International Journal of Bioinformatics and Computational Biology  
Manuscript Information
 
 
Estimating Biodiversity and the Fractal Nature of Ecosystems
International Journal of Bioinformatics and Computational Biology
Vol.5 , No. 1, Publication Date: Sep. 4, 2020, Page: 15-24
2201 Views Since September 4, 2020, 437 Downloads Since Sep. 4, 2020
 
 
Authors
 
[1]    

Wilfried Allaerts, Biological Publishing A&O and Immunology Department, Erasmus MC, Rotterdam, The Netherlands.

 
Abstract
 

The problem of biodiversity impairment not only poses a global threat to the planet’s biosphere and causes global health issues, it is also a cumbersome pièce de résistance for mathematical modeling. Already the definition of biodiversity requires knowledge of the hierarchical structure of an ecological environment and taxonomic complexity of life forms in all of their manifestations. Not only the enumeration of easily recognized, large vertebrate species, but an estimation of all living species present in a given area, forms the ultimate challenge for biodiversity estimation. This is an important goal for enabling a scientifically sound approximation of biodiversity impairment. In this paper, the analogy of the fractal geometry of nature and the fractal appearance of ecosystems is followed, in order to define a constitutive approach for estimating the local and global biodiversity of an ecosystem. Moreover, following the rationale of percolation theory and Mandelbrot’s (1983) definition of the bounds of a critical fractal dimension in a hierarchically stratified system, the notions of critical biodiversity and biodiversity resilience strength (BRS) are proposed. It is concluded that in order to understand the dynamics of biodiversity change in a stressed, stratified environment such as the global biosphere, not only the stratification into trophic levels, but also short and long distance migration effects have to be considered.


Keywords
 

Biodiversity, Biodiversity Resilience Strength, Fractal Dimension, Similarity and Mass Dimension, Uniformity Index, Percolation Probability


Reference
 
[01]    

United Nations (1992) Convention on biological diversity. In: United Nations conference on environment and development at Rio de Janeiro. Washington DC.

[02]    

Millenium Ecosystem Assessment (2005) Ecosystems and Human Well-Being. Biodiversity Synthesis. Washington DC: World Resources Institute.

[03]    

Begon M (1979) Investigating Animal Abundance: capture-recapture for biologists. London: Edward Arnold Publ. Ltd.

[04]    

de Neergaard A (2005) Soil biology and microbiology. In: de Neergaard A (ed.), Land use, land cover and soil sciences, vol VI. UNESCO: Encyclopedia of life support systems (EOLSS).

[05]    

Law R (1999) Theoretical aspects of community assembly. In: McGlade J (ed.), Advanced Theoretical Ecology: Principles and Applications, pp 143-171. London: Blackwell Science.

[06]    

Fisher RA, Corbet AS and Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12: 42-58.

[07]    

Anitas EM, Marcelli G, Szakacs Z, Todoran R and Todoran D (2019) Structural properties of Vicsek-like deterministic multifractals. Symmetry 11: 806 (doi: 10.3390/sym11060806)

[08]    

Mandelbrot BB (1983) The Fractal Geometry of Nature. San Francisco: W. H. Freeman and Company.

[09]    

Saunders PT (1992) Collected Works of A. M. Turing. vol. 3: Morphogenesis (Amsterdam, London: Elsevier Science Publishers).

[10]    

Motomura I (1932) On the statistical treatment of communities (in Japanese). Zoological Magazine (Tokyo) 44: 379-383 (fide AE Magurran, 1988)

[11]    

Magurran AE (1988) Ecological Diversity and its Measurement. Princeton, NJ: Princeton University Press.

[12]    

Preston FW (1948) The Commonness, And Rarity, of Species. Ecology 29 (3): 254-283.

[13]    

Vandermeer J and Lin BB (2008) The importance of matrix quality in fragmented landscapes: Understanding ecosystem collapse through a combination of deterministic and stochastic forces. Ecological complexity 5: 222-227 (doi: 10.1016/j.ecocom.2008.01.001).

[14]    

Allaerts W (2019a) Connectivity, continuity and distance norm in mathematical models for community ecology, epidemiology and multicellular pathway prediction. International Journal of Bioinformatics and Computational Biology 4 (1): 1-10.

[15]    

Parrilo PA and Jadbabaie A (2008) Approximation of the joint spectral radius using sum of squares. Linear Algebra and its Applications 428: 2385-2402.

[16]    

Brinkmann HW and Klotz EA (1971) Linear Algebra and analytic geometry. Reading, Menlo Park, London: Addison-Wesley Publishing Company.

