{Zebra Stripes

By Aaron Jackson and Kristin Emrich

Models the rate of change in the concentration of a gene.

S0 is a concentration of a gene signal g(t) is the concentration of the gene

The idea behind stripes

The third term tells us a lot about the equation.

A simpler version is

The sudden change from y=0 to y=1

The on/off switch

0.

Consider “dimensionless” variables

The result has 2 parameters instead of 5

Simplify the equation

We need fixed points to find bifurcations and a region that is attracting or repelling (this will indicate black or white)

Bifurcation at To graph this we were given the two

equations

Fixed points

a

b

A closer look

a

b

If then

so for , x will decrease

If then

So for , x will increase

Then a is repelling and b is attracting

Given and we must show that

Remember:

True, so graph r and s parametrically to see bifurcation curves

Bifurcation Curves

Graph of bifurcation curve

There are 3 regions Pick coordinates in each region to find

how many fixed points there are

????1

3 1

Relating more to zebra stripes

Find fixed point for

or

• Consider the patch of “zebra skin” from 0 to 4π

• For values approaching b, color the “skin” black; for those approaching 0, color it white

b

a

“Chaos Theory”