A Dynamical Model Based on the Chapman–Richards Growth Equation for Fitting Growth Curves for Four Pine Species in Northern Mexico

Round 1

Reviewer 1 Report

Tree growth models describe the growth and development of forest ecosystems by considering how the dimensions of each simulated tree change in a specified time, i.e. periodic increment of a tree in response to the life processes period and can be divided into single-tree models (dendrometric variables) and whole stand models that include stand characteristics such as age, size, density, site index and species affiliation and composition. These models are used to answer ecological questions, for example, determining the impact of the interdependence between tree species and, their environment on forest development and assessing the forest yields under certain prescribed conditions. Tree growth is a complex process and growth modelling methodologies are evolving to better describe this process.

The aim of the research presented by Protazio et al. was to quantify the putative associations between the growth parameters and to test the accuracy of a novel Chapman-Richards growth model, relative to Hossfeld, Lundqvist and Chapman-Richards GADA models.

The manuscript is written in good style, is informative and gives overview of the topic. In this review, I offer a few suggestions as to where certain points can be elaborated upon or revised in the manuscript.

Detailed comments (with reference to lines):

1.     Lines 21-57: In the introduction, emphasize the importance of your research findings for forestry practice.

2.     Figure 2; Figure 3. Please correct (enlarge) the descriptions of the X, Y  axes.

3.     Page 8, 9, 10: A much better explanation of the elements of the box-plot must appear in the figure 4, 5 and 6 caption (figures must be auto-explicative).

4.     Table 3, page 5. No reference in the text.

5.     Lines 142-144: Rewrite the sentence.  Consider deleting Kruskal-Walilis, as it is repeated

6.     Line 145: Check the table numbers in the text. There is table 5, it should be table 6.

7.     Lines 152-162: In Discussion Authors should discuss the results and how they can be interpreted in perspective of previous studies and of the working hypotheses. The findings and their implications should be discussed in the broadest context possible and limitations of the work highlighted. Future research directions may also be mentioned. The Discussion should be rewritten. Indicate (emphasize) the originality of the obtained results

8.     Reference Lines 195-261: References  not prepared in accordance with the requirements journal Forests.

9.     The authors did not provide the publication's DOI numbers or the Uniform Resource Locator (URL) in the reference list, consequently making it  difficult to find the quoted publication.

Author Response

Response to the Reviewer #1

The manuscript is written in good style, is informative and gives overview of the topic.

Many thanks!

1. Lines 21-57: In the introduction, emphasize the importance of your research findings for forestry practice.

We added (lines 58-60): “Our research results are important for forestry practice, as the new model can be used to predict future yields in a simpler way and with the same accuracy.”

2. Figure 2; Figure 3. Please correct (enlarge) the descriptions of the X, Y  axes.

It was done.

3. Page 8, 9, 10: A much better explanation of the elements of the box-plot must appear in the figure 4, 5 and 6 caption (figures must be auto-explicative).

We modified the captions, now: “Boxplot (including the minimum score, first (lower) quartile, median, third (upper) quartile, and maximum scoreo) of the Mean Absolute Percentage Error (MAPE) function calculated for the species: (a) Pinus arizonica, (b) Pinus engelmannii, (c) Pinus strobiformis and (d) Pinus teocote, for the GADA and W-H models in comparison with the CR-H method proposed in the study.”

4. Table 3, page 5. No reference in the text.

We added (lines 139-140): “Table 3 shows the results of fitting the parameters for the models presented in Table 2 and for the hybrid model (CR-H) proposed for each of the species included in the study.”

5. Lines 142-144: Rewrite the sentence.  Consider deleting Kruskal-Walilis, as it is repeated

We added (lines 141-147): Applying the Kruskal-Wallis test to the four species studied (Pinus arizonica, P. engelmannii, P. strobiformis, P. teocote), we searched for possible differences between the CR-GADA, H-GADA, L-GADA and W-H models with respect to our proposed model (CR-H). Our results show that there are no significant differences in any of the parameters (MAPE, RMSE and R²) tested to compare the models, as the p-values for the K-W test were above 0.65 in all cases. In addition, the AIC values were also similar for each model (see Tables 4 - 7 and Figures 4 - 6).

