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continuous Keller–Segel model can describe a large population of cells, and is also amenable to mathematical analysis, but does not allow for the tracking of individual cells. The agent-based model cannot handle as many cells, and is difficult to analyse, but could be utilised in order to investigate the effects of solitary mutant cells in a population of normal cells (Fig.6).
There are also cases where a single model is able to describe two or more dis- parate phenomena. For example the Keller–Segel model has been applied in order to describe the formation of galaxies. To model this process one represents the stellar material as a self-gravitating Brownian gas, a large collection of particles that move according to a random walk and also attract each other according to Newton’s law of gravity. A mathematical analysis of this situation shows that the density of stel- lar material also obeys the Keller–Segel equation.34 This example shows that there are cases of two disparate systems that are governed by analogous mechanisms and exhibit a kind of universality. A connection can be seen to the mechanical analogies used by 19th-century physicists, where the microscopic world behavedas if it was made of springs and pulleys, but in reality it was not. A healthy degree of cautiousness is therefore recommended: just because two phenomena look alike doesn’t imply that they are governed by the same mechanisms.
for making predictions on short time-scales might be utilised in order to make long- term predictions, or a model could be valid only in a certain parameter range, outside of which the approximations within the model are not valid. A concrete example of this are models of laminar flow of a fluid, which are valid only when the velocity of the fluid is small (or rather when the system is characterised by a small Reynolds number). If the velocity increases the characteristics of the flow change from laminar to turbulent, and this type of flow requires a different kind of model to be properly described.
Another more serious type of misuse occurs when a model is no longer perceived as a simplification, but is assumed to be a direct representation of reality. This situation is considerably worse since it not only might lead to false predictions, but also could lead to a completely wrong-headed conception of how the world is constituted.
An instance of this is the model of the atom devised by Niels Bohr in 1913. This model can predict several characteristics of the hydrogen atom to a high degree of accuracy, such as its spectral lines, but it contains the invalid assumption that electrons travel in circular orbits around the nucleus, much like the planets of the solar system orbit the sun. The correct description in terms of the probability distribution of the electrons around the nucleus (atomic orbitals) was discovered in the 1920s by Pauli and Schrödinger, but the image of atoms as a miniature solar systems persists.35
In contrast to the above problem we have the reverse worry where models are looked down upon by scientists and laymen alike, simply because they contain sim- plifications. That an explanation of a phenomenon is based on a model is seen as a drawback, since the trust in simplifications of reality might be low. This issue has been brought to the fore in the debate about climate change, where it has often been claimed that the models utilised by climatologists are inadequate simplifications of a system we in reality know very little about. With regard to this it might be worth pointing out that although a model might not be able to describe and explain every aspect of a phenomenon, it is still better than not knowing anything at all.
Issues that arise in connection with models are enhanced if they are used in polit- ical or societal policy making. Conclusions drawn from models might e.g. affect how environmental problems are dealt with. This issue has been discussed by Orrin Pilkey and Linda Pilkey-Jarvis in the bookUseless Arithmetic(2007), which focuses on environmental problems. Even though the book is critical of mathematical mod- elling, and unfortunately contains a number of misconceptions about the subject, it highlights a number of important topics. Certain phenomena, such a shoreline ero- sion, are so complex that we currently are unable to model them with the accuracy that society requires. In this case, should we use poor mathematical models that
35A contributing cause might be that Bohr’s model of the atom is easier to comprehend and relate to compared to the quantum mechanical description, which doesn’t describe any actual orbits, but instead provides a probabilistic description, which to each point in space assign a probability of finding an electron in that location.
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provide wrong answers, or instead make use of conceptual and qualitative models, that do not provide exact answers, but at least are open to interpretation? Or in other words: what is the best model to use if we want to describe an unknown phenomenon:
a mechanistic, quantitative model or a qualitative, phenomenological model based on experience? The answer is rarely clear cut, and depends on a number of factors, such as purpose, the existence of underlying theory, and the possibility of comparing the model with data.
In the previous chapter we discussed and analysed the structure of models, their relation to theories and phenomena, and the basics of model construction. In this chapter we will be less concerned with philosophy and instead focus on the practical application of models. In order to achieve this we have looked beyond theoretical arguments, and gone straight to the source, namely to the scientists themselves. The reason being that it is their opinions and views on modelling that actually affect how and where models are constructed, and how they later are applied.
The interviews consisted of a number of questions related to models and their application, and were conducted over the phone. Our intention was never to quantify the answers in any way, and this brief survey should therefore be labelled qualitative.
The sound recordings were transcribed, the interviewees were anonymised, and we only identify them by their respective discipline. The answers are presented, one question at a time, together with a concluding comment. We would like to highlight that the answers should not be viewed as representative for the discipline, but rather reflect the views of a single scientist.