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MATHEMATICAL MODELS Jerzy A. Filar, Centre for Industrial and Applicable Mathematics, University of South Australia, Mawson Lakes, SA, 5095, Australia. Keywords: Mathematical models, mathematical computer models, simulation models, integrated models, hybrid models, statistical models, domain knowledge, governing equations, exploratory data analysis, complex adaptive systems, principle of parsimony, model uncertainty, error propagation, computer implementations. Contents: 1. Introduction 2. Why do we resort to mathematical modelling of life support systems? 3. What kinds of life support systems can be described by mathematical models? 4. How is mathematical modelling done? 4.1. Modelling on the basis of previously established “governing equations” 4.2. Extracting models from data 4.3. Mathematical Computer Models 4.4. Hybrid mathematical models 4.5. The iterative nature of model construction 5. Understanding uncertainty accompanying mathematical models 6. The impact of the information technology “revolution” on both the practice and uses of mathematical modelling. 7. A brief guide to the theme. 1. Introduction We begin this theme with the following excerpt from the famous “Cave Allegory” by Plato (from “THE REPUBLIC” by Plato, 360 BC, translated by Benjamin Jowett). “AND now, I said, let me show in a figure how far our nature is enlightened or unenlightened: --Behold! human beings living in a underground den, which has a mouth open towards the light and reaching all along the den; here they have been from their childhood, and have their legs and necks chained so that they cannot move, and can only see before them, being prevented by the chains from turning round their heads. Above and behind them a fire is blazing at a distance, and between the fire and the prisoners there is a raised way; and you will see, if you look, a low wall built along the way, like the screen which marionette players have in front of them, over which they show the puppets. I see. And do you see, I said, men passing along the wall carrying all sorts of vessels, and statues and figures of animals made of wood and stone and various materials, which appear over the wall? Some of them are talking, others silent. You have shown me a strange image, and they are strange prisoners. Like ourselves, I replied; and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave? True, he said; how could they see anything but the shadows if they were never allowed to move their heads?

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And of the objects which are being carried in like manner they would only see the shadows? Yes, he said. And if they were able to converse with one another, would they not suppose that they were naming what was actually before them? Very true. And suppose further that the prison had an echo which came from the other side, would they not be sure to fancy when one of the passers-by spoke that the voice which they heard came from the passing shadow? No question, he replied. To them, I said, the truth would be literally nothing but the shadows of the images. That is certain.” The rationale for selecting the above excerpt from The Republic is that it describes in a visual and an emotive way what is arguably the essence of the challenge facing most of the modern era researchers involved in the mathematical modelling of life support systems. The challenge is that of creating a model whose outputs – Plato’s shadows of images – correspond very closely (under a wide spectrum of inputs) to the measurements of the outputs of the real phenomenon being studied. For instance, a sound model of the spread of an epidemic in a population should be able to estimate the sizes of the different cohorts affected by the disease, at various stages of the epidemic. And yet, the mathematical modelling cognoscenti will be conscious of the fact that even a best model of an epidemic is essentially distinct from the epidemic itself. It is more like a wooden figure of an animal in Plato’s parable than the animal itself. Despite the preceding cautionary allegory, the purpose of this contribution is to provide an introduction to the tremendous power of mathematical models – when properly applied – to provide insight to and understanding of many important phenomena. Nowadays, the success of mathematical models and their computer implementations is well documented and spans a wide spectrum of applications from image reconstruction in medical tomography, through spread of pollution in porous media, mathematical models for weather forecasting to traffic flow models or large scale production planning models. The societal reliance on mathematical models to support planning, technological innovation, engineering design, and business and development practices is greater than ever before in the history of civilisation. Furthermore, as availability of high speed computing increases, this trend can only continue. Therefore, the question addressed in this contribution is not whether mathematical modelling is valuable or desirable – that is taken as self-evident – but rather: What are the key principles of best practice when mathematical models of life support systems are developed, implemented and used? The latter can, perhaps, be best understood by examining the independent nature of mathematical systems constituting a model and the consequent limitations when outputs of the model are applied in the “real world”. Interestingly, perhaps, the spectrum of applications is far wider than the spectrum of mathematical techniques used to generate these applications. For instance, a system of ordinary differential equations can be used to adequately model a range of very disparate phenomena (eg. a population of a colony of insects, or harmonic motion of a

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mass bouncing on a spring). Therein lies one of the great efficiencies of mathematical modelling: the understanding of a relatively small number of mathematical techniques enables one, at least in principle, to model and understand a vast array of phenomena. Nonetheless, researchers employing these techniques must be vigilant to never forget that – no matter how well a model fits the observed data – at a most fundamental level it is still a mathematical object whose “allegiance” is to the internal consistency of the mathematical system and not to the external phenomenon that the researcher wishes it to model. Thanks to the above mentioned efficiency of mathematical methods, it would have been possible to structure this theme around a strictly mathematical partition of the most widely used techniques such as algebraic equations, differential equations, statistical models, probabilistic models, simulation models and others. Each technique could then be discussed in some detail and illustrated with a number of successful applications. The above approach to the Mathematical Models theme would have minimised overlaps. It may also have appealed to some mathematicians by resembling a curriculum of an undergraduate applied mathematics major, but – in all likelihood – it would have been of very limited use to the diverse community of practitioners, researchers and students interested in the modelling of various aspects of life support systems. The main drawback with such a, mathematical, approach to the theme would have arisen from a failure to communicate the context and purpose underlying the mathematical modelling undertaken in various disciplines. For the theme to be useful to a broad audience a researcher in, say, ecology needs to be able to find an article written in the language used by ecologists and addressing issues relevant to ecologists. Only then, will the mathematical models described in such an article communicate their intended meaning to the intended audience. In view of this the nine topics of the Mathematical Models theme represent broad categories of endeavour, relevant to the mission of the encyclopedia, where there is already a large body of literature that exploits mathematical modelling to study phenomena and issues relevant to these topics. Thus for each of these topics there exists a community of practitioners, scholars and users with broad interest in that topic. It is hoped that members of these communities will find it easy, informative and rewarding to scan the encyclopedia and the theme for the articles that are most relevant to them. Inevitably, the above “user oriented” approach to the Mathematical Models theme results in some inefficiencies and duplication. For instance, it would be reasonable to expect mathematical models of forest management to appear both in the topic devoted to biology and ecology and in the topic dealing with food and agricultural sciences. Furthermore, the mathematical methods used to construct these models may appear and be discussed, in necessarily similar terms, in a number of other topics as well. This is accepted as a necessary consequence of the principle that the theme is being developed to serve a wide interdisciplinary audience.

