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Modeling sapling distribution over time using a functional predictor in a generalized additive model

  • Daniel Moreno-Fernández
  • Nicole H. Augustin
  • Fernando Montes
  • Isabel Cañellas
  • Mariola Sánchez-González
Original Paper

Abstract

Key message

The effect of adult trees on sapling density distribution during the regeneration fellings is determined in a Pinus sylvestris L. Mediterranean forest using generalized additive models.

Context

Spatial pattern of adult trees determines the number of new individuals after regeneration fellings, which modify the light and air temperature under tree canopy.

Aims

We proposed a novel spatiotemporal model with a functional predictor in a generalized additive model framework to describe nonlinear relationships between the size of the adult trees and the number of saplings of P. sylvestris and to determine if the spatial pattern of the number of saplings remained constant or changed in time.

Methods

In 2001, two plots (0.5 ha) were set up in two phases of regeneration fellings under the group shelterwood method. We mapped the trees and saplings and measured their diameter and height. The inventories were repeated in 2006, 2010, and 2014.

Results

We found a negative association between the diameter of adult trees and number of saplings up to 7–8 m. Beyond these distances, the diameter of adult trees was not associated with the number of saplings. Our results indicate that the spatial pattern of the number of saplings remained quite constant in time.

Conclusion

The generalized additive models are a flexible tool to determine the distance range of inhibition of saplings by adult trees.

Keywords

Edge effect Intra-specific competition Mountain forest Shade tolerance Mediterranean areas 

1 Introduction

Two main types of models can be used to explain or predict the renewal of a forest after regeneration fellings, seed dispersion, and germination: regeneration and recruitment models. The former is related to the youngest individuals, seedlings, whereas the latter is related to larger stems, saplings, which reach or exceed a nominal size limit determined by the researcher (Vanclay 1992; Eerikäinen et al. 2007; Miina and Heinonen 2008). Since it is both difficult and expensive to obtain suitable data for modeling the regeneration, recruitment is more often modeled than regeneration. Both processes are influenced by the capacity of the soil to supply water and the amount of light that reaches the young seedlings. These are the most important factors for success in the establishment of new individuals (Kozlowski 2002). Hence, the summer drought in dry environments cause high mortality rates of seedlings over Mediterranean areas (Castro et al. 2004; Pardos et al. 2007; McDowell et al. 2008), where the water is a limited resource in the vegetative period.

Regeneration fellings can modify the effect of summer drought on seedlings and saplings by setting different target densities or spacing between remaining trees and, thus, modify the shade and the air temperature (Caccia and Ballaré 1998; Pardos et al. 2007). However, not all species can tolerate the same amount of shade and the shade tolerance behavior may vary with site conditions (Kobe and Coates 1997; Gómez-Aparicio et al. 2006). Additionally, the light requirement of plants varies with age. Indeed, the light requirement increases faster with plant age in light-demanding species than in shade tolerant species (Valladares and Niinemets 2008). This determines the density and spacing between remaining trees after regeneration fellings. Therefore, it is necessary to have a clear understanding of the effects of the density of residual trees on new individuals over the regeneration period in order to ensure the spatial continuity of the forest stand after the regeneration fellings.

In addition to the density and spacing between remaining trees after regeneration fellings, several features should be taken into account to model the number of saplings. The age of the stand should also be considered, particularly where shifts in the spatial relationship between trees and offspring over the stages of the forest renewal may occur (Wada and Ribbens 1997). Changes in spatial patterns of trees over time are determined by regeneration mechanisms, substrate characteristics, moisture and light availability as well as intra and inter specific competition (LeMay et al. 2009). Hence, the time perspective allows us to distinguish between competition and the initial spatial pattern of individuals (Wolf 2005; Getzin et al. 2006), i.e., the initial distribution of seedlings as a consequence of the dispersion and germination of the seeds can vary with the development of the seedlings and competition for resources.

