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 = -Buv^{2} + a_{1}v^{3} + lambda(1 - u)
.
v = Buv^{2} - a_{1}v^{3} - v + a_{3}w + lambda(gamma_{2} - v)
.
w = v - a_{3}w + lambda(gamma_{2} - 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 + gamma_{2} + gamma_{3} - 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 = -Buv^{2} + a_{1}v^{3} + lambda(1 - u)
.
v = Buv^{2} - a_{1}v^{3} - v + a_{3}(1 + gamma_{2} + gamma_{3} - u - v) + lambda(gamma_{3} - v)

These equations have six non-negative parameters, relating to how the experiment is set up initially. B, a_{1}, a_{3} are reaction rate constants; gamma_{2}, and gamma_{3} 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 ; a_{1} = 0.1 ; a_{3} = .02 ; gamma_{3} = 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)