resolution CONAMA 392, the fragment is at a medium stage of regeneration presenting tree species with an average dbh of between 10 and 20 cm, as well as a height between 5 and 12 m (Brasil, 2007).
The local climate type is Cwa, according to the Köppen classification. The average temperature, humidity and annual precipitation for the period from 1968 to 2015 was 21.9°C, 79% and 1,274 mm, respectively (UFV, 2016). The region of Viçosa has pedogeomorphological gradients in which the tops of the hills are occupied by dystrophic Latosols rich in aluminum, the colluvial ramps by thinner, cambic Latosols and the bottom of the valleys present mainly nutrient rich epieutrophic Cambisols (Ferreira Júnior et al., 2012).
Various disturbances have occurred over the years in the forest fragment, including the extraction of wood and planting of agricultural cultivars and of eucalypt Prognosis of the diameter distribution... (Torres et al., 2013). Regeneration of the native vegetation in the area has been underway however, for approximately 25 years.

Data collection and analysis
Twenty 10x50 m plots were inventoried in which all the stems with dbh > 5 cm cm were measured and identified botanically in 2010 and 2015. The species were classified into ecological groups according to the division proposed by Gandolfi et al. (1995) and used in other studies such as Callegaro et al. (2015), Figueiredo et al. (2013) and Figueiredo et al. (2015), in which they were presented as pioneer (P), early secondary (Si), late secondary (St) and species without classification (Sc).
In 2015, mortality, which corresponds to those stems alive on the first occasion but dead on the following occasion, and the recruitment, which correspond to those stems that reached the minimum diameter for inclusion (dbh > 5 cm) in the last measurement were determined.
To project forest growth, the transition matrix model was used, which is based on the concept of the state of the arboreal individual. The movements between states were calculated using the probability of transition: p ij = n ij / n i , in which p ij is the probability of transition; n ij is the number of stems in the diameter class j, at time t+; and n i is the total number of stems in the diameter class i, at time t (Souza and Soares, 2013).
The transition matrix was constructed based on the determination of probability of transition with diameter amplitude of 5 cm, in which the stems could assume one of the four possible states: migrate to a larger diameter class due to growth in diameter, remain in the same class in which they were found, enter into the first diameter classes or die (Vasconcelos et al., 2009).
To estimate the number of trees at time t+ for each diameter class, the chapman-Kolmogorov matrix equation: Y t+ = (G*Y it ) + C 1 , was used, in which Y t+ is the vector column for the number of trees by diameter class, at time t; and C 1 is the vector column for the stems that entered during the monitoring period (Parzen, 1962).
To estimate the number of stems that entered into the first diameter class, the adjustment of the equation referent to the model: I t+ = e 0+1*Bt + , in which I t+ is the tree recruitment per hectare in time t+;  0 and  1 are the parameters of the equation to be estimated; Bt is the base area in m 2 ha -1 in tempo t;  is the random error, was evaluated (Souza and Soares, 2013). The evaluation criteria for the adjustment were the coefficient of correlation between the observed and estimated values (r yy ) and the significance of the parameters. These procedures were necessary due to the limitation of the Markov chain in not projecting the stems that entered into the system.
Based on the observed data, diameter distribution projections were realized for 2015 and 2020 considering all the stems of the fragment as well as by ecological group (pioneer and non-pioneer species). In the projection by ecological group, those species without ecological classification were excluded. To test the statistical significance between the projected and observed diameter distributions, the Kolmogorov-Smirnov adherence test (K-S) was used, at a significance of 5% (Reis, 2017).
Tree volume in the 2015 and 2020 projections was determined by the equation: VF cc =0,000070*CC 2,204301 *H L 0,563181 *SD, in which VF cc is the volume of stems with bark, in m 3 ; CC is the center of the diameter class, in cm; H L is Lorey's height in m; SD is the stems density, in stems ha -1 (Amaro, 2010).
Biomass and the carbon stock were determined using the average basic wood density and the carbon content of the species, respectively. For this, three trees from each species were selected as well as by diameter class for the partial analysis of the trunk, which corresponds to the removal of a wood sample at a height of 1.30 m, with the help of increment borers (Torres et al., 2017). The material was then taken to the laboratory where part of it was used to determine the basic wood density for each species, according to the methodology described by Vital (1984) andNBR 11941 (ABNT, 2003), and the other part submitted to complete calcination in an muffle furnace to determine the carbon content, according to the methodology described by Torres et al. (2013).

3.RESULTS
The parameter  1 of the equations adjusted to estimate the number of stems that entered at a level of 5% significance was not significant for the forest fragment or for the ecological groups (Table 1). Therefore, the observed averages for recruitment was used for the monitoring period, equal to 355, 255 and 77 stems ha -1 for the forest fragment, and non-pioneer and pioneer species, respectively.  (Table 2).
The average basic wood density and carbon content of the pioneer species was 0.517 g cm -3 and 52.3%, respectively. For the non-pioneer species, the density was 0.526 g cm -3 and the carbon content was 53.0%. Based on this data, the biomass was calculated in Mg ha -1 , and the carbon stock in MgC ha -1 , both for the forest fragment and the ecological groups by diameter class (Table 3).