The spatial relationships between adult trees and new cohorts have previously been evaluated using different approaches. The bivariate Ripley’s K and related functions have been used to determine if stems of two mapped cohorts of trees show spatial positive, negative, or random association (see Montes and Cañellas 2007; Wild et al. 2014) by testing the spatial independence between the two cohorts. Ledo et al. (2014) used inhomogeneous Poisson process spatial models. These models allow the spatial distribution of new individuals to be defined in function of attributes of adult trees. Other authors used distance-dependent influence indices (Contreras et al. 2011) and available light under the forest canopy or the global site factor as explanatory variables in different models (Pardos et al. 2007; Moreno-Fernández et al. 2015a). Distance-dependent influence indices determine, at a given point, the influence of the tree size (such as diameter, height, or crown variables) and the distance between trees and the studied point whereas the global site factor measures the amount of light at a given point by analyzing hemispherical photographs. Influence indices and site factors can easily be entered in a time-dynamic model as additional variables (Eerikäinen et al. 2007; Manso et al. 2013). However, the temporal modeling of Ripley’s K and related functions over time is complex. LeMay et al. (2009) investigated the evolution of these functions in the regeneration of Pseudotsuga menziesii var. glauca (Mirb.) Franco over time using a random coefficient mixed model. Furthermore, specific distance dependent models implemented using packages such as SILVA or SORTIE-ND have been used in forest development simulation studies which include the regeneration establishment phase (Hanewinkel and Pretzsch 2000; Ameztegui et al. 2015). These software packages are compounded of several submodels for the biological processes operating at individual tree level. Comas (2008) and Redenbach and Särkkä (2012) adapted the growth-interaction model proposed by Renshaw and Särkkä (2001) to develop a spatiotemporal regeneration model under two regeneration methods using values taken from the literature to estimate the parameters. This approach generates marked point configurations changing over time.

Generalized additive models (GAMs) may describe a complex relationship between the response and the predictors. This is especially useful in research fields such as ecology, biology, or forestry in which simple models cannot capture the structure of the data and more complex models may be required (Faraway 2006). Whereas GAMs have been used in different areas of forest science such as wood quality (Jordan et al. 2008), annual radial growth (Moreno-Fernández et al. 2014), mortality (Barbeito et al. 2012), or species distribution (Franklin 1998), their use in regeneration or recruitment studies is relatively scarce (Rabasa et al. 2013). Augustin et al. (2009) fit spatiotemporal models within a GAM framework to monitor forest health data. However, these techniques have never been used to assess the dynamics of forest regeneration.

Pinus sylvestris L. is the most widely distributed pine species in the world (Mason and Alía 2000). It can be found throughout Eurasia, stretching from Spain in the South-West to the far east of Russia (Houston Durrant et al. 2016). This pine species is commonly considered to be a light-demanding species in Central and northern Europe (Mátyás et al. 2003). However, it has a half-shade tolerant behavior in southern locations like Spain (Montes and Cañellas 2007), partially due to the high temperatures and drought conditions present during the summer months. Whereas during the early stages P. sylvestris seedlings prefer moderate light conditions (Pardos et al. 2007; Barbeito et al. 2009), the later development of saplings is inhibited by competition from the adult trees (Montes and Cañellas 2007). The variation on shade tolerance and climate conditions across its distribution condition the regeneration method; while seed tree and clear cutting are the main methods used in Central and Northern Europe, different alternatives of the shelterwood method are commonly used in Southern Europe (Mason and Alía 2000). In general, 2000 seedlings per hectare are considered to be a sufficient natural regeneration density (Rodríguez-García et al. 2010; Hyppönen et al. 2013).

In this work, we propose a methodology to describe nonlinear relationships between the size of the adult trees and number of saplings of P. sylvestris in Mediterranean mountains as a smooth function. We carried it out analyzing data from repeated measurements of two large plots at two stages of the regeneration period where all the stems were mapped. We modeled the spatiotemporal distribution of the number of saplings using a functional predictor (see for example Wood 2011) in a GAM framework (Hastie and Tibshirani 1989; Wood 2006). The functional predictor allowed us to weight the effect of every adult tree on the number of saplings per quadrat based on the distance between adult trees and saplings. In addition, the approach can deal with spatial correlation and a spatiotemporal trend, i.e., changes in the spatial pattern of number of saplings during the development of the stand. In this regard, we fitted two models with different spatiotemporal structures to determine if the spatial pattern of the number of saplings remained constant or changed in time.

2 Material and methods

2.1 Study area and data

The study was carried out in a Scots pine forest (Pinar de Valsaín) located on the north facing slopes of the Central Range of Spain (40°49’ N, 4°01’ W). The elevation ranges from 1200 to 1600 m, the annual rainfall is about 1000 mm and the mean temperature is around 9.8 °C. Regeneration is achieved using the group shelterwood method over a 40-year regeneration period. The regeneration fellings create small gaps (0.1–0.2 ha) for the establishment of the regeneration. As regeneration appears, subsequent harvests are carried out over the regeneration period to widen the gaps. The final fellings under the group shelterwood method take place at 120–140 years but some legacy trees are left for biodiversity conservation reasons at the end of the regeneration period.

