## Dynamic modelling

It may come as a surprise to see dynamic modelling in a book on vegetation ecology. Probably the first dynamic models of the type used here served the investigation of systems other than ecological, mainly economic and industrial. Forrester (1968), in his pioneering book Principles of Systems, gives a very simple definition of the subject: 'As used here, a "system" means a grouping of parts that operate together for a common purpose.' In models of such systems, 'parts' are described by state variables such as weight of plant biomass, percentage cover of vegetation, plant nutrients per cubic decimetre of soil, population size of a species in a plot and so on. Hence, dynamic models perfectly serve the analysis of natural systems. But what is the meaning of the buzzword 'model'? Again, Forrester (1968) gives a simple explanation: 'A model is a substitute for an object or a system.' When modelling we are working with this substitute, being a system by itself, just like the real system it describes. When modelling we investigate the substitute by, for

example, performing test runs and studying how it succeeds or fails without doing harm to the real system - not even damaging the computer we use. In the early days of electronic computing everyone was fascinated by the apparently unlimited possibilities of simulation, culminating in the world model of Dennis L. Meadows, by which he justified his Limits to Growth (Meadows et al. 1972). Limits to modelling were experienced later because the computer models proved difficult to handle when complexity increased, as illustrated in an early attempt shown in Figure 10.1. Even simple models may be difficult to handle and to understand; small is often beautiful.

Dynamic modelling became popular because it is easy to understand and easy to do - even for the mathematically less well trained. The rules by which state variables change are described by one or several differential equations. Based on initial conditions given by the modeller the resulting change of the entire system in time is derived through numerical integration, all carried out by the computer. When introducing the method I start with the simplest temporal systems, comprising one state variable only, and subsequently add interacting variables, then finally extend the principle to spatial systems.