Multifractality in ecological monitoring

Nuclear Instruments and Methods in Physics Research A 502 (2003) 799–801

Multifractality in ecological monitoring D.I. Iudina,*, D.B. Gelashvilyb a

Radiophysical Research Institute, 25/14 B. Pecherskaya St., 603950 Nizhniy Novgorod, Russia b Nizhniy Novgorod State University, 23 Gagarina Av., 603146 Nizhniy Novgorod, Russia

Abstract In this paper we introduce the concept of multifractality at the problem of ecological monitoring and species diversity estimation. Ecological communities can be considered as an open and strongly nonequilibrium systems, that experience an external driving. The process of resource consumption and resource allocation in communities reflects their complex internal structure. The structure is characterized by number of species, population, links between species and extent of domination. In experiment, we deal with relative frequencies or specific numbers of individuals in species that we find in a sample. We consider every species as a separate box that contain an arbitrary number of individuals and apply boxcounting method for relative frequencies calculation. We find that box number or number of species as a function of sample population follows power law when population increases and, consequently, species distribution may be considered as a fractal set. To estimate species diversity one generally uses well-known diversity indexes, each of which simply enters a measure in space of relative frequencies. We tender multifractal generalization of this routine. An example from aquatic ecology is considered. r 2003 Elsevier Science B.V. All rights reserved. Keywords: Fractal analysis; Multifractality; Ecological monitoring

1. Introduction In this paper we introduce the concept of multifractality at the problem of ecological monitoring and species diversity estimation. Ecological communities can be considered as an open and strongly nonequilibrium systems that experience an external driving. The process of resource consumption and resource allocation in communities reflects their complex internal structure. The structure is characterized by number of species, population, links between species and extent of domination. In experiment, we deal with relative frequencies or specific numbers of individuals in *Corresponding author. E-mail address: iudin@nirfi.sci-nnov.ru (D.I. Iudin).

species that we find in a sample: pi ¼ Ni =N; where Ni is number of individuals in i-species, N is sample population, and i runs from unit up to total number of species SðNÞ that we find in sample. It is obvious that S X

pi ¼ 1:

ð1Þ

i¼1

To estimate species diversity one ordinarily uses well-known diversity indexes, each of which simply enters a measure in space of relative frequencies. Shannon index, for example, is determined as entropy measure: H¼

S X i¼1

0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-9002(03)00587-4

pi ln pi :

ð2Þ

D.I. Iudin, D.B. Gelashvily / Nuclear Instruments and Methods in Physics Research A 502 (2003) 799–801

800

Simpson’s predominance index: C¼

S X

0.7 1

p2i

0.6

ð3Þ

i¼1

2

0.5

Mq ðNÞ ¼

S X

pqi pN tðqÞ ;

ð5Þ

3 4

0.4 f (a)

represents square measure in space of relative frequencies. We may consider every species as a separate box that contain an arbitrary number of individuals and apply box-counting method for relative frequencies calculation. We find that box number or number of species as a function of sample population follows power law when population increases and, consequently, species distribution may be considered as a fractal set: ln S : ð4Þ S ¼ Nk; k ¼ ln N And what is more we may tender multifractal generalization of this routine [1]:

0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

a

Fig. 1. Multifractal spectrum seasonal behavior for benthos of Nizhny Novgorod municipal lakes in 2000. Numbers denote months from May to August.

or

i¼1

where NpqpN is a momentum order, and exponent tðqÞ characterize momentum modification with growth of population. We consider generalized dimensions [2]:   1 ln Mq ðNÞ tðqÞ : ð6Þ Dq ¼ lim ¼ N-N 1  q ln N 1q In practice instead of Eq. (6) we use Dq ¼

1 ln Mq ðNÞ : 1  q ln N

ð7Þ

Obviously for q ¼ 0 we have ln SðNÞ D0 ¼ k ¼ : ln N For q ¼ 1: P  Si¼1 pi ln pi H : ¼ D1 ¼ ln N ln N

ð8Þ

ln C ln N

ð11Þ

We may calculate total spectrum of the generalized dimensions Dq for any q in the interval N N: The generalized dimensions Dq do not depend on sample population. Function Dq ðqÞ is a nonascending sequence: ?XD1 XD0 XD1 XD2 ? : One may obtain the equality in the last expression only for case when all pi are equal with each other. After the Legendre transforms of the exponents tðqÞ we obtain aðqÞ

¼

d dq tðqÞ;

f ðaðqÞÞ

¼

qaðqÞ þ tðqÞ:

ð12Þ

Fig. 1 shows the function f ðaÞ representing the fractal dimensions for an aquatic ecology case. ð9Þ Acknowledgements

For q ¼ 2 (Eq. (3)): D2 ¼ 

1 ¼ N D2 : C

ð10Þ

This work has been partially supported by the RFBR under Project No. 01-02-17403.

D.I. Iudin, D.B. Gelashvily / Nuclear Instruments and Methods in Physics Research A 502 (2003) 799–801

References [1] B.B. Mandelbrot, in: M. Rosenblatt, C. Van Atta (Eds.), Statistical Models and Turbulence, Lecture Notes in Physics, Vol. 12, Springer, New York, 1972, p. 333.

801

[2] B.B. Mandelbrot, in: M. Rosenblatt, C. Van Atta (Eds.), Statistical Models and Turbulence, Lecture Notes in Physics, Vol. 12, Springer, New York, 1972, p. 5.