In 2001, we set up a chronosequence of six plots (0.5 ha) covering all the rotation period (see Moreno-Fernández et al. 2015b for details) to study the dynamics and structure in Mediterranean forests of P. sylvestris. This chronosequence represents the management of P. sylvestris in the study area from the beginning to the end of the rotation period (Fig. 1) and it contains six plots. The plots were as homogeneous as possible in terms of altitude, exposure and site quality. Since we aim to address the influence of the adult trees on the saplings, we selected two plots at different stages of the regeneration period: at an intermediate stage of the regeneration period (100 × 50 m, Fig. 2, ca. 19 years old) and at the end of the regeneration period (58.82 × 85 m, Fig. 3, ca. 32 years old). Young individuals with different size were spread over the youngest studied plot. In this plot, regeneration fellings were done from 2010 to 2014 removing mainly trees located in the corners of the plot (Fig. 2). At the end of the regeneration period, the arrival of new individuals has almost been completed and the crown cover is getting closer. Additionally, some legacy trees (larger trees) appear in this plot (Fig. 3). Another plot, at the first stages of the regeneration period, was available. However, the arrival of new individuals has started as consequence of the natural dynamics but the number of saplings was still quite low (Fig. 1). Therefore, we did not include this plot in the analysis.
Fig. 1

Position of adult trees (dbh ≥ 20 cm; red circles), small trees (10 ≤ dbh ≤ 20 cm; green dots) and number of saplings per quadrat (darker tones indicate larger number of saplings) of the six plots of the chronosequence in 2001. Size of adult trees is proportional to dbh

Fig. 2

Position of adult trees (dbh ≥ 20 cm; red circles), small trees (10 ≤ dbh ≤ 20 cm; green dots) and number of saplings per quadrat (black and gray squares) at intermediate stages of the regeneration period in 2001 (upper left), 2006 (upper right), 2010 (bottom left), and 2014 (bottom right). Size of adult trees is proportional to dbh

Fig. 3

Position of adult trees (dbh ≥ 20 cm; red circles), small trees (10 ≤ dbh ≤ 20 cm; green dots) and number of saplings per quadrat (black and gray squares) at the end of the regeneration period in 2001 (upper left), 2006 (upper right), 2010 (bottom left), and 2014 (bottom right). Size of adult trees is proportional to dbh

At the time the plots were set up, we carried out the first inventory in which all the stems higher than 1.30 m were labeled individually and their diameter at breast height (dbh) and height were measured. We numbered and classified the stems into the following: trees (dbh ≥ 10 cm) and saplings (height ≥ 1.30 m and dbh < 10 cm). We distinguished two cohorts of trees: adult trees (dbh ≥ 20 cm) and small trees (10 ≤ dbh < 20 cm). We mapped the position of every tree (adult and small trees) in each plot and additionally, we grouped the saplings into a 2 × 2 m quadrat grid. The coordinates of the center of each quadrat were used to determine the position of each quadrat. These measurements were repeated in 2006, 2010, and 2014.

In order to model the sapling distribution, we used the number of saplings per quadrat (Ns) in each plot as response variable. We expected Ns to be highly related to the density of surrounding trees and distance to the surrounding trees, as well as to the time since the beginning of the regeneration fellings. However, the spatial dependence between the saplings and the two cohorts, adult and small trees, varies over stand development (Montes and Cañellas 2007). Thus, we considered as predictors the dbh of the adult trees (dbh ≥ 20 cm), the distance in meters from adult tree to each sapling quadrat (considering all the adult trees within a maximum radius of 30 m from each sapling quadrat; Montes and Cañellas 2007) and the number of small trees (Nsmall; 10 ≤ dbh < 20 cm) surrounding every sampling quadrat within a radius of 10 m and the inventory year. The distribution of Ns, number of small and adult trees over inventories is shown in Figs. 1, 2, and 3. We assume that at a given distance, larger dbh of the adult trees entails greater competition between adult trees and saplings. Furthermore, we consider that this competition effect between adult trees and Ns decreases with distance. Therefore, a model in which the coefficient of the dbh depends on the distance between adult trees and the sapling quadrat would be very suitable. These requirements can be taken into account using a linear functional predictor in a GAM. Thus, this approach allowed us to weight the effect of every adult tree on the number of saplings per quadrat based on the distance between adult trees and saplings.

2.2 Edge effect correction

The quadrats close to the boundaries of the plots are affected by the edge effect and this must be corrected (Ledo et al. 2014). Thus, the number of adult and small trees which surround a quadrat within 30 and 10 m, respectively, can be underestimated because some of them may be located outside the plot (Goreaud and Pélissier 1999). Several authors (Lancaster and Downes 1998; Perry et al. 2006; Pommerening and Stoyan 2006) have investigated the edge effect and have analyzed the suitability of different edge-corrections for the calculation of the indices of spatial forest structure, Ripley’s K and related second order functions.

