# Background

This model was developed by two English chemists as a way to mathematically describe autocatalytic behavior in chemical systems. Autocatalysis is a type of behavior in which one of the reactants in a chemical process acts as a catalyst to its own reaction. This type of reaction exhibits oscillatory, or other strange behavior.

This particular system has three equations:

``` . u = -Buv2 + a1v3 + lambda(1 - u) . v = Buv2 - a1v3 - v + a3w + lambda(gamma2 - v) . w = v - a3w + lambda(gamma2 - w) ```

Where u is the first chemical, v is the autocatalytic chemical, and w is the catalyst reaction.

The above equation has three components, but if you add these equations together:

``` . . . (u + v + w) = lambda(1 + gamma2 + gamma3 - u - v - w) ```

To get an analytically solvable equation that can be solved for w in terms of u and v. Substituting this into the system, reduces the system to only 2 equations:

``` . u = -Buv2 + a1v3 + lambda(1 - u) . v = Buv2 - a1v3 - v + a3(1 + gamma2 + gamma3 - u - v) + lambda(gamma3 - v) ```

These equations have six non-negative parameters, relating to how the experiment is set up initially. B, a1, a3 are reaction rate constants; gamma2, and gamma3 are scaled parameters representing feed concentrations; and lambda represents the time each chemical stays in its particular form before changing (approximately equal to the flow rate/volume) In this project we have fixed four of them:

``` B = 15 ; a1 = 0.1 ; a3 = .02 ; gamma3 = 0.4 ```

We have allowed gamma2 to vary beween 0.1 and 0.2. Lambda is not explicitly defined because we will use it to determine steady states, and bifurcation points.

Ethan Deneault (eand@wpi.edu)