Modeling approaches

Systems sciences know different approaches to modeling. Roughly they can be categorized in respect to the three dimensions: “continual ↔ discrete”, “deterministic ↔ stochastic” and “non spatial ↔ spatial”.

ODE = ordinary differential equations (as distinguished from difference equation)

PDE = partial differential equations

CAs = cellular automata

We started this introduction to systems sciences with considerations about recursions which in the above rectangle would be positioned in the corner labeled Difference equation. In the following however, we saw that dynamics like growth or decline processes often are represented and analyzed with the help of differential equations. This moved us along the base of the rectangle to the corner labeled ODE, for ordinary differential equations. Since ODEs have a quite demanding mathematical format, systems scientists developed computer-based techniques to help them calculate and analyze dynamics that are represented as differential equations. These modeling techniques are known as equation-based modeling techniques (EBM, in contrast to agent-based modeling, or ABM, which we will get to know here)

Some solutions for ordinary differential equations, like the one for simple growth processes of the form $$\frac{dN}{dt} = \gamma N$$ are known. In many cases of more complicated or coupled differential equations however, solutions are either difficult or impossible to obtain. Although mathematicians know tricky methods like polynomial factorization or power series to approach solutions, in many practical cases of modeling one rather would resort to the numerical method of EBM for analyzing differential equations.

One such numerical method is known under the name System Dynamics. It has become the base for a wide range of software tools, program packages such as Stella, Powersim, Simulink (of Matlab) or Vensim. The last one, Vensim, offers a free version for personal or educational use, called Vensim-PLE (downloadable here).