The Brusselator (combination of Brussels and Oscillator) equations of I. Prigogine and R. Lefever are solved both continuously and stochastically.

Description

 The Brusselator equations form an oscillating system that
approaches a limit cycle. For an example of oscillating systems watch a video
of the Briggs-Rauscher Oscillating Reaction (several videos available on the
internet).

The same equations are solved stochastically using the Gillespie method.
A and B are fixed, X and Y vary, C and D are not calculated.

NOTE: to increase the number of molecules by a factor
of 10, divide k1 by 10, k2 by 1000, k3 by 100, and k4 by 10.
Be sure to make the corresponding change in the ODE model.

Compute lambda1, lambda2, ... lambda4 as shown in the code
for model StochasticBrusselator.

Sum the lambda's to calculate lambda.
Create a random exponential distribution by taking the natural
  log of uniform random numbers.
When multiplied by -1/lambda it becomes the time step when 
  another reaction takes place. The uneven time steps are
  accumulated in tau.
A second uniform random number for which reaction takes place
  is determined by the probability of the reaction which is the
  ratio of its lambda to the sum of all the other lambdas.
  We Calculate P1=lambda1/lambda, P2=(lambda1+lambda2)/lambda, ...
  For the reaction selected, the appropriate pools are 
  incremented or decremented in whole units.

Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.