Scaling, Self-Similarity, and Systems of Fractional OrderView this Special Issue
Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus
The forest new gap models via local fractional calculus are investigated. The JABOWA and FORSKA models are extended to deal with the growth of individual trees defined on Cantor sets. The local fractional growth equations with local fractional derivative and difference are discussed. Our results are first attempted to show the key roles for the nondifferentiable growth of individual trees.
Fractals had been used to describe special problems in biology and ecology [1–4] because of the measure of nature objects underlying the geometry, replacing the complex real-world objects by describing the Euclidean ideas. Fractal dimension was applied to describe the measure of the complexity in biology and ecology. In forestry, the fractal geometry had been applied to estimate stand density, predict forest succession, and describe the form of trees [5–7]. The scaling of dynamics in hierarchical structure was investigated in [8–12]. The ecological resilience example from boreal forest was presented in the context . The quantitative theory of forest structure was discussed . The allometric scaling laws in biology were proposed in the works [10, 11]. Based upon the cross-scale analysis, the geometry and dynamics of ecosystems were considered and the structure ecosystems across scales in time and space were discussed in [12, 13]. Fractal forestry was modeled by using the scaling of the testing parameters for ecological complexity.
Forest gap model (JABOWA) developed by Botkin et al. [14–16] was the first simulation model for gap-phase replacement. It was applied to describe a forest as a mosaic of closed canopies and simulate forest dynamics based upon the establishment, growth, and death of individual trees [17–20]. The JABOWA model in the form of the FORET model (called JABOWA-FORET) was further developed in [21–24]. The JABOWA model of the simulation of stand structure in a forest gap model was improved in  and the FORSKA  was proposed by Botkin et al. In [14–16], the ecological functions are continuous. In [10–13], the ecological functions were expressed across scales in time and space. However, as it is shown in Figure 1 the ecological functions distinguishing hierarchical size scales in nature, such as the measures of tree size and measure of soil fertility, are defined on Cantor sets. The above approaches do not deal with them.
Local fractional calculus theory [26–38] was applied to handle the nondifferentiable functions defined on Cantor sets. The heat-conduction, transport, Maxwell, diffusion, wave, Fokker-Planck, and the mechanical structure equations were usefully shown (see for more details [28–36] and the cited references therein). In order to simulate forest dynamics on the basis of the establishment, growth, and death of individual trees defined on Cantor sets, the aim of this paper is to present the forest new gap models for simulating the gap-phase replacement by employing the local fractional calculus.
The paper has been organized as follows. In Section 2, we review the JABOWA and FORSKA models for the forest succession. In Section 3, we propose JABOWA and FORSKA models for the fractal forest succession. Finally, Section 4 is conclusions.
2. Growth Models for Forest Gap
In this section we will revise the JABOWA and FORSKA models.
2.1. The JABOWA Model
The growth equation with difference form is given by [15, 16, 24, 39] where the function is diameter at breast height of the trees, the parameter is tree height, is a growth rate, the function is a quantity encapsulating this allometric relationship, is a quantity influencing the abiotic and biotic environment on tree growth, , and and are the maximum measures of the tree dimensions.
The parameter is simulated as follows : where is a quantity of available light, is a quantity of stand basal area, and is a quantity of the annual degree-day sum. It was referred to as Liebig’s law of the minimum .
Leaf area index reads as follows : where with a species-specific parameter and the scale leaf weight per tree to the projected leaf area .
In order to implement a new height-diameter relationship , the differential form of growth equation in the JABOWA model was suggested as follows : where the function has the following form: with the parameter and the initial slope value of the height diameter relationship .
2.2. The FORSKA Model
The FORSKA model was developed for unmanaged natural forests and tree height had relationship with the FORSKA model given by [24, 25, 39, 41] where the parameter is the initial slope value of the height diameter relationship at , is the maximum measure of the tree dimension, is diameter at breast height of the trees, and is tree height.
3. The JABOWA and FORSKA Models for the Fractal Forest Succession
In this section, based upon the local fractional calculus theory, we show the JABOWA and FORSKA models for the fractal forest succession. At first, we start with the local fractional derivative.