In order to take account of the edge effect on the number of small and adult trees, we used values per unit area, i.e., density. For each quadrat, we estimated the area of the 10 m radius circle within the plot (AreaIn10 in m2). Therefore, AreaIn10 changes with the distance between the quadrat and plot border, i.e., AreaIn10 is smaller in the quadrats closer to the plot border. Then, we obtained the density of small trees as Nsmall/AreaIn10. We corrected the edge effect on adult trees by using the dbh density as dbh/AreaIn30. AreaIn30 is the area (in m2) of the 30 m radius circle within the plot. Thus, we assume that the surrounding shelter trees outside the plot would be of similar density than within the area. AreaIn30 is the area (in m2) of the 30 m radius circle within the plot.

2.3 Statistical analysis

For each of the two plots, we modeled the expected number of saplings E(Ns ij ) = μ ij in quadrat i and jth inventory (j = 1,…,4) using the following GAM:
$$ \log \left({\mu}_{ij}\right)=\alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{dbh_{jn}}{\mathrm{AreaIn}{30}_i}\right)+{f}_2\left({X}_i,{Y}_i\right)+{\mathrm{Time}}_j $$
(1)
with Ns ij following a negative binomial distribution. This distribution is suitable for overdispersed counts such as those we are dealing with here. The variance function is V(μ ij ) = μ ij  + μij2/θ, involving the extra parameter θ to be estimated. The greater θ is, the more similar the negative binomial distribution is to the Poisson distribution. Small values for θ indicate aggregation. The parameter α is the intercept of the model, β is the unknown but estimable parameter of the number of small trees. Dist in is a matrix which contains the distances (in m) from the adult tree (n = 1, ..., N) to the ith quadrat, whereas dbh jn is the matrix of the dbh of the adult tree (n = 1,..., N). When the distance of the nth adult tree to the ith quadrat was greater than 30 m, the dbh was set to 0. \( \sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot {dbh}_{jn}/\mathrm{AreaIn}{30}_i\right) \) is functional predictor where f 1 (Dist in ) is the smooth coefficient of dbh jn . The function f2(X i , Y i ) is a spatial smooth term to account the spatial trend and spatial correlation of the number of saplings. Any spatial trend will caused by other unmeasured environmental variables and hence the spatial smooth term is a proxy for other unmeasured environmental effects. X i and Y i are the coordinates of the ith quadrat and Time j is the temporal factor referred to the jth inventory.
Model 1 above separates the effects of space and time, i.e., the two effects are additive. The model can be made more flexible by allowing the spatial smooth to change in time, i.e., this model contains a spatial smooth per jth inventory:
$$ \log \left({\mu}_{ij}\right)=\alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{dbh_{jn}}{\mathrm{AreaIn}{30}_i}\right)+{f}_2^j\left({X}_i,{Y}_i\right) $$
(2)

We used Akaike’s Information Criterion (AIC) to select the variables by using backward stepwise procedure and choosing the best spatiotemporal structure.

Functions f1 and f2 were represented using thin plate regression splines (Wood 2003). Thin plate regression splines keep the basis and the penalty of the full thin plate splines (Duchon 1977) but the basis is truncated to obtain low rank smoothers. This avoids the problems of the knot placement of the regression splines and reduces the computational requirements of the smoothing splines (Wood 2003). Penalized regression smoothers such as thin plate regression splines are computationally efficient because their basis have a relatively modest size, k. In practice, k determines the upper limit on the degrees of freedom associated with the smooth function, hence k must be chosen when fitting models. However, the actual effective degrees of freedom of the smooth function are controlled by the degree of penalization selected during fitting. The degree of penalization determines how smooth the function is. So, k should be chosen to be large enough to represent the underlying process reasonably well, but small enough to ensure reasonable computational efficiency. The exact choice of k is not critical (Wood 2006).

The spatial smooth f2(X i , Y i ) is confounded with the functional predictor term, \( \sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot {dbh}_{jn}/\mathrm{AreaIn}{30}_i\right) \), since both terms describe, in some way, the spatial pattern in the response. To avoid further confounding, we decided to include the effect of small trees in a linear form rather than a functional predictor. We used k = 10 for f1 since it was enough to represent the variation of the coefficient of dbh as the actual effective degrees of freedom for f1 was between 3 and 4—well below 10. As we are ultimately interested in estimating the f1 of the functional predictor and f2 is entered to eliminate the spatial correlation, we selected the smallest basis dimension (k) in f2 that eliminated the spatial correlation. For the different values of k in f2, we checked whether the spatial correlation had been eliminated in the model by plotting semivariograms of the model residuals per inventory with envelopes from 99 permutations under the assumption of no spatial correlation (see Augustin et al. 2009 for a description).

