URBS PANDENS is funded by the European Commission (Contract No. EVK-CT-2001-0052) and carried out by a consortium of nine partners in seven European countries.
Qualitative Differential Equations
Starting from the case studies a qualitative model (or, if necessary, a family of models) will be developed using the technique of qualitative differential equations (QDEs). Since the project has to take care of inescapable data gaps, hardly quantifyable data and peculiarities of the case studies, the approach offers some advantages. QDEs takes uncertainties about the intensity of issues or indicators (variables) into account, as well as uncertainties about driving forces, processes or dependences (constraints). Even more importantly, the approach allows to generalize single case studies to patterns of change without blurring regional peculiarities and specifics. Thus, this formal qualitative approach fills the gap between uncertain, singular quantitative prognoses and prognoses by pure argument dealing with the problem to consider all consequences of the underlying assumptions. In mathematical terms a qualitative model represents a well-defined class of quantitative models, which includes different (but similar) hypothesis about case studies, which are allowed to be coarse-grained enough to describe low levels of knowledge.
The set up of a qualitative model is performed in three steps: Identification of relevant qualitative variables, formulating qualitative hypothesis of their inter-relationships, and computing all dynamics consistent with these assumptions. To view a bigger map click on the relevant image.
1. Identification of Qualitative Variables
A variable in a qualitative model represents an issue or relevant dynamic property of a (set of) case studies. It can generalize a set of strongly correlated features of an urban area, or subsume a class of more detailed specifications in the different cases, as far as they correspond to the same concept (e.g. environmental degradation). In some cases variables can be measured by indicators. However, they need not to be specified numerically. Only their direction of change in time and their intensity relative to critical thresholds has to be known. About these well-ordered thresholds, it has only need to be known that they exist somewhere.
2. Hypothesis about inter-relationships
Once the variables are specified, so-called constraints between them are formulated. These are qualitative statements like "if x increases, so does y", which describe processes and influences between variables. The "language of QDEs" is rich enough to specify a broad variety of inter-relationships, e.g.
· Critical thresholds for the occuring of processes
· Functional relationships
· Factors enforcing trends
· Multivariate influences, multi-causality
· Algebraic relations (e.g. population in center + population in periphery = total population).3. Computation of Results
When the variables and constraints of a qualitative model are specified, we obtain a QDE, which is the input for the qualitative simulation tool QSIM from the University of Texas at Austin (US). It computes all dynamic behaviours of the qualitative variables which are consistent with the structural assumptions made explicit in the QDE. The result is a graph contains all combinations of qualitative values the variables can take due to the assumptions (called qualitative states). They are linked with arrows to indicate all possible changes of the qualitative states over time. Thus, we do not compute a unique prediction, but weak prospects for the future.
Interpretation of Results
The resulting graph can be used to
· Detect unexpected consequences of the assumptions
· Validate the model using the recent histories of urban sprawl identified in the case studies
· To reveal new emergent properties of the system
· To compare different case studies
· To identify positive or non-sustainable patterns of development
· To formulate and compare possible strategies for management