3.1. Local Fractional Derivative
We now give the local fractional calculus and the recent results.
For , the -local fractional differential of reads as [26, 27] From (14), we have approximate formula in the form Let . The local fractional integral of of order is given by [26–31] where , , and , , , , is a partition of the interval .
The -dimensional Hausdorff measure is calculated by 
3.2. The Local Fractional JABOWA Models (LFJABOWA)
Here, we structure the LFJABOWA models via local fractional derivative and difference.
Making use of the fractional complex transform  and (20), the growth equation in the JABOWA model with local fractional derivative (LFJABOWA) is suggested by where is the diameter at breast height, is the scaling coefficient, and and are the fractal dimensions. In order to illustrate the difference from the works presented in , we consider the following case: when , (21) can be integrated to give For the parameters , , the solutions of (21) with different values , , , and are, respectively, shown in Figures 2, 3, 4, and 5.
Using the fractional complex transform, (5) becomes into Comparing (22) and (23), we have where or From (25) and (26), we could get Hence, the local fractional JABOWA model (LFJABOWA) reads as follows: where is a nondifferentiable function and is the diameter at breast height.
When the fractal dimension is equal to 1, we get the generalized form of (1); namely, where
3.3. The Local Fractional FORSKA Models (LFFORSKA)
Here, we present the LFFORSKA models via local fractional derivative and difference.
The tree height is the nondifferentiable function; namely, where for , and .
Following the fractional complex transform , from (8) we have the local fractional growth equation in the following form: where is a local fractional continuous function, is the tree height, and is a parameter.
The expression (36) is the difference form of local fractional FORSKA model (LFFORSKA).
In this work we investigated the local fractional models for the fractal forest succession. Based on the local fractional operators, we suggested the differential and difference forms of the local fractional JABOWA and FORSKA models. The nondifferentiable growths of individual trees were discussed. It is a good start for solving the rhetorical models for the fractal forest succession in the mathematical analysis.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
This work is supported by the Science and Technology Commission Planning Project of Jiangxi Province (no. 20122BBG70078).
A. Fielding, “Applications of fractal geometry to biology,” Computer Applications in the Biosciences, vol. 8, no. 4, pp. 359–366, 1992.View at: Google Scholar
B. L. Li, “Fractal geometry applications in description and analysis of patch patterns and patch dynamics,” Ecological Modelling, vol. 132, no. 1, pp. 33–50, 2000.View at: Google Scholar
G. Sugihara and R. M. May, “Applications of fractals in ecology,” Trends in Ecology & Evolution, vol. 5, no. 3, pp. 79–86, 1990.View at: Google Scholar
J. H. Brown, V. K. Gupta, B. L. Li, B. T. Milne, C. Restrepo, and G. B. West, “The fractal nature of nature: power laws, ecological complexity and biodiversity,” Philosophical Transactions of the Royal Society of London B, vol. 357, no. 1421, pp. 619–626, 2002.View at: Google Scholar
N. D. Lorimer, R. G. Haight, and R. A. Leary, The Fractal Forest: Fractal Geometry and Applications in Forest Science, North Central Forest Experiment Station, Berkeley, Calif, USA, 1994.