The statistical analyses were carried out in R 3.3.3. (R Core Team 2017) using the “gam” function of the package “mgcv” (Wood 2011) for fitting the models where we used the restricted maximum likelihood option. This means that the smoothness parameters are estimated using restricted maximum likelihood estimation and a penalized iterative re-weighted least squares algorithm is used to find all other parameters, i.e., the coefficients of basis functions and coefficients of linear terms. See Wood (2011) for the theory and Augustin et al. (2015) for a functional predictor example. For model checking, we used the functions “variog” and “variog.mc.env” of the package “geoR” (Ribeiro and Diggle 2016) for estimating the semivariograms and the envelopes.

3 Results

3.1 Intermediate stages of the regeneration period

The total number of saplings was inversely related to the time whereas the mean size (dbh and height) of the saplings increased with time (Table 1). Saplings were spread around the plot except in the center and the bottom left corner (Fig. 2). Unlike the saplings, we found that the number of trees, both small and adult trees, increased with time, the mean dbh and size of this stratum decreasing with time due to ingrowth of individuals from the previous class (Table 1). During the study period, we found a great increase in small trees, especially in the lower right part of the plot (Fig. 2).
Table 1

Summary of the mean forest features in each plot during the four inventories. Trees (dbh ≥ 10 cm), adult trees (dbh ≥ 20 cm), small trees (10 ≤ dbh < 20 cm), saplings (dbh < 10 cm and height ≥ 1.30 m). Standard deviation is within brackets

Feature

2001

2006

2010

2014

Intermediate stage of the regeneration period

 Number of saplings

1861

1625

1498

1347

 Mean diameter of saplings (cm)

3.68 (2.31)

4.42 (2.29)

4.61 (2.38)

4.77 (2.36)

 Mean height of saplings (m)

4.20 (1.78)

4.86 (2.03)

5.38 (2.21)

5.76 (2.36)

 Number of adult trees

80

62

75

102

 Number of small trees

152

250

351

399

 Number of trees

232

312

426

501

 Mean diameter of trees (cm)

23.07 (14.75)

17.89 (11.30)

16.68 (8.89)

16.96 (8.72)

 Mean height of trees (m)

15.00 (6.90)

12.46 (5.32)

12.31 (4.23)

13.26 (4.16)

End of the regeneration period

 Number of saplings

558

364

208

117

 Mean diameter of saplings (cm)

5.54 (2.66)

6.43 (2.32)

6.55 (2.24)

6.97 (2.19)

 Mean height of saplings

7.03 (3.27)

7.82 (3.32)

8.61 (3.44)

8.70 (3.55)

 Number of adult trees

174

233

283

317

 Number of small trees

568

492

434

366

 Number of trees

742

725

717

683

 Mean diameter of trees (cm)

16.88 (6.04)

17.77 (5.99)

19.15 (6.45)

20.30 (6.78)

 Mean height of trees (m)

14.72 (3.06)

16.21 (3.44)

17.16 (3.13)

18.85 (3.33)

Both model 1 and model 2 explained a similar amount of deviance, almost 41%. However, the AIC of model 1 was lower than that of model 2 (Table 2). Therefore, we selected model 1, the more parsimonious model, with additive effects of space and time. This entails that the spatial distribution of the saplings remained constant over the time. The spatial smooth function (f2) and the temporal factor (Time) improved the model in terms of AIC (Table 3). Figure 4 shows the estimated spatial smooth function f2 on the scale of the linear predictor. The estimate of the aggregation parameter θ of the negative binomial distribution is 1.5 and 1.4 in model 1 and 2, respectively. Our results show that we have chosen k large enough for both functions f1 and f2, as we see that the effective degrees of freedom given in Table 2 are below k-1; the same applies to results for the other plot. The coefficients of dbh were robust to changes in k. This also applies to results of the other plot. Figure 5 shows that the spatial correlation was eliminated.
Table 2

Percentage of deviance explained, AIC (Akaike’s Information Criterion), θ parameter in the variance of the negative binomial distribution, basis dimension (k) and effective degrees of freedom (e.df) of the functional linear predictor and the spatial smooth according to model 1 and model 2 in both plots. Inventory 2001, Inventory 2006, Inventory 2010, and Inventory 2014 indicate the effective degrees of freedom of the spatial smooth during the four inventories in model 2

Feature

Intermediate stage of the regeneration period

End of the regeneration period

Model 1

Model 2

Model 1

Model 2

Deviance explained (%)

40.7

41.0

34.8

41

AIC

1 3120.98

1 3432.74

5083.48

5184.80

θ of variance

1.53

1.42

0.83

0.76

k of f1

10

10

10

10

e.df of f1

3.47

3.599

4.33

4.40

k of f2

100

100

30

30

e.df of f2 in model 1

90.42

24.50

Inventory 2001

69.60

19.47

Inventory 2006

65.64

17.30

Inventory 2010

63.42

16.28

Inventory 2014

63.11

13.15

f1 linear predictor, f2 spatial smoother

Table 3

Summary of the backward stepwise variables selection process according to the Akaike’s Information Criterion (AIC). In bold, the selected model

