File: SimulationTau.htd

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<html>
<head>
<title>Tau-Leaping</title>
</head>
<body>
<h1>Tau-Leaping</h1>

<p>
With tau-leaping we take steps forward in time. For each reaction we calculate
a predicted average propensity. We then generate Poisson deviates to determine
how many times each reaction will fire during the step. The advantage of
tau-leaping is that it can jump over many reactions and thus may be much
more efficient than exact methods. The disadvantage is that it is not an
exact method. 
</p>

<p>
There are several options for the tau-leaping solver. By default it
will use an adaptive step size and will correct negative populations.
You can also choose to not correct negative populations, the
simulation will fail if a species is overdrawn.
There is also a fixed time step option. This option is only useful for
studying the tau-leaping method. With a fixed step size it is
difficult to gauge the accuracy of the simulation.
</p>

<p>
In tau-leaping, one uses an expected value of the propensities in
advancing the solution.  The propensities are assumed to be constant
over the time step. There are several ways of selecting the
expected propensity values. The simplest is forward stepping;
The expected propensities are the values at the beginning of
the step. One can also use midpoint stepping. In this case one
advances to the midpoint of the interval with a deterministic step.
Then one uses the midpoint propensity values to take a stochastic
step and fire the reactions. Midpoint stepping is analogous to a second
order Runge-Kutta method for ODE's. One can also use higher order
approximations to determine the expected propensities. You can use
a fourth order Runge-Kutta scheme with deterministic steps to choose
the expected propensities and then take a stochastic step with these
values. Note that regardless of how you choose the expected
propensities, the tau-leaping solver is still a first-order accurate
stochastic method. That is, you can choose a first, second, or fourth
order method for calculating the expected propensities, but you still
assume that the propensities are constant when taking the stochastic
step. Thus it is a first-order stochastic method. However, using
higher order formulas for the expected propensities is typically more
accurate.
</p>

</body>
</html>