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Rocks - Papers - Scissors
MODEL
CALCULATE
GET TOOLS
Motivation
So let's consider the
We like simple models, so let's take 3 species. 2 is too less, 4 isn't necessary for a model of cyclic dominance.
A
B
C
or a little bit more
specific:
or in differential equations:
:= a+b+c
Cyclic dominance is appearing in nature:
salamander populations in California
bacteria
If you want, the circle of life is somehow a big cyclic domination

AND YOU SEE NICE PATTERNS IN EXPERIMENTS.
So as a theoretical physicist you want to describe it!
And we have many individuals of every species.
NON-SPACIAL SYSTEM
let's introduce
SPACE
to keep things simple --> space := lattice
1. Wow! Fascinating patterns.
2. There is a deterministic behaviour in this system, isn't it?
1 unstable (reactive) fixed point (biodiversity)
3 stable fixed points (one species survives)
If we start somewhere, the system soon arrives on a "triangular" plane and exibits heteroclinic orbits.
That means, you get a limit cycle which stays longer and longer in the vicinity of the stable fixed points. With fluctuations -> it ends up there.
Bacterial games with cyclic dominance
Felix Victor Münch
Ludwig-Maximilians-Universität München
Simulations with randomness
But, ok, you have to care a little bit about the parameters when increasing the system size. If you're interested, ask me later.
So we try a continuum limit.
With constant diffusion
And we get the following
We could have guessed that.
But there is a proper way to derive it.
It also delivers the relation of epsilon to D.
If you're interested...
Deterministic PDEs
dimension
Number of lattice sites
Numerical solutions
stochastic PDE
same wavelength!
same velocity!
To study the system analytically we need some tools to simplify it.
NORMAL FORMS
a method to find nonlinear coordinate transformations that delivers a simple form of the equations: the normal form.
method of
characteristics:
local
structure of normal form determined by the LINEAR part of the vector field
consider:
with x = 0 a fixed point. (We transformed it this way...)
Jordan normal form
of linearized system
nonlinear stuff
Taylor expand F(x):
Now we bowl down the nonlinearities step by step.
Introduce the transformation
First the second order
second order in y
Insert in Taylor expanded equation
NOTE!
can be written
so
with
for small y
So what is a clever
?
We want to eliminate
So we try to find a solution of
2nd order
3rd order
order r-1
This is a equation on a linear vector space: 
The space of the vector valued homogenous polynomials of degree 2.
that means all constituents 
of the polynom are of degree 2.
for example in 2 dimensions:
has a space of possible solutions that you can determine with linear algebra.

So in our example:
complementary to L
If
NICE! Second order in y vanishes totally.
If not, there remains
So in the end we have our normal form up to second order
but that is just the
we want to model the petri dish
Now we can start to analyze the system.
first: linearization of the rate equations of the non spacial system
We see, that the system goes exponentially towards the 2-dim manifold in yellow and then spirals outwards.

So we try to find this manifold and analize the system only there.

So we change our coordinates from                                          to                             .

This delivers the equations of motions:

Now we try to find
With an Ansatz that G is rotational invariant to second order you get:
With this we obtain
Looks more ugly, but we eliminated one dimension!
Now comes the trick! d(o_O)b
With the method of normal forms one can show, that for this 2-dim-system, the following normal form applies:
So we seek a nonlinear transformation z = y + h(y) which eliminates the second order terms in the rate equations for z.

And we find:
With this transformation we get the normal form above
with the constants depending on the parameters.
In polar coordinates:
limit cycle!
Now we introduce                       and rewrite our PDE with diffusion to
LINEAR ANALYSIS
TRANSFORMING TO THE CGLE
WITH HELP OF NORMAL FORMS
y... hola, yo conozco a esta chica!
It's the well known Complex Ginzburg-Landau-Equation.
Now we can easily derive characteristics of the emerging patterns using known standard tools. 

We derive the wavelength and the linear spreading velocity of the spiral waves.
DERIVE WAVELENGTH & VELOCITY
1. Get the dispersion relation
2. Get the linear spreading velocity
3. Get the wavelength
We just look at the CGLE in first order:
If we take the Fourier-Transfomation
the dispersion relation follows:
To get the linear spreading velocity of travelling waves, there is a general method, if you've got the dispersion relation:
This gives you:
this gives you k*
this gives you v*
For this we have to consider third order terms in
With a travelling wave ansatz ( z ( r, t) =             )        ) we get the dispersion relation:
With the imaginary part set to zero
Insert this:
from the well
mixed case
due to spacial structure
=
Now we compare this to the simulated model
and we see quite nice agreement!
...quite good, but not perfect, so the cycle starts again ;-)
References:
in order of subjective importance for this talk
Tobias Reichenbach, Mauro Mobilia, Erwin Frey; Self-Organization of Mobile Populations in Cyclic Competition; arXiv:0801.1798v2 [q-bio.PE]; 2008
Wiggins, S.; Introduction to Applied Nonlinear Dynamical Systems and Chaos; Chapter 2; 1990
Tobias Reichenbach, Mauro Mobilia, Erwin Frey; Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games; Nature 448, 1046-1049; 2007
Wim van Saarloos; Front propagation into unstable states; arXiv:cond-mat/0308540v2 [cond-mat.soft]; 2003
from http://www.geekology.co.za/blog/2010/02/how-to-play-rock-paper-scissors-lizard-spock/
Details for the continuum limit
Rescale in a way that spacial dependence doesn't vanish!
Expand
With that
Only the zeroth order of the other terms remains for delta-r going to zero.
Now take this and do the same with 3rd order.
Then you get
You can read this again, with examples, in the texed lecture (Part 5) or in the book by Wiggins (see references).
WE SEE:
The space of the resonant Terms (the ones with the r) is determined by J.
BUT they are not unique, because you can choose the base of your vector space complementary to L.
"The normal form theorem" says in fact, that you can do that up to any order necessary ;-)
This gives you the band of possible q.