Variables included in the alternative models during backward stepwise selection process

AIC

Intermediate stage of the regeneration period

 (1) \( \alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}\right)+{f}_2\left({X}_i,{Y}_i\right)+{\mathrm{Time}}_j \)

1 3120.98

 (2) \( \alpha +\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{dbh_{jn}}{\mathrm{AreaIn}{30}_i}\right)+{f}_2\left({X}_i,{Y}_i\right)+{\mathrm{Time}}_j \)

1 3126.06

 (3) \( \alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+{f}_2\left({X}_i,{Y}_i\right)+{\mathrm{Time}}_j \)

1 3125.11

 (4) \( \alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}\right) \)

1500.13

 (5) \( \alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}\right)+{f}_2^j\left({X}_i,{Y}_i\right) \)

1 3432.74

End of the regeneration period

 

 (1) \( \alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}\right)+{f}_2\left({X}_i,{Y}_i\right)+{\mathrm{Time}}_j \)

5083.94

 (2) \( \alpha +\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{dbh_{jn}}{\mathrm{AreaIn}{30}_i}\right)+{f}_2\left({X}_i,{Y}_i\right)+{\mathrm{Time}}_j \)

5083.48

 (3) α + f2(X i , Y i ) + Time j

5120.71

 (4) \( \alpha +\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{dbh_{jn}}{\mathrm{AreaIn}{30}_i}\right) \)

5387.18

 (5) \( \alpha +\beta \cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}+\sum \limits_{n=1}^N\left({f}_1\left({\mathrm{Dist}}_{\mathrm{in}}\right)\cdot \frac{N{\mathrm{small}}_{ij}}{\mathrm{AreaIn}{10}_i}\right)+{f}_2^j\left({X}_i,{Y}_i\right) \)

5184.80

Fig. 4

Estimated f2(X i , Y i ) spatial smooth function (continuous black contour lines) and standard errors (dashed red and green contour lines) on the scale of the linear predictor at intermediate stages of the regeneration period. Large values of f2(X i , Y i ) indicate large number of saplings

Fig. 5

Semivariograms (circles) and envelopes (dashed lines) of the Pearson residuals from the sapling distribution model at intermediate stages of the regeneration in 2001 (upper left), 2006 (upper right), 2010 (lower left), and 2014 (lower right)

Removing the term relating to the density of small trees increased the AIC (Table 3). The β of the density of small trees was negative (β = − 0.0004) pointing towards competition between small trees and saplings. Furthermore, Fig. 6 shows the smooth coefficient of dbh (f1) of adult trees over Dist. More saplings are expected to be found when the product of the smooth coefficient and the dbh is large, that is, the model predicts the greatest number of saplings for the largest trees located at the distances to which f1 is highest. f1 varied smoothly across the distances with significantly negative values from 0 m up to 7 m. From 7 m, f1 is not statistically different from zero as the 95% confidence intervals contained zero. This suggests competition between adult trees and saplings at shorter distances (< 7 m) and no relationship at larger distances between these two cohorts. From 13 to 20 m, the mean value of f1 turned positive and significant reaching the largest values of the smooth function. Beyond 20 m, the smooth function f1 started decreasing and it was not statistically different from zero.
Fig. 6

Estimated f1(Dist in ) smooth coefficient function of the diameter at breast height of adult trees over the distance between adult trees and saplings (continuous lines) and 95% confidence intervals (dashed lines) at intermediate stages (upper) and the end (lower) of the regeneration period. Positive values of f1(Dist in ) indicate positive effects of the diameter at breast height of adult trees on the number of saplings

3.2 End of the regeneration period

The dynamics of the saplings and the trees followed the same trends as at intermediate stages of the regeneration period: the number of saplings decreased and their mean size increased with time. The number of trees decreased but the mean dbh and height increased over the four inventories because of fellings. However, in this plot, there were less saplings and their mean size was larger than in the youngest studied plot. Additionally, there were more trees overall at the end of the regeneration period than in the previous stages. Nevertheless, Table 1 shows that the number of small trees reduced with time whereas the number of adult trees increased.