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In this introduction to the Mathematical Models theme, we shall address three main questions: 1. Why do we resort to mathematical modelling of life support systems? 2. What types of life support systems can be described by mathematical models? 3. How is mathematical modelling done in, at least, the most broad conceptual

terms? Arguably, the discussion of the above three questions will shed light only on what might be called the “classical” view of mathematical modelling. However, we live in an era where most educated people have easy access to many tools of mathematical modelling embedded in personal computers on their desks. Furthermore, it is an era where interdisciplinary teams regularly develop large mathematical models on a scale that would have been unthinkable until very recently. In some cases, the models themselves generate mathematical expressions that may or may not be detailed in the conventional way of being written down in a published book, manuscript, or even a technical manual. In recent years, the terms “computer models” and “numerical models” have been frequently used to name some of these modern models; thereby suppressing the fact that, at least internally, they consist of (possibly many) mathematical models. The technological advances that made these new classes of models possible open up many exciting opportunities as well as some inherent dangers. While it will be seen that advantages of the technological progress clearly outweigh disadvantages, it will also be clear that we have entered an era where new issues concerning the nature and practice of mathematical modelling need to be examined and some of the old issues need to be re-examined. It is in this context that we shall also discuss the very important issues of: 4. Understanding and managing uncertainty accompanying the use of mathematical

models, and 5. The impact of the information technology “revolution” on both the practice and

uses of mathematical modelling. In subsequent sections items 1-5, listed above, will be discussed in more detail. 2. Why do we resort to mathematical modelling of life support systems? In the words of R. Isaacs (in: On Applied Mathematics. Journal of Optimization Theory and Applications, 27 (1) p37) “The human mind is incapable of thinking other than about models”. Irrespective of whether one completely agrees with the preceding statement, it would be hard to argue against a proposition that the desire to use models to help reduce the complexity of situations we face is fundamental to our way of thinking and analysis. It appears to be an innate human trait. It is sufficient to observe children playing with toy cars, dolls or soldiers to be convinced of how natural it is for us to desire simple models of complex things that we encounter.

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However, even if we accept as innate the need to develop models so as to simplify complexity, this does not necessarily make the case for the use of mathematical models. After all, there are many other kinds of models that help reduce complexity such as architects’ physical models of new cities, conceptual models such as those often used by psychologists, or evolutionary models used to explain the origins of the Homo sapiens (eg. single versus multiple origin theories). These are all instances of very useful models of complex life support systems, interpreted broadly, which make little use of mathematics. The rationale for resorting to mathematical modelling probably stems from the underlying “dual nature” of mathematics as the science of relations as well as the science of quantity. Thus whenever it is desirable to study both the quantifiable magnitude of effects and their relationship to one another (if any), the use of mathematics is almost an inevitable consequence. Of course, there is a legitimate argument that the role of mathematics in modelling phenomena is “merely” that of a language used to describe the knowledge and understanding of those who observe and study these phenomena. Indeed, this is the case with many mathematical models of physical phenomena. Furthermore, if that were the only role of mathematics in modelling, then it would be possible to argue that as it becomes increasingly easy to encode understanding of relations (eg., with the help of logical statements of any computer language) and quantities in computer files, that the future role of mathematics in modelling will be greatly diminished. However, the fact that mathematics is also the science of relations means that the role of mathematics in modelling is deeper than that. Thanks to the latter, it is possible to take an initial mathematical statement that merely describes other scientists’ knowledge and transform it by a sequence of logically consistent operations to arrive at a new statement, or a model. The latter will be as true as the initial one but may exhibit almost a very new insight into the modelled phenomenon. In effect, by this process, the mathematical description acquires a status of a “theory” in the sense that all logical consequences of the initial description become available as tools to either support or reject the theory. It is, perhaps, inevitable that as computer systems continue to evolve to imitate the processes of mathematical analysis and algebraic manipulation (eg., as the many currently available symbolic manipulators already attempt), software packages will emerge that will – to varying degrees – automate the above process of creating and transforming a mathematical model. We choose to call this new generation of models mathematical computer models, as distinguished from mathematical models merely implemented on a computer; and we shall discuss this issue in more detail later on. In addition to the previously mentioned capability to reduce complexity of real phenomena by extracting only certain essential features that are of interest, some mathematical models also have the capability to idealise “imperfect” real phenomena. Perhaps, one of the most trivial examples of this is the equation of a circle:

x2 + y2 = r2,

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where r is the radius. While, conceptually we may define a circle as a set of points equidistant from the centre point, we are incapable of constructing such an object or, indeed, of observing it in nature except to a certain degree of accuracy. And yet, the preceding equation constitutes a perfect, absolutely precise, description of a circle. In this case, if the above equation is thought of as the model, then it is an idealised model and all real world “circles” are, at best, approximations to such an “ideal” circle. Up to this point we attempted to point to certain powerful and generic features that mathematical modelling has to offer. However, perhaps, the single most compelling rationale – though not independent of the preceding ones - for the use of mathematical models is historical. Mathematics is the universal language of science, engineering and increasingly of other disciplines. Much of the recorded knowledge of physics and engineering is written in the form of mathematical models. These mathematical models form the foundations of our understanding of the universe we live in. Furthermore, nearly all of the existing technology, in one way or another, rests on these models. To the extent that we are surrounded by evidence of the technology working and being reliable, human confidence in the validity of the underlying mathematical models is all but unshakeable. Even when revolutions in physics such as Einstein’s discovery of relativity take place, they merely re-emphasise that the old Newtonian models work exceedingly well in the parameter ranges in which we could normally wish to use them. However, it is not only the long history of success of mathematical models in sciences that provides strong support for their continued use. The rationale for their continued and even much expanded role is stronger than ever because of the revolution in high speed computing. The latter has been accompanied by a revolution in algorithms for numerically solving larger and more complex systems of equations which in turn opens up the opportunity of modelling increasingly complex phenomena. In recent years we have witnessed the emergence of very sophisticated models of both local, short term, weather prediction and of coupled ocean-atmosphere models of global climate change. The numerical solution of the underlying systems of equations would have been completely impossible not long ago but has now been facilitated by the combination of powerful computers and powerful algorithms. This combination of algorithmic and computational power is providing a tremendous impetus for a much expanded role of mathematical modelling and brings with it many opportunities as well as some dangers that will be discussed later on in this introduction and in many other places across the theme. To summarise this section we claim that the following are among the key drivers supplying the rationale for continued and even expanded role of mathematical modelling of life support systems:

• The dual nature of mathematics as the “science of relations” as well as the “science of magnitude”

• The inherent ability of a mathematical model to provide a “theory” describing the modelled phenomenon

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• The inherent ability of mathematics to reduce complexity by modelling only a few of the most relevant characteristics of the problem

• The ability of mathematics to create idealised models that serve as benchmarks for physical entities

• The historical role of mathematics as the universal language of science and engineering

• The opportunities presented by the combined power of algorithms and high speed computing.