6. Line 145: Check the table numbers in the text. There is Table 5, it should be table 6.

It was done.

7. Lines 152-162: In DiscussionAuthors should discuss the results and how they can be interpreted in perspective of previous studies and of the working hypotheses. The findings and their implications should be discussed in the broadest context possible and limitations of the work highlighted. Future research directions may also be mentioned. The Discussion should be rewritten. Indicate (emphasize) the originality of the obtained results.

We modified the discussion und conclusions (lines 154-170): “....The accuracy of our GADA and H models was similar to the GADA models of other studies (e.g., [17,29,30,31]).

The main advantage of the widely used GADA site index models over ADA is that they can be polymorphic and have multiple asymptotes caused by more than one site-specific parameter to create functions of χ0 (i.e., one unobservable independent variable which describes site productivity, as a summary of management regimes, soil, climatic and ecological factors) [29,30]. This leads to more flexible dynamic models [31]. However, if these site-specific parameters are highly correlated to each other, as we have shown (Figure 2), these several parameters could be replaced by the parameter easiest to estimate in the forest practice, making it mathematically more attractive due to the smaller number of parameters, which makes a complex multiparametric χ0 function obsolete, without compromising the fitting capability of the model (Tables 4-7) [26,22]. I. e., the main advantage of this novel H model over GADA is the simplification described above by reducing the number of site-specific parameters to the the only one most easily measurable site-specific parameter, while statistically maintaining the predictive quality of the model. Moreover, in contrast to the CR-GADA factor χ0, when θ2 is assumed to be site-specific [18], the CR-H has always a closed-form solution.”

            ....

(lines 186-189) “Furthermore, sufficiently high correlations of the site-specific parameter θ1, θ2 and θ3, therefore, the accuracy of these simplified dynamic H-models should be tested when environmental changes and competition parameters are incorporated into the models that effectively perturb the growth of individuals.”

8. Reference Lines 195-261: References not prepared in accordance with the requirements journal Forests. The authors did not provide the publication's DOI numbers or the Uniform Resource Locator (URL) in the reference list, consequently making it difficult to find the quoted publication.

The reference presentation standard, according to Forest MDPI itself, should be: Journal Articles: 1. Author 1, A.B.; Author 2, C.D. Title of the article. Abbreviated Journal Name Year, Volume, page range; Books and Book Chapters: 2. Author 1, A.; Author 2, B. Book Title, 3rd ed.; Publisher: Publisher Location, Country, Year; pp. 154–196. 3;. Author 1, A.; Author 2, B. Title of the chapter. In Book Title, 2nd ed.; Editor 1, A., Editor 2, B., Eds.; Publisher: Publisher Location, Country, Year; Volume 3, p. 154–196.

Reviewer 2 Report

The paper entitled “A Dynamical Model based on the Chapman-Richards Growth Equation for fitting Growth Curves for four Pine Species in northern Mexico” reflects the development of applied research, the topic is interesting and the manuscript has an approach innovative. However, the methods, results and discussion need to be improved. Thus, major changes are recommended.

 

Comments

1) Line 35 – Prior to use an acronym it should be defined. In the text diameter at breast height should be used instead of dbh.

2) Lines64-65 – “The diameter (dbh in cm) at 1.30 m height” please replace by The diameter at breast height, measures at 1.30 m (dbh in cm).

3) Line 67 – “wood tissue” or cores?

4) table 4, caption – tree number or number of trees?

5) Table 2 – In the caption the meaning of some variables is missing. For example, dr, l₀, θ_i (where i=1 to 3) is missing.

6) Lines 88-89 – Please revise English.

7) Section 2.5 – It is no clear why and for what this test was used. Please revive the text.

8) Lines 132-133 – Is the equation correct?

9) Lines 137-139, 142-144, 148-150 – Results of what?

10) 3. Results – This section need to be rewritten. It is not clear which is the best model and why.

11) 4. Discussion – This section needs further improvements. The results should be discussed is comparison to other published studies. The three studies presented are not enough for a detailed discussion.

