Abstract and Applied Analysis

Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus

Chun-Ying Long, Yang Zhao, and Hossein Jafari

Full-text: Open access

Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 782393, 6 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425047786

Digital Object Identifier
doi:10.1155/2014/782393

Mathematical Reviews number (MathSciNet)
MR3186979

Zentralblatt MATH identifier
07023060

Citation

Long, Chun-Ying; Zhao, Yang; Jafari, Hossein. Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 782393, 6 pages. doi:10.1155/2014/782393. https://projecteuclid.org/euclid.aaa/1425047786


Export citation

References

  • A. Fielding, “Applications of fractal geometry to biology,” Computer Applications in the Biosciences, vol. 8, no. 4, pp. 359–366, 1992.
  • 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.
  • G. Sugihara and R. M. May, “Applications of fractals in ecology,” Trends in Ecology & Evolution, vol. 5, no. 3, pp. 79–86, 1990.
  • 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.
  • 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.
  • B. Zeide, “Analysis of the 3/2 power law of self-thinning,” Forest Science, vol. 33, no. 2, pp. 517–537, 1987.
  • G. D. Peterson, “Scaling ecological dynamics: self-organization, hierarchical structure, and ecological resilience,” Climatic Change, vol. 44, no. 3, pp. 291–309, 2000.
  • 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.
  • 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.
  • 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.
  • C. S. Holling, “Cross-scale morphology, geometry, and dynamics of ecosystems,” Ecological Monographs, vol. 62, no. 4, pp. 447–502, 1992.
  • G. Peterson, C. R. Allen, and C. S. Holling, “Ecological resilience, biodiversity, and scale,” Ecosystems, vol. 1, no. 1, pp. 6–18, 1998.
  • 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.
  • 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.
  • 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.
  • R. Chen and R. R. Twilley, “A gap dynamic model of mangrove forest development along gradients of soil salinity and nutrient resources,” Journal of Ecology, vol. 86, no. 1, pp. 37–51, 1998.
  • R. W. Hall, “JABOWA revealed-finally,” Ecology, vol. 75, no. 3, p. 859, 1994.
  • 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.
  • H. H. Shugart and I. R. Noble, “A computer model of succession and fire response of the high-altitude Eucalyptus forest of the Brindabella Range, Australian Capital Territory,” Australian Journal of Ecology, vol. 6, no. 2, pp. 149–164, 1981.
  • 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.
  • 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.
  • M. Lindner, R. Sievänen, and H. Pretzsch, “Improving the simulation of stand structure in a forest gap model,” Forest Ecology and Management, vol. 95, no. 2, pp. 183–195, 1997.
  • 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.
  • 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.
  • 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.
  • 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.
  • Y. Zhao, D. Baleanu, C. Cattani, D. F. Cheng, and X.-J. Yang, “Maxwell's equations on Cantor Sets: a local fractional approach,” Advances in High Energy Physics, vol. 2013, Article ID 686371, 6 pages, 2013.
  • 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.
  • X.-J. Yang, H. M. Srivastava, J. H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,” Physics Letters A, vol. 377, no. 28–30, pp. 1696–1700, 2013.
  • K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998.
  • A. Carpinteri, B. Chiaia, and P. Cornetti, “Static-kinematic duality and the principle of virtual work in the mechanics of fractal media,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 1-2, pp. 3–19, 2001.
  • F. Ben Adda and J. Cresson, “About non-differentiable functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 721–737, 2001.
  • A. Babakhani and V. Daftardar-Gejji, “On calculus of local fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 66–79, 2002.
  • 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.
  • A. D. Moore, “On the maximum growth equation used in forest gap simulation models,” Ecological Modelling, vol. 45, no. 1, pp. 63–67, 1989.
  • A. C. Risch, C. Heiri, and H. Bugmann, “Simulating structural forest patterns with a forest gap model: a model evaluation,” Ecological Modelling, vol. 181, no. 2-3, pp. 161–172, 2005.
  • D. A. Coomes and R. B. Allen, “Effects of size, competition and altitude on tree growth,” Journal of Ecology, vol. 95, no. 5, pp. 1084–1097, 2007. \endinput