3. What kinds of life support systems can be described by mathematical models? The deceptively simple question in the heading of this section touches on some very deep issues related to the foundations of philosophy of science as well as to mathematics. Readers familiar with Heisenberg’s Uncertainty Principle and with Gödel’s Undecidability Theorem will be aware of certain fundamental limitations on the power of mathematics to arrive at the sort of unambiguous, exact conclusions that are normally so characteristic of mathematical reasoning. Fortunately, these fundamental limitations arise in situations so rare that – from the point of view of modelling life support systems – they can be disregarded as anomalies that a practical modeller will “never” encounter. It is for that reason and for the sake of imposing reasonable boundaries on the scope of this contribution that these fundamental limitations will not be discussed here and were mentioned only for completeness. In the same vein it is necessary to clearly state that – because of the mission of the encyclopedia – we shall concern ourselves only with mathematical models that might be useful in describing “real” life support systems. Thus we exclude from considerations mathematical modelling that is inspired purely by a mathematician’s understanding of strictly mathematical concepts. For instance, a mathematical entity known as a Banach space can be viewed as modelling certain properties not captured by another mathematical entity known as a Hilbert space. Certainly, the creative reasoning process that led to the development of Banach spaces may have involved important aspects of mathematical modelling. However, we judge these to be outside the scope of this contribution and focus only on situations where mathematical modelling techniques are brought to bear on a situation or a phenomenon that exists independently of mathematics. Thus the use of Banach space methods to describe some characteristic of such an “external” situation lies well within the scope of our theme, however, the process of generalising Hilbert spaces to Banach spaces has been excluded from our considerations, by the preceding argument. With the above restrictions in mind, this section is devoted to a discussion of minimal requirements that the phenomena and situations studied should possess in order to make the models meaningful. The characterisation of these requirements will be neither precise nor exhaustive; as the boundaries of what can be achieved with mathematical models are constantly pushed back. Nonetheless, there is still a small number of principles capturing what might be called an applied mathematician’s “common sense” which, if violated, raise questions as to whether mathematical modelling is meaningful in these situations.

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Recalling the dual nature of mathematics as the science of relations as well as of magnitude we shall claim that the following are, perhaps, the minimal requirements for a meaningful application of mathematical modelling techniques: 1. The presence of (at least some) variables that can be quantified and (ideally)

observed and measured, or of data from which such variables could be extracted 2. The presence of some understanding of relations between quantifiable variables

or of, at least, a need to discover such relations empirically 3. The presence of data, experimental designs, or other procedures to be used to

validate the model. The above requirements may be regarded as being part of “domain knowledge” of a phenomenon or situation being modelled. They may appear to be so obvious as to be taken for granted and yet, there are many important situations where one or more of these requirements are very difficult to satisfy. The main problem appears to be with the third requirement. For instance, the concept of a “nuclear winter” that might follow a nuclear exchange between Russia and USA is one that has been around for many years and is clearly critical to life support systems. There is an existing literature on the subject and models that attempt to estimate population and crop losses have been developed during the “cold-war” era. The need for understanding the consequences of such a catastrophic event with the help of modelling has been and remains urgent. And yet, it is clear that the third requirement stated above is extremely difficult to satisfy. The situation to be modelled is one that has never occurred and hopefully, will never occur. The complexity of the ecological consequences of an all out nuclear war is so enormous, with such a multitude of possible feedbacks, that it would be naive to think that even the most sophisticated models could capture more than just a few trends of this cataclysm. Hence, consensus opinion of “experts” constitutes perhaps the only benchmark against which a mathematical model could be validated. Of course, with the help of simulations, many alternative scenarios of a model could be run. However, once again, we would need to rely on expert opinion to select a “most likely” scenario. The latter is usually done with the help of experts even though, in a certain sense, for such an unprecedented situation there are no experts. One strategy, which only partially overcomes this difficulty, is to have independent teams develop independent models which could then be compared and tested against one another. Even requirements 1 and 2 listed above may, at times, be hard to satisfy. Many practicing applied mathematicians are familiar with cases of unreasonable client expectations. For instance, at a recent industrial mathematics conference a problem was presented to a group of mathematicians that involved the development of a mathematical model. The latter was to explain (and ultimately help reduce) the noise pollution caused by squeaky train wheels on a portion of a journey that involved the train climbing up a hill close to a residential area. This was a situation where a client approached a group of mathematicians at a stage where the underlying physics and engineering of the situation have not yet been investigated. Thus the expert domain knowledge concerning the physical situation was missing and the mathematicians had

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to hypothesise about possible relations between a number of key variables, for which there were no available data. To summarise, the main point of this section is that whenever researchers or practitioners undertake the development of a mathematical model of a phenomenon or a situation that is not itself a mathematical entity a certain, minimal, amount of domain knowledge is required to make meaningful progress. Generally, this domain knowledge needs to come from those who are most familiar with the situation modelled. In some cases, it can be extracted, for instance via statistical techniques, from data generated by the phenomenon but in such a case these data need to be available or, at least, amenable to being simulated in a credible way. Once again, the determination of what would be a “credible data simulator” is part of prerequisite domain knowledge. In short, a meaningful mathematical model of a real life support system cannot be created out of vacuum; domain knowledge is needed to provide a “reality check”. 4. How is mathematical modelling done? Once again, the question stated in the heading of this section is probably too complex to answer fully. Nowadays, mathematical modelling is so diverse and so widely used that it may be nearly impossible to give a simple and yet exhaustive classification of all types of mathematical modelling techniques. The Mathematical Models theme includes topic 6.3.1 entitled Basic Principles of Mathematical Modeling which is further subdivided into seven articles representing the following sub-topics that can be viewed as corresponding to some of the major issues that mathematicians interested in modeling have considered: 6.3.1.1. Classification of Models (Continuum, Discrete, Stochastic, etc) 6.3.1.2. Basic Methods of the Development and Analysis of Mathematical Models 6.3.1.3. Measurements in Mathematical Modeling and Data Processing 6.3.1.4. Controllability, Observability, Sensitivity and Stability of Mathematical Models 6.3.1.5. Identification, Estimation and Resolution of Mathematical Models 6.3.1.6. Mathematical Theory of Data Processing in Models (Data Assimilation

Problems) 6.3.1.7. Complexity, Pattern Recognition and Neural Models. However, the above research areas do not attempt to capture the main - philosophically different - approaches to mathematical modelling that have evolved over the years. Instead, we categorize these approaches as: A. Modelling on the basis of previously established “governing equations” ; B. Extracting models from data; C. Mathematical Computer Models; D. Hybrid mathematical models. Below, we shall briefly discuss each of these approaches to mathematical modelling.

