# Multifractals in Ecology Using R - Day 1

## Introduction to fractals and multifractals

What are fractals?

**Highly irregular**: fractal objects tend to be highly irregular and fill the space in which it is embedded.**Self-similarity**: an object that displays the same basic pattern at all scales. The simplest fractals are deterministic, and are generated using recursive or iterative procedures.**Fractal Dimension**: the characteristic are captured by a dimension that is a measure of complexity of the object.

## Ecological fractals

Fractal behavior can be observed looking at different scales. In the next figure (modified from Solé & Bascompte 2006) a beetle species walks on the surface of a trunk with lichens carrying lichens on its back.

## Deterministic fractals

*The Sierpinsky gasket*: Starting with an equilateral triangle, the procedure consist on removing from the central portion an upside down equilateral triangle with half the side length of the starting triangle.

## Deterministic fractals 1

*The Coch curve*: A segment of length 1 is divided into thirds. The center one is replaced by the other two sides of an equilateral triangle of length 1/3.The curve occupies a definite space, but its length \(L\) goes to infinity.

We can compute \(L_n\) at different steps \(n\)

\(L_0=1\)

\(L_1=4/3\)

\(L_2=(4/3)^2\)

At an arbitrary step \(L_n=(4/3)^n\) that goes to infinity as \(n\) grows.

Why \(L_n=(4/3)^n\) ?

## Dynamic fractals

*Cellular automata*(CA) are discrete time, discrete space and discrete state dynamical models. We will consider a one dimensional CA with N sites. We can think that each site contains one individual of one species \(S_i(t)\) for \(i=1,…,N\)Each time step all elements are updated following a rule table:

\(S_i(t+1) = \Phi \left( S_{i-1}(t),S_{i}(t),S_{i+1}(t) \right)\)

The state of each unit change according to its own state and the state of some neighborhood.

The simplest case is that we have only one species: the possible states are 0 and 1.

## Exercise: play with 1D CA

Rule 22 (monogamy)

`current pattern 111 110 101 100 011 010 001 000 ----------------- ----- ----- ----- ----- ----- ----- ----- ----- new state 0 1 1 0 1 0 0 0`

Starting with the following initial configurations

`a) 1 0 1 0 1 0 1 0 1 0 b) 0 1 1 0 1 0 1 1 0 0`

But the rule 22 does not generate the Sierpinsky triangle, the following rule generates it:

`current pattern 111 110 101 100 011 010 001 000 ----------------- ----- ----- ----- ----- ----- ----- ----- ----- new state 0 1 0 1 1 0 1 0`

Starting with the following initial configurations

`a) 0 0 0 0 1 0 0 0 0 0`

## Random Fractals

All the previous fractals constructions have random analogues. In the Von Koch curve we replace the middle third by the sides of an equilateral triangle, we might toss a coin to determine the position of the new part above or below the removed segment.

## Statistical self similarity

The pattern of random fractals is self-similar in the statistical sense.

A given property \(L(r)\), which can be length, mass, population abundance or number or species, measured at some scale of resolution \(r\).

Then we look at a different scale \(r'=\alpha r\). If \(\alpha < 1\) then is a finer resolution, else a coarser resolution.

Statistical self similarity means that \(L(r)\) is proportional to \(L(\alpha r)\)

\(L(\alpha r) = k L(r)\)

where k is a constant.

This definition implies that the statistical features of a fractal set are the same when measured at different scales.

## Scaling laws

Statistical self similar patterns can be analyzed by power laws or scaling laws

**Zipf’s law**: one of the best known scaling lawsThe fraction of cities \(N(n)\) with \(n\) inhabitants shows a power law dependence:

\(N(n) \propto n^{-r}\) with \(r \approx 2\)

An example of an ecological scaling law is the frequency distribution of biomass, the plot shows the cumulative distribution \(N(>n)\) against biomass

Scaling in the cumulative biomass distribution of all organisms in lake Konstanz (from Gaedke 1992).

For a scaling law \(N(n) \propto n^{-r}\) we get \(N(>n) \propto n^{-r+1}\)

## Power laws are scale invariant

To show that power laws are scale invariant we can see the effect of a scale transformation.

