Flow lines in Murray's model of pattern formation

In order to understand analytically what instability leads to formation of patterns, it's important to understand what happens to the model when there is no spatial variation. In this case, write down the model described here. If there is no spatial variation, then f(u,v) and g(u,v) tell you how much u and v, respectively, change in time. If we start off at a particular value of u and v, say u_init and v_init, then we can see how these will evolve in time.

The code hw4/flow_lines/eqn_flow.py shows what happens for a range of initial trajectories. By looking at the final position after a long time, what are u and v? Do these depend on initial conditions? What do these values represent? Look at this analysis of pattern formation here Links to an external site. and give your answer in terms of variables used there.

Now change the code so that ArrowsOn=True. The direction of the arrows tells you how u and v will change in time. The arrows tell you the direction the trajectory in, (u,v) space, will take. You can change the region that is being plotted by altering the line above "quiver". This gives you another way to visualize the behavior of this differential equation.