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# Introduction

Mathematical models form an important aid in understanding complex systems. They also help problem solvers to capture and reason about the essential features and dynamics of such systems. Constructing mathematical models is not an easy task, however, and many disciplines have contributed approaches to automate it. Compositional modelling [10,20] is an important class of approaches to automated model construction. It uses predominantly knowledge-based techniques to translate a high level scenario into a mathematical model. The knowledge base usually consists of generic fragments of models that provide one of the possible mathematical representation of a process that occurs in one or more components. The inference mechanisms instantiate this knowledge base, search for the most appropriate selection of model fragments, and compose them into a mathematical model. Compositional modelling has been successfully applied to a variety of application domains ranging from simple physics, over various engineering problems to biological systems.

The present work aims at a compositional modelling approach for building model repositories of ecological systems. In the ecological modelling literature, a range of models have been devised to formally characterise the various phenomena that occur in ecological systems. For example, the logistic growth [43] and the Holling predation [16] models describe the changes in the size of a population. The former expresses changes due to births and deaths and the latter changes due to one population feeding on another. A compositional model repository aims to make such (partial) models more generally usable by providing a mechanism to instantiate and compose them into larger models for more complex systems involving many interacting phenomena.

Thus, the input to a compositional model repository is a scenario describing the configuration of a system to be modelled. A sample scenario may include a number of populations and various predation and competition relations between them. The output is a mathematical model, called a scenario model, representing the behaviour of the system specified in the given scenario. A set of differential equations describing the changes in the population sizes in the aforementioned scenario due to births, natural deaths, deaths because of predation, available food supply or competition would constitute such a scenario model.

This application domain poses three important new challenges to compositional modelling. Firstly, the processes and components of an ecological system that are to be represented in the resulting composed model depend on one another and on the ways they are described. In population dynamics for example, models describing the predation or competition phenomena between two populations rely on the existence of a population growth model for each of the populations involved in the phenomenon. This inhibits the conventional approach of searching for a consistent and adequate combination of partial models, one for each component in the scenario. This approach provides an adequate solution for physical systems because these are comprised of components implementing a particular functionality that can be described by one or multiple partial models. Although the seminal work on compositional modelling [10] recognised the existence of more complex interdependencies in model construction in general, it provided only a partial solution for it: all the conditions under which certain modelling choices were relevant had to be specified manually in the knowledge base.

Secondly, the domain of ecology lacks a complete theory of what constitutes an adequate model. Most existing compositional modellers are based on a predefined concept of model adequacy. They employ inference mechanisms that are guaranteed to find a model that meets such adequacy criteria. However, criteria to determine how adequate an ecological model may be vary between ecological domains and even between the ecologists that require the model within the same domain. Therefore, the compositional modeller requires a facility to define the properties that the generated ecological models must satisfy.

Thirdly, it is not possible to express all the criteria imposed on the scenario model in terms of hard requirements. Often, ecological models that describe mechanisms and behaviours are only partially understood. In such cases, the choice of one model over another becomes a matter of expert opinion rather than pure theory. Therefore, in the ecological domain, modelling approaches and presumptions are, to some extent, selected based on preferences. Existing compositional modellers are not equipped to deal with such user preferences and this paper presents the very first compositional modeller that incorporates them.

Generally speaking, the above three issues are tackled in this paper by means of a method to translate the compositional modelling problem into an activity-based dynamic preference constraint satisfaction problem (aDPCSP) [21]. An aDPCSP integrates the concept of activity-based dynamic constraint satisfaction problem (aDCSP) [27,31] with that of order-of-magnitude preferences [21]. The attributes and domains of this aDPCSP correspond to model design decisions, with constraints describing the restrictions imposed by consistency requirements and properties and order-of-magnitude preferences describing the user's preferences on modelling choices. The translation method brings the additional advantage that compositional modelling problems can now be solved by means of efficient aDCSP techniques. As such, compositional modellers can benefit from recent and future advances in constraint satisfaction research.

The remainder of this paper is organised as follows. Section 2 introduces the concept of an aDPCSP, a preference calculus that is suitable to express subjective user preferences for model design decisions and to be integrated with the general framework of aDPCSPs. It also gives a solution algorithm for aDPCSPs. Next, section 3 presents the compositional model repository and shows how such an aDPCSP is employed for automated model construction. These theoretical ideas are then illustrated by means of a large example in section 4, applying the compositional model repository to population dynamics problems. Section 5 concludes this paper with a summary and an outline of further research.

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Jeroen Keppens 2004-03-01