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4.1. Modelling on the basis of previously established “governing equations” This is, undoubtedly, the oldest form of mathematical modelling of phenomena with its roots in antiquity. The ancient Egyptians were able to anticipate eclipses of the sun on the basis of their understanding of astronomy that must have involved a mathematical description of that understanding which in turn permitted a fairly precise numerical calculation. Certainly, the role that Archimedes played in the defense of Syracuse in the 3rd century BC indicates that he used models based on his understanding of the geometry of a parabola to place his catapults in strategic locations. In addition, his famous use of mirrors held by soldiers appropriately positioned to focus sun rays on the attacking Roman ships to set them on fire demonstrates that he had modeled a complex physical situation involving the shape of the shoreline and the incidence and reflection angles of sun rays. As the greatest mathematician and physicist of that time he naturally combined his physical insight with mathematical models to derive correct results that were implemented in an effective, if rather deadly, manner. As scientific knowledge accumulated over the centuries, scientists in general but physicists in particular relied extensively on mathematical descriptions of the understanding they gained. Thus whenever a physical situation is modeled in which the underlying physical processes are – at least approximately – understood, there is likely to be a set of mathematical expressions, often called the “governing equations”, already available which can immediately form a basis of a mathematical model of that situation. For instance, if we wanted to model the flow of water in a stream, we could call upon the famous Navier-Stokes equation. The latter can be derived from the physical insight contained in Newton’s second law of motion which states that the sum of the forces acting on a particular volume of fluid equals the mass of that fluid multiplied by its acceleration. An application of this to an “element” of fluid and a series of algebraic manipulations lead to the following set of equations

0,ug,uuuu=⋅∇+∇+−∇=⎟

⎠⎞

⎜⎝⎛ ∇⋅+∂∂ ρµρ 2p

t

where u denotes the three dimensional vector of fluid velocity, p is the fluid pressure, ρ is the fluid density, µ is the fluid viscosity and g is the gravitational force. These equations are in fact the Navier-Stokes equations for an incompressible Newtonian fluid. The latter equations represent a fluid in which the velocity of a fluid element is proportional to an applied force, that is, if you double the force with which you push the fluid, its velocity should double. The power of the above equation and its variants stems from the fact that it can be used to model a diverse set of phenomena ranging from flow of air around a wing of an aircraft, through flow of water in a pipe to the movement of micro-organisms in a fluid medium. Furthermore, in addition to being a theoretically derived mathematical model, the Navier-Stokes equations have been empirically verified in a very large set of applications. In the cases where they fail it is often necessary to consider non-Newtonian fluids. In addition, a well understood model such as the Navier-Stokes equation can be used as a building block for developing the governing equations of more complex

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phenomena. For instance, the theory of multiple fluid interaction can be used to model the dispersion of an oil slick in a lake or river system. In summary, when modelling a situation or a phenomenon for which a set of well founded governing equations can be written down researchers find themselves in a strong starting position. That is, they are able to draw on existing theories and properties associated with the governing equations used to describe the modelled situation. 4.2. Extracting models from data Of course, phenomena that can be fully modelled with the help of governing equations require a significant amount of prior domain knowledge, in the sense introduced in Section 3, above. In modelling many life support systems, especially those involving close interactions with human societies, this detailed level of domain knowledge is often unavailable. Fortunately, a lot of information may be available in the form of data that may have been collected in a variety of ways. The manner by which these data are collected is very important, as will be seen below. However, the mere possibility of collecting data generated by the phenomenon of interest opens up the possibility of invoking statistical methods to develop mathematical models describing associations or relationships between important variables. For instance, with the help of a statistical technique known as “experimental design” it is possible to develop a model capturing the effects of a number of new treatments of tomato seeds on the eventual size and yield of tomato plants. In such a case, it would usually be assumed that the researchers have the luxury of designing an experiment that involves planting a prescribed number of tomatoes in a prescribed number of plots, each having a prescribed selection of treatments performed on the seeds. The untreated seeds would, generally, be assigned to the individual groups based on a suitable random procedure. Embedded in the above statistical technique is an underlying (usually linear) mathematical model relating the yield of tomato plants to the application of certain treatments to tomato seeds. Indeed, statistical models such as the above – and the more generic regression analysis models - have acquired a profound role as tools of scientific inquiry. They permit researchers to identify both unanticipated relationships among variables and, in some cases, confirm the presence of anticipated relationships. The latter leads us to an important distinction between two rather different types of data: experimental and observational. The preceding, hypothetical, example concerning tomato plant seeds clearly described a situation where the collected data came from a carefully designed experiment. Such data are usually called experimental and permit statisticians to perform rigorous statistical tests on the validity of the underlying mathematical model. These statistical analyses are sometimes called confirmatory because they confirm the validity of some statements about the model, or about some of its parameters. Generally, confirmatory analyses require experimental data.

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Regrettably, there are many phenomena and situations where the available data are historical, or collected by methods that were not based on proper statistical experimental design. These data may well contain a wealth of valuable information but still preclude many of the confirmatory statistical analyses from being applied. Data describing the paths of observed tropical cyclones are an instance of this. They are usually called observational data. When faced with the task of modelling a phenomenon for which only observational data are available a statistician may still resort to a body of techniques known as exploratory data analysis, (EDA, for short) that was pioneered by John Tukey in the 1970’s. One critical difference between exploratory and confirmatory statistical analyses, in the context of extracting a mathematical model from data, is that the former may only suggest a plausible model for a phenomenon but cannot confirm its validity, in the classical statistical sense. Nonetheless, EDA has an extremely important role to play as a method for formulating worthwhile hypotheses: a key step in scientific inquiry (On the role of description in statistical inquiry). It is very much in the Baconian spirit of what science is all about. Sir Francis Bacon saw himself as the inventor of a method which would kindle a light in nature - "a light that would eventually disclose and bring into sight all that is most hidden and secret in the universe". This method involved the collection of data, their judicious interpretation, the carrying out of experiments, thus to learn the secrets of nature by organized observation of its regularities. Bacon's proposals had a powerful influence on the development of science in seventeenth century Europe. 4.3. Mathematical Computer Models As indicated in Section 2, in recent years a new generation of models that we shall call Mathematical Computer Models have began to emerge. Once again, there are already many different types of models that could be classified under this heading. However, in what follows we shall briefly discuss three generic types of mathematical computer models; namely, (i) Models created with the help of high level languages (ii) Integrated mathematical computer models (iii) Complex adaptive systems.