[17]    

Mandelbrot BB (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 155: 636-638, (Reproduced in: BB Mandelbrot [1983] The Fractal Geometry of Nature, pp 25-33. W. H. Freeman and Company).

[18]    

Hausdorff F (1919) Dimension und äusseres Mass. Mathematische Annalen 79: 157-179 (fide BB Mandelbrot 1983).

[19]    

Besicovitch AS (1935) On the sum of digits of real numbers represented in the dyadic system (On sets of fractional dimensions II). Mathematische Annalen 110: 321-330 (fide BB Mandelbrot 1983).

[20]    

Von Koch H (1904) Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Arkiv för Matematik, Astronomi och Fysik 1: 681-704 (fide BB Mandelbrot 1983).

[21]    

Marcelli G (2019) Structural properties of two-phases deterministic multifractals (Bachelor thesis). Sapienza - University of Rome.

[22]    

Tricot C (1995) Curves and Fractal Dimension. Berlin: Springer Verlag.

[23]    

Shante VKS and Kirkpatrick S (1971) An introduction to percolation theory. Advances in Physics 20: 325-357.

[24]    

Perelson AS (1989) Immune network theory. Immunol Rev 110: 5-36.

[25]    

Allaerts W (1999) The biological function paradigm applied to the immunological self-non-self discrimination: Critique of Tauber’s phenomenological analysis. Journal for General Philosophy of Science 30 (1): 155-171.

[26]    

Boswell GP, Britton NF and Franks NR (1998) Habitat fragmentation, percolation theory and the conservation of a keystone species. Proceedings Royal Society London B 265: 1921-1925.

[27]    

Peano G (1890) Sur une courbe, qui remplit une aire plane. Mathematische Annalen 36: 157-160 (Translated in G Peano [1973], Selected works, Ed. HC Kennedy, Toronto University Press) (fide BB Mandelbrot 1983).

[28]    

MacArthur RH and Wilson EO (1967) The Theory of Island Biogeography. Princeton: Princeton University Press.

[29]    

Williams CB (1964) Patterns in the balance of nature and related problems in quantitative ecology. London: Academic Press.

[30]    

World Health Organization (2015). Connecting Global Priorities: Biodiversity and Human Health (A State of Knowledge Review). WHO and Secretariat of the Convention on Biological Diversity (NLM classification: WD 600).

[31]    

Bekenstein JD (1973) Black holes and Entropy. Physical Review D, 7 (8): 2333-2346.

[32]    

Brillouin L (1953) Negentropy Principle of Information. Journal of Applied Physics 24 (9): 1152-1163.

[33]    

Morton T (2013) Hyperobjects: Philosophy and Ecology After the End of the World. Minneapolis: University of Minnesota Press.

[34]    

Heil Ch (2018a) Metrics, Norms, Inner Products and Operator Theory. (Suppl. Chapter 8): Integral Operators. Birkhäuser/Springer, Cham, and http://people.math.gatech.edu.

[35]    

Reinhardt F and Soeder H (1980) Atlas zur Mathematik. München: Deutscher Taschenbug Verlag Gmbh & Co.

[36]    

Heil Ch (2018b) A Brief Review of Lebesgue Measure and Integration. Source: http://people.math.gatech.edu/~heil/metricnote.

[37]    

Allaerts W (2008) De appel van Alan Turing. (Of de wording van een moderne mythe). bi-logical, 1 (1): 10-25.

[38]    

Frantzikinakis N (2017) Ergodicity of the Liouville system implies the Chowla conjecture. Discrete Analysis 19, arXiv: 1611.09338v4 [math.NT] (doi: 10.19086/da.2733).

[39]    

Jackson ST and Sax DF (2009) Balancing biodiversity in a changing environment: extinction debt, immigration credit and species turnover. Trends in Ecology and Evolution 1196 (doi: 10.1016/j.tree.2009.10.001).

[40]    

Murray JD (1989) Mathematical Biology. Berlin: Springer Verlag.

[41]    

Allaerts W (2019b) Can lichens form a barrier for Fungi marching on? International Journal of Environment & Agricultural Science 3 (2): 022.

[42]    

van der Linden F (2009) Fractalen en een heel gewoon bodemplantje. Letters to: bi-logical 2 (2): 19-23.

[43]    

Swinton J (2013) Turing, Morphogenesis and Fibonacci Phyllotaxis: Life in Pictures. In: SB Cooper and J van Leeuwen (Eds), Alan Turing: His Work and Impact, pp. 834-849. Amsterdam, Boston, London: Elsevier.





 
  Join Us
 
  Join as Reviewer
 
  Join Editorial Board
 
share:
 
 
Submission
 
 
Membership