Self similarity implies:

\(\frac{L(r)}{L(\alpha r)}=k\)

Let us assume that \(L(r)\) follows a power law

\(L(r)=A r^\eta\)

then

$ = = k $

## Fractal dimension

Let us consider different geometric objects:

A line \(\Omega_1\) of length \(L\)

A square \(\Omega_2\) of area \(L^2\)

A cube \(\Omega_3\) with volume \(L^3\)

We want to cover these with a set of identical non-overlaping segments/squares/cubes of side \(\epsilon L\) with \(\epsilon < 1\).

The number of segments required to cover \(\Omega_1\) will be

\(N(\epsilon) = \frac{L}{\epsilon L} =\epsilon^{-1}\)

For the squares \(\frac{L^2}{(\epsilon L)^2} =\epsilon^{-2}\)

In general

\(N(\epsilon) = \epsilon^{-d}\)

Where \(d=dim(\Omega_d)\)

## Fractal dimension 1

Thus we can define a dimension taking logarithms

\[d = -\lim_{\epsilon \to 0}\frac{\log N(\epsilon)}{\log \epsilon}\]

Why we need the limits?

We can apply it to the Sierpinsky gasket:

For the first step we need 1 triangle of side \(\epsilon_0=1\)

For the second step we need \(N_1(\epsilon)=3\) of side \(\epsilon_1=1/2\)

In general \(N_n(\epsilon)=3^n\) triangles of side \(\epsilon_n=(1/2)^n\)

The fractal dimension

\[d = -\lim_{n\to \infty}\frac{\log 3^n}{\log (1/2)^n}=\frac{log 3}{log(1/2)}=1.5849\]

This is a non-integrer value between a line dim=1 and a surface dim=2. In general fractal objects have a dimension below of the dimension of the space that contains it.

Exercise: what is the dimension of the Koch Curve

- \(-\frac{log 4}{log(1/3)}\)

## Estimation of fractal dimension

How to compute fractal dimensions for natural objects that display statistical self similarity?

The box counting algorithm

We cover the object with square non-overlaping boxes of size \(\epsilon^2\) and repeat the procedure using a range of \(\epsilon\) values

This range will be limited by the resolution scale \(\epsilon_m\) the pixels of our system, and the system size \(\epsilon_M\)

For each \(\epsilon\) in our range the number of boxes \(N_b(\epsilon)\) containing at least one part of the object will be counted

Following the definition of dimension we can see that \(N_b\) will approximately scale as

\(N_b(\epsilon) \thicksim \epsilon^{-d}\)

in practice \(d\) is estimated by the slope of the scaling relation

\(-\log(N_b(\epsilon))/\log(\epsilon)\)

## An ecological example

The fine scale movement patterns of the ocean sunfish Mola mola (From Seuront 2009). The inset is the detail of the diurnal and nocturnal (shaded) movements.

## Mola mola swimming

- The fractal dimension was calculated for diurnal and nocturnal movement paths and they were different.
*lower D during daylight*suggest individuals move in more directed manner.*Higher D In the night*the movements were more complex suggesting individual interact with environmental heterogeneity on a finer scale.An increase in the complexity of spatial movements should indicate an increase in foraging or searching effort.

## Characteristic features of fractals

Mandelbrot Originally defined fractals as sets that have fractal dimension strictly greater than its topological dimension.

There is no hard and fast definition but a list of properties.

We refer to

*F*as fractal if:*F*has a fine structure: i.e. detail on small scales.*F*is too irregular to be described by traditional geometrical language*F*has some form of self-similarity, perhaps approximate or statisticalUsually the fractal dimension of

*F*is greater than its topological dimension

## Paper to read

- Sugihara G, May RM (1990) Applications of fractals in ecology. Trends in Ecology & Evolution 5: 79–86.

## Bibliography

```
Gaedke U (1992) The size distribution of plankton biomass in a large lake and its seasonal variability. Limnology and Oceanography 37: 1202–1220.
Seuront L (2009) Fractals and Multifractals in Ecology and Aquatic Sciences.
Taylor & Francis.
```