(i) We shall define models created with the help of high level languages as those models where the mathematical expressions constituting the model are created only as part of the model code. In these cases the mathematical description of the phenomenon has not been fully written down or analysed in a traditional mathematical way. Instead, the mathematical components to be solved or analysed exist only “internally” in the model’s source code. There are now many software packages that provide convenient tools for generating precisely such mathematical computer models. These include, high level optimisation modelling languages, statistical packages and matrix manipulators. Frequently, the models created by these tools have such large dimensions that a traditional “pencil and paper” description is practically impossible. In the area of

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energy modelling at a national level, the large scale linear programming models known as MARKAL models are good examples of such mathematical computer models. These models attempt to capture the economic costs and benefits – at a national level - of a country relying on a given mix of fuels (eg., fossil, nuclear, solar etc). They aim to find an optimal mix with respect to specified sets of objective function(s)/targets and include constraints representing permissible impacts on different sectors of industry. Understandably, such models are so large that only a macro level description in a report format is practical. The precise functional relationships are embedded in the code. (ii) By contrast, we shall define integrated mathematical computer models as models of complex phenomena consisting of a number of coupled modules, where each module is a mathematical model of some aspect of that phenomenon. Once again, the essential point is that, irrespective of whether the individual modules are traditional mathematical models or mathematical computer models in the sense of (i) above, the entire (coupled) integrated model has not been analysed in a traditional mathematical way. The European global model TARGETS - Tool to Assess Regional and Global Environmental and health Targets for Sustainability – is an example of such a model. In these cases, it may not only be the high dimensionality of the model that prevents classical mathematical analysis, but also the complexity of the coupling of the different modules. Thus it may happen that, as mathematical objects, individual modules may be simple to analyse. However, the coupled modules may constitute a much more difficult mathematical system. (iii) Since the 1980’s a new family of models known as complex adaptive systems gained prominence. Typically, these models attempt to first reproduce and then simulate into the future the behaviour of "complex" phenomena in the social, life, physical or decision sciences. An essential feature of these models is the introduction of individual “agents” in the situation under investigation. The characteristics and activities of these agents are the building blocks of these models. These characteristics are dynamic in the sense that they change over time, as the agents adapt to their environment, learn from their experiences, or experience natural selection in an evolutionary process. The main objective of these models is to capture the macro-level “emergent behaviour” that results from the local decisions by individual agents that reflect the assumptions of the local adaptation rules. The agents may be organised into groups that may also influence how the system evolves over time. At first sight, these models may appear to be entirely different from classical mathematical models in the sense that they do not attempt to describe the underlying phenomenon with the help a set of mathematical expressions. However, we have chosen to include complex adaptive systems into our classification of mathematical computer models for two main reasons. Firstly, the adaptation rules that determine the essential system dynamics, the feedbacks that the agents receive as a result of their activities and the built in “learning” functions can be thought of as constituting mathematical models in their own right. Secondly, computer simulations that are almost invariably used to generate the outputs of these models are also mathematical models in the sense that they rely on iterating deterministic or stochastic recursive relations. Consequently, the resulting emergent behaviour that these simulation models generate and which is the main object of interest depends both on the

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underlying mathematical models of the dynamics of the process and on the mathematics embedded in the simulator. Perhaps, one of the most attractive aspects of complex adaptive systems is that they often require only minimal domain knowledge and very few assumptions. They aim to reproduce complex aggregate behaviour patterns with the help of rather simple modelling of actions of individual agents. The emergent complexity is seen as a result of replication of these simple actions by many agents and of repetition over many time periods. As such, they resemble the properties of chaotic nonlinear mathematical systems the iterative solutions of which often produce the complex fractal patterns. Of course, a weakness of complex adaptive systems stems from the inherent difficulty of validating these models. It is intuitively clear that, in principle, more than one set of adaptation rules can generate very similar emergent behaviour. If so, why should we prefer one over the other? To answer the latter question, one may need to return to one of the more classical mathematical modelling approaches. Indeed, it is often hoped that the analysis of the dynamics and emergent behaviour of these simulation models can lead to new mathematical models, new hypotheses and new real-world experiments. 4.4. Hybrid mathematical models It should now be clear that there are no formal barriers to developing mathematical models that contain elements of one or more of the approaches discussed in Sections 4.1-4.3. We may think of there as being hybrid mathematical models in the sense that they exploit more than one philosophical foundation for different components of a model. This frequently occurs in the case of integrated models discussed in the previous section. For instance, the well known Integrated Model for the Assessment of Greenhouse Effect (IMAGE) can be thought of as such a hybrid model. This model includes physics and atmospheric chemistry modules that are based on governing equations of energy balance and chemical reactions as well as global deforestation and socioeconomic impact on (The Netherlands) modules that include essentially statistical and simulation sub-models. Clearly, the appeal of these hybrid models is based on their potential to provide a more holistic description of complex problems (e.g, regional acid rain and global climate change) to decision-makers. These models combine different strands of knowledge representing the insight of experts from particular disciplines that are relevant to the problem studied. However, their limitations also stem precisely from their hybrid nature. Whereas models based on governing equations, or statistical models, or even simulation models have already evolved certain methodologies for testing and validation, these techniques usually rest on the philosophical foundations underlying these models. Since a hybrid model rests on more than one foundation, the problem of developing suitable validation techniques is more formidable (see Basic Principles of Mathematical Modelling).

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4.5. The iterative nature of model construction We conclude this introductory section on how mathematical modelling is done with a brief discussion of what we shall refer to as the iterative process of model simplification and refinement. Arguably, this process constitutes what some mathematicians call the “art of modelling” (see Basic Principles of Mathematical Modelling). It aims to arrive at the simplest possible model that adequately explains the aspects of the phenomenon modelled that are of interest to the researcher.

Examine the"systems"

Identify thebehavior and

makeassumptions

Can youformulate a

model?

Yes Can yousolve themodel?

Validate themodel

Are theresults precise

enough?