4.DISCUSSION
The non-significance for the parameter  1 shows that there was no relation between the basal areas of the parcels and tree recruitment. Despite this, the independent variables that denote the conditions of the stand, such as the basal area, are the most recommendable to estimate the number of trees recruited in tropical forest (Rossi et al., 2007a). Therefore, the observed average for tree recruitment during the monitoring period was used as input for the transition matrix. However, it is only possible to infer how other variables, for example, the occurrence of clearings (Rossi Table 1-Basal area (Bt), in m 2 ha -1 , parameters of the equation ( 0 e  1 ) and correlation coefficient (r yy ) for pioneer (P) and non-pioneer (NP) species and for the forest fragment (T). Tabela 1-Área basal (Bt), em m 2 ha -1 , parâmetros da equação ( 0 e  1 ) e coeficiente de correlação (r yy ) para as espécies pioneiras (P), não pioneiras (NP) e para o fragmento florestal (T).* significant at the 5% level of significance ns not significant GE Bt   Prognosis of the diameter distribution... et al., 2007b), fires (Ribeiro et al., 2012) and pests (Trumbore et al., 2015) modify the environment and influence tree recruitment with the long term evaluation.
The number of stems between the monitoring period (2010 and 2015), increased from 1,526 to 1,692. This is mainly explained by the trees recruitment (335 stems ha -1 ) having been greater than mortality (169 stems ha -1 ). Additionally, there was a significant migration of stems between the diameter classes due to the accumulation of biomass by the trees (Hanberry et al., 2016;Poorter et al., 2016).
in terms of tropical forest (Batista et al., 2016). Another point that contributed to the efficacy of the projection was the absence of disturbances in the fragment, given that this can affect the prognosis and generate imprecise and tendentious estimations (Xie et al., 2013). However, the absorbent state was observed in the center of the 47.5 cm class, indicating an accumulation of stems in this diameter class due to presenting a probability equal to zero for transition to the next class. However, this did not invalidate the use of the transition matrix for short periods, since it is a stationary matrix not recommended over long evaluation periods (Dalla Lana et al., 2015).
Based on the demonstration of the efficacy of the Markov chain, the diameter distribution for 2020 was projected, which could reach 1,841 stems ha -1 , taking into account the recruitment of 335 stems ha -1 , and a mortality of 180 stems ha -1 . Therefore, due to the elevated number of young plants that will substitute the adult plants in the future, we can infer that the fragment will continue to evolve in a sustainable manner over the coming years (Batista et al., 2016).
In terms of the ecological groups, a reduction in the number of pioneer species in the forest fragment between the monitoring periods, dropping from 485 to 476 stems ha -1 , was observed. For these species, mortality was greater than tree recruitment, corresponding to 86 and 77 stems ha -1 , respectively. By contrast, the number of non-pioneer species increased over the years. In 2010, 1,030 stems ha -1 were counted increasing to 1,203 stems ha -1 in 2015. Tree recruitment for this species was 255 stems ha -1 while mortality was 82 stems ha -1 .
This predominance of non-pioneer species in the forest fragment mainly occurred due to the successional advance (Souza et al., 2012). Therefore, it is presumed that there is a reduction of luminosity due to the tree canopy, which suppressed shade intolerant species and benefited recruitment of ombrophil species (Chazdon, 2012). In addition to this abiotic condition, factors such as the existence of a source of propagules and dispersers near to the forest may have favored the substitution of the species throughout the natural succession process (Shono et al., 2007;Chazdon, 2012;. Just as with the forest fragment as a whole, the Markov chain was able to satisfactorily project the diameter distribution when divided into ecological groups. For the pioneer species, the number of estimated and observed stems per hectare for 2015 was 476 (D Calc = 0.015 < D Tab = 0.062). For the non-pioneer species, the estimated and observed number was 1,203 stems ha -1 (D Calc = 0.002 < D Tab = 0.039). When projecting the diameter distribution for 2020, 472 stems ha -1 were observed for the pioneer species and 1,362 stems ha -1 for the non-pioneer species. Based on these results, it was observed that the forest fragment could continue in successional advancement over the coming years if there was no disturbance that affected the forest's growth dynamic.
In terms of the carbon stock, a 5.69 McC ha -1 increase was observed in the period from 2015 to 2020, taking into account the whole forest fragment. This increase was also verified for the ecological groups over this same period, with the carbon stock of the pioneer species increasing from 13.28 MgC ha -1 to 14.80 MgC ha -1 and the non-pioneer species from 29.41 MgC ha -1 to 33.42 MgC ha -1 This greater contribution to the carbon stock by the non-pioneer species is mainly due to the greater number of trees per hectare, according to the projection by the Markov chain, as well as due to these species presenting greater density in comparison with the pioneer species which directly affects the carbon stock (Poorter, 2008;Fonseca et al., 2011).
Secondary forests in tropical regions and those undergoing successional advance, have a fundamental importance in providing ecosystem services, mainly as carbon sinks for the atmosphere (Chazdon, 2012). It is estimated that, in the period from 2000 to 2007, these forests stored a total of 1.6 ± 0.5 PgC year -1 , with this estimate being higher than for some primary tropical forests due to the rapid accumulation of biomass under succession (Pan et al., 2011). Therefore, it is expected that the forest fragment can stock a greater quantity of carbon with successional advance and contribute to attenuating the negative effects stemming from climate change.

5.CONCLUSIONS
By 2020, the diameter structure of the fragment will reach a total of 1,841 stems ha -1 following the "Jinverted" distribution. The forest will continue to act as a carbon sink for the atmosphere over the coming years, reaching a carbon stock of 48.97 MgC ha -1 , with the greater contribution coming from non-pioneer species.

6.ACKNOWLEDGEMENTS
We would like to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Prognosis of the diameter distribution... the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and the Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) for the research funding. We would also like to thank the study group in economy and forest management (GEEA) for their assistance with data collection.