B. Zeide, “Fractal geometry in forestry applications,” Forest Ecology and Management, vol. 46, no. 3, pp. 179–188, 1991.View at: Google Scholar
B. Zeide, “Analysis of the 3/2 power law of self-thinning,” Forest Science, vol. 33, no. 2, pp. 517–537, 1987.View at: Google Scholar
G. D. Peterson, “Scaling ecological dynamics: self-organization, hierarchical structure, and ecological resilience,” Climatic Change, vol. 44, no. 3, pp. 291–309, 2000.View at: Google Scholar
G. B. West, B. J. Enquist, and J. H. Brown, “A general quantitative theory of forest structure and dynamics,” Proceedings of the National Academy of Sciences, vol. 106, no. 17, pp. 7040–7045, 2009.View at: Google Scholar
G. B. West, J. H. Brown, and B. J. Enquist, “A general model for the origin of allometric scaling laws in biology,” Science, vol. 276, no. 5309, pp. 122–126, 1997.View at: Google Scholar
B. J. Enquist, G. B. West, E. L. Charnov, and J. H. Brown, “Allometric scaling of production and life-history variation in vascular plants,” Nature, vol. 401, no. 6756, pp. 907–911, 1999.View at: Google Scholar
C. S. Holling, “Cross-scale morphology, geometry, and dynamics of ecosystems,” Ecological Monographs, vol. 62, no. 4, pp. 447–502, 1992.View at: Google Scholar
G. Peterson, C. R. Allen, and C. S. Holling, “Ecological resilience, biodiversity, and scale,” Ecosystems, vol. 1, no. 1, pp. 6–18, 1998.View at: Google Scholar
D. B. Botkin, J. F. Janak, and J. R. Wallis, “Rationale, limitations, and assumptions of a northeastern forest growth simulator,” IBM Journal of Research and Development, vol. 16, no. 2, pp. 101–116, 1972.View at: Google Scholar
D. B. Botkin, J. F. Janak, and J. R. Wallis, “Some ecological consequences of a computer model of forest growth,” The Journal of Ecology, vol. 60, no. 3, pp. 849–872, 1972.View at: Google Scholar
D. B. Botkin, Forest Dynamics: An Ecological Model, Oxford University Press, New York, NY, USA, 1993.
G. L. Perry and N. J. Enright, “Spatial modelling of vegetation change in dynamic landscapes: a review of methods and applications,” Progress in Physical Geography, vol. 30, no. 1, pp. 47–72, 2006.View at: Google Scholar
R. W. Hall, “JABOWA revealed-finally,” Ecology, vol. 75, no. 3, p. 859, 1994.View at: Google Scholar
M. I. Ashraf, C. P. A. Bourque, D. A. MacLean, T. Erdle, and F. R. Meng, “Using JABOWA-3 for forest growth and yield predictions under diverse forest conditions of Nova Scotia, Canada,” The Forestry Chronicle, vol. 88, no. 6, pp. 708–721, 2012.View at: Google Scholar
H. H. Shugart and D. C. West, “Development of an Appalachian deciduous forest succession model and its application to assessment of the impact of the chestnut blight,” Journal of Environmental Management, vol. 5, pp. 161–179, 1977.View at: Google Scholar
H. H. Shugart, A Theory of Forest Dynamics. The Ecological Implications of Forest Succession Models, Springer, New York, NY, USA, 1984.
H. Bugmann, “A review of forest gap models,” Climatic Change, vol. 51, no. 3-4, pp. 259–305, 2001.View at: Google Scholar
X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, China, 2011.
X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,” Thermal Science, vol. 17, no. 2, pp. 625–628, 2013.View at: Google Scholar
X. J. Yang, D. Baleanu, and J. H. He, “Transport equations in fractal porous media within fractional complex transform method,” Proceedings of the Romanian Academy A, vol. 14, no. 4, pp. 287–292, 2013.View at: Google Scholar
C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,” Thermal Science, vol. 17, no. 3, pp. 715–721, 2013.View at: Google Scholar
A. M. Yang, Y. Z. Zhang, and Y. Long, “The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar,” Thermal Science, vol. 17, no. 3, pp. 707–713, 2013.View at: Google Scholar
X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximate solutions for diffusion equations on Cantor space-time,” Proceedings of the Romanian Academy A, vol. 14, no. 2, pp. 127–133, 2013.View at: Google Scholar
H. K. Bugmann, X. Yan, M. T. Sykes et al., “A comparison of forest gap models: model structure and behaviour,” Climatic Change, vol. 34, no. 2, pp. 289–313, 1996.View at: Google Scholar
A. D. Moore, “On the maximum growth equation used in forest gap simulation models,” Ecological Modelling, vol. 45, no. 1, pp. 63–67, 1989.View at: Google Scholar