Yes

Yes

Make predictionsand/or

explanations

Apply resultsto the

systemExit

No,

refine

No, simplify

Unacceptable results

No, simplify

Figure 1. Iterative model construction process. [Reprinted from Giordano, Frank R. and Weir, Maurice D. (1985). A first course in mathematical modeling, with permission from the Brooks/Cole Publishing Company.] Formally, or informally, most practicing applied mathematicians are familiar with this iterative model construction process. Figure 1 above, depicts the logical flow chart of the process in a form that can be used to explain modelling to students. It points out that, typically, a modeller would first examine the system studied, identify the behaviour of interest and make assumptions and then attempt to propose a model. If he or she failed to formulate a model, or if the model formulated is too difficult to solve, then it is desirable to simplify. The latter is usually achieved by deleting variables, replacing some variables with constants (often averaged values over some range), aggregating groups of variables, or assuming simple relationships, or even restricting the problem under investigation. However, if a model can be formulated and solved but the results fail to reproduce the observed data or are not precise enough, then it is desirable to refine. The latter is usually achieved by introducing additional variables, assuming more sophisticated (often higher order) relationships among variables, or by expanding the scope of the problem by including some new, or previously excluded, aspects of the phenomenon studied. 5. Understanding uncertainty accompanying mathematical models Whenever scientists attempt to model a life support phenomenon they need to be cognisant of both the nature and the magnitude of the uncertainties that will, invariably, be embedded in the model's outputs/forecasts. Upon some reflection, it is possible to classify the nature of the uncertainties into five distinct types (see Figure 2, below):

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1. Uncertainties caused by errors, Eo, in observations. These errors may occur prior to the construction of the model but, to the extent that the observations will be used to estimate the model's parameters, they will also affect the model's eventual outputs.

2. Uncertainties caused by errors, Ep, in the estimation of parameters. These errors

occur because whatever statistics are used to estimate parameters of interest, these statistics incur some errors.

3. Uncertainties caused by errors, Es, due to the solution algorithm. Nowadays,

most models are solved numerically, rather than analytically, and the numerical algorithms almost always involve a degree of approximation.

4. Uncertainties caused by errors, Ec, due to the computer implementation. These

errors occur due to the way that computers perform arithmetic operations and can be strongly influenced by the manner in which various steps of a solution algorithm are programmed.

5. Uncertainties caused by errors, Em, in the modelling. These occur simply because

whatever model is constructed, it is only an approximate representation of the underlying natural system that is being modelled.

Figure 2. Sources of uncertainty in the outputs of mathematical models.

Observation Phenomenon

Mathematical Model ∫ ∑ .,,),( etcxf

Parameter Estimation

Solution Algorithm

∑∫ ∆= xxfdxxf )(lim)( etc.

Computer Implementation ...1010011011011...

Output Data

sE

cE

pE

mE

oE

Input Data

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It can be argued that, of all of the above types of uncertainties, it is the last, or the modelling uncertainty Em that is most often overlooked or misunderstood. This is because, to a significant extent, each of the first four types of uncertainties is associated with the mainstream activities of a specific professional group whose practitioners have developed methods for estimating bounds on the corresponding types of errors. Thus, designers of observational equipment (eg. gauges, barometers etc) can tell us about the precision of various instruments, that is place bounds of Eo. Similarly, statisticians can usually tell us about the bounds on Ep, or the estimation errors, numerical analysts can frequently supply bounds on Es and computer scientists can bound Ec. However, in the case of the uncertainties caused by the modelling errors Em, there appears to be no professional group that specialises in their analyses. Perhaps, this is because the understanding of this kind of uncertainty requires an apriori admission from the creators of a model that their creation is, somehow, "flawed". To be fair to the modelling community, it should be said that, nowadays, most widely used models are based on sound scientific or economic principles and have been calibrated against the available data. Indeed, in the case of many models such a calibration will be accompanied by a statistical measure of the "goodness of fit" of the model's outputs to the observed data. This, in turn, can provide an estimate of the bounds on Em, if the model is used in the data ranges comparable to the ranges used for calibration purposes. What makes the modelling error especially challenging in many models is the fact that, often, such models are producing outputs in the ranges which are very different from those of the data that were used for model fitting and/or calibration. In the case of global change models, the outputs that reach 100, 200 or 1000 years into the future clearly step far outside of these ranges. Thus a fundamental question arises: Even if the modelling error Em is small initially, how does the uncertainty due to this error propagate over time, when the model is run over large time horizons? The inherent difficulty of the preceding question is that the analyses that might provide an answer are intimately linked to the mathematical formulation of the model. Thus a model described with the help of partial differential equations may require a different analysis of the uncertainty propagation than one that is described with the help of ordinary differential equations. Furthermore, the form of the functions used to describe various quantities will also affect the manner in which this uncertainty propagates. Since many modellers choose the model description (including functional forms) on the basis of what they perceive is the best current understanding of the phenomenon being modelled, the notion that this description may need to be modified as a result of the uncertainty propagation analysis, may appear as pure heresy.

Fortunately, the above difficulties are usually not insurmountable, once importance of the problem is recognised. In particular, there are many rigorous techniques - primarily statistical or probabilistic in nature – that are potentially adaptable to the

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problem of estimating both the size of the modelling error and its propagation over time. 6. The impact of the information technology “revolution” on both the practice

and uses of mathematical modelling.

Throughout this introduction there have been many references to the impact of advances in information technology on both the nature and the practice of mathematical modelling. This is not surprising because this impact has had a profound effect on the subject; an effect that is yet to be understood fully. Consequently, this section is devoted to a brief discussion of some of the benefits and dangers that advances in information technology have brought to the science and to the art of mathematical modelling. The benefits are almost self-evident: computational power, ubiquity, transportability and power to communicate a model’s results. The impact of computational power has, implicitly, already been discussed in Section 4.3 when mathematical computer models were introduced. Certainly, most of these sophisticated models would not have been developed were it not for the ready availability of enormous computing power. However, we have not yet commented on the impact of the ubiquity of mathematical models and their transportability, as facilitated by the information technology revolution. The latter have been brought about by the ready availability of many tools of mathematical modelling both on personal computers and through the web. Most people in the developed as well as many in the developing countries have easy access to the standard spreadsheet packages such as Excel® or Lotus®. While generally, these packages are not used by professional modellers, they contain a considerable selection of reliable mathematical modelling tools. It is possible, using just such ubiquitous packages, to evaluate relatively complex mathematical functions, plot their graphs, perform common matrix operations, statistical analyses and even optimisation routines. Furthermore, all of the above can be done within familiar operating systems and with little technical support other than the on-line help manuals. Hence, increasingly frequently simple mathematical models – even when they are not called by that name - are appearing in routine reports written not only by scientists and students but also by physicians, accountants, government officials, managers and even lawyers. Whenever a report contains a graph with a projected trend of some variable there is every reason to believe that some sort of mathematical model, possibly a very simple one, had been used. Furthermore, the Internet provides immediate access to a wide range of much more sophisticated modelling tools that are either in public domain or available for purchase. The above mentioned ubiquity of mathematical models is also fuelled by their transportability. Models produced with the help of common packages are easily transported on disks, in E-mail messages and via the world wide web. This serves not only a useful communication function but also opens many of these models to scrutiny by a wide range of users; thereby providing an informal but possibly very rigorous validation process. For instance, it is reasonable to assume that a government report that projects a deterioration in the ambient air quality of a particular city on the basis of a model is likely to be challenged and the underlying

