1. The equations of Turing and Murray are easier to understand in one dimension. This would correspond to constant values in one direction, or stripes. In this project, you will analyze how the frequency of the stripes can be understood, The code hw4/1d/1d_patterns.py is a one dimensional version of the 2d simulation described in problem 2 below.

We can seed the one dimensional patterns with random noise. What happens as you evolve the simulation? What happens as a function of gamma?

Compare this with the one dimensional analysis of in Evelyn Sander's writeup.

You can also start the patterns by adding small cosine terms of different spatial frequencies, instead of random noise. Uncomment and comment out the appropriate code and look at the evolution as a function of frequency k. What happens to the evolution as you change k. Note that machine precision is finite and this can introduce small amounts of "noise". So be careful about interpreting the results as t becomes very large.

What do the results for cosine perturbations tell you about the case of random noise?

This code illustrates a very important principle in pattern formation: the response to small density perturbations. Normally you think of these in terms of sine waves. To understand what's happening above, try to think about this conceptually. Try to see how you'd expect fast, medium, and slow wiggles will evolve in time.

2. Now use hw4/2d/patterns.py to simulate the reaction diffusion equation as described here and look at explore its behavior as a function of parameters. Periodic boundary conditions are being employed. What kind of shaped animal would this correspond to?

The simulation takes place on a grid of size nx by ny. In the script they are both set to 128 You can make this smaller, say 64 if the simulation is taking too long to converge.

Note the initial conditions used. Random noise has been added. Why?

As a function of the parameter gamma, how do the evolved patterns change? Try values for example, from 0.5 to 12. Make sure that the time step dt doesn't get too large or the iteration scheme used will cease to converge.

Now change nx to make it a lot smaller, say 8 and repeat your investigation for different values of gamma. What do you find now?

In understanding this, it's important to understand how redefining length scales, and time scales in the differential equations, leads to a more universal equation independent of parameters. You should see if you can rescale the equations to understand if this explains your numerical findings. Importantly, you don't have to solve the equations, just rescale independent variables. And example of this would be to see how rescaling time in the equation for a mass on a spring so that its angular frequency is constant. This will eliminate both the spring constant and mass in the differential equation, but alter the clock speed. Again, this is an important and power technique that should be emphasized a lot more!

Note that the code is currently set to use finite differences between neighboring values of the fields. This leads to some anisotropy in the patterns for large gamma. If you are interested, you can uncomment the code using Fourier transforms instead. It runs a bit slower but leads to more isotropic patterns.

In both these problems, make sure to run the code for long enough!

3. Now you can explore the biological relevance of these kind of models.

Give some biological examples of where the same general mechanism proposed by Turing are thought to be used.
What other mechanisms have been found to produce the complex structures and functions found in organisms?
How do the creation of complex structures in biology differ from those made by human technology?
What are the advantages and disadvantages that you see in Turing's mechanism as opposed to others that have been found?
Plants often show self similar morphology. Give some examples of this.
Can you think of, or find a viable theory, different to the Turing mechanism, or DLA, for how such self similar structures are produced?