# Basic Theory - 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.

The spatial distribution of low canopy areas (less than 15m) in a rainforest in Panama (BCI) where clusters of many different sizes can be observed

## 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 laws

The 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

$\frac{A r^\eta}{A (\alpha r)^\eta} = \frac{1}{\alpha^\eta} = k$

## 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:

1. F has a fine structure: i.e. detail on small scales.

2. F is too irregular to be described by traditional geometrical language

3. F has some form of self-similarity, perhaps approximate or statistical

4. Usually the fractal dimension of F is greater than its topological dimension