As in the youngest studied plot, the model with the additive spatiotemporal structure (model 1), which assumes a constant spatial distribution of the saplings over the studied period, showed a lower AIC than model 2 (Table 2). We also found a significant effect of the spatiotemporal terms (f2 and Time) in terms of AIC (Table 3). The map of the contour lines (Fig. 7) represents well the spatial distribution of the saplings during the last stages of the regeneration period (Fig. 3). The semivariograms showed that the spatial structure eliminated the spatial correlation (Fig. 8). The estimate of the aggregation parameter θ of the negative binomial distribution is smaller than in the other plot, it is around 0.8 (Table 2). This indicates that saplings were more aggregated at the end of the regeneration period than in the previous stage, which is also confirmed by the visual inspection of the spatial distribution of saplings (Figs. 2 and 3) showing a more homogenous spread of saplings in earlier stages of the regeneration process.
Fig. 7

Estimated f2(X i , Y i ) spatial smooth function (continuous black contour lines) and standard errors (dashed red and green contour lines) on the scale of the linear predictor at the end of the regeneration period. Large values of f2(X i , Y i ) indicate large number of saplings

Fig. 8

Semivariograms (circles) and envelopes (dashed lines) of the Pearson residuals from the sapling distribution model at the end of the regeneration period in 2001 (upper left), 2006 (upper right), 2010 (lower left), and 2014 (lower right)

In this plot, the β of density of small trees did not reduce the AIC whereas the rest of the terms reduced the AIC significantly (Table 3). Table 2 shows the effective degrees of freedom of the basis functions. The coefficient of the dbh of adult trees (f1) took significant negative values from 0 to 8 m (Fig. 6). From 8 m, the 95% confidence intervals contained the zero, and therefore we can state that the coefficient is not statistically different from 0. This suggests competition between saplings and adult trees at very small distances and no effect beyond 8 m.

4 Discussion

We fitted a GAM with a functional predictor in the model to describe the influence of size of trees on the number of saplings by distance in two stages of the regeneration period. We have confirmed that the functional predictor is useful to achieve this aim. In GAMs explanatory, variables may enter the model in many different forms: as variables with linear effects, smooth terms, tensor products of several variables, with varying coefficients or as functional predictors. Additionally, alternative response distribution families and link functions can be selected (see for example Wood 2006). Therefore, all this makes the approach employed suitable to be used in other fields of forestry or ecology in which the response variable depends on the size and distance of the neighbors. For instance, this approach could be useful to fit growth or mortality models instead of using competition indices in parametric models (Contreras et al. 2011).

In this work, we have studied the last stages of the renewal of a forest after regeneration fellings. Other authors have modeled the whole renewal of the forest using multistage models. For instance, Manso et al. (2014) proposed a multistage model based on partial studies or submodels in order to predict the regeneration occurrence of Pinus pinea L. in space and time. They considered different stages such as seed dispersal, seed germination, post-dispersal predation, and seedling survival. Multistage models provide deeper ecological understanding than ours but the implementation is harder and requires stronger ecological hypotheses. However, our approach shows great flexibility and might be used to determine the effects of a limited number of factors on sapling distribution without making any assumptions about other factors involved on dispersion and survival processes.

As mentioned above, this methodology allows different types of variables to be included in the model. In this work, we included the density of small trees as a linear term and the spatiotemporal structure. It might be useful to use variables driving the regeneration as predictors in the model, like shrub cover, soil characteristics, cover, and depth of litter or grass in each quadrat. However, gathering this data on large plots requires a great effort and the influence of these variables on the seedlings of P. sylvestris has already been studied at smaller scales (González-Martínez and Bravo 2001; Pardos et al. 2007; Barbeito et al. 2011; Moreno-Fernández et al. 2015a). On the other hand, new individuals of P. sylvestris are expected to be more affected by soil moisture than by other microsite characteristics (Barbeito et al. 2009; Moreno-Fernández et al. 2015a). However, because youngest individuals—seedlings—are less resistant to drought than older—saplings (Maseda and Fernández 2006; Rodríguez-García et al. 2011; Manso et al. 2014)—it seems it is more necessary to include environmental variables in models dealing with seedlings rather than in those dealing with older individuals—saplings. Moreover, it is likely the distance to adult trees is confounded with other local factors. Any residual spatial trend in a model without a spatial smooth term is caused by missing (unmeasured) environmental variables. Furthermore, the residual spatial trend could be due to the seedling spatial structure that would result from past dispersal events from adjacent mother trees. We have included the spatial smooth term as a proxy for effects of unmeasured environmental variables and for the spatial pattern of the new individuals during previous stages of the forest renewal. We have investigated goodness-of-fit thoroughly, and found that we did not have any spatial trend in residuals or residual spatial correlation. This means that the models fit well and there was no model mis-specification. Although we have only results from two plots, it is striking that the estimated functions f1 (of the effect of dbh) shown in Fig. 6 are very similar.

Our approach allows to test whether the spatial pattern remained constant over time by comparing model 1 which assumes a constant spatial pattern with model 2 which allows for a spatial pattern changing in time in the model selection. In our case, model 1 was selected suggesting that the spatial pattern of the saplings remained constant over time. If model 2 had been selected, the spatial pattern of saplings would have changed over the time. Due to the gradual low intensity fellings regime, which avoids damaging the established saplings clumps, these clumps persist and the spatial structure of the saplings remains fairly constant in each plot during the 15-year measurement period. On the other hand, if we were dealing with faster-growth species the saplings could move to the next cohort faster and then change the spatial pattern. Our results are consistent with LeMay et al. (2009) who reported that the spatial pattern of the new individuals of P. menziesii did not change very much over time.