Author Response

Response to the Reviewer #2

• Line 35 – Prior to use an acronym it should be defined. In the text diameter at breast height should be used instead of dbh.

It was done.

• Lines 64-65 – “The diameter (dbh in cm) at 1.30 m height” please replace by The diameter at breast height, measures at 1.30 m (dbh in cm).

It was done.

• Line 67 – “wood tissue” or cores?

Cores. It was corrected.

• Table 4, caption – tree number or number of trees?

number of trees. It was corrected.

• Table 2 – In the caption the meaning of some variables is missing. For example, dr, l₀, θ_i (where i=1 to 3) is missing.

We added (after line 85): “…GADA and hybrid equations, where d0 is the dominant diameter (m) at age t0 (years), dr is the diameter of the individual at the time t = 0, χ0 is a site-specific parameter and θi is the parameter associated for each model.”

• Lines 88-89 – Please revise English.

Was improved.

• Section 2.5 – It is no clear why and for what this test was used. Please revive the text.

We modified the section, now (lines 115-120): “The Kruskal-Wallis test is a non-parametric test used here to indicate whether there is any significant difference between the residuals obtained for each of the models presented in the study (Table 2) compared to the proposed hybrid method (CR-H). It is an extension of the Wilcoxon-Mann-Whitney test and is used to test the null hypothesis that all populations have identical distribution functions, against the alternative hypothesis that at least two of them have different distribution functions [24].”

• Lines 132-133 – Is the equation correct?

Correct!

• Lines 137-139, 142-144, 148-150 – Results of what? Results – This section need to be rewritten. It is not clear which is the best model and why.

We modified Results and added (lines 141-147): “Applying the Kruskal-Wallis test to the four species studied (Pinus arizonica, P. engelmannii, P. strobiformis, P. teocote), we searched for possible differences between the CR-GADA, H-GADA, L-GADA and W-H models with respect to our proposed model (CR-H). Our results show that there are no significant differences in any of the parameters (MAPE, RMSE and R²) tested to compare the models, as the p-values for the K-W test were above 0.65 in all cases. In addition, the AIC values were also similar for each model (see Tables 4 - 7 and Figures 4 - 6).”

• Discussion – This section needs further improvements. The results should be discussed is comparison to other published studies. The three studies presented are not enough for a detailed discussion.

We improved the discussion and added (lines 154-170): “....The accuracy of our GADA and H models was similar to the GADA models of other studies (e.g., [17,29,30,31]).

The main advantage of the widely used GADA site index models over ADA is that they can be polymorphic and have multiple asymptotes caused by more than one site-specific parameter to create functions of χ0 (i.e., one unobservable independent variable which describes site productivity, as a summary of management regimes, soil, climatic and ecological factors) [29,30]. This leads to more flexible dynamic models [31]. However, if these site-specific parameters are highly correlated to each other, as we have shown (Figure 2), these several parameters could be replaced by the parameter easiest to estimate in the forest practice, making it mathematically more attractive due to the smaller number of parameters, which makes a complex multiparametric χ0 function obsolete, without compromising the fitting capability of the model (Tables 4-7) [22, 26]. I. e., the main advantage of this novel H model over GADA is the simplification described above by reducing the number of site-specific parameters to the the only one most easily measurable site-specific parameter, while statistically maintaining the predictive quality of the model. Moreover, in contrast to the CR-GADA factor χ0, when θ2 is assumed to be site-specific [18], the CR-H has always a closed-form solution.”

and

(lines 186-189) “Furthermore, sufficiently high correlations of the site-specific parameter θ1, θ2 and θ3, therefore, the accuracy of these simplified dynamic H-models should be tested when environmental changes and competition parameters are incorporated into the models that effectively perturb the growth of individuals.”

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The paper entitled “A Dynamical Model based on the Chapman-Richards Growth Equation for fitting Growth Curves for four Pine Species in northern Mexico” has improved in the second version of the manuscript. The authors revised the manuscript according to the suggestions. There is only one remark:

1) Lines 91-92 – “maximum dbh reached when the upper asymptote is reached in the individual”. Please check English.

It is recommended to accept the manuscript after minor changes.