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model carefully examined; especially if the model is sufficiently simple. The latter is a crucially important point because it ties in with the recurrent theme in this introduction, namely, that unnecessary model complexity implies risks that must be minimised. In addition to facilitating widespread use of mathematical models and their transportability, information technology has also revolutionised the way that the results of these models are communicated both to researchers and to the public. Whereas, in the past, mathematical models typically supplied a number or a table of numbers, nowadays, they can readily supply a variety of visually attractive graphs, simulations, sensitivity analyses, coloured maps and even animation. This has given the modellers a set of powerful communication tools with which to communicate the meaning of a model’s outputs; even to those who may not understand the underlying structure of that model. In summary, both the science and the art of mathematical modelling benefited greatly from advances in information technology that yielded: • Increased computational power • Widespread access to many tools of mathematical modelling • Easy transportability of mathematical models via electronic media • Increased power to communicate the findings of mathematical models to a large

audience. However, accompanying the preceding benefits are a number of difficulties and dangers that information technology also brings to mathematical modelling. These can, perhaps, be classified under the headings of: (i) misinformed use of mathematical modelling tools, (ii) a threat to the “principle of parsimony” and (iii) shadows of computer implementations. In the remainder of this section we shall briefly discuss each of these issues. (i) Misinformed use of mathematical modelling tools. One of the unintended but, perhaps, unavoidable consequences of the widespread availability of mathematical modelling tools, is that they are more likely to be used incorrectly or used unscrupulously to arrive at inaccurate or false conclusions. This, of course, has always been a danger, even in the days of classical mathematical modelling. However, nowadays these risks are magnified as people with limited understanding of mathematics plunge into “menu driven” mathematical modelling. Furthermore, due to the enhanced capability to communicate models’ findings, flawed conclusions can reach large constituencies and thrive for some time under the aura of “scientific respectability”. Ultimately, it is reasonable to expect that problems arising from such misuse will be minor. The public and decision makers can be expected to quickly develop a healthy dose of scepticism about poorly substantiated claims based on poorly constructed or purposely misleading models. Furthermore, just as social scientists and health practitioners have become sophisticated users of statistical concepts and software packages, it is reasonable to expect that many professionals will become sophisticated users of other mathematical modelling tools. Indeed, many engineers, economists, physicists and chemists have been such users for many years now.

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(ii) A threat to the “principle of parsimony”. This issue presents a more difficult challenge because it stems from a cultural gap between mathematicians and many researchers from other disciplines. Mathematicians and statisticians are trained to strive for simplicity of description and constantly warned of the dangers of “over fitting”. They are trained in the application of the “principle of parsimony” which implies that, in general, if two models of a phenomenon yield comparable results and one of the models is simpler, then this is the preferred model. On the other hand, researchers in many other disciplines are trained to strive for completeness of understanding. Hence if a simple model adequately describes three out of five variables of interest, it is natural for many researchers to strive to refine the model by adding new variables and equations in an attempt to capture the behaviour of all five variables of interest. The idea of constructing a separate model to explain the behaviour of the two remaining variables is usually very unappealing to those interested in the understanding of the whole phenomenon. This is natural, desirable and commendable. However, most mathematicians are aware of situations where two very robust, simple, stand alone models may lose their desirable properties when coupled; especially so if the coupling is via nonlinear equations. In the past this cultural gap did not create much difficulty because of the limitations on our ability to solve complex systems of equations. The availability of powerful computers and of all purpose numerical solvers has changed all that. The temptation to model ever more complex phenomena by building ever more complex mathematical models with the goal of increased “realism” seems almost irresistible. Unfortunately, these complex models may suffer from a variety of problems such as the error propagation issue discussed in Section 5 or difficulties with testing and validation methods already mentioned in Section 4.4. Many of these models are constructed by interdisciplinary teams of researchers and modellers working under the auspices of institutes or agencies. As the models are frequently updated, the detailed documentation and source codes are frequently unavailable to those who are not members of the research team. The latter, together with the software and hardware requirements and the underlying complexity of the models may ensure that their outputs are never independently reproduced. Hence in contrast to the simple models that are regularly tested by many users, the most sophisticated models of life support systems (e.g. models of national economies or global change models) may be very rarely subjected to extensive independent testing. To some extent this problem has been addressed by comparison and consistency testing between different teams developing complex models of important phenomena. Such comparisons are clearly valuable but need to be monitored to ensure that the emerging consensus does not come at the expense of objective testing and validation, whenever these are possible. Even when reliable data are available for validation of complex models the problem of excessive “degrees of freedom” often arises. This can be a direct consequence of violating the principle of parsimony. Namely, in models that have many parameters that can be adjusted, experienced modellers can usually find a configuration of parameter values that will produce outputs in close agreement with the observed data. A lot of care needs to be taken to check that such parameter configurations agree with

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the domain knowledge of the underlying phenomenon and are not selected merely out of modelling expediency. Finally, the construction process of complex models is more likely to “overlook” the presence of inherently chaotic mathematical equations that may exhibit behaviour that is totally unrelated to the underlying phenomenon of interest. This issue is discussed below. (iii) Shadows of computer implementations. It is, perhaps, appropriate to conclude this introduction to the mathematical models theme by returning to the Cave Allegory that we used in the opening section. In that section we implied that mathematical models of life support phenomena were like Plato’s wooden or clay animals and the models’ outputs were like the shadows on the cave wall. However, when dealing with outputs of mathematical models that are produced by a computer we need to worry about yet another, potential, distortion. This is the distortion that may be generated by what might be called a “computer phenomenon” arising in response to the mathematical model that is being solved but not in response to the life support system modelled. It is as though yet another model of an animal was produced from the wooden animal, and the shadows on Plato’s cave wall were generated by that new model; hence the name: shadows of computer implementations. The point - well known to many mathematicians and some modellers from other disciplines - is that certain mathematical expressions are ill suited for the type of large scale recursive numerical computations that computer implementations thrive on. If a model is sufficiently simple these expressions will, in all likelihood, be identified and the problem avoided. If, however, a model is sufficiently complex or the problematic expressions were automatically generated by higher level automated modelling tools, then their presence may be overlooked. Thus vigilance is needed to prevent this from arising. We shall illustrate the seriousness of the problem with the following classical example. Consider one of the simplest “population” models:

x(t+1) = 3.8x(t)[1 - x(t)]; x(1) = 0.3; t = 1, 2, 3, …., where x(t) denotes the fraction of the population of an insect colony, at generation t, that is made up by one particular variety of that insect and it is assumed that this variety constitutes 0.3 of the initial insect population. Assume, for the moment that this very simple model is indeed a very good description of the biological situation of interest. The solution algorithm is trivial: one merely multiplies three numbers at each iteration, that is, x(1)=.3, x(2)=3.8(.3)(.7)= .798, and so on. The correctness of the numbers generated by this algorithm seems beyond any doubt. However, if we wanted an estimate of the fraction of population made up by this special variety of insect at the 97th generation, the quantity x(97) would need to be calculated by implementing the preceding simple algorithm on a computer. In principle, almost any computer language or package could be used for this purpose.

