{ Zebra Stripes By Aaron Jackson and Kristin Emrich

 Models the rate of change in the concentration of a gene.  S 0 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 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 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”