Ventana software support forum

John Wolfendon is correct that system dynamics models are solved
numerically by approximating the underlying continuous
(differential/integral) equations with difference equations. This is
because for all but trivial models, no analytic solutions are known, so
numerical integration is the only practical way to determine the
solution of the equations. Often system dynamics models use Euler
integration (the difference equation approach), though often
higher-order integration schemes such as variable step size 4th order
Runge-Kutta are used (these higher order integration methods are
standard in many simulation software packages).

Let us not confuse the numerical integration method with the conceptual
issue of continuous versus discrete time in our models. Most SD models
are continuous time models, not difference equation models. The
numerical integration method should be chosen to be accurate enough, and
the time period between successive updates of the state vector, usually
known as DT (delta time), short enough so that integration error (the
discrepancy between the numerical and true analytic solutions) is
tolerably low for the purpose at hand.

This is quite different from a difference equation model in which the
time period between state updates is large, as this latter assumption
implies that there is a nontrivial delay in all the feedback loops, and
often leads to the introduction of spurious dynamics. Further, in
difference equation models of the type John illustrates, the delay in
the transmission of information in all feedback loops is the same (or an
integer multiple of the basic time step). This is rarely a good
assumption in practice. Often in difference equation models, especially
in economics, the time period between periods is determined by the
frequency with which the data are published (monthly, quarterly, or
annually), rather than as it should be by consideration of the time
delays or other structural elements in the system. Even more seriously,
many difference equation models fail to specify the calendar time
corresponding to the interval between periods. Without specifying
whether the time between periods is a day, month, quarter, or year,
these models are vacuous and without empirical content, for their
parameters cannot be estimated and their behavior cannot be compared to
any real life data. Lest you think this is a rookie error, consider
that Paul Samuelsons famous multiplier/accelerator model of the
business cycle, and a host of similar famous economic models, make
exactly this error: they formulate the economy as a set of difference
equations without ever specifying how long the time between periods is.
They then go on to observe that the model can oscillate and assume that
the fluctuations correspond to the business cycle.

As George Richardson has documented in his book Feedback Thought in
Social Science and Systems Theory, those who work with difference
equation models as opposed to continuous time models have gotten trapped
in these and other conceptual and technical difficulties which have
confounded their ability to draw useful inferences from their models.
Good system dynamics practice is to formulate your model in continuous
time. If there are delays in the reporting of information (such as
quarterly corporate earnings), in decisionmaking or in the effects of
actions these should be modeled explicitly.

John Sterman
jsterman@mit.edu