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In Figure 3 below we demonstrate the results of two “identical” implementations of the previous algorithm but in different, yet very standard, computer languages: FORTRAN and MATLAB®. Some would expect that we shall obtain the same answers; after all, that’s just like saying “Good Morning” in two different languages. The meaning should still be the same. However, it turns out that while the two implementations indeed yield almost identical results up to about the 90th generation, thereafter they rapidly start diverging significantly. By the 97th generation there is no resemblance between the two values. Indeed, after sufficiently many generations (and with that particular starting point) the numbers generated by the two implementations seem no longer connected to the original mathematical model, let alone to the insect population that this model was supposed to describe. They are, indeed, shadows of computer implementations rather than of the mathematical model. Figure 3. Outputs of the MATLAB and FORTRAN generated trajectories, with values of x(t) plotted on the vertical axis and the iteration index, t, plotted on the horizontal. What is disconcerting is that there appears to be no valid way of accurately calculating the true values of x(t) for sufficiently large values of t, no matter what computer, language or software is employed for their calculations. This is the nature of chaos. In most practical models of life support systems it should be possible to avoid such phenomena and still provide a good description of the behaviour of the variables of interest. However, it is important to be aware of the limitations of certain mathematical models and of the distortions that could occur in computer implementations of these models. 7. A brief guide to the theme. The Mathematical Models theme of this encyclopedia is divided into nine broad topics reflecting the versatility of the science of mathematical modelling:

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6.3.1. Basic Principles of Mathematical Modeling 6.3.2. Mathematical Models in Water Science 6.3.3. Mathematical Models in Energy Sciences 6.3.4. Mathematical Models of Climate and Global Change 6.3.5. Transport Models in Soils: Surface and Subsurface Hydrology 6.3.6. Mathematical Models in Food and Agricultural Sciences 6.3.7. Mathematical Models of Biology and Ecology 6.3.8. Mathematical Models in Medicine and Public Health 6.3.9. Mathematical Models of Society and Development. With the exception of the first topic dealing with basic principles of mathematical modelling, the remaining eight represent a broad categorisation of those components of life support systems where mathematical models have been extensively used. As mentioned earlier, there is already a large body of literature that exploits mathematical modelling to study phenomena and issues relevant to these topics. In fact, each of the nine topics is further divided in a number of articles that are devoted to more specific subtopics. In addition, topics 6.3.2-6.3.9 of the theme span the “four pillars” of global sustainable development as conceptualised in the EOLSS content organisation chart. The four pillars were: Water, Energy, Environment and Food and Agriculture. In particular, we note that 6.3.2 , 6.3.3 , 6.3.7 and 6.3.6 correspond nearly directly to these four pillars. Topic 6.3.5 dealing with transport models in soils cuts across the environment and water areas. The climate and global change topic 6.3.4 is technically specialised because of the sophistication of its models but it impacts on all four of these areas. Similarly, topic 6.3.8 addressing mathematical models used in medicine and public health is of obvious and universal importance to those aspects of global sustainable development that relate to human health. Finally, topic 6.3.9 spans a wide range of applications that cover societal activities influencing both economic development and social and natural environment. Acknowledgments: The views expressed in this paper reflect the evolution of the author’s thoughts about mathematical modelling over the past twenty years. During that period he had benefited from interactions with many mathematicians, engineers, operations researchers and environmental scientists. In particular, the author wishes to acknowledge the strong influence of his mathematical teachers and mentors (the late) E. Conway, J.S. Maritz, T.E.S. Raghavan and A.J. Goldman. In particular, Alan Goldman had helped shape the author’s philosophy about the role of mathematics in modelling phenomena and some of his recent thoughts on this subject have been incorporated into the text. In addition, the author is indebted to his colleagues, at the University of South Australia, especially Stephen Lucas, Ross Frick and Bruce Brown for their help and useful comments. Finally, Lynne Scott, who has been acting as a research assistant on this particular project has also made a number of valuable contributions. Lynne has been supported by a divisional research grant from the University of South Australia and that support is also gratefully acknowledged.

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Annotated Bibliography

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Batchelor, G. K. (1994). An introduction to fluid dynamics. Cambridge: Cambridge University Press. [A classic textbook on the fundamental theory of fluid mechanics.]

Beltrami, E. J. (1987). Mathematics for dynamic modeling. Boston: Academic Press. [An introduction to mathematical tools needed for the understanding of equilibrium, stability, limit cycles, bifurcation and chaos and other concepts.]

Bender, J. D. (1978). An introduction to mathematical modelling. New York: John Wiley and Sons. [A textbook designed to teach students how to formulate, analyse and criticise mathematical models.]

Fielding, Alan H. (ed.) (1999). Machine learning methods for ecological applications. Boston: Kluwer Academic Publishers. [A collection of papers dealing with the topic of machine learning methods in ecological modelling.]

Finch, P.D. (1981). On the role of description in statistical inquiry. Philosophy of Science, 32. [Contains an insightful discussion of the role of exploratory data analysis in Statistics.]

Forrester, Jay W. (1971). World Dynamics. Cambridge, Mass: Wright-Allen Press. [One of the classical, and much criticised, global models simulating the world’s economic, social, political and environmental systems.]

Giordano, Frank R. and Weir, Maurice D. (1985). A first course in mathematical modeling. Monterey, Ca: Brooks/Cole Publishing Company. [A comprehensive introduction to mathematical modelling techniques in a wide range of contexts.]

Haefner, James W. (1996). Modeling Biological Systems - Principles and Applications. New York: Chapman and Hall. [A textbook on modelling biological systems with emphasis on simulations.]

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problems with the help of high-level modelling languages, in particular the Logical Programming Language (LPL).]

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Saaty, Thomas L. and Alexander, Joyce M. (1981). Thinking with models: mathematical models in the physical biological and social sciences. Oxford: Pergamon. [A textbook on mathematical modelling which emphasises the model formulation phase.]

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Tukey, John W. (1977). Exploratory Data Analysis. Reading, Mass.: Addison-Wesley Publishing Co. [A seminal treatise on exploratory data analysis.]

Wan, Frederic Y. M. (1989). Mathematical models and their analysis. New York: Harper & Row. [An advanced undergraduate textbook on mathematical modelling.]