Although we only analyzed data from two large plots (0.5 ha), we re-measured the plots four times, leading to four observations per plot. Large-sized plots with few sampling over time are common in regeneration studies describing spatial processes. These kind of plots have been used in tropical (Ledo et al. 2015), temperate (McDonald et al. 2003) and Mediterranean forests (Montes and Cañellas 2007; Ledo et al. 2014). Additionally, we modeled the number of saplings per 4 m2 quadrat, i.e., we used 5000 quadrats covering different competition conditions to fit every model. Moreover, the models presented in this work were fitted for explanatory purposes rather than predictive purposes. If we had aimed to fit a predictive model, we would have needed more temporal measurements to cover all the regeneration period.

The underlying process studied here is the competition between trees and saplings. Our findings are in concordance with other studies: the saplings of P. sylvestris require high light conditions for successful development (Montes and Cañellas 2007). In Mediterranean areas, P. sylvestris seedlings require microsites with moderate light conditions (Pardos et al. 2007). These microsites ensure higher soil moisture than in open canopies but conserve enough level of sun radiation. In this regard, Castro et al. (2005) analyzed the growth of P. sylvestris seedlings in southern Spain under different light and water conditions concluding that the effects of water addition on seedlings growth are more evident in lightly microsites. Moreover, once the seedlings have stepped into saplings, the maintenance costs increase with size (Falster and Westoby 2003) and higher minimum light levels are required for survival (Williams et al. 1999). Additionally, their roots can reach deeper soil layers with more water availability (Ritchie 1981). Considering this, it seems it is necessary to reduce the canopy to favor the development of the saplings after seedling establishment under moderate light conditions in Mediterranean areas. However, the shade tolerance of P. sylvestris differs among regions. In northern locations where the summer drought is not a limiting factor for seedling development, natural regeneration takes place in open canopies by using the seed tree method (Hyppönen et al. 2013). In these latitudes, the negative spatial association between P. sylvestris adult trees and saplings may be even more pronounced than in our study.

The establishment of the new stand has been achieved successfully at the end of the regeneration period, the number of saplings decreased and the arrival of new individuals is no longer expected. Hence, the mean dbh of the saplings is getting close to 10 cm, the lower limit for small trees. In this plot, the number of small trees decreased over time due to the mortality as well as the growth and consequent reclassification of trees as adult trees. Most of the trees in this plot were not mother trees of the saplings but rather new cohorts of trees established at the first and intermediate stages of the regeneration period, such as those in our youngest studied plot. Therefore, the spacing between saplings and adult trees is a consequence of the competition between trees of different sizes.

5 Conclusions

We show that functional predictor in GAMs is a useful tool for modeling these kind of data as they allow to model nonlinear and linear relationships. In addition, they allow to take account of the spatiotemporal structure of the data by inclusion of spatial and spatiotemporal smooth predictors. The methodology proposed has not been employed in forestry or ecology and can be broadly used in regeneration studies or in other fields of forestry or ecology dealing with spatiotemporal data. Therefore, this methodology is potentially applicable in future ecological studies because of its flexibility. Additionally, this model can be used as a first step for a predictive model when more temporal data is available. We found that once the seedlings have become established, the density of the adult trees must be reduced heavily to allow the saplings to grow under high light conditions. In Mediterranean stands of P. sylvestris, the radius of the gaps created during the regeneration fellings under the group shelterwood should be always larger than 7–8 m in order to minimize the competition between adult trees and saplings; whereas if the radius is between 13 and 20 m the number of saplings will be maximized.

Notes

Acknowledgements

We wish to thank everybody who participated in the field work, especially Ángel Bachiller, Estrella Viscasillas, and Enrique Garriga. We appreciate the comments made by the referees and the editors of Annals of Forest Science during the revision process. We also thank Adam Collins for revising the English writing.

Funding

DMF has been funded by Spanish Ministry of Education, Culture and Sport thorough the FPU program (FPU13/02113) and by the short-term visiting program (EST15/00242). This work has been funded by AGL2013-46028-R and BOSSANOVA-CM (S2013/MAE-2760).

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Copyright information

© INRA and Springer-Verlag France SAS, part of Springer Nature 2018

Authors and Affiliations

  1. 1.INIA-CIFORMadridSpain
  2. 2.MONTES (School of Forest Engineering and Natural Resources)Universidad Politécnica de MadridMadridSpain
  3. 3.Department of Mathematical SciencesUniversity of BathBathUK

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